Modern Mathematics Education for Engineering Curricula in Europe

Authors Christian Mercat Seppo Pohjolainen Sergey Sosnovsky Tuomas Myllykoski

Plaintext

Seppo Pohjolainen
Tuomas Myllykoski
Christian Mercat
Sergey Sosnovsky
Editors

Modern
Mathematics
Education for
Engineering
Curricula in Europe
Modern Mathematics Education for Engineering
Curricula in Europe
Seppo Pohjolainen • Tuomas Myllykoski •
Christian Mercat • Sergey Sosnovsky
Editors

Modern Mathematics
Education for Engineering
Curricula in Europe
A Comparative Analysis of EU, Russia,
Georgia and Armenia
Editors
Seppo Pohjolainen Tuomas Myllykoski
Laboratory of Mathematics Laboratory of Mathematics
Tampere University of Technology Tampere University of Technology
Tampere, Finland Tampere, Finland

Christian Mercat Sergey Sosnovsky
Université Lyon 1 Utrecht University
IREM de Lyon, Bâtiment Braconnier Utrecht, The Netherlands
Villeurbanne Cedex, France

This project has been funded with support from the European Commission. This publication
reflects the views only of the author, and the Commission cannot be held responsible for any
use which may be made of the information contained therein.

ISBN 978-3-319-71415-8 ISBN 978-3-319-71416-5 (eBook)
https://doi.org/10.1007/978-3-319-71416-5

Library of Congress Control Number: 2018943911

Mathematics Subject Classification (2010): 97Uxx

© The Editor(s) (if applicable) and The Author(s) 2018. This book is an open access publication.
Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 Inter-
national License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation,
distribution and reproduction in any medium or format, as long as you give appropriate credit to the
original author(s) and the source, provide a link to the Creative Commons license and indicate if changes
were made.
The images or other third party material in this book are included in the book’s Creative Commons
license, unless indicated otherwise in a credit line to the material. If material is not included in the book’s
Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the
permitted use, you will need to obtain permission directly from the copyright holder.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, express or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.

Printed on acid-free paper

This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered
company Springer International Publishing AG part of Springer Nature.
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface

Modern science, technology, engineering and mathematics (STEM) education is
facing fundamental challenges. Most of these challenges are global; they are not
problems only for the developing countries. Addressing these challenges in a timely
and efficient manner is of paramount importance for any national economy.
Mathematics, as the language of nature and technology, is an important subject in
the engineering studies. Despite the fact that its value is well understood, students’
mathematical skills have deteriorated in recent decades in the western world. This
reflects in students’ slow progressing and high drop-out percentages in the technical
sciences.
The remedy to improve the situation is a pedagogical reform, which entails
that learning contexts should be based on competencies, engineering students
motivation should be added by making engineering mathematics more meaningfully
contextualized, and modern IT-technology should be used in a pedagogically
appropriate way so as to support learning.
This book provides a comprehensive overview of the core subjects comprising
mathematical curricula for engineering studies in five European countries and
identifies differences between two strong traditions of teaching mathematics to
engineers. It is a collective effort of experts from a dozen universities taking a critical
look at various aspects of higher mathematical education.
The two EU Tempus-IV projects—MetaMath (www.metamath.eu) and Math-
GeAr (www.mathgear.eu)—take a deep look at the current methodologies of
mathematics education for technical and engineering disciplines. The projects aim at
improving the existing mathematics curricula in Russian, Georgian and Armenian
universities by introducing modern technology-enhanced learning (TEL) methods
and tools, as well as by shifting the focus of engineering mathematics education
from a purely theoretical tradition to a more application-based paradigm.
MetaMath and MathGeAr have brought together mathematics educators, TEL
specialists and experts in education quality assurance from 21 organizations across
6 countries. A comprehensive comparative analysis of the entire spectrum of math
courses in the EU, Russia, Georgia and Armenia has been conducted. Its results
allowed the consortium to pinpoint issues and introduce several modifications in

v
vi Preface

their curricula while preserving the overall strong state of the university mathemat-
ics education in these countries. The methodology, the procedure, and the results of
this analysis are presented here.
This project has been funded with support from the European Commission. This
publication reflects the views only of the authors, and the Commission cannot be
held responsible for any use which may be made of the information contained
therein.
This project has been funded with support from the European Commission. This
publication reflects the views only of the author, and the Commission cannot be held
responsible for any use which may be made of the information contained therein.

Tampere, Finland Seppo Pohjolainen
Tampere, Finland Tuomas Myllykoski
Villeurbanne, France Christian Mercat
Utrecht, the Netherlands Sergey Sosnovsky
Contents

1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
2 Methodology for Comparative Analysis of Courses . . . . . . . . . . . . . . . . . . . . 33
Sergey Sosnovsky
3 Overview of Engineering Mathematics Education
for STEM in Russia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 39
Yury Pokholkov, Kseniya Zaitseva (Tolkacheva), Mikhail Kuprianov,
Iurii Baskakov, Sergei Pozdniakov, Sergey Ivanov, Anton Chukhnov,
Andrey Kolpakov, Ilya Posov, Sergey Rybin, Vasiliy Akimushkin,
Aleksei Syromiasov, Ilia Soldatenko, Irina Zakharova,
and Alexander Yazenin
4 Overview of Engineering Mathematics Education for STEM
in Georgia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 55
David Natroshvili
5 Overview of Engineering Mathematics Education for STEM
in Armenia.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 63
Ishkhan Hovhannisyan
6 Overview of Engineering Mathematics Education for STEM
in EU . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 69
7 Case Studies of Math Education for STEM in Russia .. . . . . . . . . . . . . . . . . 91
8 Case Studies of Math Education for STEM in Georgia . . . . . . . . . . . . . . . . 141
9 Case Studies of Math Education for STEM in Armenia . . . . . . . . . . . . . . . 169
10 Overview of the Results and Recommendations . . . .. . . . . . . . . . . . . . . . . . . . 185
Sergey Sosnovsky, Christian Mercat, and Seppo Pohjolainen

vii
About the Editors

Seppo Pohjolainen is a (emeritus) professor at the Laboratory of Mathematics of
the Tampere University of Technology. His research interests include mathematical
control theory, mathematical modeling and simulation, development and the use of
information technology to support learning. He has led several research projects and
written a number of journal articles and conference papers on all above-mentioned
fields.
Tuomas Myllykoski is a Master of Science and a Teacher of Mathematics at the
Laboratory of Mathematics of the Tampere University of Technology. His focus
has been on the development and use of learning tools in mathematics, and his
current research interests are in the fields of data science, educational psychology
and personality.
Christian Mercat is professor in the laboratory Sciences, Société, Historicité,
Éducation et Pratiques (S2HEP, EA 4148) and director of the Institute for Research
on Mathematics Education (IREM). He trains mathematics teachers at the École
Supérieure du Professorat et de l’éducation (ESPE). His main interest is mathemat-
ics teaching that respects the creative potential of students. He took part in several
research projects on technology-enhanced learning and creative mathematical
thinking such as, lately, the mcSquared project (http://www.mc2-project.eu/). He
published a number of articles and gave conference talks on the subject, as well as
on his main mathematical specialty: discrete differential geometry.
Sergey Sosnovsky is an assistant professor of software technology for learning
and teaching at the Institute of Information and Computing Sciences of Utrecht
University. He has co-authored more than 90 peer-reviewed publications on topics
related to technology-enhanced learning and adaptive information systems. Dr.
Sosnovsky served on the programming committees of several conferences and
workshops dedicated to Adaptive and Intelligent Educational Technologies. He
has built up a strong record of participation in research projects supported by US
and EU funding agencies on various aspects of developing AI-based educational
technologies, including adaptive support of learning processes and semantic access

ix
x About the Editors

to instructional resources. Dr. Sosnovsky holds a M.Sc. degree in Information
Systems from Kazan State Technological University (Kazan, Russia) and a PhD
degree in Information Sciences from University of Pittsburgh (Pittsburgh, PA,
USA). He is a receiver of the EU Marie-Curie International Incoming Fellowship.
He coordinated the projects MetaMath and MathGeAr, which are at the basis of this
book.
Chapter 1
Introduction

1.1 Mathematics Education in EU for STEM Disciplines

Seppo Pohjolainen ()
Tampere University of Technology (TUT), Laboratory of Mathematics, Tampere,
Finland
e-mail: seppo.pohjolainen@tut.fi

Good competency in mathematics is important in science, technology and
economy; mathematics can be considered as the language of nature and technology
and also is an important methodology in economics and social sciences. A study
by Hanushek and Wößman [21] shows that the quality of education has a strong
positive influence on economic growth. In their research, students’ skills were
measured using 13 international tests, which included mathematics, science, and
reading. An OECD report on mathematics in industry [42] states that the remarkable
development of the natural sciences and of engineering since the Renaissance
is a consequence of the fact that all nature’s known laws can be expressed as
mathematical equations. The Financial Times outlined their news on February 13th,
2006, as “Mathematics offers business a formula for success”.
Despite the fact that the value of mathematics in society and economics is
understood, in recent decades students’ mathematics skills have deteriorated in
western countries. The report “Mathematics for the European Engineer” [50] by
the European Society for Engineering Education SEFI1 states that this phenomenon
prevails in Europe. According to the SEFI report, universities in the western world

1 SEFI (http://www.sefi.be).

© The Author(s) 2018 1
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_1
2 1 Introduction

have observed a decline in mathematical proficiency among new university students
and have taken action to remedy the situation. The most common measures are:
reducing syllabus content; replacing some of the harder material with more revisions
of lower level work; developing additional units of study; establishing mathematics
support centres. But sometimes one does nothing.
The decline in mathematical competency may have serious consequences as
Henderson and Broadbridge [22] point out. Their message is that industry can
only be internationally competitive through mathematical know-how. The number
of students majoring in mathematics e.g., in Australia, has decreased, while the
number of positions requiring mathematical skills has increased.
The union of Academic Engineers and Architects in Finland2 (TEK) published
a report in 2009 with the recommendation “Knowledge of mathematics and natural
sciences must be emphasized more strongly as part of common cultivation and their
appreciation should be improved in the society”. The report also points out that
good command of mathematics and natural sciences is one of the strongest features
in engineering studies.
As mathematical proficiency is a prerequisite for studying technical sciences,
weak mathematical skills slow down studies. For instance, in Germany the drop-out
rate of students sometimes goes up to 35% and one of the primary reasons is the lack
of mathematical skills. This caused the industrial Arbeitgeberverband Gesamtmetall
to raise an alarm. Drop-out rates in engineering studies are high Europe-wide.3
For example, less than 60% of B.Sc. students starting their studies in Finland
at Tampere University of Technology (TUT) in 2005 had completed all mandatory
first year mathematics courses in four and a half years. Students who had progressed
fastest in their studies had typically completed first year mathematics courses
according to the recommended schedule. Students who faced problems in studying
mathematics more often progressed slowly with their studies in general.
The problems universities are facing with their enrolling students’ mathematical
proficiency are partly due to school mathematics. The level of school mathematics
is being assessed internationally by PISA (The Programme for International Stu-
dent Assessment), and TIMMS (Trends in International Mathematics and Science
Study). PISA is an internationally standardised assessment for 15-year-olds in
schools testing literacy in reading, mathematics and science. TIMSS collects
educational achievement data at the 4th and 8th grades to provide information on
quantity, quality, and content of instruction. The test results from 2012 [44] and
2011 [37] confirm that East-Asian countries are on the top but they are criticised
for teacher centred education, large amounts of homework, rote learning etc. EU-
countries are doing relatively well, but lagging behind the East-Asian nations, and
developing countries can be seen at the bottom.

2 http://www.tek.fi/.
3 http://ec.europa.eu/information_society/apps/projects/factsheet/index.cfm?project_ref=ECP-

2008-EDU-428046.
1 Introduction 3

Learning outcomes in mathematics are not dependent solely on good teaching,
sufficient resources or other external considerations with bearing on learning.
Factors with bearing on what the student does include attitudes: orientations,
intentions and motivations. In order to achieve learning objectives, activity on
the part of the learner is required. As student’s attitudes and motivational factors
are individual, good teaching should take into account student’s different learning
styles [25].
The recent report ‘Mathematics in Europe: Common Challenges and National
Policies’ by EURYDICE [17] points out that many European countries are con-
fronted with declining numbers of students of mathematics, science and technology,
and they face a poor gender balance in these disciplines. The report gives recommen-
dations on how to increase motivation to learn mathematics and encourage the take-
up of mathematics-related careers. The report also suggests that the mathematics
curriculum should be broadened from contents to competences. Student motivation
should be increased by demonstrating and finding evidence how mathematics is used
in industry and society, in students’ everyday life, and in their future career. New
teaching approaches, such as problem-based learning and inquiry-based methods,
should be taken into use. Addressing low achievement is important to decrease
the drop-out figures, and gender issues should be considered to make mathematics
more tempting to female students. Education and professional development of
mathematics teachers also plays a key role in this reform.
The European Society for Engineering Education (SEFI), mentioned above, is
an international non-profit organisation established in 1973 in Belgium and founded
by 21 European Universities. It is an association directly linking the institutions of
higher engineering education as an international forum for discussing problems and
identifying solutions relating to engineering education. Today, SEFI is the largest
network of institutions of higher engineering education, individuals, associations
and companies in Europe. Its mission is to contribute to the development and
improvement of engineering education in Europe and to the enhancement of the
image of both engineering education and engineering professionals in society.
SEFI has set up several working groups on developing engineering education.
Among them is the Working Group on Mathematics and Engineering Education,
established in 1982. The major outcome of the group resulted in a “Core Curriculum
in Mathematics for the European Engineer”, first published 1992 and then revised
in 2002 as “Mathematics for the European Engineer” [50], and updated 2013 as “A
Framework for Mathematics Curricula in Engineering Education” [49].
These documents clearly reflect the European understanding of what the math-
ematics is that engineers need, and how it should be learned and taught. The 1992
version of the Core Curriculum answers mainly the question: what should be the
contents of mathematics courses for engineers? It presented a list of mathematical
topics, which are itemised under the headings of Analysis and Calculus, Linear
Algebra, Discrete Mathematics and Probability and Statistics.
The SEFI 2002 document identifies four content levels defined as Cores 0, 1,
2, 3. The entry level Core 0 and Core level 1 comprise the knowledge and skills
which are necessary and essential for most engineering areas and they should be
4 1 Introduction

mandatory for all engineering education, whereas from the other two different parts
(Cores 2 and 3) contents will be chosen for the various engineering disciplines. The
document also specifies learning outcomes for all the topics and contains additional
comments on teaching mathematics.
The most recent report:“ A Framework for Mathematics Curricula in Engineering
Education”, SEFI 2013 [49], proposes a pedagogical reform for engineering
mathematics to put more emphasis on what students should know instead of
what they have been taught. The learning goals are described as competencies
rather than learning contents. Contents should be embedded in a broader view of
mathematical competencies that the mathematical education of engineers strives
to achieve. Following the Danish KOM project [39], SEFI recommends that the
general mathematical competence for engineers is “the ability to understand, judge,
do, and use mathematics in a variety of intra- and extra-mathematical contexts
and situations in which mathematics plays or could play a role”. The general
mathematical competence can be divided into eight sub-competencies which are:
thinking mathematically, reasoning mathematically, posing and solving mathe-
matical problems, modelling mathematically, representing mathematical entities,
handling mathematical symbols and formalism, communicating in, with, and about
mathematics, and making use of aids and tools.
Following the SEFI 2013 document we briefly introduce the eight
subcompetencies:
Thinking Mathematically This competency comprises the knowledge of the kind
of questions that are dealt with in mathematics and the types of answers mathematics
can and cannot provide, and the ability to pose such questions. It includes the recog-
nition of mathematical concepts and an understanding of their scope and limitations
as well as extending the scope by abstraction and generalisation of results. This also
includes an understanding of the certainty mathematical considerations can provide.
Reasoning Mathematically This competency includes the ability to understand
mathematical argumentation (chain of logical arguments), in particular to under-
stand the idea of mathematical proof and to understand its the central ideas. It
also contains the knowledge and ability to distinguish between different kinds
of mathematical statements (definition, if-then-statement, iff-statement etc.). On
the other hand it includes the construction of logical arguments and transforming
heuristic reasoning into unambiguous proofs (reasoning logically).
Posing and Solving Mathematical Problems This competency comprises on the
one hand the ability to identify and specify mathematical problems (pure or applied,
open-ended or closed) and the ability to solve mathematical problems with adequate
algorithms. What really constitutes a problem is not well defined and it depends on
personal capabilities.
Modelling Mathematically This competency has two components: the ability to
analyse and work with existing models and to perform mathematical modelling (set
up a mathematical model and transform the questions of interest into mathematical
questions, answer the questions mathematically, interpret the results in reality and
1 Introduction 5

investigate the validity of the model, and monitor and control the whole modelling
process).
Representing Mathematical Entities This competency includes the ability to
understand and use mathematical representations (symbolic, numeric, graphical and
visual, verbal, material objects etc.) and to know their relations, advantages and
limitations. It also includes the ability to choose and switch between representations
based on this knowledge.
Handling Mathematical Symbols and Formalism This competency includes the
ability to understand symbolic and formal mathematical language and its relation to
natural language as well as the translation between both. It also includes the rules
of formal mathematical systems and the ability to use and manipulate symbolic
statements and expressions according to the rules.
Communicating in, with, and about Mathematics This competency includes the
ability to understand mathematical statements (oral, written or other) made by others
and the ability to express oneself mathematically in different ways.
Making Use of Aids and Tools This competency includes knowledge about
the aids and tools that are available as well as their potential and limitations.
Additionally, it includes the ability to use them thoughtfully and efficiently.

In order to specify the desired cognitive skills for the topical items and the sub-
competences, the three levels described in the OECD PISA document [43] may be
used. The levels are: the reproduction level, where students are able to perform the
activities trained before in the same contexts; the connections level, where students
combine pieces of their knowledge and/or apply it to slightly different situations;
and the reflection level, where students use their knowledge to tackle problems
different from those dealt with earlier and/or do this in new contexts, so as to have
to reflect on what to use and how to use their knowledge in different contexts.
While the necessity of a pedagogical reform is well understood, there still are
not enough good pedagogical models scalable to universities’ resources available.
Modern information and communication technology (ICT) provides a variety of
tools that can be used to support students’ comprehension and pedagogical reform.
Teachers may run their courses using learning platforms like Moodle. In these
environments they may distribute course material, support communication, collabo-
ration, and peer learning and organise face-to-face meetings with videoconferencing
tools. Students can get feedback on their mathematical skills’ from their teacher,
peers, and also by using carefully chosen computer generated exercises, which
are automatically checked by computer algebra systems (Math-Bridge, System for
Teaching and Assessment using a Computer algebra Kernel4 (STACK)). There

4 http://www.stack.bham.ac.uk/.
6 1 Introduction

exist mathematical programs like MATLAB5 and Mathematica,6 which support
mathematical modelling of real world problems.
Math-Bridge is one of the learning platforms available to study mathematics. It
offers online courses of mathematics including learning material in seven languages:
German, English, Finnish, French, Dutch, Spanish and Hungarian. The learning
material can be used in two different ways: in self-directed learning of individuals
and as a ‘bridging course’ that can be found at most European universities (Math-
Bridge Education Solution7).
Internet contains a variety of open source mathematical tool programs
(R, Octave, Scilab), computational engines (Wolfram alpha), various visualisations,
and apps that can be used alongside the studies. Links to external resources can
be easily used to show real world applications. Some universities have started to
provide Massive Open Online Courses (MOOC’s) available world-wide for their
on- and off-campus students.
Information and communication technology can be used to support the learning
process in many ways, but great technology cannot replace poor teaching or lack of
resources. The use of technology does not itself guarantee better learning results;
instead it can even weaken the student performance. This obvious fact has been
known for a long time. Reusser [46] among many other researchers, stated that
the design of a computer-based instructional system should be based on content-
specific research of learning and comprehension and a pedagogical model of the
learner and the learning process. In designing computer-based teaching and learning
environments real didactic tasks should be considered. One should thoroughly
consider what to teach and how to teach. Jonassen [26] has presented qualities of
“meaningful learning” that the design and use of any learning environment should
meet. The list has been complemented by Ruokamo and Pohjolainen [48].
In a recent report by OECD [41], it was discovered that in those countries
where it is more common for students to use the Internet at school for schoolwork,
students’ average performance in reading declined, based on PISA data. The impact
of ICT on student performance in the classroom seems to be mixed, at best. In
fact, PISA results show no appreciable improvements in student achievement in
reading, mathematics or science in the countries that had invested heavily in ICT
for education. One interpretation of these findings is that it takes time and effort to
learn how to use technology in education while staying firmly focussed on student
learning.
As mathematics is a universal language, the problems in teaching and learning
are globally rather similar. The importance of mathematics is internationally well
understood and deterioration in the students’ skills is recognised. Pedagogical
reforms are the way EU is going and pedagogically justified use of information
technology and tools will play an important role here.

5 http://mathworks.com.
6 http://www.wolfram.com/mathematica.
7 http://www.math-bridge.org.
1 Introduction 7

1.2 TEMPUS Projects MetaMath and MathGeAr

Sergey Sosnovsky
Utrecht University, Utrecht, The Netherlands
e-mail: s.a.sosnovsky@uu.nl

1.2.1 Introduction

The world-wide system of STEM8 (science, technology, engineering and math)
education faces a range of fundamental challenges. Addressing these challenges
in a timely and efficient manner is paramount for any national economy to stay
competitive in the long range. It is worth noticing that most of these problems
are truly global; they are actual not only for the developing countries, but also
for countries characterised by stable engineering sectors and successful educational
systems (such as, for example, Germany and USA). Experts identify three main
clusters of factors characterising the change in the requirements towards the global
system of STEM education [40]:
• Responding to the changes in global context.
• Improving perception of engineering subjects.
• Retention of engineering students.

1.2.1.1 Responding to the Changes in Global Context

The speed of renewal of engineering and technical knowledge and competencies
is ever growing. Most engineering sectors observe acceleration of the cycle of
innovation, i.e. the time between the birth of a technology and its industrial
mainstreaming. Technical skills evolve rapidly, new competencies emerge, old
competencies dissolve. In 1920, the average half-life of knowledge in engineering
was 35 years. In 1960, it was reduced to 10 years. In 1991, it was estimated
that an engineering skill is half-obsolete in 5 years. Nowadays, “IT professional
would have to spend roughly 10 h a week studying new knowledge to stay current
(or upskilling, in the current lingo)” [13]. This essentially means that modern
engineering programmes have to teach students how to obtain and operate skills
that are not yet defined and how to work at jobs that do not yet exist. Continuous
education and retraining is common already and will only widen in the future. This
increase in intensity of education and diversity of educational contexts renders the

8 A similar term has been suggested in the German literature as well—MINT (“Mathematik,

Informatik, Naturwissenchaft un Technik”).
8 1 Introduction

traditional system of STEM education inadequate. New forms of education powered
by new educational technologies are required.
At the same time, the engineering and technical problems themselves are
changing. Technology has become an integral part of most sectors of a modern
economy, technical systems have become more complex and interconnected, even
our daily life activities have been more and more penetrated by technology. Finding
solutions for this kind of problems and management of this kind of systems requires
new approaches that take into account not only their technical side, but also their
relations to the social, ecological, economical and other aspects of the problem.
Effective teaching of future engineers to deal with these interdisciplinary problems
by applying more comprehensive methods should require significant redesign of
STEM courses and curriculae. One more group of factors influencing STEM
education come from the ever growing globalisation of economic and technological
relations between courtiers and companies. Markets and manufacturers have inter-
nationalised and the relations between them have become more dynamic. Solutions
for typical engineering problems have become a service, basic engineering skills
and competencies have transformed into a product that has a market value and can
be offered by engineers of other nations. Countries that have not invested in own
STEM education and do not possess strong enough engineering workforce will be
forced to pay other courtiers for engineering problems by outsourcing.

1.2.1.2 Improving Perception of Engineering Subjects

While the demand for engineer professionals is increasing world-wide, the number
of engineering graduates is not growing, and in some countries has even dropped
over the last several years. Potential students often consider engineering and
technical professions less interesting. Those driven by financial stimuli do not
consider engineering as an attractive, money-making career and choose business-
oriented majors. Young people motivated rather by a social mission and a public
value of their future profession also seldom believe that STEM answers their life
goals and remain in such fields as medicine and humanities. Such beliefs are
clearly misconceptions as engineering professions are both well paid and societally
important. Yet, if the appeal of STEM careers is often not apparent to potential
students it is the responsibility of STEM programme administrations and teachers
to properly advertise their fields. In addition to that, many students simply consider
STEM education too complex and formal, and, at the end, boring. Changing such an
image of STEM education is an important practical task that every national system
of engineering education must address.

1.2.1.3 Retention of Engineering Students

One of the biggest problems for engineering and science education is the high drop-
out rates (especially among freshmen and sophomores). For example, in American
1 Introduction 9

universities, up to 40% of engineering students change their major to a non-
technical one or simply drop out from college [40]. The situation in Europe is
similar. For instance, in Germany over the last 15 years, the number of students
who do not finish their university programmes has grown by about 10% for most
engineering specialisations. Now, depending on the program, this number fluctuates
between 25 and 30% nation-wide. For degrees that include an intensive mathematics
component, the rate goes up to 40% of all enrolling students [23]. Similar trends can
be observed in other developed EU nations (e.g., The Netherlands, Spain, United
Kingdom). One of the reasons for this is the fact the traditional structure of
engineering education does not enable students to develop their engineer identity for
several academic semesters. All engineering programmes start with a large number
of introductory “101” courses teaching formal math and science concepts. Only on
the second or third year, the actual “engineering” part of the engineering educational
programs starts. For a large percentage of students this comes too late. How can
engineering students be sooner exposed to the competencies, requirements, and
problems and use cases of engineering is a big methodological problem. Should
the structure of the engineering curriculum be modified to gradually introduce
engineering subjects in parallel with the introductory science courses, or should the
structure of the “101” courses be changed to become more attractive to students and
include more engineering “flavour”, or, maybe, can new educational technologies
make these subjects more engaging and help relieve the student retention problem?

1.2.1.4 National Problems of Engineering Education

The three countries addressed in this book (Russia, Georgia and Armenia) have
inherited the strong system of school and university STEM education developed in
Soviet Union. It was developed to support industrial economy and has used many
unique methodological innovations [29]. Yet, after the collapse of Soviet Union, the
educational systems of these countries went through a significant transformation.
This process has been characterised by several trends, including attempts to resolve
the disproportion of the old Soviet education systems that emphasised formal
and technical subjects while overlooking the humanities; closing gaps in largely
fragmented inherited national educational systems (mostly, the case of Georgia and
Armenia); introduction and implementation of elements of the Bologna process and
ECTS. Despite the economic and political turmoil of the 1990s, significant progress
has been achieved. Yet, there still exist a number of problems impeding further
development of these countries’ systems of education. This book takes a closer look
at the problems pertaining to the mathematics component of STEM education.

1.2.1.5 Role of Mathematics in STEM Education

All the problems mentioned above are especially important when it comes to
the mathematics component of STEM education. Math is a key subject for all
10 1 Introduction

technical engineering and science programs without exception. In many respects
mathematics serves as a lingua franca for other more specialised STEM subjects.
The level of math competencies of a student is critical for successful engineering
college education, especially in the beginning, when possible learning problems are
amplified and math represents a large share of his/her studies. Differences between
the requirements of school and university math education can be rather large,
especially since different schools can have very different standards of math training.
In addition, students themselves often underestimate the volume of math knowledge
required to succeed in an engineering university program. In Georgia and Armenia,
these problems are especially actual. There is a massive gap between the level of
technical competencies of GE/AM school graduates and the requirements they face
once enrolling in universities. This gap has emerged as a result of asynchronous
reforms of secondary and tertiary education in these countries. According to the
statistics of the National Assessment and Examinations Centre of Georgia, about
a third of the university entrants fail the national exam. In Armenia, the data of
the Ministry of Education and Science shows that the average score of school
graduates in math reaches only about 50%. In Russia, the situation is not that
drastic. Yet these problems also exist due to unique national circumstances. After the
introduction of the unified state exam (USE) in the beginning of the 2000s, Russian
universities have abolished the common habit of year-long preparatory math courses
that many potential students took. This resulted in a considerable decrease in math
competences of freshmen in provincial universities (more prestigious universities of
Moscow and St. Petersburg are less affected, as they can select stronger students
based on the result of their USE). Finally, from the organisational perspective, a
deficit of STEM students coupled with increased market demands for engineering
specialists makes many universities loosen enrollment standards, especially with
regards to mathematics. Georgia provides a particular example of this situation.
Several years ago, when Georgian national tests did not stipulate mandatory
subjects, it was a common practice for students with weak school math grades, to
not take a math test, yet get accepted to engineering programs. Such a practice not
only reduces the overall level of students but also adds an extra load on university
teachers. At the end, this will unavoidably result in a decrease in the quality of
engineering programme graduates [1].

1.2.2 Projects MathGeAr and MetaMath

The complete titles of the projects are:
• MetaMath: Modern Educational Technologies for Math Curricula in Engineering
Education of Russia;
• MathGeAr: Modernisation of Mathematics curricula for Engineering and Natural
Sciences studies in South Caucasian Universities by introducing modern educa-
tional technologies.
1 Introduction 11

They both have been supported under the 6th call of the Tempus-IV Program
financed by the Education, Audiovisual and Culture Executive Agency (EACEA) of
EU. The project have been executed in parallel from 01/12/2013 until 28/02/2017.

1.2.2.1 Objectives

MetaMath and MathGeAr projects aimed to address a wide spectrum of the listed
problem of math education in engineering programs of Russian, Georgian and
Armenian universities. To solve these problems, the projects rely on a comprehen-
sive approach including studying international best practices, analytical review and
modernisation of existing pedagogical approaches and math courses. The objectives
of the projects include:
• Comparative analysis of the math components of engineering curricula in Russia,
Georgia, Armenia and EU and detection of several areas for conducting reforms;
• Modernisation of several math courses within the selected set of programs
with a special focus on introduction of technology-enhanced learning (TEL)
approaches.
• Localisation of European TEL instrument for partner universities, including
digital content localisation with a focus on the introduction of the intelligent
tutoring platform for mathematical courses Math-Bridge [51].
• Building up technical capacity and TEL competencies within partner universities
to enable the application of localised educational technologies in real courses.
• Pilot evaluation of the modernised courses with real students validating the
potential impact of the conducted reform on the quality of engineering education.
• Disseminate results of the projects.

1.2.2.2 Consortia

Projects consortia consisted of organisation from EU and partner countries (Russia
for MetaMath and Georgia/Armenia for MathGeAr). The set of EU partners was the
same for the two projects and included:
• Universität des Saarlandes—USAAR (Saarbrücken, Germany),
• Université Claude Bernard Lyon I—UCBL (Lyon, France),
• Tampere University of Technology—TUT (Tampere, Finland),
• Deutsches Forschungszentrum für Künstliche Intelligenz—DFKI (Saarbrücken,
Germany),
• Technische Universität Chemnitz—TUC (Chemnitz Germany).
Additionally, the MetaMath consortium contained six more partners form Russia:
• the Association for Engineering Education of Russia—AEER (Tomsk, Russia),
• Saint Petersburg Electrotechnical University—LETI (St. Petersburg, Russia),
12 1 Introduction

• Lobachevsky State University of Nizhni Novgorod—NNSU (Nizhni Novgorod,
Russia),
• Tver State University—TSU (Tver, Russia),
• Kazan National Research Technical University named after A.N. Tupolev—
KNRTU (Kazan, Russia),
• Ogarev Mordovia State University—OMSU (Saransk, Russia).
The MathGeAr consortium, instead of Russian participants, contained five organi-
sation from Georgia and four from Armenia:
• Georgian Technical University—GTU (Tbilisi, Georgia),
• University of Georgia—UG (Tbilisi, Georgia),
• Akaki Tsereteli State University—ATSU (Kutaisi, Georgia),
• Batumi Shota Rustaveli State University—BSU (Batumi, Georgia),
• Georgian Research and Educational Networking Association—GRENA (Tbilisi,
Georgia),
• National Centre for Educational Quality Enhancement—NCEQE (Tbilisi,
Georgia),
• National Polytechnic University of Armenia—NPUA (Yerevan, Armenia),
• Armenian State Pedagogical University after Khachatur Abovian—ASPU (Yere-
van, Armenia),
• Armenian National Centre For Professional Education and Quality Assurance—
ANQA (Yerevan, Armenia),
• Institute for Informatics and Automation Problems of the National Academy of
Sciences of the Republic of Armenia—IIAP (Yerevan, Armenia).
Partner organisations had different roles in the projects based on their main
competencies. Figures 1.1 and 1.2 graphically represent the structure of the two
consortia including the main role/expertise for each organisation. This book is a
collective effort of all partners from the two consortia.

1.2.2.3 Execution

The projects have been conducted in three main phases. The results of the first phase
are essentially the subject of this book. It included the following tasks:
• Development of the methodology for comparative analysis of math courses.
• Pairwise comparative analysis of math courses between EU and partner
universities.
• Development of recommendations for the consequent reform of structural,
pedagogical/technological/administrative aspects of the target courses.
• Identification of areas where TEL would bring about the most impact, and
selection of TEL instruments to lead to this impact.
The second phase built up on the results of the previous one. It included the
following activities:
1 Introduction 13

Fig. 1.1 Structure of the MetaMath project consortium

Fig. 1.2 Structure of the MathGear project consortium

• Modification of a set of math courses taught to students of engineering programs
in partner universities.
• Localisation of the Math-Bridge platform and its content into Russian, Georgian
and Armenian.
• Training of teaching and technical personnel of partner universities to use Math-
Bridge and other TEL tools for math education.
14 1 Introduction

Finally, the third phase contained:
• Implementation of the (parts of) modified courses into the Math-Bridge platform.
• Planning and conduction of a large-scale pedagogical experiment across three
countries and eleven universities examining the effect of the modernised courses
on different learning parameters of engineering students.
• Analysis and dissemination of the project results.

1.2.3 Learning Platform Math-Bridge

Both projects plans have been especially focussed on applying TEL approaches.
This decision has been motivated by recent advancements in developing intelligent
and adaptive systems for educational support, such as Math-Bridge, especially
for STEM subjects. For example, the use of computers to improve students’
performance and motivation has been recognised in the final report of the Math-
ematical Advisory Panel in the USA: “Research on instructional software has
generally shown positive effects on students achievement in mathematics as com-
pared with instruction that does not incorporate such technologies. These studies
show that technology-based drill and practice and tutorials can improve student
performance in specific areas of mathematics” [18]. As the main TEL solution,
the projects rely on Math-Bridge, which is an online platform for teaching and
learning courses in mathematics. It has been developed as a technology-based
educational solution to the problems of bridging courses taught in European
universities. As its predecessor—the intelligent tutoring system ActiveMath [36]—
Math-Bridge, has a number of unique features. It provides access to the largest
in the world collection of multilingual, semantically annotated learning objects
(LOs) for remedial mathematics. It models students’ knowledge and applies several
adaptation techniques to support more effective learning, including personalised
course generation, intelligent problem solving support and adaptive link annotation.
It facilitates direct access to LOs by means of semantic search. It provides rich
functionality for teachers allowing them to manage students, groups and courses,
trace students’ progress with the reporting tool, create new LOs and assemble
new curricula. Math-Bridge offers a complete solution for organizing TEL of
mathematics on individual, course and/or university level.

1.2.3.1 Math-Bridge Content

The Math-Bridge content base consists of several collections of learning material
covering the topics of secondary and high school mathematics as well as several
university-level subjects. They were originally developed for teaching bridging
courses by mathematics educators from several European universities. Compared
to the majority of adaptive e-learning applications, Math-Bridge supports a mul-
1 Introduction 15

Fig. 1.3 Hierarchy of LO types in Math-Bridge

titude of LO types. The OMDoc language [30] used for representing content in
Math-Bridge defines a hierarchy of LOs to describe the variety of mathematical
knowledge. On the top level, LOs are divided into concept objects and satellite
objects. Satellite objects are the main learning activities; they structure the learning
content, which students practice with: exercises, examples, and instructional texts.
Concept objects have a dualistic nature: they can be physically presented to a
student, and she/he can browse them and read them; at the same time, they are
used as elements of domain semantics, and, as such, employed for representing
knowledge behind satellite objects and modelling students’ progress. Figure 1.3
provides further details of the types of LOs supported in Math-Bridge.

1.2.3.2 Learning Support in Math-Bridge

The Math-Bridge platform provides students with multilingual, semantic and adap-
tive access to mathematical content. Its interface consists of three panels (Fig. 1.4).
The left panel is used for navigation through learning material using the topic-based
structure of the course. The central panel presents the math content associated with
the currently selected (sub)topic. The right panel provides access to the details of
the particular LO that a student is working with, as well as some additional features,
such as semantic search and social feedback toolbox.
Math-Bridge logs every student interaction with learning content (e.g., loading
a page or answering an exercise). The results of interactions with exercises
(correct/incorrect/partially correct) are used by the student-modelling component
of Math-Bridge to produce a meaningful estimation of the student’s progress. For
every math concept the model computes the probabilities that the student has
mastered it. Every exercise in Math-Bridge is linked with one or several concepts
(symbols, theorems, definitions etc.) and the competencies that the exercise is
training for these concepts. A correct answer to the exercise is interpreted by the
system as evidence that the student advances towards mastery and will result in the
increase of corresponding probabilities. Math-Bridge implement three technologies
for intelligent learning support:
16 1 Introduction

Fig. 1.4 Math-Bridge student interface

• Personalised courses. The course generator component of Math-Bridge can
automatically assemble a course optimised for individual students’ needs and
adapted to their knowledge and competencies.
• Adaptive Navigation Support. Math-Bridge courses can consist of thousands
of LOs. The system helps students find the right page to read and/or the right
exercise to attempt by implementing a popular adaptive navigation technique—
adaptive annotation [9]. The annotation icons show the student how much
progress she/he has achieved for the corresponding part of learning material.
• Interactive Exercises and Problem Solving Support. The exercise subsystem
of Math-Bridge can serve multi-step exercises with various types of interactive
elements and rich diagnostic capabilities. At each step, Math-Bridge exercises
can provide students with instructional feedback, ranging from mere flagging the
(in)correctness of given answers to presenting adaptive hints and explanations.

1.2.4 Conclusion

Now that both MetaMath and MathGeAr have finished, it is important to underline
that their success was in many respects dependent on the results of the com-
parative analyses conducted during their first phase. Although the overall project
approach was defined in advance, individual activities have been shaped by the
findings of projects partners contrasting various aspects of math education in
Russia/Georgia/Armenia and the EU. The rest of this book presents these findings
in detail.
1 Introduction 17

1.3 Perceptions of Mathematics

Christian Mercat and Mohamed El-Demerdash and Jana Trgalova
IREM Lyon, Université Claude Bernard Lyon 1 (UCBL), Villeurbanne, France
e-mail: christian.mercat@math.univ-lyon1.fr; jana.trgalova@univ-lyon1.fr
Pedro Lealdino Filho
Université Claude Bernard Lyon 1 (UCBL), Villeurbanne, France
e-mail: pedro.lealdino@etu.univ-lyon1.fr

1.3.1 Introduction

The global methodology of this comparative study project is based on the analysis
of the proposed curriculum and of the actual way this curriculum is implemented
in the classroom, in order to identify venues for improvements and modernisations,
implement them and study their effect.
In the literature, the philosophical features of the scientific spirit are evident in
the sciences which need more objectiveness, chiefly mathematics [11]. From the
ingenuous perception of a phenomenon, a pre-scientific spirit needs to overcome a
set of epistemological obstacles to reach a scientific stage. We consider this scientific
stage an important factor in order to learn, acquire and improve mathematical
competencies. The definition of mathematical competence on this project follows
the one used in the Danish KOM project and adapted in the SEFI Framework. It is
defined as “the ability to understand, judge, do, and use mathematics in a variety of
intra- and extra-mathematical contexts and situations in which mathematics plays
or could play a role”. The attitude towards mathematics is a long standing strand of
research and uses reliable measuring tools such as the seminal ATMI [52]. But our
research identified dimensions which we find specific for engineering, especially
through its relationship with reality.
Mathematics is considered as the foundation discipline for the entire spectrum of
Science, Technology, Engineering, and Mathematics (STEM) curricula. Its weight
in the curriculum is therefore high [1]. In Armenia, Georgia and Russia, all
university students pursuing this kind of curriculum are obliged to take a three
semester standard course in higher mathematics. Special studies in Europe suggest
that a competencies gap in mathematics is the most typical reason for STEM
students to drop out of study [4, 5, 10, 24, 31].
Several research studies show that students’ perceptions of mathematics and of
mathematics teaching have an impact on their academic performance of mathemat-
ics [12, 38], and a positive attitude and perceptions toward the subject will encourage
an individual to learn the subject matter better. In a broader sense, perceptions
towards mathematics courses are also important to take into account in order to
grasp the cultural differences between all the different institutions, from the point of
view of the students of the course, of their teachers and of the engineers themselves.
18 1 Introduction

The fact that culture does influence these beliefs, while seemingly obvious, is not
widely studied. Therefore this study fills a gap in the literature. Without assuming
cultural determination, we do show significant differences between institutions. We
present in this section a study that investigates these issues, the methodology of data
acquisition, the main themes that the study investigates, and the main results.
In order to evaluate students’ perceptions of mathematics we elaborated an online
survey spread out over all participant countries (Armenia, France, Finland, Georgia
and Russia). The survey was elaborated to investigate the three main dimensions of
the mathematical courses:
• The usefulness of mathematics.
• Mathematical courses in engineering courses—Contents and Methods.
• Perception of Mathematics.
The survey was spread with a web tool and translated into each partner language to
ensure that the meaning of the questions was adequately taken into account. A total
of 35 questions were answered by 1548 students from all participant countries.
After collecting the data from the online survey we used the statistical package
R to analyse the data and draw conclusions.
We performed a Principal Component Analysis to verify whether there were
some patterns in the students’ responses. Using the graphical representation of
data, we can propose the hypothesis that the methodology of teaching mathematics
of each country shapes the average students’ perception towards mathematics.
Thus, the first conclusion of this analysis is that in the European universities
the mathematics are taught as tools to solve problems, that is, mathematics by
practicing, while in the non-European universities, the mathematics are taught
focussing on proofs and theorems, that is, mathematics by thinking.

1.3.2 Theoretical Background and Research Question

Furinghetti and Pehkonen [19] claim that students’ beliefs and attitudes as regards
mathematics have a strong impact on their learning outcomes. Mathematics-related
perceptions are referred to as a belief system in the literature [15, 19]. Furinghetti
and Pehkonen point out that there is a diversity of views and approaches to the study
of beliefs in the field of mathematics education, and they conclude that the definition
of the concept of belief itself remains vague. Some researchers acknowledge that
beliefs contain some affective elements [35], while others situate beliefs rather
on the cognitive side [53]. Furinghetti and Pehkonen (ibid.) bring to the fore a
variety of concepts used by researchers to address issues related to beliefs, such
as conceptions, feelings, representations or knowledge. Some authors connect these
concepts, for example beliefs and conceptions, as Lloyd and Wilson [34]: “We use
the word conceptions to refer to a person’s general mental structures that encompass
knowledge, beliefs, understandings, preferences, and views”. Others distinguish
clearly between the two concepts, as Ponte [45]: “They [beliefs] state that something
1 Introduction 19

is either true or false, thus having a propositional nature. Conceptions are cognitive
constructs that may be viewed as the underlying organizing frame of concepts. They
are essentially metaphorical.” Furinghetti and Pehkonen (ibid.) attempt to relate
beliefs or conceptions and knowledge by introducing two aspects of knowledge:
“objective (official) knowledge that is accepted by a community and subjective
(personal) knowledge that is not necessarily subject to an outsider’s evaluation”
(p. 43).
The purpose of this study is not to contribute to the theoretical discussion on
these concepts, but rather to study how engineering students view mathematics and
its teaching in their schools. We therefore adopt the word ‘perceptions’ to address
these students’ views and opinions.
Breiteig, Grevholm and Kislenko [8] claim that “there are four sets of beliefs
about mathematics:
• beliefs about the nature of mathematics,
• beliefs about teaching and learning of mathematics,
• beliefs about the self in context of mathematics teaching and learning,
• beliefs about the nature of knowledge and the process of knowing.”
Our interest has thus been oriented toward such perceptions of mathematics
found with students in engineering courses. These students, engaged in the sciences,
have nevertheless different positions, whether philosophical, practical or epistemo-
logical, towards mathematics.
The study thus investigates the following question:
“How far do the students’ perceptions of mathematics in engineering courses
regarding the usefulness of mathematics in real life, the teaching of mathematics
(contents and methods) and the nature of mathematics knowledge differ in terms
of university, country (France, Finland, Russia, Georgia, Armenia), region (Cau-
casian, European, Russian) and gender (female, male)?”

1.3.3 Method and Procedures

Drawing on prior studies [16, 20, 38] related to students’ mathematics perceptions,
and in particular the four sets of beliefs about mathematics suggested by Breiteig
et al. [8], we have designed a questionnaire to gather and assess students’ percep-
tions of mathematics and their mathematics courses, and to get concrete indicators
of their beliefs about the following:
1. Usefulness of mathematics.
2. Teaching of mathematics in engineering schools, its contents and methods.
3. Nature of mathematical knowledge.
Given the target audience, namely students in engineering courses, we decided
not to address beliefs about ‘the self in context of mathematics teaching and
20 1 Introduction

Table 1.1 Questionnaire dimensions and numbers of corresponding items
Questionnaire dimension Number of questions
1. Usefulness of mathematics 8
2. Teaching of mathematics in engineering schools, contents and 15
methods
3. Nature of mathematics knowledge 12
Total 35

Table 1.2 Items related to the dimension “usefulness of mathematics”
Item n Usefulness of mathematics
1 Mathematics has an interest, especially for solving concrete problems.
2 Mathematics is used mostly in technical domains.
3 Mathematics is useless in everyday life.
4 Only applied mathematics is interesting.
5 Mathematics serves no purpose in human sciences.
6 Natural phenomena are too complex to be apprehended by mathematics.
7 Mathematics can be applied to man crafted objects and much less to objects found in
nature.
8 Learning mathematics in early classes serves mostly the purpose to help children get
around in life.

learning’ because this perception is not our focus here: mathematics cannot be
avoided and has to be confronted with.
Based on the three above-mentioned dimensions of the questionnaire, we
developed 35 questions that cover these dimensions as shown in Table 1.1.
The first dimension of the questionnaire (Table 1.2) explores the students’ beliefs
about the usefulness of mathematics. Chaudhary [14] points out that mathematics is
useful in everyday life: ‘since the very first day at the starting of the universe and
existence of human beings, mathematics is a part of their lives’ (p. 75) and in some
professions, such as architects who ‘should know how to compute loads for finding
suitable materials in design’, advocates who ‘argue cases using logical lines of
reasons; such skill is developed by high level mathematics courses’, biologists who
‘use statistics to count animals’, or computer programmers who develop software
‘by creating complicated sets of instructions with the use of mathematical logic
skills’ (p. 76). In line with such a view of the utility of mathematics, we proposed
eight items addressing the utility of mathematics in everyday life (items 1, 3, 8),
as well as in technical (item 2), natural (items 6, 7) and human (item 5) sciences.
Item 4 questions the perception of the usefulness of mathematics in relation with
the distinction between pure and applied mathematics.
A higher education evaluation conducted in France in 2002 [6] focussed, among
others, on the mathematics teaching in engineering schools. The study concludes
that mathematics takes a reasonable place among the subject matters taught: the
amount of mathematics courses is 16% of all courses in the first year of study, in
the second year 10% and in the third one only 6%. Another result of this study
1 Introduction 21

Table 1.3 Items related to the dimension “teaching mathematics in engineering schools”
Item n Teaching mathematics in engineering schools
9 In engineering school, mathematics are mostly pure and abstract.
10 We need more applied mathematics in engineer training.
11 In engineering school, theory are taught without taking into account their
applications.
12 There is almost no connection between math teaching and the engineer job reality.
13 Math teaching does not try to establish links with other sciences.
14 Mathematics weigh too much in engineer training.
15 Mathematics cannot be avoided in engineer training.
16 A teacher’s only purpose is to bring knowledge to students.
17 The structure of math courses does not allow learning autonomy.
18 With new means available to students, learning is no longer required; one just has to
quickly find solutions to problems that are encountered.
19 Courses have not changed in the last decades when the world is evolving greatly and
fast.
20 The mathematics courses are extremely theoretic.
21 The mathematics courses are not theoretical enough.
22 The mathematics courses are extremely practical.
23 The mathematics courses are not practical enough.

pointed out that engineering students are mostly taught basic mathematics and do
not encounter enough applications. Based on these results, we wished to gather
students’ perceptions about the teaching of mathematics they are given in their
engineering schools. Thus, in the second dimension of the questionnaire (Table 1.3),
we decided to address the students’ perceptions of the place mathematics takes in
their engineering education (item 14 and 15), of the balance between theoretical and
practical aspects of mathematics (items 9, 10, 11, 20, 21, 22, and 23) and of the links
established between mathematics and other subject matters (item 13). In addition,
we wanted to know whether the students feel that the mathematics teaching they
receive prepares them for the workplace (item 12). Finally, a set of items addressed
the students’ perceptions of the methods of teaching of mathematics (items 16, 17,
18, and 19).
The third dimension attempted to unveil the students’ implicit epistemology of
mathematics, i.e. their perceptions of “what is the activity of mathematicians, in
what sense it is a theoretical activity, what are its objects, what are its methods,
and how this all integrates with a global vision of science including the natural
sciences” [7]. The items related to this dimension (Table 1.4) addressed the relations
of mathematics with reality (items 27 and 30), with the truth (items 28, 29 and 32),
with other sciences (item 25), with creativity (items 24 and 26), and with models
(item 31). Moreover, they questioned the students’ perceptions of the nature of
mathematical objects (item 35) and the accessibility of mathematical knowledge
(items 33 and 34).
22 1 Introduction

Table 1.4 Items related to the dimension “perception of mathematics”
Item n Perception of mathematics
24 In mathematics, there is nothing left to discover.
25 Mathematics are only a tool for science.
26 There is no room for creativity and imagination in mathematics.
27 Mathematics raised from concrete needs.
28 There is no room for uncertainty in mathematics.
29 Only math can approach truth.
30 Mathematics is only an abstraction, it does not deal with reality.
31 A mathematical model is necessarily limited.
32 A mathematical theory cannot be refuted.
33 Mathematics cannot be the subject of a conversation (contrarily to literature or
philosophy).
34 Mathematics is better left to experts and initiated people.
35 Mathematics is a human construction.

Table 1.5 Participants of the study
Country Number of students Number of students with completed responses
Armenia 24 12
Finland 189 112
France 430 245
Georgia 285 179
Russia 612 410
Total 1548 958

We calculated the reliability coefficient (Cronbach’s alpha) by administering an
online version of the questionnaire to a sample of 1548 students from all participant
countries (see the Sample section). Students’ responses were analysed to calculate
the scores of each student. The reliability coefficient (Cronbach’s alpha) of 0.79 is
high enough to consider our questionnaire as a whole construct a reliable measuring
tool.
The experimental validity of the questionnaire as an estimation of the tool
validity is also calculated by taking the square root of the test reliability coefficient
[3]. Its value of 0.89 shows that the questionnaire has a high experimental validity.
An operational definition of students’ perception of their engineering courses is
therefore defined for us as a random variable taking vector values represented by the
Likert score of the students on the 35 items of the prepared questionnaire on a 1–6
Likert-type scale.
The population on which we base this study are students from partner universities
in two Tempus projects, MetaMath in Russia and MathGeAr in Georgia and
Armenia, and French and Finnish students on the European side. Within this
population a sample of 1548 students filled in the survey with 958 complete
responses (incomplete surveys discarded)—see Table 1.5.
1 Introduction 23

1.3.4 Data Analysis

After collecting the data from the online survey we used the statistical package R
to analyse the data and draw preliminary conclusions. We performed a Principal
Component Analysis (PCA) [27, 47, 54] to investigate patterns in the students’
responses. Although the students’ responses are not strictly speaking continuous but
are a Likert scale between 1 and 6, Multiple Factor Analysis, where different Likert
values are not numerically linked but used as simply ordered categories, did not
yield finer results. PCA uses a vector space transform to reduce the dimensionality
of large data sets giving some interpretation to variability. The original data set,
which involves many variables, can often be interpreted by projecting it on a few
variables (the principal components).
We used PCA to reveal patterns in students’ responses. Using the two first
principal components, explaining almost a quarter of the variability, we identify the
main common trends and the main differences. Principal components are described
in Table 1.6 and Fig. 1.5. In particular, the main result is that we can verify the
hypothesis that the methodology of teaching mathematics of each partner, and in
particular each country, shapes the average students’ perception of mathematics.
During students’ interviews and study visits in the project, we could point out
the main trends in the way mathematics is taught in the partner institutions. Thus
it appeared that mathematics in Europe is taught as a sophisticated tool addressing

Table 1.6 Importance of the components
PC1 PC2 PC3 PC4
Standard deviation 2.3179 1.69805 1.39846 1.2321
Proportion of variance 0.1535 0.0823 0.05588 0.0433
Cumulative proportion 0.1535 0.2358 0.29177 0.3351

4
Variances

0
Components

Fig. 1.5 Variance of the first 10 principal components
24 1 Introduction

Fig. 1.6 PCA grouped by countries

real engineer’s issues; it stands out with respect to a more theoretical approach in
the East. This fact does show in the data.
The analysis shows that all engineering students responding to the questionnaire
(15.2% of the variability) feel that mathematics teaching is too theoretical, is not
practical enough and does not have enough connection with other sciences and the
reality of an engineer’s job. Therefore, modernised curricula for engineers should
address these issues. On the other hand, we identify that Finnish and French students
(Fig. 1.6) share most of their perceptions, while the Caucasian students notably
differ from them, the Russian students lying in between with a broader variability
even given their size. The semantic analysis of the second principal component
(8.6% of variability) reveals that in the European universities, mathematics is taught
as a tool to solve problems, that is to say, by practicing of applying mathematics
to problems, while in the Caucasian universities, mathematics is taught focussing
on theorems and proofs, that is to say, mathematics is an abstract subject matter.
The Caucasian students tend to perceive of mathematics as consisting of knowledge
rather than competencies, mainly of theoretical interest, with a discrepancy between
early practical mathematics and theoretical engineering mathematics (Fig. 1.7).
The European students feel that advanced mathematics is useful, that the role
of a teacher is more to help students to apply mathematics than to only transmit
knowledge. The Russian students fall in between the two groups and are more
diverse in their opinions [32, 33].
Apart from the country and the institution, which do explain a lot of the
variability, we looked for characters separating students into groups in a statistically
1 Introduction 25

Fig. 1.7 PCA grouped by regions

significant way. In engineering courses, gender is a major differentiating trait
[2, 28]. And, to our surprise, the partner’s institution explains much better the
differences between students than gender: male and female students have very
similar responses, only 6 out of 35 questions are statistically distinguishable
(p-value < 0.05) and we have no clear-cut semantic explanation of the slight
differences: male students tend to disagree a little bit more strongly to the proposal
that mathematics can be applied more easily to man crafted objects than to objects
found in nature, while female students tend to find slightly more that mathematics
courses are practical enough. But the differences are much higher between partner
institutions than between genders: there are statistically greater differences between
the answers of a student in St Petersburg Electrotechnical University (LETI) and
another in Ogarev Mordovia State University (OMSU), both in Russia (much lower
p-values, with 16 out of 35 being less than 0.05) than between a male and a
female student in each university (Figs. 1.8 and 1.9). And the differences are even
higher between institutions belonging to different countries. We have to look at
the seventh principal component, which is almost meaningless, in order to get a
dimension whose interpretation of the variability relates to gender. The same relative
irrelevance with respect to age appears: students’ perceptions depend on the year of
study, but to an extent much lower than the dependency on the institution. We find
these results remarkable.
The main finding of this analysis is that there are indeed great differences
between students’ responses in partners’ higher education institutions, with homo-
geneous European universities tending to see engineering mathematics as a profes-
26 1 Introduction

Fig. 1.8 PCA grouped by institutions

Fig. 1.9 PCA grouped by gender
1 Introduction 27

sional tool on the one side; homogeneous Caucasian universities on the other, where
advanced mathematics are felt as dealing with abstraction, and Russian universities
in between.

1.3.5 Conclusions

In this study, we observed that European countries on the one hand and South
Caucasian countries on the other are quite aligned. However, Russian students’
perception is more spread out and in between those of the European and South
Caucasian students. The country factor has a large influence but within these
differences, institutions can be more finely differentiated and this difference is
higher than most other criteria, like gender: a student can be linked to his/er
university in a more confident way than to his/her gender or his/her year of study.
Comparison with other institutions would be interesting.
The main implication for the MetaMath and MathGeAr projects from this study
is that if the European way is to be promoted, the project should put forward the
applications of advanced mathematics and focus on competencies rather than on
transmission of knowledge.
This questionnaire has some limitations. For instance, its item-internal con-
sistency reliability was not high enough regarding the three dimensions of the
questionnaire which we identified a priori. The item-internal consistency reliabilities
measured by Cronbach’s alpha are 0.52, 0.65, and 0.62, which tells us that reality is
more complex than our question choices based on epistemology. It evokes the need
to further study to qualify the questionnaire with a bigger homogeneous sample
and/or redesign of the current questionnaire by adding more items related to these
dimensions or qualify these dimensions better.
Because perceiving mathematics in a positive way would influence students’
motivation and performance, it is desirable to change the mathematics contents
and the way we teach it in order to address the negative aspects of the perceptions
identified here, for instance teaching mathematics as a powerful modelling tool not
abstractly, but in actual students’ projects.
We might as well try to directly modify students’ perceptions by better informing
them about some aspects of mathematics; its usefulness in engineer’s profession
for example. Therefore, we need to know which type of mathematics in-service
engineers do use in a conscious way, and what their perceptions are of the
mathematics they received during their education.
The current study suggests further investigation avenues: the first one is to
study deeper the influence of engineering students’ perceptions on mathematics
performance for each partner institution. The second one is the elaboration of
questionnaires targeting engineers in order to study the perceptions and actual
usage of mathematics by professionals. Because the link between students and
engineers goes through teachers, we need to study as well the perceptions of teachers
themselves. We have already adapted this questionnaire in order to address these two
28 1 Introduction

targets and it will be the subject of subsequent articles. This study is only the first
real size pilot of a series of further studies to come.

References

1. ACME. (2011). Mathematical needs: the mathematical needs of learners. Report. Advisory
Committee on Mathematics Education.
2. Alegria, S. N., Branch, E. H. (2015). Causes and Consequences of Inequality in the STEM:
Diversity and its Discontents. International Journal of Gender, Science and Technology, 7(3),
pp. 321–342.
3. Angoff, W. H. (1988). Validity: An evolving concept. In H. Wainer, & H. Braun (Eds.) Test
validity (pp. 19–32), Hillsdale: Lawrence Erlbaum Associates.
4. Araque, F., Roldán, C., Salguero, A. (2009). Factors influencing university drop out rates.
Computers & Education, 53(3), pp. 563–574.
5. Atkins, M., Ebdon, L. (2014). National strategy for access and student success in higher
education. London: Department for Business, Innovation and Skills. Accessed 25 Octo-
ber 2016 from https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/
299689/bis-14-516-national-strategy-for-access-and-student-success.pdf
6. Benilan, P., Froehly, C. (2002). Les formations supérieures en mathématiques orientées vers les
applications - rapport d’évaluation. Comité national d’évaluation des établissements publics à
caractère scientifique, culturel et professionnel, France. Accessed 26 October 2016 from http://
www.ladocumentationfrancaise.fr/var/storage/rapports-publics/054000192.pdf
7. Bonnay, D., Dubucs, J. (2011). La philosophie des mathématiques. Published online.
Accessed 26 October 2016 from https://hal.archives-ouvertes.fr/hal-00617305/file/bonnay_
dubucs_philosophie_des_mathematiques.pdf
8. Breiteig, T., Grevholm, B, Kislenko, K. (2005). Beliefs and attitudes in mathematics teaching
and learning, In I. Stedøy (Ed.), Vurdering i matematikk, hvorfor og hvor? : fra småskoletil
voksenopplæring : nordisk konferanse i matematikkdidaktikk ved NTNU (pp. 129–138),
Trondheim: Norwegian University of Science and Technology.
9. Brusilovsky, P., Sosnovsky, S., Yudelson, M. (2009). Addictive links: the motivational value of
adaptive link annotation. New Review of Hypermedia and Multimedia, 15(1), pp.97–118.
10. Byrne, M., Flood, B. (2008). Examining the relationships among background variables and
academic performance of first year accounting students at an Irish University. Journal of
Accounting Education, 26(4), pp. 202–212.
11. Cardoso, W. (1985). Os obstáculos epistemológicos segundo Gaston Bachelard. Revista
Brasileira de História da Ciência, 1, pp. 19–27.
12. Chaper, E. A. (2008). The impact of middle school students’ perceptions of the classroom
learning environment on achievement in mathematics. Doctoral Dissertation. University of
Massachusetts Amherst.
13. Charette, Robert N. “An Engineering Career: Only a Young Person’s Game?” Institute
of Electrical and Electronics Engineers website (Accessed from http://spectrum.ieee.org/
riskfactor/computing/it/an-engineering-career-only-a-young-persons-game).
14. Chaudhary, M. P. (2013). Utility of mathematics. International Journal of Mathematical
Archive, 4(2), 76–77. Accessed 26 October 2016 from http://www.ijma.info/index.php/ijma/
article/view/1913/1133.
15. De Corte, E., Op’t Eynde, P. (2002). Unraveling students’ belief systems relating to mathe-
matics learning and problem solving. In A. Rogerson (Ed.), Proceeding of the International
Conference “The humanistic renaissance in mathematics education”, pp. 96–101, Palermo,
Italy.
1 Introduction 29

16. Dogan-Dunlap, H. (2004). Changing students’ perception of mathematics through an inte-
grated, collaborative, field-based approach to teaching and learning mathematics. Online
Submission. Accessed 25 October 2016 from http://files.eric.ed.gov/fulltext/ED490407.pdf.
17. EURYDICE (2011). Mathematics in Europe: Common Challenges and National Policies,.
(http://eacea.ec.europa.eu/education/eurydice).
18. The Final Report of the National Mathematics Advisory Panel (2008). U.S. Department of
Education.
19. Furinghetti, F., Pehkonen, E. (2002). Rethinking characterizations of beliefs. In G. C. Leder, E.
Pehkonen, G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 39–57).
Springer Netherlands.
20. Githua, B. N. (2013). Secondary school students’ perceptions of mathematics formative
evaluation and the perceptions’ relationship to their motivation to learn the subject by gender
in Nairobi and Rift Valley Provinces, Kenya. Asian Journal of Social Sciences and Humanities,
2(1), pp. 174–183.
21. Hanushek E. A., Wößmann L. The Role of Education Quality in Economic Growth. World
Bank Policy Research Working Paper No. 4122, February 2007. (http:/info.worldbank.org/
etools/docs/library/242798/day1hanushekgrowth.pdf)
22. Henderson S., Broadbridge P. (2009). Engineering Mathematics Education in Australia, MSOR
Connections Vol. 9, No 1, February - April.
23. Heublein, U., Schmelzer, R., Sommer, D. (2008). Die Entwicklung der Studienabbruchquote
an den deutschen Hochschulen: Ergebnisse einer Berechnung des Studienabbruchs auf der
Basis des Absolventenjahrgangs 2006. Projektbericht. HIS Hochschul-Informations-System
GmbH. Accessed 12 April 2013 from http://www.his.de/pdf/21/20080505_his-projektbericht-
studienabbruch.pdf.
24. Heublein, U., Spangenberg, H., & Sommer, D. (2003). Ursachen des Studienabbruchs: Analyse
2002. HIS, Hochschul-Informations-System.
25. Huikkola M., Silius K., Pohjolainen S. (2008). Clustering and achievement of engineering stu-
dents based on their attitudes, orientations, motivations and intentions, WSEAS Transactions
on Advances in Engineering Education, Issue 5, Volume 5, May, pp. 343–354.
26. Jonassen, D. H. (1995). Supporting Communities of Learners with Technology: A Vision for
Integrating Technology with Learning in Schools, Educational Technology, 35, 4, pp. 60–63.
27. Johnson, R. A., Wichern, D. W. (1999). Applied multivariate statistical analysis (6th ed). Upper
Saddle River, New Jersey: Prentice-Hall. Accessed 25 October 2016 from http://www1.udel.
edu/oiss/pdf/617.pdf
28. Jumadi, A. B., Kanafiah, S. F. H. M. (2013). Perception towards mathematics in gender
perspective. In Proceedings of the International Symposium on Mathematical Sciences and
Computing Research 2013 (iSMSC 2013) (pp. 153–156). Accessed 25 October 2016 from
http://lib.perak.uitm.edu.my/system/publication/proceeding/ismsc2013/ST_09.pdf.
29. Karp, Alexander, Bruce Ramon Vogeli (2011). Russian mathematics education: Programs and
practices. Vol. 2. World Scientific.
30. Kohlhase, M. (2006). OMDoc – An open markup format for mathematical documents [version
1.2]. Berlin/Heidelberg, Germany: Springer Verlag.
31. Lassibille, G., Navarro Gómez, L. (2008). Why do higher education students drop out?
Evidence from Spain. Education Economics, 16(1), pp. 89–105.
32. Lealdino Filho, P., Mercat C., El-Demerdash, M. (2016). MetaMath and MathGeAr projects:
Students’ perceptions of mathematics in engineering courses, In E. Nardi, C. Winsløw T.
Hausberger (Eds.), Proceedings of the First Conference of the International Network for
Didactic Research in University Mathematics (INDRUM 2016) (pp. 527–528). Montpellier,
France: University of Montpellier and INDRUM.
33. Lealdino Filho, P., Mercat C., El-Demerdash, M., Trgalová, J. (2016). Students’ perceptions of
mathematics in engineering courses from partners of MetaMath and MathGeAr projects. 44th
SEFI Conference, 12–15 September 2016, Tampere, Finland.
30 1 Introduction

34. Lloyd, G. M., Wilson, M. (1998). Supporting Innovation: The Impact of a Teacher’s Concep-
tions of Functions on His Implementation of a Reform Curriculum. Journal for Research in
Mathematics Education, 29(3), pp. 248–274.
35. McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualization.
In D. A. Grouws (Ed). Handbook of research on mathematics teaching and learning (pp. 575–
596). New York, NY, England, Macmillan Publishing Co.
36. Melis, E., Andres, E., Büdenbender, J., Frischauf, A., Goguadze, G., Libbrecht, P., Ullrich,
C. (2001). ActiveMath: a generic and adaptive Web-based learning environment. International
Journal of Artificial Intelligence in Education, 12(4), pp. 385–407.
37. Mullis I., Martin M., Foy P., Arora A., (TIMSS 2011). Trends in International Mathematics
and Science Study, International Results in Mathematics. (Accessed from http://timssandpirls.
bc.edu/).
38. Mutodi, P., Ngirande, H. (2014). The influence of students’ perceptions on mathematics
performance. A case of a selected high school in South Africa. Mediterranean Journal of Social
Sciences, 5(3), p. 431.
39. Niss, M., Højgaard T., (Eds.) (2011). Competencies and Mathematical Learning, Ideas and
inspiration for the development of mathematics teaching and learning in Denmark, English
edition, October 2011, Roskilde University, Department of Science, Systems and Models,
IMFUFA P.O. Box 260, DK - 4000 Roskilde. (Accessed from http://diggy.ruc.dk/bitstream/
1800/7375/1/IMFUFA_485.pdf).
40. NSF Report (2007). Moving forward to improve engineering education #NSB-07-122.
41. OECD (2015). Students, Computers and Learning: Making the Connection, PISA, OECD
Publishing. (Accessed from http://dx.doi.org/10.1787/9789264239555-en).
42. OECD (2008). Report on Mathematics in Industry, Global Science Forum, July. (Accessed
from https://www.oecd.org/science/sci-tech/41019441.pdf).
43. PISA (2009). Assessment Framework: Key competencies in reading, mathematics and science.
(Accessed from http://www.oecd.org/pisa/pisaproducts/44455820.pdf).
44. PISA (2012). The Programme for International Student Assessment. (Accessed from http://
www.oecd.org/pisa/keyfindings/pisa-2012-results.htm).
45. Ponte, J. P. (1994). Knowledge, beliefs, and conceptions in mathematics teaching and learning.
In L. Bazzini (Ed.), Proceedings of the 5th International Conference on Systematic Cooperation
between Theory and Practice in Mathematics Education (pp. 169–177). Pavia: ISDAF.
46. Reusser, K., (1995). From Cognitive Modeling to the Design of Pedagogical Tools. In S.
Vosniadou, E. De Corte, R. Glaser, H. Madl (Eds.). International Perspectives on the Design of
Technology-Supported Learning Environments, Hillsdale, NJ: Lawrence Erlbaum, pp. 81–103.
47. Richardson, M. (2009). Principal component analysis, Accessed 25 October 2016 at http://
www.sdss.jhu.edu/szalay/class/2015/SignalProcPCA.pdf
48. Ruokamo H., Pohjolainen S. (1998). Pedagogical Principles for Evaluation of Hypermedia
Based Learning Environments in Mathematics, Journal for Universal Computer Science, 4 (3),
pp. 292–307, (Accessed from http://www.jucs.org/jucs_4_3/)
49. SEFI (2013), A Framework for Mathematics Curricula in Engineering Education. (Eds.)
Alpers, B., (Assoc. Eds) Demlova M., Fant C-H., Gustafsson T., Lawson D., Mustoe L.,
Olsson-Lehtonen B., Robinson C., Velichova D. (Accessed from http://www.sefi.be).
50. SEFI (2002). Mathematics for the European Engineer. (Eds.) Mustoe L., Lawson D. (Accessed
from http://sefi.htw-aalen.de/Curriculum/sefimarch2002.pdf).
51. Sosnovsky, S., Dietrich, M., Andrés, E., Goguadze, G., Winterstein, S., Libbrecht, P., Siek-
mann, J., Melis, E. (2014). Math-Bridge: Closing Gaps in European Remedial Mathematics
with Technology Enhanced Learning. In T. Wassong, D. Frischemeier, P. R. Fischer, R.
Hochmuth, P. Bender (Eds.), Mit Werkzeugen Mathematik und Stochastik lernen - Using Tools
for Learning Mathematics and Statistics. Berlin/Heidelberg, Germany, Springer, pp. 437–451
52. Tapia, M., Marsh, G. E. (2004). An Instrument to Measure Mathematics Attitudes. Academic
Exchange Quarterly, 8(2), pp. 16–21.
1 Introduction 31

53. Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In
D. A. Grouws (Ed), Handbook of research on mathematics teaching and learning: A project
of the National Council of Teachers of Mathematics (pp. 127–146). New York, NY, England:
Macmillan Publishing Co.
54. Tipping, M. E., Bishop, C. M. (1999). Probabilistic principal component analysis. Journal of
the Royal Statistical Society: Series B (Statistical Methodology), 61(3), pp. 611–622.

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give appropriate
credit to the original author(s) and the source, provide a link to the Creative Commons license and
indicate if changes were made.
The images or other third party material in this chapter are included in the chapter’s Creative
Commons license, unless indicated otherwise in a credit line to the material. If material is not
included in the chapter’s Creative Commons license and your intended use is not permitted by
statutory regulation or exceeds the permitted use, you will need to obtain permission directly from
the copyright holder.
Chapter 2
Methodology for Comparative Analysis
of Courses

Sergey Sosnovsky

2.1 Introduction

One of the main goals of the projects MetaMath and MathGeAr was to conduct a
range of comparative case studies that would allow the consortium to understand
the set and the magnitude of differences and commonalities between the ways
mathematics is taught to engineers in Russia, Georgia, Armenia and in the EU.
Equipped with this knowledge the consortium was able to produce recommenda-
tions for modernisation of existing Russian, Georgian and Armenian mathematics
courses during the project.
However, before such comparative analyses can even begin, one needs a clear
set of criteria that would determine the nature and the procedure of the planned
comparison. Identification of these criteria was the goal of this methodology
development. The results of this activity are described in this chapter.
The methodology has been developed by the EU partners (especially, Saarland
University and Tampere University of Technology) in consultation with Russian,
Georgian and Armenian experts. Dedicated methodology workshops have been
organised by the Association for Engineering Education of Russia in the Ministry
of Education and Science in Moscow, Russia, and in the National Center for
Educational Quality Enhancement in Tbilisi, Georgia, in June, 2014. During these
workshops, the first draft of the comparison criteria has been discussed with the
partners and invited experts. It has been amended in the future versions of the
Methodology to better match the realities of the university education in Russia,
Georgia and Armenia and provide a more objective picture of it with regards to
mathematics for engineers. Also, during the workshop, the SEFI Framework for

S. Sosnovsky ()
Utrecht University, Utrecht, the Netherlands
e-mail: s.a.sosnovsky@uu.nl

© The Author(s) 2018 33
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_2
34 S. Sosnovsky

Mathematical Curricula in Engineering Education [1] has been introduced to the
Russian, Georgian and Armenian partners as the unified instrument to describe and
compare the content of the corresponding courses.

2.2 Criteria for Conducting Comparative Case Studies

2.2.1 University/Program Profile

When comparing courses, it is not enough to choose courses with similar titles. The
goals and the purpose of the courses should align and the focus of the courses and
their roles within the overall curricula should be comparable. Even the character of
the university where a course is offered can make a difference. A classic university
and a university of applied sciences can have very different perspectives on what
should be the key topics within courses with the same name. In a larger university
a professor can have a much richer set of resources than in a smaller one; at the
same time, teaching a course to several hundreds of students puts a much bigger
strain on a professor than teaching it to several dozens of them. The overall program
of studies that the target course is part of is equally important for similar reasons.
Therefore, the first set of criteria characterising a course profile focus on the general
description of the university and the program (major) where the course is taught.
These parameters include:
• Criterion A: University profile
Classic or applied
Overall number of students
Number of STEM disciplines
Number of STEM students
• Criterion B: Program/discipline profile
Theoretical or applied
Number of students
Role/part of mathematics in the study program

2.2.2 Course Settings

The next set of criteria describes the context of the course including all its
organisational settings and characteristics not directly related to pedagogical aspects
or the content. This is the metadata of the course, which allows us to easily identify
whether the two courses are comparable or not. For example, if in one university a
course is taught on a MSc level and in another on a BSc level, such courses are not
directly comparable, because the levels of presentation of the course material would
differ much. If in one university a course’s size is 3 ECTS credits and in the other 7
2 Methodology for Comparative Analysis of Courses 35

ECTS, such courses are not the best candidates for comparison either, because the
amount of work students need to invest in these two courses will be very different
even if the titles of the courses are similar. Sometimes, we had to relax some of these
conditions if for particular universities best matches cannot be found. The complete
list of course characteristics include:
• Criterion C: Course type
Bachelor or Master level
Year/semester of studies (1/2/. . .)
Selective or mandatory
Theoretical/applied
• Criterion D: Relations to other courses in the program
Prerequisite courses
Outcome courses
If the course is a part of a group/cluster (from which it can be selected), other
courses in the group
• Criterion E: Department teaching the course
Mathematical/graduating/other
• Criterion F: Course load
Overall number of credits according to ECTS regulations
Number of credits associated with particular course activities (lec-
tures/tutorials/practical work/homework/etc.)

2.2.3 Teaching Aspects

In order to describe how the teacher organises the course, we identify three
important criteria: use of any particular didactic approach (such as project-based
learning, inquiry-based learning, blended learning, etc.), organisation of course
assessment (how many tests and exams, what form they take, how they and the
rest of the course activity contribute to the final grade) and the resources available
to a teacher—from the help of teaching assistants to the availability of computer
labs. Teaching aspects include:
• Criterion G: Pedagogy
Blended learning
Flipped classroom
MOOC
Project-based learning
Inquiry-based learning
Collaborative learning
Game-based learning
• Criterion H: Assessment
Examinations (how many, oral/written/test-like)
Testing (how often)
36 S. Sosnovsky

Grade computation (contribution of each course activity to the final grade,
availability extra credits)
• Criterion I: Teaching resources
Teaching hours
Preparatory hours
Teaching assistants (grading/tutorials)
Computer labs

2.2.4 Use of Technology

One of the aims of the MetaMath and MathGeAr projects is to examine and
ensure the effective use of modern ICT in math education. Therefore, a dedicated
group of criteria has been selected to characterise the level of application of these
technologies in the target courses. There are two top-level categories of software
that can be used to support math learning: the instruments that help students
perform essential math activities and the tools that help them to learn mathematics.
The former category includes such products as MATLAB, Maple, Mathematica,
or SPSS. These are, essentially, the systems that a professional mathematician,
engineer or researcher would use in everyday professional activity. Using them
in a course not only helps automate certain computational tasks but also leads to
mastering these tools, which is an important mathematical competency on its own.
The latter systems are dedicated educational tools. They help students understand
mathematical concepts and acquire general mathematical skills. In MetaMath and
MathGeAr projects, we apply a particular tool like this—an intelligent education
platform Math-Bridge. In both these categories, the number and diversity of
available systems is very large. The focus of the criteria in this set is to detect
whether any of these systems are used and to what degree, namely what the role
in the course is.
• Criterion J: Use of math tools
Name of the tool(s) used (MATLAB, Maple, MathCAD, Mathematica, SPSS,
R, etc.)
Supported activities (tutorials, homework)
Overall role of the tool (essential instrument that must be learnt or one way to
help learn the rest of the material)
• Criterion K: Use of technology enhanced learning (TEL)-systems
Name and type of tool used (Geogebra—math simulation; STACK—
assessment software; Math-Bridge—adaptive learning platform; etc.)
Supported activity (assessment, homework, exam preparation)
Role on the course (mandatory component/extra credit opportunity/fully
optional supplementary tool)
2 Methodology for Comparative Analysis of Courses 37

2.2.5 Course Statistics

Another important aspect of the course is the data collected about it over the years.
It shows the historic perspective and evolution of the course, and it can also provide
some insights into the course’s difficulty and the profile of a typical student taking
a course. Although by itself this information might be not as important for course
comparison, combined with other criteria it can provide important insights.
• Criterion L: Course statistics
Average number of students enrolled in the course
Average percentage of students successfully finishing the course
Average grade distribution
Percentage of international students
Overall student demographics (gender, age, nationality, scholarships, etc.)
Average rating of the course by students

2.2.6 Course Contents

Finally, the most important criterion is the description of the learning material taught
in the course. In order to describe the content of the analysed courses in a unified
manner that would allow for meaningful comparison we needed a common frame
of reference. As the context of mathematical education in this project is set for
engineering and technical disciplines, we have decided to adopt a “Framework for
Mathematics Curricula in Engineering Education” prepared by the Mathematics
Working Group of the European Society for Engineering Education (SEFI) [1]. This
report is written about every 10 years; and the current edition formalises the entire
scope of math knowledge taught to engineering students in EU universities in terms
of competences. The competencies are broken into four levels, from easier to more
advanced, and allow for composite representation of any math course. As a result,
every course can have its content described in terms of atomic competencies and
two similar courses can easily be compared based on such descriptions.
• Criterion M: Course SEFI competency profile
Outcome competencies of the course (what a student must learn in the course)
Prerequisite competencies of the course (what a student must know before
taking the course)
38 S. Sosnovsky

2.3 Application of the Criteria for the Course Selection
and Comparison

This set of criteria should be used (1) for selecting appropriate courses for the
comparison and (2) for conducting the comparison itself. During the selection
process, Criteria A and B will ensure that only universities and study programs
with matching profiles are selected. Criteria C, D, E and F will help to select the
courses that correspond in terms of their metadata. Criteria G, H and I will help to
filter out courses that utilise unconventional pedagogical approaches or differ too
much in terms of assessment organisation and teaching resources available.
At this point, if a pair of courses passed the screening, they can be safely
compared; all criteria starting J contribute to the comparison. One needs to note that,
in some cases, the strict rules of course selection might not apply, as a particular
partner university sometimes presents a very unique case. In such situations, the
selection rules can be relaxed.

Reference

1. SEFI (2013). A Framework for Mathematics Curricula in Engineering Education. (Eds.) Alpers,
B., (Assoc. Eds) Demlova M., Fant C-H., Gustafsson T., Lawson D., Mustoe L., Olsson-
Lehtonen B., Robinson C., Velichova D. (http://www.sefi.be).

Yury Pokholkov, Kseniya Zaitseva (Tolkacheva), Mikhail Kuprianov,
Iurii Baskakov, Sergei Pozdniakov, Sergey Ivanov, Anton Chukhnov,
Andrey Kolpakov, Ilya Posov, Sergey Rybin, Vasiliy Akimushkin,
Aleksei Syromiasov, Ilia Soldatenko, Irina Zakharova, and Alexander Yazenin

Y. Pokholkov · K. Zaitseva (Tolkacheva)
Association for Engineering Education of Russia (AEER), Tomsk, Russia
e-mail: pyuori@mail.ru
M. Kuprianov · I. Baskakov · S. Pozdniakov · S. Ivanov () · A. Chukhnov · A. Kolpakov ·
V. Akimushkin
Saint Petersburg State Electrotechnical University (LETI), St. Petersburg, Russia
e-mail: mskupriyanov@mail.ru; bosk@bk.ru; sg_ivanov@mail.ru
I. Posov
Saint Petersburg State Electrotechnical University (LETI), St. Petersburg, Russia
Saint Petersburg State University (SPbU), St. Petersburg, Russia
e-mail: i.posov@spbu.ru
S. Rybin
Saint Petersburg State Electrotechnical University (LETI), St. Petersburg, Russia
ITMO University, Department of Speech Information Systems, St. Petersburg, Russia
A. Syromiasov
Ogarev Mordovia State University (OMSU), Department of Applied Mathematics,
Differential Equations and Theoretical Mechanics, Saransk, Russia
e-mail: syal1@yandex.ru
I. Soldatenko · A. Yazenin
Tver State University (TSU), Information Technologies Department, Applied Mathematics
and Cybernetics Faculty, Tver, Russia
e-mail: soldis@yandex.ru; Yazenin.AV@tversu.ru
I. Zakharova
Tver State University (TSU), Mathematical Statistics and System Analysis Department,
Applied Mathematics and Cybernetics Faculty, Tver, Russia
e-mail: zakhar_iv@mail.ru

© The Author(s) 2018 39
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_3
40 Y. Pokholkov et al.

3.1 Review of Engineer Training Levels and Academic
Degrees

Higher education in general and engineering education in particular are divided into
three levels in Russia. The first level is the Bachelor’s degree course; the second
level is the specialist and Master’s degree program; the third level is the postgraduate
training program.
“The Bachelor’s degree” training level was introduced in Russia in 1996. The
standard training period to get the Bachelor qualification (degree) is at least 4
years with the total study load volume of 240 credits. The Bachelor qualification
is conferred based on the results of a presentation of the graduate thesis at a session
of the State Certifying Commission.
In 1993 the term of Master was established in Russia as the qualification of the
graduates of educational institutions of higher professional education. The standard
period of the Master training program (for the intramural form of study) is currently
2 years with the credit value of the educational program of 120 credits. Before
that, however, the student is to complete the Bachelor (4 years) or specialist (5–6
years) training program. The Master’s qualification (academic degree) is conferred
only after presentation of the Master’s thesis at a session of the State Certifying
Commission.
The Bachelor’s and Master’s qualifications (degrees) were historically preceded
by the specialist qualification presupposing 5–6 years of continuous learning. In
the Soviet Union it was the only possible qualification; but then the gradual
transition to the Bachelor and Master levels took place. At present the specialist
qualification has been preserved. When a prospective student applies documents to
a university he or she may be admitted to a Bachelor or specialist’s training program
(depending on the selected department, future profession, etc.). Today, however,
the specialist qualification is comparatively rare, having receded in favor of the
Bachelor’s training program. It is conferred based on the results of the presentation
of a graduation project or graduation thesis at the session of the State Certifying
Commission and gives the right to enter the Master’s Degree course (although like
the Master’s Degree course the specialist degree course is the second level of higher
education) and the postgraduate training program.
The postgraduate program is one of the forms of training of top-qualification
personnel. Before September 1, 2013 the postgraduate program was one of major
forms of training of the academic and scientific personnel in the system of
postgraduate professional education. Since September 1, 2013 (the date when the
Federal Law No. 273-FZ dated December 29, 2012 “On Education in the Russian
Federation” came into force) the postgraduate program was referred to the third
level of higher education. The person who has completed the postgraduate program
and presented a thesis receives the academic degree of a candidate of the sciences.
In the USSR, the Russian Federation (RF) and in a number of Commonwealth
of Independent States (CIS) countries this degree corresponds to the Doctor of
Philosophy degree (PhD) in western countries. Presentation of a candidate thesis
3 Overview of Engineering Mathematics Education for STEM in Russia 41

is public and takes place in a special dissertation council in one or several related
scientific specialties. In many cases the thesis is presented outside of the higher
educational institution where the applicant for the degree studied.
A Doctor of Sciences is the top academic degree (after the Candidate of
Sciences). In Russia the degree of a Doctor of Sciences is conferred by the Presidium
of the Higher Attestation Commission (VAK) of the Ministry of Education and
Science of the Russian Federation based on the results of the public presentation
of the doctorate thesis in a specialized dissertation council. The applicant for the
degree of a Doctor of Sciences is to have the academic degree of a Candidate of
Sciences. An approximate analog of the Russian doctoral degree accepted in Anglo-
Saxon countries is the degree of a Doctor of Sciences (Dr. Sc.) or the German Doctor
habilitatis degree (Dr. habil.).

3.2 Forms of Studies of Engineering Students

Three forms of obtaining education have been traditionally established in Russia:
intramural (full-time), part-time (evening time), extramural.
In the intramural form of education the student attends lectures and practical
classes every day or almost every day (depending on the timetable). Most classes
are held in the morning or in the afternoon (hence the second name).
The evening time form of studies is primarily designed for those students
who work during the day. In this connection classes are held in the evening
hours. Accordingly, less time is provided for classroom studies and the volume of
unsupervised activities increases.
The extramural form of studies presupposes that students meet teachers only
during the examination periods which take place 2–3 times a year and are 1–2 weeks
long each. During these periods students have classroom studies and take tests and
exams; besides, students get assignments which should be done in writing by the
beginning of the next examination period. Thus, in the extramural form of studies
the contact work with the teacher is minimal and the volume of unsupervised work
is maximum.
The new law of education (to be further discussed below) includes new forms
of organization of education: on-line learning and remote learning. Network
learning presupposes that a student studies some subjects in one higher educational
institution and other subjects in another and then gets a “combined” diploma of both
institutions. In remote learning the students communicates with the teacher mostly
by means of Internet and classes are conducted in the form of a video conference.
42 Y. Pokholkov et al.

3.3 Statistics and Major Problems of Engineering Education
in Russia

The Russian Federation has 274 engineering higher educational institutions, training
1,074,358 students. With account of comprehensive universities also admitting
students for engineering training programs the total number of higher educational
institutions where a student can obtain engineering education is 560. The total
number of engineering students is about a million and a half. The distribution of
engineering students by regions of Russia1 is shown on the map in Fig. 3.1. The
numbers on the map are described in Table 3.1.
The number of students per 10 thousand inhabitants of the population varies
from 78 (East-Siberian region) to 295 (Northwest region) but in other regions the
distribution is more uniform, and it varies between 150 and 200 students per 10,000
inhabitants.
The problems of Russian engineering education include:
• Disproportionality between the distribution of higher educational institutions by
regions of Russia and the territorial distribution of production facilities.
• Low quality of admission (weak school knowledge of many prospective stu-
dents).
• Low level of Russian domestic academic mobility.
• Seclusion from international educational networks.

Fig. 3.1 Geographical overview of Russian engineering education

1 Statistics (2016), http://aeer.cctpu.edu.ru/winn/ingobr/tvuz_main.htm.
3 Overview of Engineering Mathematics Education for STEM in Russia 43

Table 3.1 Statistics of Russian engineering education
Number of engineering
students in higher Number of
# Region educational institutions engineering students
1 North region 12 25, 481
2 Northwest region 24 123, 386
3 Central region 92 338, 002
4 Volga-Vyatka region 7 31, 455
5 Central Black Earth region 10 38, 819
6 Volga region 34 114, 324
7 North Caucasus region 21 81, 650
8 Ural region 23 103, 071
9 West-Siberian region 24 119, 207
10 East-Siberian region 20 66, 462
11 Far East region 6 28, 039
12 Kaliningrad region 1 4462
Total 274 1, 074, 358

3.4 Regulatory Documents

The system of state standardization of higher education program, acting from the
mid-1990s, since the introduction of federal state standards (FSES 3), is relaxing
strict regulation of the contents of education in the form of a specified set of subjects
with a fixed amount of credits (state educational standards SES-1, SES-2), is now
developing towards regulation of the structures of educational programs, conditions
of implementation and results of learning (FSES 3, FSES 3+, in the long term FSES
4). For example SES-2 contained a cyclic structure:
• GSE cycle—general humanities and social-economic subjects;
• EN cycle—general mathematical and science subjects;
• OPD—general professional subjects;
• DS—specialization subjects;
• FTD—optional subjects.
The central place in SES-2 is taken by section 4 “Requirements to the compulsory
minimal contents of the basic educational training program”. This section includes
a list of compulsory subjects for every cycle, their credit values in academic hours
and a mandatory set of didactic units.
44 Y. Pokholkov et al.

Convergence of the national systems of education within the frames of the
European Union is an important landmark in the global development of the
higher school in the twenty-first century. The official date of the beginning of the
convergence and harmonization process in higher education of European countries
with a view to creation of harmonized higher education is considered to be June
19, 1998 when the Bologna Declaration was signed. Russia joined the Bologna
process in 2003. As a result of this joining the educational process in most European
countries is currently in the process of reforming. Higher educational institutions
have set the task of not to unify but to harmonize their educational programs with
others. In this connection, the state educational standards are undergoing rethinking
and considerable changes.
With the introduction of the federal state standards of the third generation (FSES
3, 2011) Russian higher educational institutions gain greater independence in the
formation of the major educational programs, choice of the learning contents, forms
and methods, which enables them to compete on the market of educational services,
and to respond to the demands of the labor market.
One of the major distinguishing features of the new standards is the competency-
based approach. The essence of this approach is that the focus of the educational
process is transferred from the contents of education to the results of studies which
should be transparent, i.e. clear to all the stakeholders: employers, teachers, and
students. The results of training are described by means of a system of competencies
being a dynamic combination of knowledge, aptitudes, skills, abilities and personal
qualities, which the student can demonstrate after completion of the educational
program. The federal state standards of the third generation have inherited a cyclic
structure. A major specific feature of the FSES of the higher professional education
was the use of credits as a measure of the credit value of educational programs. The
indicators of the credit value of educational programs, general speaking, the credit
value of the cycles of subjects, are set in educational standards in credit units. For
example, the aggregate credit value of the bachelor training is set at 240 credit units,
Master training 120 credit units, specialist 300 credit units.
Just as “an academic hour”, a “credit unit” is a unit of measurement of the credit
value of academic work, but much more consistently oriented towards the work of
the student rather than to the teacher. In all international and national systems there
is a correspondence between credits units and hours. The method recommended by
the Ministry of Education of Russia in 2002 establishes the equivalent of 1 credit
unit to correspond to 36 academic hours.
3 Overview of Engineering Mathematics Education for STEM in Russia 45

The central place of these standards was taken by the section with a list of study
cycles, mandatory subjects for every cycle which regulated the credit value of every
cycle in credit units and code of competencies formed in studying the subjects.
Another specific feature of FSES 3+ is the introduction of the postgradu-
ate program (postgraduate military course), residency training and assistantship–
traineeship into the levels of higher education.
Pursuant to 273-FZ, dated 29.12.2012, in developing the main curriculum an
educational organization independently determines the distribution of the learning
material by subjects and modules and establishes the sequence of their study.
The competency-based approach demanded comprehensive restructuring and
modernization of the existing education system. Effective use of the competency-
based approach is unthinkable without an adequate system of appraisal of every
formed competency of the student determined by the state standard as mandatory
for the particular educational program. Accordingly, there is need for development
and introduction of the fund of means of appraisal allowing such an appraisal to
provide a qualified conclusion regarding the conformity of the educational process
to regulatory requirements. The need of the availability of such a fund with every
educational organization is unequivocally enshrined in the Order of the Ministry
of Education and Science of the Russian Federation dated 19.12.2013 No.1367
(revised on 15.01.2015): “20. Appraisal means are presented in the form of the fund
of appraisal means for midterm assessment of the learners and for final (state final)
assessment. 21. The fund of appraisal means for midterm assessment of learners in
the subject (module) or practice included, respectively, into the steering program of
the subject (module) or program of practice contains a list of competencies stating
the stages of their formation in the process of study of the educational program;
description of the indicators and criteria of appraising the competencies at different
stages of their formation; description of the appraisal scales, standard assignments
for submission or other materials necessary to appraise the knowledge, aptitudes,
skills and (or) experience of activities characterizing the stages of formation of the
competencies in the process of study of the educational program; guidance materials
determining the procedures of appraisal of the knowledge, skills and (or) experience
of activities characterizing the stages of formation of the competencies. For every
result of study in a subject (module) or practice the organization determines the
indicators and criteria of appraising the formedness of competencies at different
stages of their formation and the appraisal procedures.” The list of universal
competencies has been approved by the Ministry of Education and Science of the
RF. Universal competencies within the frames of the concept of modern education
form the level of development of a specialist distinguishing a specialist with higher
education from a specialist of a lower level.
46 Y. Pokholkov et al.

Besides, the methodological recommendations of the Ministry of Education and
Science of the Russian Federation dated 22.01.2015 gear educational organizations
to taking account of the requirements of the relevant professional standards in the
creation of the basic educational programs.
Initially, FSES 3+ was to point to conformity to professional standards. The
professional standard of a characteristic of the qualification necessary for an
employee to perform a particular kind of professional activities. A professional
standard is actually a document containing requirements to:
• the level of the employee’s qualification;
• the experience of practical activities, education and learning;
• the contents and quality of the activities;
• the conditions of performance of the labor activities.
As of the moment of issue of FSES 3+ the professional standards in most
fields of professional activities had not yet been approved; therefore, FSES 3+
could not formulate the graduate’s professional competencies oriented towards
generalized labor functions (kinds of professional activities) set by a concrete
professional standard (PS). Analysis of the structure of the already approved PS has
shown the impossibility to establish a mutually equivocal correspondence between
fields of professional activities and educational fields. Therefore, “the core” of
training has been identified in FSES 3+ in the form of universal (general culture)
competencies and general professional competencies (independent of the particular
kind of professional activities for which the learner is preparing and the focus
(specialization) of the program). “The core” of training determines the “basic”
part of the educational program which is quite fundamental and unalterable. “The
variative part” of the program is oriented towards particular generalized labor
functions or kind (kinds) of professional activities set by professional standards
(if available). This part of the program is to be easily renewable and adaptable to
new demands of the labor market. Figure 3.2 presents the structure of the list of
education areas in the RF stating the number of consolidated groups of specialties
and specializations included in every area of education and the number of such
specializations.
Higher educational institutions are currently facing a crucial task: development of
educational programs with account of the available professional standards, creation
of adequate funds of means of appraisal. Of interest in this connection will be
the available experience of the leading Russian universities in this area gained in
the implementation of the international project 543851-TEMPUS-1-2013-1-DE-
TEMPUS-JPCR (MetaMath) “Modern educational technologies in development of
the curriculum of mathematical subjects of engineering education in Russia” and
the Russian scientific-methodological projects “Scientific-methodological support
3 Overview of Engineering Mathematics Education for STEM in Russia 47

Fig. 3.2 List structure of education areas

of development of exemplary basic professional educational programs (EBPEP)
by areas of education”, “Development of models of harmonization of professional
standards and FSES of higher education by fields of study/specialties in the field
of mathematical and natural sciences, agriculture and agricultural sciences, social
sciences, humanities and levels of education (Bachelor’s, Master’s, specialist degree
programs)”. Within the frames of grants working groups of Russian higher educa-
tional institutions developed exemplary educational programs of higher education
in the modular format under the conditions of the introduction of “framework”
federal state educational standards—FSES 3+ and in the long term FSES-4. The
developers give practical recommendations for implementation of the competency-
based approach in designing and implementing the educational programs.
48 Y. Pokholkov et al.

3.5 Comparison of Russian and Western Engineering
Education

Given below is the comparison of Russian and western systems of education in
terms of several formal features (such as the number of academic hours allotted for
study of the Bachelor’s degree program). This analysis certainly cannot be called
complete; the contents of the educational program and the teaching quality depend
on the particular higher educational institution and department. Nevertheless, such
a comparison is to emphasize the greatest similarities and differences between
educational systems in Russia and Europe.
The information as regards the structure of domestic educational programs is
presented by the example of the curriculum of the field of Software Engineering of
Ogarev Mordovia State University (OMSU) as one of the typical representatives of
the Russian system of education. An example of a European technical university
is Tampere University of Technology, TUT (Finland). The information has been
taken from the TUT Study Guide; the authors were oriented towards the degrees of
a Bachelor of Science in Technology and a Master of Science (Technology) in the
field of Information technology (Tables 3.2, 3.3, 3.4, 3.5, 3.6, 3.7 and 3.8).

Table 3.2 Academic hours and credit units
Russia (OMSU) European Union (TUT)
1 academic hour = 45 min 1 academic hour = 45 min
1 credit unit (CU) = 36 h (adapted to a 1 credit unit (credit, cu, ECTS) = 26 2/3 h in
18-week term, weekly credit amount is 1.5 Finland (25–30 h in different countries of the
CU) EU)
CU includes all kinds of student’s work, ECTS includes all kinds of student’s work,
including independent studies including independent studies
Credit value of 1 year 60 CU Credit value of 1 year 60 ECTS
The credit value of a subject is a whole Formally the credit value of a subject is a
number (at least half-integer) of CU, i.e. a whole number of ECTS, although it may be a
multiple of 36 or 18 h rounded off number (Discrete Mathematics): 4
ECTS = 105 h)
3 Overview of Engineering Mathematics Education for STEM in Russia 49

Table 3.3 Freedom in choosing subjects
Russia (OMSU) European Union (TUT)
When entering the university the student When entering the university the student
chooses both the department and the field of chooses the study program. Orientation is
study. Thereby the student invariably chooses organized for first-year students before the
the greater part of subjects: both in terms of first week of studies. The student must (by
quantity and volume of hours the selected means of a special online-instrument) draw
subjects make at least 1/3 of the volume of the up a Personal Study Plan approved by the
variative part. On average this is about 1/6 of department. Most subjects are mandatory but
the total volume of academic hours (the many courses can be freely chosen; the order
mandatory and variative parts are of their study is recommended but can be
approximately equal in terms of volume). modified by the student. The student enrolls
Choosing the educational program for the courses to get the necessary amount of
specialization the student automatically makes credits (180 for the Bachelor’s degree
the decision on all positions of the program, 120 more for the Master’s degree).
professional subjects as chosen Some courses or minors can be taken from
another higher educational institution (these
may be the so-called minor studies—see
below). The students have to annually
confirm their plans to continue their education
at the TUT (see below)

Table 3.4 Organization of the educational process
Russia (OMSU) European Union (TUT)
There are lectures and practical classes Courses may contain only lectures or also
stipulated in all subjects. The total share of contain practical exercises, laboratory work
lectures in every cycle of the subjects (GSE, or work in groups. These exercises sometimes
EN, OPD + DS + FTD) is not more than 50% are mandatory
The details of organization of the educational The details of organization of the educational
process in every subject are given at one of the process in every subject are given at the
first classes introductory lecture
Lectures are delivered to the whole class A lecture is delivered to all students that have
(students of one specialization of training); enrolled for the course. This also refers to
practical studies are conducted with students practical classes (if they are provided)
of one group
There is a possibility to be a non-attending There is a possibility to be a non-attending
student student
A well-performing student is promoted to the A student must annually confirm the desire to
next course automatically continue education
An academic year starts on 1 September and is An academic year can start in late August
divided into two terms (autumn and spring). At early September and is divided into 2 terms
the end of every term there is an examination (autumn and spring); every term is divided
period. The “net” duration of a term is 18 into two periods (8–9 weeks each). The “net”
weeks duration of a term is 17–19 weeks
50 Y. Pokholkov et al.

Table 3.5 Grading system
Russia (OMSU) European Union (TUT)
Rating system (max 100 points). 70% of the The rating system is not introduced at the
grade are gained during the term. The university as a whole; the current and
aggregate rating is converted into the final midterm performance is not always taken into
grade account (see above: there may be no practical
classes but sometimes they are stipulated, just
as mandatory exercises)
A 4-point scale is applied: 6-point (from 0 to 5) scale is consistent with
the European ECTS:
• Excellent: not less than 86 points out of 100.
• Good: 71–85.9 points out of 100. • Excellent (ECTS—A): 5 points
• Satisfactory: 51–70.9 points out of 100. • Very good (ECTS—B): 4 points
• Unsatisfactory: not more than 50.9 points • Good (ECTS—C): 3 points
out of 100. • Highly satisfactory (ECTS—D): 2 points
• Satisfactory (ECTS—E): 1 point
Some Russian higher educational institutions • Unsatisfactory (ECTS—F): 0 points
(primarily in Moscow, for example, MEPI)
introduced the ECTS grade system in their
institutions tying it to the student’s rating. The
specific weight of the term may differ from
70% accepted at OMSU (for example, 50% at
KVFU)
There are subjects in which “pass” and “fail” There are subjects in which “pass” and “fail”
grades are given grades are given
If there are no grades below “good”, with at If the weighted grade average is not below 4
least 75% “excellent” grades and an and the Master’s thesis is passed with the
“excellent” grade for the FSA a “red” diploma grade not below 4, graduation with distinction
(with honors) is issued. Tests (all passed) are is issued. Tests are not taken into account in
not taken into account in calculation of the the calculation of the average grade point
grades
3 Overview of Engineering Mathematics Education for STEM in Russia 51

Table 3.6 Organization of final and midterm assessment
Russia (OMSU) European Union (TUT)
Exams
The timetable of exams is drawn up for a Exams may be taken at the end of every
group. The entire group takes the exam on the academic period. Students enter for an exam
same day individually
After a subject has been studied, the group is In the case of desire to take an exam, the
to take an exam in it automatically student is to enter for it in advance (at least a
week before it is to be taken)
The “net” duration of an exam for every The “net” duration of an exam for every
student is generally about 1 h student is generally about 3 h
An exam is generally an oral answer An exam is a written work made on special
forms
The grade for an exam is determined right A teacher has a month to check the
after it is taken examination papers
An exam is generally administered by the The exam is conducted by a special employee
teacher delivering lectures in the subject of the university (invigilator); the teacher
(maybe together with an assistant conducting delivering the course is not present. But all the
practical classes). He is present at the exam remarks made to the student during the exam
and gives the grade are recorded by the invigilator on his forms
Final assessment
The FSA (final state assessment) consists of a In addition to Bachelor’s thesis an exam in
state exam (at the option of the university) and the specialization (matriculation exam) is
a graduation qualification paper taken and thesis is presented in a seminar.
Bachelor’s thesis may be carried out as group
work; in this case it is necessary to state the
contribution of every student in the
performance of the assignment
In the Master’s degree program the graduation In addition to a Master’s thesis a
work is prepared in the form of a Master’s matriculation exam in the specialization is
thesis taken and participation in a Master’s seminar
The final state assessment in the Master’s
degree program may include a state exam
A foreign language is included in the Foreign language is included as mandatory
mandatory part of the Bachelor’s and Master’s part of Bachelor’s degree
degree educational program
52 Y. Pokholkov et al.

Table 3.7 Bachelor’s degree program
Russia (OMSU) European Union (TUT)
Period of studies 4 years Period of studies 3 years (3–4 years in EU
countries)
Total credit value of the main curriculum = Total credit value of the main curriculum =
240 CU = 8640 h 180 ECTS = 4800 h
B.1 GSE cycle: 35–44 CU, with the basic part The core studies (basic or central subjects)
of 17–22 CU MSU: 38 CU, with the basic are the mathematical and natural science sub-
part of 20 CU jects as well as other basic courses. The
B.2 MiEN cycle: 70–75 CU, with the basic objective is to familiarize the student with
part of 35–37 CU MSU: 75 CU, with the basic notions in his field, to give the neces-
basic part of 36 CU sary knowledge for further studies: 90–100
B.3 Prof cycle: 100–105 CU, with the basic ECTS.
part of 50–52 CU MSU: 105 CU, with the Pre-major studies (introduction to the spe-
basic part of 52 CU cialization): not more than 20 ECTS
B.4 Physical education: 2 CU Major subject studies are the subject deter-
B.5 Practical training and on-the-job training: mining the future qualification, including the
12–15 CU MSU: 12 CU Bachelor’s thesis: 20–30 ECTS
FSA: 6–9 CU Minor subjects (other subjects) are additional
subjects but consistent with the future Bach-
elor’s qualification: 20–30 ECTS.
Elective studies are not mandatory if the stu-
dent has fulfilled the minimal requirements of
admission for the study program.
Practical training is by the decision of the
department, not more than 8 ECTS.
Bachelor’s thesis is an analog of the FSA: 8
ECTS
There are subjects having no direct influence There are no humanities and physical
on the future professional skills: the GSE education. The university trains a specialist
cycle, physical education. The university is only; all education is subordinate to this
considered to train both a specialist and a objective. Matriculation examination can be
cultured person written in Finnish, Swedish or English
Most subjects are fixed in the curriculum Most subjects are chosen by the student one
way or another
Practical training is a mandatory part of the The decision about the need for practical
program training is made by the department
The volume of hours just for training of a All hours are allocated for training of a
specialist (without GSE and physical specialist, their volume is 180 cu = 4800 h
education): 200 CU = 7200 h
The volume of the MiEN cycle: 2520–2700 h The volume of Core studies (analog of the
MiEN): 2400–2666.7 h
The volume of the Prof cycle: 3600–3780 h The volume of Pre-Major + Major + Minor
(analog of Prof): 1066.7–2133.3 h
Conclusion: Despite the availability of humanities the Russian Bachelor’s degree program contains
the same volume of basic knowledge (comparison of MiEN and Core studies) but substantially
outstrips the Finnish by the volume of professional training
3 Overview of Engineering Mathematics Education for STEM in Russia 53

Table 3.8 Master’s degree program
Russia (OMSU) European Union (TUT)
Period of studies 2 years Period of studies 2 years
Total credit value of the curriculum = 120 CU Total credit value of the curriculum = 120
= 4320 h ECTS = 3200 h
M.1 General science cycle: 23–26 CU, with the Common Core studies: 15 ECTS, including
basic part of 7–8 CU compulsory 7 ECTS, complementary 8 ECTS.
M.2 Professional cycle: 33–36 CU, with the There are several complementary subjects to
basic part of 10–12 CU choose from; it is just necessary to get 8
M.3 Practical training and research: 48–50 CU ECTS.
M.4 FSA: 12 CU Major study (basic specialization): 30 ECTS
(in this case several variants to choose from)
Minor study (other subjects close to the spe-
cialization): 25 ECTS (in this case without any
choice).
Elective studies: 20 ECTS (this includes stud-
ies which cannot be included in other section,
for example, the English language).
Master’s thesis—analog to the FSA: 30 ECTS
There are “general” subjects. Their volume There are “general” subjects. Their volume
(M.1 cycle): 828–936 h (common core + elective studies): 933.3 h
Volume of the subjects of the prof. cycle: Volume of the subjects of the prof. cycle
1188–1296 h (major + minor): 1466.7 h
Volume of Research work + FSA (M.3 + M.4): Volume of research work + FSA (Master’s
2160–2232 h thesis): 800 h
Conclusion: In the EU a Master’s degree program more time is spent on training than in Russia
(possibly because the volume of the Bachelor’s degree program is bigger in Russia and Finnish
Masters are still to be educated to the level of the Russian Bachelor’s degree level). The research
component in the degree is much stronger in Russia

David Natroshvili

4.1 Introduction

The three-cycle higher education (HE) system has been introduced in Georgia in
2005, when Georgia became a member of the Bologna Process at the Bergen
Summit. Bachelor, Master and Doctoral programs have already been introduced
in all stately recognised higher education institutions (HEIs), as well as ECTS
and Diploma Supplement. All students below doctoral level are enrolled in a two-
cycle degree system (except for certain specific disciplines such as medicine and
dental medicine education)—one cycle education and with its learning outcomes
corresponding to the Master’s level.
There are three types of higher education institutions; see Table 4.1 and Fig. 4.1.
University—a higher education institution implementing the educational pro-
grammes of all three cycles of higher education and scientific research;
Teaching University—a higher education institution implementing a higher
education programme/programmes (except for Doctoral programmes). A Teaching
University necessarily implements the second cycle—the Master’s educational
programme/programmes;
College—a higher education institution, implementing only the first cycle
academic higher education programmes.
HEIs can be publicly or privately founded, but the quality assurance criteria are
the same despite the legal status of the institution.
Bachelor’s Programme is a first cycle of higher education, which lasts for
4 years and counts 240 ECTS; after completion of this programme students are
awarded the Bachelor’s Degree (Diploma).
Master’s Programme is a second cycle of higher education, which lasts for
2 years with 120 ECTS; after completion of the program students are awarded the

D. Natroshvili ()
Georgian Technical University (GTU), Department of Mathematics, Tbilisi, Georgia

© The Author(s) 2018 55
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_4
56 D. Natroshvili

Table 4.1 Table of higher HEI State Private Total
education institutions in
Georgia University 12 16 28
Teaching university 7 24 31
College 1 13 13
Total 20 53 73

EDUCATION SYSTEM IN GEORGIA

ACADEMIC EDUCATION

Doctoral Degree

Master’s Degree
Dental Medcine Education
Medical Education

VOCATIONAL EDUCATION
AND TRAINING
Bachelor’s Degree

Intermediate V Level of VET
Qualification

IV Level of VET

X-Xll
Secondary Education III Level of VET

II Level of VET

I Level of VET

VII-IX Basic Education

I-VI Primary Education

Pre-school Education

Compulsory Education

Fig. 4.1 The Georgian education system
4 Overview of Engineering Mathematics Education for STEM in Georgia 57

Master’s Degree (Diploma). Students with Bachelor’s Degree Diplomas are required
to pass Unified Master’s Examinations. The Doctoral Programme is a third cycle
of higher education with a minimum duration of 3 years with 180 ECTS; after
completion of this programme students are awarded the Doctor’s Degree Diplomas.
The precondition of entering the third cycle is the completion of the second cycle.
Medical education covers 6 years of studies and counts 360 ECTS.
Dental Medicine education covers 5 years of studies with 300 ECTS.
Medical and dental medicine education is one cycle education and with its
learning outcomes corresponds to/equals the Master’s level. After completion of
these programmes students are awarded diplomas in Medicine and Dental Medicine.
Grading System: There is a unified grading system with the highest 100 score
at national level.
Admission—One of the main achievements of the higher education reform in
Georgia was the establishment of a system of the Unified National Examinations.
The state took a responsibility for students’ admission to the first and second cycle
of higher education through creating of a centralized, objective system and ensuring
the principles of equity and meritocracy.
The Quality Assurance System in Georgia consists of internal and external
quality assurance (QA) mechanisms. Internal self-evaluation is carried out by
educational institutions commensurate with the procedure of evaluation of their
own performance and is summarised in an annual self-evaluation report. The
self-evaluation report is the basis for external quality assurance. External QA is
implemented through authorization and accreditation. Authorisation is obligatory
for all types of educational institutions in order to carry out educational activities
and issue an educational document approved by the State. Program Accreditation
is a type of external evaluation mechanism, which determines the compatibility of
an educational program with the standards. State funding goes only to accredited
programmes. Accreditation is mandatory for doctoral programmes and regulated
professions as well as for the Georgian language and Liberal Arts. Authorisation
and accreditation have to be renewed every 5 years.
The national agency implementing external QA is the Legal Entity of Public
Law—National Centre for Educational Quality Enhancement (NCEQE).

4.2 Georgian Technical University

As an example of Georgian educational system a more detailed description of the
Georgian Technical University (GTU) is given. The GTU is one of the biggest
educational and scientific institutions in Georgia.
The main points in the history of GTU include:
1917—the Russian Emperor issued the order according to which was set up
the Polytechnic Institute in Tbilisi, the first Higher Educational Institution in the
Caucasian region.
1922—The Polytechnic faculty of Tbilisi State University was founded.
1928—The Departments of the polytechnic faculty merged into an independent
Institute, named the Georgian Polytechnic Institute (GPI).
58 D. Natroshvili

In the 1970s—The Institute consisted of 15 full-time and 13 part-time faculties.
1985–1987—For the volume of the advanced scientific research and work carried
out by the students, the Polytechnic Institute first found a place in the USSR higher
educational institutions. During this period, the Institute became the largest higher
educational institution in the Caucasian region as regards the number of students
(total 40,000) and academic staff (total 5000).
1990—The Georgian Polytechnic Institute was granted the university status and
named the Georgian Technical University.
1995—Due to reforms and restructuring of the curriculum, GTU gradually began
installation of new training standards introducing Credit System 120 of the UK
credits.
2001—GTU became a full member of the European University Association—
EUA.
2005—GTU joined the Bologna process and introduced 60 ECTS credits.
2005—Due to the reorganisations conducted at GTU, eight Faculties were set up.
2007—GTU was awarded accreditation by the National Center for Educational
Accreditation.
2014—Due to the reorganisations conducted at GTU, ten Faculties were set up.
Educational programmes at GTU include:
• the Bachelor’s Programme—240 ECTS;
• the Master’s Programme—120 ECTS;
• the Doctoral Programme—180 ECTS;
• the Vocational Programme—150 ECTS.
There are four languages of study: Georgian, Russian, English and German.
GTU has licensed TELL ME MORE language training software, which includes
American English, British English, Dutch, French, German, Italian, and Spanish.
Learning programmes are also suited to meet user needs, as the software offers
Complete Beginner, Intermediate and Advanced levels.
There are ten faculties at GTU:
• the Faculty of Civil Engineering;
• the Faculty of Power Engineering and Telecommunications;
• the Faculty of Mining and Geology;
• the Faculty of Chemical Technology and Metallurgy;
• the Faculty of Architecture, Urban Planning and Design;
• the Faculty of Informatics and Control Systems;
• the Faculty of Transportation and Mechanical Engineering;
• the Business-Engineering Faculty;
• the International Design School;
• the Faculty of Agricultural Sciences and Biosystems Engineering.
Altogether 20,000 undergraduate students study in these faculties. There are also
1251 Master’s students, 640 PhD students and 795 students in the Professional
stream; see Table 4.2. The total amount of academic personnel at GTU is 1228, with
505 full professors, 533 associate professors, 190 assistant professors, 279 invited
professors, 369 teachers and 2176 technical staff.
4 Overview of Engineering Mathematics Education for STEM in Georgia 59

Table 4.2 Table of GTU higher education
Number of Number of Number of Number of
undergraduate postgraduate doctoral vocational
Faculty students students students students
Civil Engineering 789 90 58 221
Power Engineering and 1448 163 95 120
Telecommunications
Mining and Geology 533 32 39 235
Chemical Technology 850 47 66 140
and Metallurgy
Transportation and 1493 73 85 716
Mechanical
Engineering
Architecture, Urban 397 54 42 15
Planning and Design
Business-Engineering 4195 183 48 0
Informatics and 3413 267 152 15
Control Systems

4.3 STEM Programs at GTU

The GTU has been offering Engineering Degrees for decades with special attention
to the following STEM fields: Computer Sciences, Computer Engineering, Energy
and Electrical Engineering, Civil Engineering, Food Industry, and Forestry. The
GTU participates in the Millennium Challenge Corporation (MCC) project for
STEM Higher Education Development in Georgia. The project objectives are to
build up capacity in Georgian public universities and to offer international standard
US degrees and/or ABET (Accreditation Board for Engineering and Technology)
accreditation in the STEM fields.
Three finalist consortium universities have been selected through an open
competition: San Diego State University of California (SDSU); Michigan State
University and University of Missouri; North Carolina State University and Auburn
University.
The programme is being funded by a $29 million grant that SDSU was awarded
by the MCC that entered into an agreement with the government of Georgia to
improve its educational systems and infrastructure.
SDSU was one of 28 universities that competed for funding from the U.S.
Millennium Challenge Corporation (MCC) to create a joint higher education
programme in Georgia.
Finally, SDSU is approaching this project in partnership with Georgian Technical
University, Ilia State University and Tbilisi State University—the three premier pub-
lic universities in Georgia. Indicative STEM programmes include Electrical (Power)
Engineering, Computer Engineering, Computer Science, Chemical Engineering and
Civil Engineering fields at Georgian Technical University.
60 D. Natroshvili

For this purpose, strengthening ABET-Accredited Georgian Degree Programmes
at GTU secures better understanding of educational needs of the next generation
of engineers, scientists and educators; achieving ABET accreditation of these
programmes is a tangible milestone of quality improvement, providing quality of
education and student outcomes of these programmes.

4.4 Mathematics at GTU

Mathematics has played and still plays nowadays a fundamental role in engineering
education in GTU. In the Georgian Polytechnic Institute the Chair of Mathematics
was founded in 1928. Many worldwide well-known Georgian mathematicians had
been working and delivering lectures in the Georgian Polytechnic Institute, such
as worldwide well-known scientists academicians Niko Muskelishvili, Ilia Vekua,
Viktor Kupradze, Boris Khvedelkdze etc.
During the Soviet period all Polytechnic Institutes were forced to follow a unified
mathematical curriculum with a sufficiently rich pure theoretical part. It should
be mentioned that the level of the school mathematics at that time was very high
in the Soviet Union and the entrants were well prepared to start learning of high
mathematics, containing a very wide spectrum of courses starting from analytical
geometry and classical calculus to boundary value problems for partial differential
equations and theory of measure and Lebesgue integrals along with the theory of
probability and mathematical statistics.
However, such a high fundamental educational level in mathematics never gave
the expected and desired progress in technology and engineering. There was a big
gap between theoretical preparation of students and their skills in applied practical
aspects. This was one of the main drawbacks of the Soviet educational system.
In 2007, the Department of Mathematics was founded at GTU on the basis of the
existing three chairs of high mathematics. The Department of Mathematics belongs
to the Faculty of Informatics and Control Systems.
The staff of the Department of Mathematics consists of 20 full-time professors,
21 full-time associate professors, 3 full-time assistant professors, 7 teachers, 16
invited professors, and 5 technical specialists.
From 2008 the BSc, MSc, and PhD accredited programmes in pure and applied
mathematics have been launched at the Department of Mathematics (it should
be mentioned that presently the mathematical programmes are free of charge—
from 2013 the Georgian Government has covered all expenses for 20 educational
programmes; among them is mathematics).
4 Overview of Engineering Mathematics Education for STEM in Georgia 61

The Department of Mathematics delivers lectures in high mathematics for all
engineering students of GTU. Depending on the specific features of the engineering
educational programmes the content of mathematical syllabuses varies and reflects
mainly those parts of mathematics which are appropriate for a particular engineering
specialisation.

Ishkhan Hovhannisyan

The higher education system in Armenia consists of a number of higher education
institutions (HEIs), both state and private. State higher education institutions operate
under the responsibility of several ministries, but most of them are under the
supervision of the Ministry of Education and Science. At present there are 26 state
and 41 private higher education institutions operating in the Republic of Armenia,
of which 35 are accredited institutions, 6 are non-accredited institutions, 3 are
branches of state HEIs and 4 are branches of private HEIs from the Commonwealth
of Independent States (CIS). Higher education is provided by many types of
institutions: institutes, universities, academies and a conservatory.
The University HEIs are providing higher, postgraduate and supplementary edu-
cation in different branches of natural and sociological fields, science, technology,
and culture, as well as providing opportunities for scientific research and studies.
The institute HEIs are conducting specialized and postgraduate academic pro-
grams and research in a number of scientific, economic and cultural branches.
The Academy (educational) HEI’s activity is aimed at the development of
education, science, technology and culture in an individual sphere; it conducts
programs preparing and re-training highly qualified specialists in an individual field,
as well as postgraduate academic programs.
The preparation of specialists for Bachelor’s, Diploma Specialist’s and Mas-
ter’s degrees, as well as postgraduate degrees, including a PhD student program
(research) is implemented within the framework of Higher Education. The main
education programs of higher professional education are conducted through various
types of teaching: full-time, part-time or external education. The academic year, as
a rule, starts on September, ending in May and is comprised of 2 semesters with
16–22 weeks of duration. There are mid-term exams and a final exam at the end of
each semester.

I. Hovhannisyan ()
National Polytechnic University of Armenia (NPUA), Faculty of Applied Mathematics and
Physics, Yerevan, Armenia

© The Author(s) 2018 63
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_5
64 I. Hovhannisyan

ARMENIAN EDUCATION SYSTEM DIAGRAM

Doctor of science = PhD
›
Doctoral thesis defense, awarded
Candidate of science = PhD
›
PhD thesis defense, awarded
Researcher (3 years)

Trainings and complementary education
›
AGE

Master (2 years)

›
21 Bachelor (4 years)
20
19 diploma

18
College-Diploma (2 years) with honor ›
State Entrance Examination
17 College-Certificate (3 years) High School-attestat (3 years)
16
15
› ›
14 General Education-Certificate (5 years)
13
12
11
10
›
9 Primary Education (4 years)
8
7
6

Fig. 5.1 The Armenian education system

The formal weekly workload (contact hours) that students are expected to carry
out depends on the type of programs and differs considerably from institution to
institution within the country, but common practices are as follows: for Bachelor
programs 28–32 h per week (sometimes up to 36), for Master programs 16–18 h and
for postgraduate (Doctorate) programs 4–8.
Starting from 2008 all educational programs in Armenia are based on the ECTS.
Internal systems of student evaluation and assessment are regulated by the HEIs
themselves. Students’ learning outcomes are assessed on the basis of examinations
5 Overview of Engineering Mathematics Education for STEM in Armenia 65

and tests, which are conducted in writing or orally. The results of examinations are
assessed by grading systems varying considerably among institutions (5-, 10-, 20-
or 100-point marking scales, 4 scale A–F letter grading, etc.). A final evaluation of
graduates in state HEIs is conducted by state examination committees both through
the comprehensive examination on specialization and defense of graduation work
(diploma project, thesis or dissertation) or schemes only one of them is used.
A diagram of the Armenian education system can be seen in Fig. 5.1.

5.1 National Polytechnic University of Armenia

The National Polytechnic University of Armenia (NPUA) is the largest engineering
institution in Armenia.
The University has a leading role in reforming the higher education system in
Armenia. NPUA was the first HEI in RA that introduced three level higher education
systems, implemented the European Credit Transfer System (ECTS) in accordance
with the developments of the Bologna Process.
The University is a member of European University Association (EUA), the
Mediterranean Universities Network, and the Black Sea Universities Network. It
is also involved in many European and other international academic and research
programs. The University aspires to become an institution, where the entrance and
educational resources are accessible to diverse social and age groups of learners, to
both local and international students, as well as to become an institution which is
guided by a global prospective and moves toward internationalization and European
integration of its educational and research systems.

5.2 Overview of Mathematics Education at NPUA

The NPUA Faculty of Applied Mathematics and Physics is responsible for major
and minor mathematical education at the University. It was established in 1992
by uniting University’s 3 chairs of Higher Mathematics. Academician of National
Academy of Sciences Prof. Vanik Zakaryan is the founder-Dean of the Faculty.
Nowadays the faculty is one of the top centers of Mathematics and Physics in the
country and the biggest faculty in the University, having more than 90 full-time
faculty members (12 professors and 48 associate professors). The student body of
the faculty consists of approximately 200 majors (all programs) and more than 3000
minors. The faculty offers the following programs as majors:
• Bachelor in Informatics and Applied Mathematics;
• Bachelor in Applied Mathematics and Physics;
• Master in Informatics and Applied Mathematics;
• PhD in Mathematics.
66 I. Hovhannisyan

In addition to these major programs, the Faculty caters to the mathematics and
physics subsidiary (minor) courses in other BSc and MSc programs of the University
with specializations in Engineering, Industrial Economics and Management. It also
renders services of its full-time faculty to teach elective courses of mathematics at
MSc and PhD programs.

5.3 Mathematics Courses at NPUA

The Bachelors of Science in Engineering (BSE) degree at NPUA involves complet-
ing 240 credit hours of courses in various categories, which include: a module of
languages, Economics and Humanities module, a module of General Engineering
courses, Module of Specialization courses, and a module of Math and Natural
Science courses. The University requires that all engineering students, regardless
of their proposed engineering major, complete specific courses in the core subjects
of mathematics which are listed in Table 5.1 with the number of ECTS credits for
each course.

Table 5.1 BSc mathematics courses at NPUA
Course Semester Credits Hours
Mathematical Analysis 1 (An 1 5 Lectures 32,
introduction to the concepts of tutorials 32
limit, continuity and derivative,
mean value theorem, and
applications of derivatives such as
velocity, acceleration,
maximization, and curve
sketching)
Mathematical Analysis 2 2 5 Lectures 32,
(introduction to the Riemann tutorials 32
integral, methods of integration,
applications of the integral,
functions of several variables,
partial derivatives, line, surface
and volume integrals)
Analytic Geometry and Linear 1 4 Lectures 32,
Algebra (vector spaces and matrix tutorials 16
algebra, matrices and
determinants, systems of linear
equations)
Theory of Probability and 3 4 Lectures 32,
Statistical Methods (probability tutorials 16
space axioms; random variables
and their distributions,
expectation values and other
characteristics of distributions)
Discrete Mathematics 2 2 Lectures 32,
tutorials 16
5 Overview of Engineering Mathematics Education for STEM in Armenia 67

Table 5.2 MSc mathematics courses at NPUA
Course Semester Credits Hours
Discrete Mathematics 1 5 Lectures 48,
practices 16
Numerical Methods 1 5 Lectures 48,
practices 16
Mathematical Programming 1 5 Lectures 48,
practices 16
Linear Algebra 1 5 Lectures 48,
practices 16
Theory of Probability and 1 5 Lectures 48,
Statistical Methods practices 16
Functions Approximation by 1 5 Lectures 48,
Polynomials practices 16

In addition to the core mathematics courses, the Master of Science in Engineering
(MSE) degree requires students to complete at least 5 credits of advanced mathe-
matics elective courses. Table 5.2 contains the list of advanced mathematics elective
courses for MSE students.

6.1 Engineering Mathematics Education in Finland

Tuomas Myllykoski and Seppo Pohjolainen ()
Tampere University of Technology (TUT), Laboratory of Mathematics, Tampere,
Finland
e-mail: tuomas.myllykoski@tut.fi; seppo.pohjolainen@tut.fi

The Finnish higher education system consists of two complementary sectors:
polytechnics and universities. The mission of universities is to conduct scientific
research and provide undergraduate and postgraduate education based on it. Poly-
technics train professionals in response to labor market needs and conduct R&D
which supports instruction and promotes regional development in particular.
Universities promote free research and scientific and artistic education, provide
higher education based on research, and educate students to serve their country
and humanity. In carrying out this mission, universities must interact with the
surrounding society and strengthen the impact of research findings and artistic
activities on society.
Under the new Universities Act, which was passed by Parliament in June 2009,
Finnish universities are independent corporations under public law or foundations
under private law (Foundations Act). The universities have operated in their new
form from 1 January 2010 onwards. Their operation is built on the freedom of
education and research and university autonomy.
Universities confer Bachelor’s and Master’s degrees, and postgraduate Licentiate
and Doctoral degrees. Universities work in cooperation with the surrounding
society and promote the social impact of research findings. The higher education

© The Author(s) 2018 69
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_6
70 6 Overview of Engineering Mathematics Education for STEM in EU

system, which comprises universities and polytechnics, is being developed as an
internationally competitive entity capable of responding flexibly to national and
regional needs. There are technological universities in Helsinki, Tampere and
Lappeenranta, and technological faculties in Oulu, Vaasa and Turku. Approximately
4000–5000 students begin their studies in one of these universities annually. The
Finnish education system diagram is in Fig. 6.1.
The system of polytechnics is still fairly new. The first polytechnics started to
operate on a trial basis in the beginning of 1990s and the first was made permanent
in 1996. By 2000 all polytechnics were working on a permanent basis. Polytechnics
are multi-field regional institutions focusing on contacts with working life and on
regional development. The total number of young and mature polytechnic students is
130,000. Polytechnics award over 20,000 polytechnic degrees and 200 polytechnic
Master’s degrees annually. The system of higher degrees was put in place after a
trial period in 2005 and the number of polytechnic Master’s programs is expected
to grow in the coming years.

Fig. 6.1 The Finnish education system (http://www.oph.fi/english/education_system)
6 Overview of Engineering Mathematics Education for STEM in EU 71

6.1.1 Tampere University of Technology

Tampere University of Technology (TUT) is Finland’s second-largest university
in engineering sciences. TUT conducts research in the fields of technology and
architecture and provides higher education based on this research. TUT is located
in Tampere, the Nordic countries’ largest inland city, some 170 km north of the
capital Helsinki. TUT’s campus in the suburb of Hervanta is a community of
10,500 undergraduate and postgraduate students and close to 2000 employees.
Internationality it is an inherent part of all the University’s activities. Around 1500
foreign nationals from more than 60 countries work or pursue studies at TUT.
TUT offers its students an opportunity for a broad, cross-disciplinary education.
Competent Masters of Science of Technology and Architecture as well as Doctors
of Technology and Philosophy graduated from TUT are in high demand among
employers.
The University combines a strong tradition of research in the fields of natural
sciences and engineering with research related to industry and business. Technology
is the key to addressing global challenges. The University’s leading-edge fields
of research are signal processing, optics and photonics, intelligent machines,
biomodeling and the built environment.
TUT generates research knowledge and competence for the benefit of society.
The University is a sought-after partner for collaborative research and development
projects with business and industry and a fertile breeding ground for innovation and
new research- and knowledge-based companies.
In 2013 the total funding of TUT Foundation, which operates as Tampere
University of Technology, was 157.6 million euros. Close to 50% of the University’s
funding was external funding, such as revenue from The Finnish Funding Agency
for Technology and Innovation (Tekes), industry, the Academy of Finland and EU
projects.
TUT started operating in the form of a foundation in the beginning of 2010. The
independence of a foundation university and the proceeds of the 137 million euro
foundation capital further promote the development of research and education at
TUT.

6.1.2 Overview of Mathematics Education at TUT

The Department of Mathematics is responsible for teaching core mathematics to all
engineering students at Tampere University of Technology, and it offers courses and
degree programs at the Bachelor’s, Master’s, and Postgraduate level for studies in
mathematics. The faculty of the department conducts research in mathematics and
its applications at an internationally competitive level.
72 6 Overview of Engineering Mathematics Education for STEM in EU

The Department of Mathematics offers mathematics and statistics expertise for
research and development projects in the private and public sectors. Research
services and collaboration can range from informal working relationships to shorter-
or longer-term research contracts.
The teaching of mathematics for the following degree programs is the responsi-
bility of the Department of Mathematics on the Bachelor and Master level:
• Automation Engineering
• Biotechnology
• Civil Engineering
• Signal Processing and Communications Engineering
• Electrical Engineering
• Environmental and Energy Technology
• Industrial Engineering and Management
• Information and Knowledge Management
• Information Technology
• Materials Engineering
• Mechanical Engineering
• Science and Engineering
Generally, the mandatory amount of mathematics is 27 ECTS in the Bachelor’s
degree. The mandatory mathematics consists of Engineering Mathematics 1–3
courses (altogether 15 ECTS) (Table 6.1), studied during the first year, and three
elective courses 4 ECTS each on the first and second year (Table 6.2). The degree
program makes the recommendation on the elective courses suitable for their
students. The degree program can place one or two mandatory mathematics courses
in their Master’s program. Then these courses are not part of the BSc program
but belong to the MSc program. The degree programs may recommend students to
include additional mathematics courses in their study plan. In this case the amount
of mathematics exceeds 27 ECTS.

Table 6.1 Mandatory engineering mathematics courses at TUT
Course ECTS Year Contents
Engineering 5 1 Set theory and mathematical logic; real functions;
Mathematics 1 (EM1) elementary functions; limits; derivative; complex
numbers; zeros of polynomials
Engineering 5 1 Vectors in Rn spaces; linear equations, Gauss’
Mathematics 2 (EM2) elimination; vector spaces; matrices,
eigenvectors, determinants; orthogonality
Engineering 5 1 Indefinite and definite integral; first and second
Mathematics 3 (EM3) order differential equations; sequences; series
6 Overview of Engineering Mathematics Education for STEM in EU 73

Table 6.2 Elective engineering mathematics courses at TUT
Course ECTS Year Contents
Engineering 4 1 Multivariable functions,limit, continuity,partial
Mathematics 4 (EM1) derivatives,gradient; vector valued functions,
matrix derivative, chain rule; maxima, minima,
Lagrange’s method; plane and space integrals
Algorithm Mathematics 4 2 Set theory, methods of proofs; relations and
(AM) functions; propositional and predicative logic;
induction and recursion; Boolean algebra
Discrete Mathematics 4 1, 2 Step-, impulse-, floor-, ceiling- signum functions;
(DM) Z-transform, difference equations; number
theory; graph theory
Fourier Methods (FM) 4 2 Real Fourier series; linearity, derivation; complex
Fourier series; discrete Fourier transform
Operational Research 4 2 Linear optimization; Simplex-method, sensitivity;
(OA) dual problem; transport model with applications;
warehouse models; game theory
Probability Calculus 4 2 Random variable and probability, Bayes’
(PC) formula; distributions and their parameters; joint
distributions, central limit theorem
Statistics (MS) 4 2 Descriptive statistics, samples; hypothesis testing,
parametric and nonparametric cases
Vector Analysis (VA) 4 2 Gradient, divergence; line integrals; conservative
vector field; surfaces, area, surface integral, flux
and Gauss’ theorem
The study program recommends the selection of (at least) three of these courses

6.1.2.1 Major/Minor in Mathematics (BSc, MSc)

In addition to the courses provided for all degree programs, the department also has
a well-rounded group of students who study mathematics as their major. The courses
provided for these students are often based on the research topics of the department,
as this further develops the department’s strategy.
The Department of Mathematics offers courses and degree programs at the
Bachelor’s, Master’s, and at the Postgraduate level. Doctoral studies can be carried
out in the main research areas of the department.
Currently there are two majors in the Master’s program of Science and Engi-
neering fully given in English: Mathematics with Applications and Theoretical
Computer Science, both by the Department of Mathematics. These offer the
uniquely useful combination of strong mathematical modeling and tools of logical
and algorithmic analysis. There is also a minor subject in Mathematics. As per
agreement, it may also be included in other international Master’s programs at TUT.
74 6 Overview of Engineering Mathematics Education for STEM in EU

The optional focus areas in Mathematics with Applications are Analysis, Discrete
Mathematics, and Mathematical and Semantic Modeling. The subject can also be
chosen as an extended one.
The postgraduate studies program leads to the PhD degree or the degree of
Doctor of Technology. Subjects of the postgraduate studies at the department usually
follow the research topics of the research groups. There is a local graduate school,
which can provide financial support for doctoral studies.
On the Bachelor level, the minor or major in mathematics consists of 25 ECTS in
mathematics in addition to the 27 ECTS studied by all students. For the major, the
students will also complete a Bachelor’s thesis worth 8 ECTS. If the student chooses
not to take mathematics as a major, then they must complete an additional 10 ECTS
of mathematics for a total of 60 ECTS. In the Master phase of studies, the students
will complete either 30 or 50 ECTS of mathematics for their major, depending on
their choice of minor studies. Those meaning to graduate from the Master’s program
with a major in mathematics are required to write the Master’s thesis in mathematics
that is worth 30 ECTS.

6.1.2.2 Teacher Studies at TUT in Mathematics

The students are offered the possibility of studying mathematics with the goal
of attaining competency for teaching mathematics at Finnish schools. Students
must study a minimum of 50 ECTS of mathematics in the university to be
able to apply for the program. After applying, the students are evaluated by a
board of academics at University of Tampere. Those who pass evaluation are
given the possibility of studying at University of Tampere to obtain 60 ECTS
of mandatory pedagogical studies. Students will study both at TUT and at the
Tampereen Normaalikoulu—high school—where they work as real teachers with
real students. The major in mathematics for teacher students is 120 ECTS, a minor
is 60 ECTS of mathematics. Teacher students will also complete a 60 ECTS minor
in pedagogical studies. Students have a possibility of studying multiple sciences in
their teacher studies, with mathematics being accompanied by chemistry, physics
and information technology. It is often suggested that students choose multiple
sciences in order to further develop their possibilities in the future when looking
for a job. The overall structure of teacher studies is depicted in Fig. 6.2.
6 Overview of Engineering Mathematics Education for STEM in EU 75

Basic Studies 91 cr (contains 19-27 cr of Mathematics)

Directing Studies 18 cr
B A C H E L O R

Major 25 cr Minor 20-30 cr

Bachelor Thesis 8 cr Extra-curricular studies

Common Program Core Studies (0-8 cr of mathematics)
M A S T E R

Minor * 20-30 cr
Major 30 cr OR 50 cr * Minor is not required in
case major is 50-60 cr.

MSc thesis and
Optional Studies 0-36 cr
Proficiency Test

Fig. 6.2 Teacher studies in mathematics at TUT
76 6 Overview of Engineering Mathematics Education for STEM in EU

6.2 Engineering Mathematics Education in France

Christian Mercat and Mohamed El-Demerdash
IREM Lyon, Université Claude Bernard Lyon 1 (UCBL), Villeurbanne, France
e-mail: christian.mercat@math.univ-lyon1.fr

The Higher Education (HE) System in France training future engineers is
organized not only around universities (the left half and the green vertical line
in Fig. 6.3), but much more around Engineering Schools (blue line to the right)
which have a special status. Pre-engineering schemes such as University Diploma
of Technology (DUT) or usual Licence trainings are as well used as stepping
stones for engineering degrees (16% of engineers access their school with a DUT,
6% from a general academic scientific licence). The entrance selection scheme
(represented by a red S bar in Fig. 6.3 is at the beginning of either the first year, right
after Baccalaureate graduation, or the third year. A continuous selection weeds out
failing students at the end of each year. Most of the data exploited in this chapter
comes from Higher Education & Research in France, facts and figures, 9th edition,
November 2016, freely available for inspection on the following governmental
website1 :

Fig. 6.3 French education system

1 http://publication.enseignementsup-recherche.gouv.fr/eesr/9EN/.
6 Overview of Engineering Mathematics Education for STEM in EU 77

6.2.1 Universities

There are 73 universities in France, for a total of 1.5 million students, which
represent 60% of the number of Higher Education students; see Fig. 6.4. The number
of students has increased by a factor of 8 in the last 50 years to reach 2.5 million,
the proportion of Baccalaureate holders increasing from a third in 1987 to two-
thirds of a generation in 1995 and three-quarters nowadays. The demographical
increase is expected to make the numbers of HE students steadily grow in 10 years
to reach 2.8 M. Short technician diplomas, BTS and DUT, are mainly responsible
for this increase. These short technical diplomas follow the creation of vocational
and technological Baccalaureates; see Fig. 6.5.
Whereas any European freshmen can enter French university (there is no entrance
selection), a significant number of HE students will never go to university and it

University and other HE courses
3

2,5

2
Million Students

1,5 Other courses
1 University

0,5

91 93 5 97 99 01 03 05 07 09 11 13 15
0- 2- 4-9 6- 8- 0- 2- 04- 6- 08- 10- 2- 14-
9 9 9 9 0 0 0 1
19 19 1 99 19 19 20 20 20 20 20 20 20 20

Fig. 6.4 The number of students in Higher Education

90,00%
80,00%
70,00%
60,00% Vocational
50,00% Technological
40,00% General

30,00%
20,00%
10,00%
0,00%
50 954 958 962 966 970 974 978 982 986 990 994 998 002 006 010 014
19 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

Fig. 6.5 Percent of a generation with Baccalaureate
78 6 Overview of Engineering Mathematics Education for STEM in EU

Fig. 6.6 Degrees per sector
of Higher Education Outer: Bachelor
Inner: Master

Law
Economics
Humanities
Sciences
Healthcare

is especially true for engineers. There are around 63,000 permanent HE teachers
in France, half of them in science and technology departments. Around 50,000
are researchers as well (a fixed half and half loads) and usually belong to a
research institute accredited by one of the French Research Institute such as CNRS,
INRIA or CEA.2 Around 13,000 permanent teachers are full time teachers and
are not supposed to do research, their level is attested by the French Agrégation
or equivalent. These Professeurs Agrégés (PRAG) are especially numerous in
engineers schools. Around 24,000 teachers have non-permanent teaching positions.
The curriculum is accredited by the ministry of higher education and research. These
research institutes, as well as the programs and degrees delivered by universities are
evaluated by an independent body, HCERES.3
The plot Fig. 6.6 of degrees per sector of higher education shows that, in
proportion with other sectors, many more students leave Sciences with a simple
Bachelor’s degree and will not achieve a Master’s degree. The reasons are twofold,
a good and a bad one: first, a Bachelor’s degree in science is sufficient to get a
job compared to humanities or law for example, especially the BTS and DUT,
and, second, science students are more likely to drop out earlier. Two-thirds of the
scientific and technical Masters are in fact engineering degrees.
While the number of engineer’s degrees has slightly increased, the lower pre-
engineering degrees of Higher Technician Diploma (BTS) and University Diploma
of Technology (DUT) have increased much more, following, respectively, the
vocational and technological Baccalaureates; see Fig. 6.7.
Foreign students account for about 15% of the university students (DUT
included) with a sharp increase in the first years of this century, from 8% in
2000. Some courses, such as preparatory schools (CPGE) and Technical University

2 National Center for Scientific Research http://www.cnrs.fr/; National Institute for computer

science and applied mathematics http://www.inria.fr; French Alternative Energies and Atomic
Energy Commission http://www.cea.fr.
3 High Council for Evaluation and Research and Higher Education, http://www.hceres.fr/.
6 Overview of Engineering Mathematics Education for STEM in EU 79

600

500

PhD
400 Master
Business
300 Bachelor
DUT
200 BTS
Engineer
100

0
98 999 000 001 002 003 004 005 006 007 008 009 010 011 012 013
⎮
⎮
90

19 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2
19

Fig. 6.7 Numbers of degrees (in thousands)

Diplomas (DUT), having no foreign counterparts, do not attract foreign students
although they are competitive and channel a lot of engineers in France, especially
the top engineering schools which have de facto mainly French students. Parallel
admission systems allow for the inclusion of foreign students. It is especially
the case for foreign students already holding a Master’s degree and training a
specialization in a French engineering school; see Fig. 6.8.
These students come traditionally mainly from Africa, notably Morocco, Algeria,
Tunisia and Senegal, but more and more from Asia, remarkably from China.
Germany and Italy are the main European partners; see Fig. 6.9.

18,00%
16,00%
14,00%
12,00%
Business
10,00% Engineer
8,00% University
DUT
6,00%
CPGE
4,00%
2,00%
0,00%
99 00 01 3 4 5 6 7 8 9 0 1 2 3 4 5
02

8- 99- 00- -0 -0 -0 -0 -0 -0 -0 -1 -1 -1 -1 -1 -1
02 003 004 005 006 007 008 009 010 011 012 013 014
-
01

9 0
19 19 20
20

2 2 2 2 2 2 2 2 2 2 2 2 2

Fig. 6.8 Percentage of foreign students
80 6 Overview of Engineering Mathematics Education for STEM in EU

Fig. 6.9 Origin of foreign AFRICA
students ASIA,
OCEANIA
EUROPE
AMERICAS

Fig. 6.10 Population and
gender repartition in
Scientific Preparatory 8 115
Schools (CPGE) in 2016
Boys
Girls

18 726

6.2.2 Preparatory Courses (CPGE and Internal)

The main entrance scheme to engineering schools (more than 80%) is not via
university but via preparatory schools. The two first years of the engineering training
are preparatory courses, done either in special preparatory schools (CPGE) or
already integrated within the school. Around 27,000 students (93% from scientific,
7% from technical Baccalaureate) go through CPGE each year; see Fig. 6.10. There
are great geographical and demographical disparities: a third of the students studies
in Paris and a third are female. Teachers in CPGE, around 86,000 people, belong to
the ministry of secondary education, they do not conduct any research and are not
affiliated with a university or a higher education institute.
The main networks of engineers schools have their own “students shuffling”
system, allowing students to allocate, at the completion of their two first internal
preparatory years, according to their accomplishments and desires, especially the
Polytech and INSA networks.
At the end of these 2 years, competitive exams, national or local, rank students,
in different ranking systems. Different schools unite in consortia and networks,
sharing the same examinations, allowing themselves to rank internally students
in order for them to express their choices and allocate places according to offer
6 Overview of Engineering Mathematics Education for STEM in EU 81

and demand. Some of these examinations are incompatible with one another, so
students have to choose which schools they want to apply to. The three main public
competitive examinations are: Polytechnique-Écoles Normales-ESPCI, Centrale-
Supélec, Mines-Ponts. These exams are open to students having followed 2 years of
preparatory schools, whether at university (a small proportion) or in special training
schools (CPGE).

6.2.3 Technical University Institutes (IUT) and Their Diploma
(DUT)

With a Technical Baccalaureate, students can enter university and become higher
technicians in Technical University Institutes (113 IUT in France and one in UCBL),
which are limited to 2 years training, eventually followed by a professional license.
More than 70% of them receive a University Diploma in Technology (DUT),
an intermediate degree of the LMD system. This diploma allows for immediate
employability and about 90% of the DUT students after graduation do get a
permanent position within the first year. But only 10% of the graduates choose to
do so! DUT is somehow “hijacked” by 90% of students that in fact are looking for
further training, and especially as engineers. The French job market lacks skilled
technicians (1500 euro/month median first salary): students want to invest in studies
in order to get a Master (1900 euro/month median first salary) or an engineer’s salary
(2700 euro/month median first salary) for the same 90% students placement success
rate. This situation has to be kept in mind when analyzing the French Engineering
School system.

6.2.4 Engineering Schools

Around 100,000 engineers are being trained in 210 French engineering schools
today, leading to 33 thousand graduations a year, a fifth of them in schools included
in a university. Unlike the general Bachelor program at university, the entrance to
these schools is competitive and the dropout rate is very low.
An engineering school has to be accredited by the Ministry of Higher Education
and Research, after an inquiry, every 6 years by a special quality assessment
body, the Engineering Accreditation Institution (CTI); this requirement has existed
since 1934. Students training, students job placement, recruitment of the personnel,
industrial and academic partnerships, and self quality assessment are among the
main criteria. CTI belongs to the European Association for Quality Assurance in
Higher Education (ENQA) and the European Consortium for Accreditation (ECA).
These schools are often independent and usually do not belong to a university.
In particular, engineering school training is limited up to the fifth year after national
82 6 Overview of Engineering Mathematics Education for STEM in EU

25
Thousands

0
⎪ ⎪
90 995
00

11
98

13
19
20

20
19

20
1

Fig. 6.11 Number of engineering degrees (in thousands)

Baccalaureate Degree or equivalent, that is to say, the equivalent of Master degrees,
and these institutions are not permitted to deliver Doctoral diplomas. In order to do
so, they have to establish a partnership with a university. A fair amount of their staff
usually belongs to a research laboratory hosted by a university. Therefore, compared
to partner countries such as Finland or Russia, most universities are “classical”
universities, while engineering schools take the role of “technical” universities.
Moreover, let us recall that in most of these schools, specialized work begins in
the third year, the first two being preparatory years, taught internally or in another
institution (CPGE). All in all, engineering degrees represent around two-thirds of
all the scientific and technical Masters in France.
The 210 accredited engineering schools in France fall into three main categories.
First, there are schools integrated inside a university, Polytech is an example of
such schools. Then there are schools with integrated preparatory years, INSA is
an example. Then there are schools which train only for 3 years after 2 years of
preparatory school (CPGE), École Centrale is an example. We will further describe
these paradigmatic examples below. Around a third of the engineering schools are
private, for a fourth of engineering students. As shown in Fig. 6.11 the number of
engineering degrees awarded each year has been increasing by 3.4% a year over the
last 5 years, reaching 32,800 in 2015.
Around 4% of engineering students continue their studies with a PhD. There are
around 1% of the students doing so in private schools and professional Masters,
compared with 14% in research Masters.
The engineering schools are often part of networks such as the Polytech4 or the
INSA5 networks. The Polytech network is constituted of 13 public schools, training
more than 68,000 students and leading to 3000 graduations per year. The INSA

4 http://www.polytech-reseau.org/.
5 http://www.groupe-insa.fr/.
6 Overview of Engineering Mathematics Education for STEM in EU 83

network of six schools trains 2300 engineers per year for 60 years, amounting to
more than 80,000 engineers on duty today.
Around 100,000 students are being trained in 210 French engineering schools
today, for 33 thousand graduations a year, a fifth of them in schools included in a
university. Unlike universities, the entrance to these schools is competitive. Their
curricula and diplomas are continuously assessed and validated by the Commission
of Engineer Title (CTI). This commission is a member of the European Association
for Quality Assurance in Higher Education and belongs to the French Ministry of
Higher Education and Research.
The entrance selection scheme is either at the beginning of the first year, right
after Baccalaureate graduation, or in the third year. A continuous selection weeds
out failing students at the end of each year. The two first years are preparatory
courses, done either in special preparatory schools (CPGE) or already integrated
within the school.

6.2.5 Engineer Training in Lyon

Higher Education players in Lyon are gathered into the Université de Lyon consor-
tium, which comprises 129,000 students and 11,500 researchers; see Fig. 6.12.
Université Claude Bernard Lyon is the science and technology university of the
Université de Lyon. There are 3000 researchers for 68 research laboratories and
40,000 students in 13 teaching departments. The mathematics research laboratory
is the Institut Camille Jordan (ICJ UMR 5208 CNRS, 200 members). Most of
the mathematics teachers from the neighboring Engineers Schools, INSA, École
Centrale, Polytech, which are as well researchers, belong to this research institute.
Lyon has another smaller research institute in the École Normale Supérieure (UMPA
UMR 5669 CNRS, 50 members) hosting much less applied mathematicians and no
engineers trainer. CPE researchers belong to the Hubert Curien laboratory.

6.2.5.1 Civil Engineering

Course Description: The course Civil Engineering and Construction is the third
year of Mechanics License—Civil Engineering. It is administratively attached to
the Department of Mechanics Faculty of Lyon 1.
Training Duration: 2 semesters. Number of hours of training at the University:
600 h. Period of internship: 6 weeks. According to the student profile, specific
modules and differentiated lessons are implemented.
Course Overview: The aim of this license is to provide an operational, flexible
and scalable framework combining scientific and technological knowledge in the
area of Building and Public Works. All major areas of construction are discussed:
Drawing, Work Management, Energy, Structures, Soil Mechanics, Topography,
Materials, etc.
84 6 Overview of Engineering Mathematics Education for STEM in EU

Fig. 6.12 Université de Lyon Consortium

6.2.5.2 Master Program

Courses offered at the Master’s degree level satisfy a dual objective of preparing
students for research and providing courses leading to high level professional
integration. The Master’s degree is awarded after acquisition of 120 credits after the
“license” (Bachelor’s degree) on the basis of training organized in four semesters.
The first 60 credits (M1) can, by request of the student, receive an intermediate
level national “maîtrise” diploma, a heritage of the previous French HE system. The
remaining credits lead to the awarding of the national “Master” diploma.

6.2.5.3 Polytech

UCBL hosts the engineers school Polytech. It belongs to a network of 13 engineer-
ing schools embedded into universities. It represents a pretty new trend in France,
the school in Lyon was founded in 1992 and joined the Polytech network in 2009.
It grew to become quite an alternative to more classical engineering schools. The
recruitment of Baccalaureate students (aged around 18) is done at a national level
through a common procedure, shared with other 29 engineering schools: the Geipi
Polytech competitive exam awarding more than 3000 students a ranking into the
affiliated schools from which to make a choice. Polytech Lyon majors are chosen
by around 200 of them per year (Fig. 6.13).
6 Overview of Engineering Mathematics Education for STEM in EU 85

Fig. 6.13 The network of Polytech universities

The two first years are preparatory, following the program of other classical
preparatory schools but at the university. The actual choice of the engineer school
is done at the completion of these 2 years. A reshuffling of the students, according
to their choice of major topic and achievements, is performed inside the Polytech
network. A student having begun studying in Lyon might very well end up finishing
her/his studies in Polytech Grenoble, for example, because she/he grew interested
in the Geotechnic major prepared there. But majors in Lyon are attractive so the
reverse is the case: from 120 preparatory students, numbers jump to around 200 in
the third year (L3) when majors begin. These majors are selective and it fuels the
competition between students at a national level in the network. The sex ratio is
around 20% of girls, but a great effort is being made in that respect, the preparatory
school ratio being now about 30%.
There are six majors in Polytech Lyon, rooted in the scientific workforce in
UCBL:
• Biomedical engineering;
• Computer science;
• Materials sciences;
• Modeling and applied mathematics;
• Mechanical engineering;
• Industrial engineering and robotics.
Every student from the third year on belongs to one of them. These majors are
backed up by research laboratories of the UCBL to which the teachers belong as
researchers. Most of these laboratories are associated with CNRS (French National
86 6 Overview of Engineering Mathematics Education for STEM in EU

Agency for Research). For mathematics, it is the Institut Camille Jordan UMR
CNRS 5208 (ICJ). Two 6-month internships in industry or research laboratory are
performed in the fourth and fifth year of study. International training is mandatory,
whether as a student or as an intern.

6.2.5.4 INSA

The seven National Institutes for Applied Sciences (INSA) form the largest network
of engineer schools in France, amounting to 12% (10,200 students) of all the trained
French engineers. The first institute was open in Lyon in 1957 and had a special
emphasis on social opportunities, humanities and international cooperation: special
cycles prepare engineer students in the spirit of cultural openness and bilingualism
such as EURINSA, ASINSA, AMERINSA, NORGINSA, SCAN, respectively, for
European, Asian, South-American, Nordic and English speaking students. French
students and foreign students are mixed half and half. Each learns the language
of the other group during the two first preparatory years, followed by a 1 month
industrial internship in the alien culture. The cultural and ethical dimensions of
science and technology, and their teaching, are therefore a distinctive point of these
institutes. Just as in the case of Polytech, the two first years are preparatory and
students can or may have to move from one INSA to another, at the end of the second
year, in order to find a major adapted to their achievements and desires. The INSA
in Lyon counts among its 5400 students over 5 years of training, 20% of foreign
students, 32% of female students and 31% of grant students, which is a distinctive
mark of its social openness, compared to other more socially discriminative schools.
Among its 660 teachers, more than half are researchers as well and depend on a
laboratory. In mathematics, among the 45 members of the math Pole, 13 belong to
ICJ, and another 15 to other laboratories (computer science, acoustics, energy, civil
engineering).
There are 12 majors taught in Lyon, linked to research laboratories. They begin
in the third year:
• Biochemistry and Biotechnologies
• Bio-computer Sciences and Modeling
• Civil Engineering and Urbanism
• Electrical Engineering
• Energetic and Environmental Engineering
• Mechanic Conception Engineering
• Mechanic Development Engineering
• Plastic Process Mechanic Engineering
• Industrial Engineering
• Computer Science
• Material Sciences Engineering
• Telecommunications
6 Overview of Engineering Mathematics Education for STEM in EU 87

6.2.5.5 École Centrale de Lyon

This engineer school was founded in 1857 and belongs to the “Centrale group” of
eight schools, five of which are in France, others in Beijing (China), Casablanca
(Morocco) and Mahindra (India). This network selects students after 2 years of
preparatory school. Teaching therefore only lasts 3 years. The national competitive
exam, common to 10 schools (the five French Centrale and five other schools)
is called Concours Centrale-Supélec, is one of the three top competitive exams
in France with Polytechnique-ÉNS-ESPCI and Mines-Ponts. The best students in
French preparatory schools take these three selection exams. There are around 1000
students in this 3-year school, completed with around 100 Master students and 200
PhD and post-doctoral students. Half of the 200 teachers are as well researchers.
The 3 years of study are divided into 2 years of common core followed by seven
majors lasting for one semester:
• Civil Engineering and Environment;
• Mathematics and Decision;
• Aeronautics;
• Transportation and Traffic;
• Computer Science and Communication;
• Energy;
• Micro- and Nano-biotechnologies.
The study is structured around three main collaborative projects, one each year, of
9 months in the first 2 years and of 6 months in the third year. Internships punctuate
the education as well, of increasing complexity, from an execution internship of
1 month in the first year, application of 3–4 months in the second year, to the
study internship of 6 months in the third year. Sabbatical leave during the school
is promoted, for personal projects, such as industrial internship or academic study
abroad or professionally oriented projects. Once graduated, students can continue
their study with complementary Masters such as innovative design, management or
numerical methods in mathematics, with jointly accredited diplomas such as “Maths
in Action: from concept to innovation”. A fair proportion of students go on with PhD
studies, whether within one of the six research laboratories present inside the school,
or in a joint research laboratory where most of the teachers perform their research.
In mathematics, all researchers (8) belong to the ICJ laboratory.

6.2.5.6 Curriculum Details

The two first preparatory years in UCBL are representative of the amount of
mathematics followed by students. We chose to detail the Electrical Engineering
stream in INSA Lyon.
88 6 Overview of Engineering Mathematics Education for STEM in EU

S1:
Algebra 1: Foundations of logic, ensembles, maps, arithmetics, complex numbers,
R2 .
Calculus 1: Reals, real functions, sequences, limits, derivation, minimization,
maximization, inf, sup, derivation of an implicit function, higher order derivation,
convexity, l’Hôpital law, differential equations of first order, primitive.
S2:
Algebra 2: Linear algebra, polynomials, rational fractions, vector spaces of finite
and countable dim, linear applications, matrices, determinant.
Calculus 2: Integration and approximation, change of variable, simple elements,
circular and hyperbolic formulas, differential equations of second order, indefi-
nite integrals, applications in probability and statistics, Landau notation, limits,
Taylor polynomials and series, Taylor–Lagrange remainder.
S3:
Algebra 3: Diagonalization, groups, determinants, eigenspaces, spectral decompo-
sitions, Cayley–Hamilton, powers, exponential of a matrix.
Calculus 3: Several variables functions, differential calculus, applications, con-
vexity, Lagrange multipliers, implicit functions, Euler equations, isoperimetric
problems.
S4:
Algebra 4: Geometric algebra, bilinear forms, scalar products, rank, kernel, Gauss
orthogonalization, adjoint, spectral decomposition of self-adjoint operators,
quadratic forms, Sylvester theorem, affine geometry, conics, quadrics, O(p,q).
Calculus 4: Series and sequences, Cauchy, d’Alembert, uniform convergence, Abel
theorem, trigonometric and Fourier series, entire series, integrals depending on a
parameter, differentiation and continuity, eulerian functions, Laplace transform,
applications to differential equations, geometry and differential calculus, curves
and surfaces, geometry, parametric curves, curvature, Frenet frame, tangent and
normal spaces, vector spaces, differentials, line, surface and multi-dimensional
integrals, Stokes theorem, Green formula.
Applied algebra: Gröbner basis, Perron–Frobenius theorem and web indexation.
Student project.
S5:
Algebra 5: Groups and morphisms, Lagrange theorem, cyclic groups, morphism,
image, kernel, Euler index, Z/nZ ring, prime numbers, quotient groups, dihedral
groups, group action, orbits, stabilizer, Bernside, Sylow theorems, SO_3 sub-
groups, Platonian solids.
Numerical analysis: Linear algebra, Gauss method, iterative methods, conditioning,
spectral problems, power method. Nonlinear equations, Newton, secant method,
remainder estimation. Interpolation, approximation, polynomial interpolation,
mean squared, numerical integration, discrete Fourier transform, Cooley–Tuckey
fast Fourier transform, differential equations, Cauchy problem numerical solu-
tion, Euler method, Runge–Kutta, implicit and explicit methods.
Topology: Metric space, normed space, topological space, continuity, Baire lemma,
Banach fixed point, Bolzano–Weierstrass theorem, Ascoli theorem, Stone–
6 Overview of Engineering Mathematics Education for STEM in EU 89

Weierstrass theorem, Lebesgue integral, Riemann vs. Lebesgue, Lebesgue dom-
inated convergence theorem, Fubini theorem, Lebesgue measure on Rn , random
variables, measures.
Starting with the third year, the curriculum depends on which program we are
talking about: INSA, Polytech, CPE, Centrale Lyon. We chose to detail here
the Electrical Engineering program of INSA. The amount of mathematics is
decreasing with time; most of the technical prerequisites are put in the first year
of major study:
S6: 96 + 88 = 184 h of math for 8 ECTS. Hilbert spaces, Fourier series,
distributions, interpolation, approximation, numerical integration, solving linear
systems. Fourier, Laplace, Z-transforms, ODEs, linear systems, convolution,
difference equations using Matlab.
S7: 83 h for 4 ECTS. Complex analysis, PDE, probabilities. Numerical Analysis:
Non linear equations, PDEs Using Matlab.
S8: 60 h 3ECTS. Probability and statistics, conditional probabilities, statistics,
moments, central limit theorem, estimation, hypothesis tests, regressions, exper-
imentation plans.
From then on, mathematics is only used as a tool integrated in other teaching
units.
The peculiar fact about the preparatory school in UCBL is that, unlike most other
preparatory schools, it is organized by units that delivers ECTS: Each semester
weights 30 ECTS which come from five teaching units of 6 ECTS. The two first
years of the curriculum are followed by all students going to engineer schools at
UCBL, but there are some other non-math options shaping them in three different
groups: physics and math, computer science and math, and mechanics and math.6

6 More details can be found (in French) on http://licence-math.univ-lyon1.fr/doku.php?id=

mathgeneappli#premiere_annee_du_parcours; see also http://licence-math.univ-lyon1.fr/doku.
php?id=mathgenappliprog. The details for INSA can be found at the following URL: http://www.
insa-lyon.fr/formation/offre-de-formation2/g/d/?gr=ING.
Chapter 7
Case Studies of Math Education
for STEM in Russia

7.1 Analysis of Mathematical Courses in KNRTU-KAI

Ildar Galeev () and Svetlana Novikova and Svetlana Medvedeva
Kazan National Research Technical University named after A.N. Tupolev—KAI
(KNRTU-KAI), Automated Systems of Information Processing and Control Depart-
ment, Kazan, Russia
e-mail: monap@kstu.ru; sweta72@bk.ru; pmisvet@yandex.ru

7.1.1 Kazan National Research Technical University named
after A.N. Tupolev—KAI (KNRTU-KAI)

Kazan National Research Technical University named after A.N. Tupolev—KAI
(KNRTU-KAI) was established in 1932. The history of the University is closely
related to the progress of Russian aeronautics. Fundamental education and profound
scientific research are the distinguishing features of the university, which make it
very attractive for a great number of young people. Until recent times, it was known
as Kazan Aviation Institute (KAI).
In 1973, the Institute was named after Andrey N. Tupolev, the prominent
aircraft designer. In 1992, it obtained the status of State Technical University. In
2009, KNRTU-KAI became 1 of 12 universities selected among all the Russian
universities (from about 900 state HE institutions and 2000 private ones) which was
awarded the prestigious title of the “National Research University”.
KNRTU-KAI is a member of the European Universities Association EUA (2008)
and of the European Association of Aerospace Universities PEGASUS (2009).

© The Author(s) 2018 91
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_7
92 7 Case Studies of Math Education for STEM in Russia

KNRTU-KAI is the largest multidisciplinary educational and scientific complex
of the Republic of Tatarstan and the Volga region. The structure KNRTU-KAI
includes five institutes, one faculty, five branches in the Republic of Tatarstan
(Almetyevsk, Zelenodolsk, Naberezhnye Chelny, Chistopol, Leninogorsk). Despite
having the institute of Economics, Management and Social Technologies in its
structure, KNRTU should be considered as a technical university. The other four
of five main institutes and one faculty are completely devoted to engineering and
computer science specializations.
Today KNRTU-KAI is one of the leading Russian institutions in aircraft
engineering, engine- and instrument-production, computer science and radio- and
telecommunications engineering.
The university includes the following institutes and faculties:
• Institute of Aviation, Land Vehicles & Energetics
• Institute of Automation & Electronic Instrument-Making
• Institute of Technical Cybernetics & Informatics
• Institute of Radio-Engineering & Telecommunications
• Institute of Economics, Management and Social Technologies
• Physics & Mathematics Faculty
There is also the German–Russian Institute of Advanced Technologies (GRIAT).
German Partners of this institute are Ilmenau Institute of Technology, TU Ilmenau,
Otto-von-Guericke-Universität Magdeburg, OVGU, and Deutscher Akademischer
Austauschdienst, DAAD.
Currently there are more than 12,000 students in the university. Most of them
(more than 10,000) are technical (STEM) students. In total at six institutes and
one faculty there are 29 Bachelor and 30 Master STEM programs (including 9
Master STEM programs of the Institute GRIAT), also there is six Specialist (specific
Russian 5-year grade).
The mathematical training at the University is carried out by Department of
Higher Mathematics, Department of Special Mathematics, and Department of
Applied Mathematics and Computer Science.
Departments of Higher Mathematics (the total number of teachers is 8) and
Special Mathematics (the total number of teachers is 18) carry out classical
mathematical training in classical engineering programs.
The department of Applied Mathematics and Computer Science trains Bachelors
and Masters in the following study programs:
• “Informatics and Computer Science” (B.Sc., MSc);
• “Mathematics and Computer Science” (B.Sc., MSc);
• “Software Engineering” (B.Sc., MSc).
It also provides specialized courses for other areas within the university and in
the Institute of Computer Technology and Information Protection. The educational
process is realized by the department under the innovation program “Industrial
production of software and information technology tools” in accordance with state
educational standards of the Russian Federation and international standards on
7 Case Studies of Math Education for STEM in Russia 93

Computing Curricula and on professional standards in the field of information
technology. The staff of this department consists of 6 full-time professors, 12 full-
time associate professors, 2 full-time assistant professors, 2 teachers and 6 technical
specialists.
There are most general courses in all areas of training; the Institute of Computer
Technology and Information Protection provides courses in mathematics: algebra
and geometry, calculus, discrete mathematics, probability theory and mathematical
statistics, computational mathematics.
The Institute of Computer Technology and Information Protection has six
chairs:
• Department of Applied Mathematics and Computer Science
• Department of Information Security Systems
• Department of Computer Aided Design
• Department of Automated Data Processing Systems and Management
• Department of Computer Systems
• Department of Process Dynamics and Control
In 2015, unified training Bachelor’s plans were accepted at the university, in
which a block of mathematical courses is the same for all STEM programs.
The block of mandatory mathematics for all STEM programs includes the
following courses: Calculus, Algebra and Geometry, Probability Theory and Math-
ematical Statistics (semesters 1–4). The courses Discrete Mathematics and Math-
ematical Logic and Theory of Algorithms are included in the block of manda-
tory courses for the training of IT-professionals. The course of Computational
Mathematics (semester 5) is in the block of selective courses for the training
of IT-professionals. Other mathematical courses, such as Differential Equations,
Mathematical Physics, Complex Analysis and Functional Analysis in the selective
block, are courses for training of specialists by an in-depth study of mathematics.
An example of this Bachelor program is the program of Mathematics and Computer
Science.

7.1.2 Comparative Analysis of “Probability Theory
and Mathematical Statistics”

“Probability Theory and Mathematical Statistics” is a theoretical course with
approximately 3000 students. There are around 240 second year students, from 6
Bachelor and 2 Specialist of Institute of Computer Technology and Information
Protection (ICTIP) programs, who study this course at the Department of Applied
Mathematics and Computer Science. A comparison of this course was conducted
with the corresponding courses “Probability Calculus” and “Statistics” by Tampere
University of Technology (TUT). The course outlines are presented in Table 7.1.
94 7 Case Studies of Math Education for STEM in Russia

Table 7.1 Outlines of probability theory and statistics courses at KNRTU-KAI and TUT
Course information KNRTU-KAI TUT
Bachelor/master level Bachelor Bachelor
Preferred year 2 2
Selective/mandatory Mandatory Mandatory
Number of credits 6 4+4
Teaching hours 90 84
Preparatory hours 108 132
Teaching assistants 1 1–4
Computer labs Available Available
Average number of students on the course 60 200
Average pass % 85% 90%
% of international students None None

“Probability Theory and Mathematical Statistics” is a mandatory Bachelor
level course on second year—Probability Theory (third semester) and Mathemat-
ical Statistics (4th semester). Its prerequisite courses are: Calculus (first, second
semesters), Algebra and Geometry (second semester), Discrete Mathematics (sec-
ond, third semesters), and Mathematical Logics and Theory of Algorithms (third
semester). Prerequisite courses at TUT are Engineering Mathematics 1–4. The
follow-up courses are Theory of Stochastic Processes and System, Programming
and various special courses of all six ICTIP Bachelor programs.
In 2011, KNRTU-KAI has started to use LMS Blackboard, which is a learning
management system supporting e-learning. The content of educational materials is
included in the LMS Blackboard in a form suitable for e-learning. For Probability
Theory and Mathematical Statistics an e-learning course was developed for the
LMS Blackboard environment. It contains the mandatory e-learning components—
course schedule, lecture notes, guidelines for tutorials, computer labs, as well as
independent student work. There is also a mandatory component—a test for students
on theoretical items of the course. Thus, KNRTU-KAI uses a blended form of
learning types that combines classroom instruction with students’ independent work
using the LMS Blackboard environment.
The size of the course is 6 credits, which means on average 216 h of student’s
work (36 h for each credit). The credits are divided among different activities as
follows: lectures 36 h, tutorials 54 h, computer labs 18 h, homework 72 h and 36 h
for preparing to the examination.
There are about 240 students studying the course every year at the Institute of
Computer Technology and Information Protection. About 10% of them are foreign
students and about 20% are female.
The lectures are theoretical, but application examples for every theorem and
algorithm are shown as well. Tutorial classes are completely devoted to problem
solving, generally using paper and pen, but also MATLAB and Excel are used in
solving some problems. Computer labs use our own computer textbook “Introduc-
7 Case Studies of Math Education for STEM in Russia 95

tion to Mathematical Statistics” in real-time. This allows the students to generate a
sample from a given distribution, to build and explore random functions and their
density distributions, as well as to build and explore their numerical characteristics,
and to carry out and explore the algorithms for testing statistical hypothesis and
one-dimensional regression analysis.
Generally, our students have to pass four tests and complete four individual
laboratory workshops during the two semesters. When all this work has successfully
been done, they are allowed to take an exam. In the exam a student has to answer
thoroughly two questions from random topics, and to briefly answer some additional
questions. Prior to this, all the students of a group had to take a pen and paper test of
15 test items, which allows the teacher to determine the readiness of students for the
exam. The final grade is determined by the examiner. The grade depends on how
successfully all the parts of the exams were passed, and it takes into account the
test results, students’ practical work and laboratory work during the semester. The
final grade is mathematically dependent on the number of points in accordance with
the score-rating system, adopted by the university. The score-rating systems is the
following:
• less than 50 points is unsatisfactory (grade “2”),
• from 50 to 69 points is satisfactory (grade “3”),
• from 70 to 84 points is good (grade “4”),
• more than 84 points is excellent (grade “5”).
The course is supported by the following educational software and TEL tools:
• MATLAB and Excel are used in tutorials for solving some problems.
• The computer textbook “Introduction to Mathematical Statistics” is used for con-
struction and study of statistical estimations of distributions and their parameters,
and also for testing the knowledge and monitoring and evaluating the skills of
students in mathematical statistics.
• The e-Learning Systems LMS Blackboard platform is used for testing the
students’ knowledge on our course.
Since September 2016, we have used the Math-Bridge system for training in and
monitoring of students’ abilities in solving problems in the theory of probability.
The piloting operation of the Math-Bridge system has been done as a part of the
MetaMath project.

7.1.2.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Table 7.2. Outcome competencies are given in Tables 7.3, 7.4, and 7.5.
96 7 Case Studies of Math Education for STEM in Russia

Table 7.2 Core 0 level prerequisite competencies of probability theory and statistics courses at
KNRTU-KAI and TUT
Core 0
Competency KNRTU-KAI TUT
Data handling With some exceptionsa X
Probability With some exceptionsb X
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Functions and their inverses X X
Sequences, series, binomial expansions X X
Logarithmic and exponential functions X X
Indefinite integration X X
Definite integration, applications to areas and volumes X X
Proof X X
Sets X X
a Interpret data presented in the form of line diagrams, bar charts, pie charts; interpret data presented

in the form of stem and leaf diagrams, box plots, histograms; construct line diagrams, bar charts,
pie charts, stem and leaf diagrams, box plots, histograms for suitable data sets; calculate the mode,
median and mean for a set of data items
b Define the terms “outcome”, “event” and “probability”; calculate the probability of an event

by counting outcomes; calculate the probability of the complement of an event; calculate the
probability of the union of two mutually exclusive events; calculate the probability of the union
of two events; calculate the probability of the intersection of two independent events

Table 7.3 Core 0 level outcome competencies of probability theory and statistics courses at
KNRTU-KAI and TUT
Core 0
Competency KNRTU-KAI TUT
Calculate the mode, median and mean for a set of data items X X
Define the terms ‘outcome’, ‘event’ and ‘probability’ X X
Calculate the probability of an event by counting outcomes X X
Calculate the probability of the complement of an event X X
Calculate the probability of the union of two mutually exclusive events X X
Calculate the probability of the union of two events X X
Calculate the probability of the intersection of two independent events X X

7.1.2.2 Summary of the Results

The comparison shows that the two courses cover generally the same topics and
competences. A difference was observed in the place of the course in the curriculum
of the degree program. The course on Probability theory and mathematical statistics
is studied in the third and fourth semesters in KNRTU-KAI, but at TUT this course
is studied during the fourth and fifth semesters, depending on the given study
program. Before MetaMath project there were more differences: in KNRTU-KAI
this course was studied in the second semester (first year of training). No difference
was observed in the total number of hours.
7 Case Studies of Math Education for STEM in Russia 97

Table 7.4 Core 1 level outcome competencies of the probability theory and statistics courses at
KNRTU-KAI and TUT
Core 1
Competency KNRTU-KAI TUT
Calculate the range, inter-quartile range, variance and standard X X
deviation for a set of data items
Distinguish between a population and a sample X X
Know the difference between the characteristic values (moments) of a X X
population and of a sample
Construct a suitable frequency distribution from a data set X X
Calculate relative frequencies X X
Calculate measures of average and dispersion for a grouped set of data X X
Understand the effect of grouping on these measures X X
Use the multiplication principle for combinations X X
Interpret probability as a degree of belief X X
Understand the distinction between a priori and a posteriori X X
probabilities
Use a tree diagram to calculate probabilities X X
Know what conditional probability is and be able to use it (Bayes’ X X
theorem)
Calculate probabilities for series and parallel connections X X
Define a random variable and a discrete probability distribution X X
State the criteria for a binomial model and define its parameters X X
Calculate probabilities for a binomial model X X
State the criteria for a Poisson model and define its parameters X X
Calculate probabilities for a Poisson model X X
State the expected value and variance for each of these models X X
Understand what a random variable is continuous X X
Explain the way in which probability calculations are carried out in the X X
continuous case
Relate the general normal distribution to the standardized normal X X
distribution
Define a random sample X X
Know what a sampling distribution is X X
Understand the term ‘mean squared error’ of an estimate X X
Understand the term ‘unbiasedness’ of an estimate X X

Thus, as a result of the modernization, the number of hours in the course
“Probability theory and mathematical statistics” now fully coincide: in the third
semester 108 h (3 credit units in the second year) and “Mathematical Statistics”
108 h in the fourth semester (3 credit units in the third year). Studying the course
has been shifted from the first to the second year.
A comparison on the use of information technology in the course “Probability
Theory and Mathematical Statistics” was carried out in the two universities. The
use of information technology in teaching this course is on a high level.
98 7 Case Studies of Math Education for STEM in Russia

Table 7.5 Core 2 level outcome competencies of the probability theory and statistics courses at
KNRTU-KAI and TUT
Core 2
Competency KNRTU-KAI TUT
Compare empirical and theoretical distributions X X
Apply the exponential distribution to simple problems X X
Apply the normal distribution to simple problems X X
Apply the gamma distribution to simple problems X X
Understand the concept of a joint distribution X X
Understand the terms ‘joint density function’, ‘marginal distribution X X
functions’
Define independence of two random variables X X
Solve problems involving linear combinations of random variables X X
Determine the covariance of two random variables X X
Determine the correlation of two random variables X X
Realize that the normal distribution is not reliable when used with X X
small samples
Use tables of the t-distribution X X
Use tables of the F-distribution X X
Use the method of pairing where appropriate X X
Use tables for chi-squared distributions X X
Decide on the number of degrees of freedom appropriate to a particular X X
problem
Use the chi-square distribution in tests of independence (contingency X X
tables)
Use the chi-square distribution in tests of goodness of fit X X
Set up the information for a one-way analysis of variance X X
Derive the equation of the line of best fit to a set of data pairs X X
Calculate the correlation coefficient X X
Place confidence intervals around the estimates of slope and intercept X X
Place confidence intervals around values estimated from the regression X X
line
Carry out an analysis of variance to test goodness of fit of the X X
regression line
Interpret the results of the tests in terms of the original data X X
Describe the relationship between linear regression and least squares X X
fitting
Understand the ideas involved in a multiple regression analysis X X
Appreciate the importance of experimental design X X
Recognize simple statistical designs X X
7 Case Studies of Math Education for STEM in Russia 99

Table 7.6 Outlines of optimization courses at KNRTU-KAI and TUT
Course information KNRTU-KAI TUT
Bachelor/master level Master Master
Preferred year 2 2
Selective/mandatory Mandatory Selective
Number of credits 3 5
Teaching hours 43 50
Preparatory hours 65 60
Teaching assistants 1 1
Computer labs Available Available
Average number of students on the course 20 30
Average pass % 95% 90%
% of international students – 60%

7.1.3 Comparative Analysis on “Optimisation Methods”

“Optimisation Methods” is a MSc level theoretical course with approximately 20
students (1 academic group). Most of them (80%) are young men. Their ages vary
from 20 to 28 years. Most of them are 21 or 22 years old. The course is studied as the
second Master’s course in the fall semester (third semester). A special feature is that
these students have no education in IT—Informatics as their second competence.
Teaching the course is carried out at the Department of Applied Mathematics and
Computer Science at the Institute of Computer Technology and Information Protec-
tion (CTIP). Comparison of this course was conducted with a similar course “Opti-
misation Methods” by TUT. Outlines of both the courses are presented in Table 7.6.
Optimisation Methods is a mandatory Master’s course in the second year of study
of the Master’s. Prerequisite courses for the Optimisation Methods are: Calculus 1–
3, Linear Algebra, Probability theory and Mathematical Statistics, Graph Theory.
These courses are part of the Bachelor’s degree.
Since 2011, KNRTU-KAI has started to use LMS Blackboard, which is a
learning management system supporting e-learning. The content of educational
materials are included in the LMS Blackboard in a form suitable for e-learning.
An e-learning course on Optimisation Methods was developed in the LMS
Blackboard environment. Mandatory e-learning components—course schedule,
lecture notes, guidelines for tutorials, Computer labs, as well as independent student
work. There is also a mandatory component—a test for students on theoretical items
of the course. Thus, KNRTU-KAI uses a blended form of learning that combines
classroom instruction with students’ independent work using the LMS Blackboard
environment.
The size of the course is 3 credits, which means on average 108 h of work
(36 h for each credit). The credits are divided among different activities as follows:
lectures 10 h, computer labs 20 h, homework 35 h, 30 h for exam preparation and 3 h
for exam.
100 7 Case Studies of Math Education for STEM in Russia

About 20 students study this course every year at the Institute of Computer
Technology and Information Protection. About 20% of them are female.
The lectures are theoretically based, but application examples on all the methods
and algorithms are shown also, as well. Computer labs use our own computer
tutorial “Optimisation methods”, which exists on LMS Blackboard, MS Excel and
MATLAB are also in use.
Generally, students have to pass one test and complete five individual laboratory
workshops during the semesters. When all this work is successfully done, they are
allowed to enter an exam. In the exam a student has to answer thoroughly to two
questions from random topics, and briefly to answer some additional questions.
Prior to this, all the students of a group had to take a pen and paper test of 40
test items, which allows the teacher to determine the readiness of the students for
the exam. The final grade is determined by the examiner. The grade depends on how
successfully all the parts of the exams were passed, and it takes into account the test
results, students’ practical work and laboratory work during the semester. The final
grade is mathematically dependent on the number of points in accordance with the
score-rating system, adopted by the university.

7.1.3.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Table 7.7. Outcome competencies are given in Tables 7.8 and 7.9.

Table 7.7 Core 0 level prerequisite competencies of the courses on optimization at KNRTU-KAI
and TUT
Core 0
Competency KNRTU-KAI TUT
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their inverses X X
Logarithmic and exponential functions X X
Rates of change and differentiation X X
Stationary points, maximum and minimum values X X
Definite integration, applications to areas and volumes X X
Proof X X
Data handling X
Probability X
7 Case Studies of Math Education for STEM in Russia 101

Table 7.8 Core 1 level outcome competencies of the courses on optimization at KNRTU-KAI and
TUT
Core 1
Competency KNRTU-KAI TUT
Rational functions X
Hyperbolic functions X
Functions With some exceptionsa X
Differentiation X X
Solution of nonlinear equations X X
Vector algebra and applications X X
Matrices and determinants X X
a Obtain the first partial derivatives of simple functions of several variables; use appropriate

software to produce 3D plots and/or contour maps

Table 7.9 Core 2 level outcome competencies of the courses on optimization at KNRTU-KAI and
TUT
Core 2
Competency KNRTU-KAI TUT
Ordinary differential equations X
Functions of several variables With some exceptionsa X
a Definea stationary point of a function of several variables; define local maximum, local minimum
and saddle point for a function of two variables; locate the stationary points of a function of several
variables

Unfortunately SEFI Framework does not describe learning outcomes suitable
for “Optimisation Methods”. After successful completion of the course at KNRTU-
KAI, a student should:
• Have experience in using optimization techniques on a PC with Microsoft
Windows operating system.
• Be able to formulate basic mathematical optimization problems, depending on
the type of quality criteria and availability limitations.
• Be able to solve the problem by the classical method of unconstrained minimiza-
tion of functions with one and several variables.
• Be able to use the simplex method of linear programming to solve related
problems;
• Be able to apply basic numerical methods for solving nonlinear programming in
practice.
• Know the fundamentals of modern technologies to develop mathematical models
and find optimal solutions.
• Know the classification of extreme problems and methods for solving them.
• Know the main analytical and numerical methods for unconstrained minimiza-
tion of functions of one and several variables.
• Know the basic methods for solving nonlinear programming problems.
• Know the main algorithms for solving linear programming problems.
102 7 Case Studies of Math Education for STEM in Russia

7.1.3.2 Summary of the Results

The comparison has shown the following similarities and differences: In both the
universities we use blended learning, however, at TUT significantly more hours are
devoted to lectures.
The content of the courses in the universities is very similar: the students acquire
knowledge on the main topics of conditional and unconditional optimization and
linear programming. In Kazan University the dimensional optimization problem is
studied separately.
Concerning the use of computer technology. Both universities use learning
management systems for presentation of the lecture material, in TUT is Moodle
is used, and KNRTU-KAI uses LMS Blackboard. Both systems provide similar
opportunities for students. In both universities laboratory work is conducted in
specialized programs: in TUT MATLAB is used, in Kazan a specially designed
program. The course “Optimisation methods” does not require extensive modern-
ization. To improve the course the best way is to use Math-Bridge technology.

7.1.4 Comparative Analysis on “Discrete Mathematics”

The course “Discrete Mathematics” is designed for Bachelor program “Software
Engineering”, training profile “Development of software and information systems”.
The course “Discrete Mathematics” is taught in semesters 2 and 3. In the current
academic year 25 students are enrolled on this program on the 2 semester (19 males
and 6 females) and 17 students on the 3 semester (11 males and 6 females). The
course has been modified to meet the requirements of professional standards of
SEFI [1] and implemented in e-learning system Math-Bridge. Modification and
development of the e-learning course was carrie out in DFKI by employees group of
KNRTU-KAI, who were specially sent there for training and for e-learning course
development.
This course was compared with two courses “Discrete Mathematics” and
“Algorithm Mathematics” at Tampere University of Technology (TUT), Tampere,
Finland. The course outlines can be found in Table 7.10.
Discrete Mathematics is mandatory course for Bachelors of the first and second
year of study (2 and 3 semesters). Prerequisite courses for the Discrete Mathematics
are: Calculus 1–3, Linear Algebra. Since 2011, KNRTU-KAI started to use learning
management system LMS Blackboard. The course material for the e-learning
courses are in the LMS Blackboard environment. For Discrete Mathematics an e-
learning course was developed into the LMS Blackboard environment. Mandatory
e-learning components—course schedule, lecture notes, guidelines for tutorials,
computer labs, as well as for independent work of students. There is also a
mandatory component—test items for students on the theoretical material of course.
Thus, KNRTU-KAI uses a blended form of education that combines classroom
instruction with independent work of students in LMS Blackboard.
7 Case Studies of Math Education for STEM in Russia 103

Table 7.10 Outlines of discrete and algorithm mathematics courses at KNRTU-KAI and TUT
Course information KNRTU-KAI TUT
Bachelor/master level Bachelor Bachelor
Preferred year 1 and 2 2
Selective/mandatory Mandatory Mandatory
Number of credits 12 4+4
Teaching hours 180 49+42
Preparatory hours 252 65+65
Teaching assistants 1 1–3
Computer labs No
Average number of students on the course 50 150
Average pass % 85% 85%
% of international students 10%

The course size is 12 credits, which means on average 432 h of work (36 h for
each credit). The credits are divided among different activities as follows: lectures
108 h, tutorials 72 h, homework 252 h, and 72 h for exam preparation.
About 50 students are learning for these course every year at the Institute of
Computer Technology and Information Protection. About 20% of them are female.
The lectures are theoretically based, but application examples of every method
and algorithm are shown as well. Computer labs use LMS Blackboard, MS Excel
and MATLAB.
Generally our students have to pass 4 tests and execute 36 tutorials during the
two semesters. When all this work is successfully done, they are allowed to pass an
exam. The students have to answer in detail two questions from random topics, and
briefly answer some additional questions. Prior to this, all the students have written
answers to a test of 15 test items, which allows one to determine the readiness of
a student for the exam. The final grade is determined by the examiner depending
on how successfully all the parts of the exams were passed. It takes into account
the results of the tests, practical work and laboratory work during the semester. The
final grade is mathematically dependent on the number of points in accordance with
the score-rating system, adopted by the university. The score-rating system is the
following:
• less than 50 points is unsatisfactory (grade “2”),
• from 50 to 69 points is satisfactory (grade “3”),
• from 70 to 84 is good (grade “4”),
• more than 84 is excellent (grade “5”).
Our course is supported by the following educational software and TEL tools:
MATLAB and Excel are used in tutorials to solving some problems and e-Learning
Systems LMS Blackboard platform for testing the students’ knowledge on our
course.
In the next academic year (from fall semester 2017) we will use the international
intellectual Math-Bridge system for training and monitoring abilities to solve
104 7 Case Studies of Math Education for STEM in Russia

problems of Discrete Mathematics in the trial operation of the system in accordance
with the project MetaMath.

7.1.4.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Table 7.11. Outcome competencies are given in Tables 7.12, 7.13, 7.14,
and 7.15.

Table 7.11 Core 0 level prerequisite competencies of the discrete and algorithm mathematics
courses at KNRTU-KAI and TUT
Core 0
Competency KNRTU-KAI TUT
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their inverses X X
Logarithmic and exponential functions X X
Rates of change and differentiation X X
Stationary points, maximum and minimum values X X
Definite integration, applications to areas and volumes X X
Proof X X
Data handling X X
Probability X –

Table 7.12 Core 0 level outcome competencies of the discrete and algorithm mathematics courses
at KNRTU-KAI and TUT
Core 0
Competency KNRTU-KAI TUT
Sets X X

Table 7.13 Core 1 level outcome competencies of the discrete and algorithm mathematics courses
at KNRTU-KAI and TUT
Core 1
Competency KNRTU-KAI TUT
Mathematical logic X X
Sets With some exceptionsa X
Mathematical induction and recursion X X
Graphs X X
Combinatorics X
a Compare the algebra of switching circuits to that of set algebra and logical connectives; analyze

simple logic circuits comprising AND, OR, NAND, NOR and EXCLUSIVE OR gates
7 Case Studies of Math Education for STEM in Russia 105

Table 7.14 Core 2 level outcome competencies of the discrete and algorithm mathematics courses
at KNRTU-KAI and TUT
Core 2
Competency KNRTU-KAI TUT
Number system With some exceptionsa X
Algebraic operators X
Recursion and difference equations With some exceptionsb X
Relations X X
Graphs X X
Algorithms With some exceptionsc X
a Carry out arithmetic operations in the binary system
b Define a sequence by a recursive formula
c Understand when an algorithm solves a problem; understand the worst case analysis of an

algorithm; understand the notion of an NP-complete problem (as a hardest problem among NP
problems)

Table 7.15 Core 3 level Core 3
outcome competencies of the
Competency KNRTU-KAI TUT
discrete and algorithm
mathematics courses at Combinatorics X
KNRTU-KAI and TUT Graph theory X

7.1.4.2 Summary of the Results

In KNRTU-KAI for the study of the course “Discrete Mathematics” four times
more hours allotted than at the Tampere University of Technology. This is due to
the fact that this item in KNRTU-KAI is studied for two semesters and only one
semester in TUT. As a consequence, many of the topics that are covered in KNRTU-
KAI not covered in TUT. In the course “Discrete mathematics” a lot of attention
in both universities given to sections related to mathematical logical expressions
and algebraic structures. This is useful for students who will continue to study
subjects such as “Theory of Algorithms” and “Logic”. The TUT students are being
prepared with the help of the Moodle e-learning environment. Moodle also helps to
gather feedback from students after the course. In KNRTU-KAI it is not used. This
experience is very useful.
In general, course “Discrete Mathematics” by KNRTU-KAI is rather good and
meets the requirements for IT-students teaching. But it will be useful to make the
course some more illustrative by using computer-based training systems. As a result
the course “Discrete Mathematics” has been modernized and the e-learning training
course was developed in Math-Bridge system.
106 7 Case Studies of Math Education for STEM in Russia

7.2 Analysis of Mathematical Courses in LETI

Mikhail Kuprianov and Iurii Baskakov and Sergey Pozdnyakov and Sergey Ivanov
and Anton Chukhnov and Andrey Kolpakov and Vasiliy Akimushkin
Saint Petersburg State Electrotechnical University (LETI), Saint Petersburg, Russia
e-mail: mskupriyanov@mail.ru; bosk@bk.ru; sg_ivanov@mail.ru
Ilya Posov
Saint Petersburg State Electrotechnical University (LETI), Saint Petersburg, Russia
Saint Petersburg State University (SPbU), Saint Petersburg, Russia
Sergey Rybin
Saint Petersburg State Electrotechnical University (LETI), Saint Petersburg, Russia
ITMO University, Department of Speech Information Systems, Saint Petersburg,
Russia

7.2.1 Saint Petersburg State Electrotechnical University
(LETI)

Saint Petersburg State Electrotechnical University (LETI) was founded in 1886 as a
technical college of the post and telegraph department. In 1891 it gained the state of
an institute. The first elected university director was the Russian-born scientist A.S.
Popov, who is one of the people credited for inventing the radio. From its founding
LETI was a center of Russian electrical engineering. Many outstanding Russian
scientists have worked there over the years. After the October Revolution, in 1918,
the institution was renamed Leningrad Electrotechnical Institute (LETI).
In the 1920s, LETI played a significant role in the development of electrification
plans of Russia. After the Second World War, intensive developments of new
scientific fields were started at the institute: radio electronics and cybernetics,
electrification and automatization of industrial equipment, automatics and teleme-
chanics, computer science and optoelectronics.
In 1991 the institute was renamed after the city and became Saint-Petersburg
Electrotechnical Institute. However, it kept Lenin’s name in its title. In early 1990s
the Faculty of Humanities was founded, and the Institute was granted university
status and was renamed again into “V.I.Ulyanov (Lenin) Saint-Petersburg State
Electrotechnical University”. In 1998 it was renamed for the current name.
Despite having a faculty of humanities and faculty of economics and manage-
ment in its structure, LETI should be considered as a technical university. The other
five of seven main faculties are completely devoted to engineering and computer
science specializations.
There is also an Open Faculty (OF) for evening and correspondence students
for all specializations. Other faculties, which are the Faculty of Military Education
and the newly created Faculty of Common Training do not have their own students;
7 Case Studies of Math Education for STEM in Russia 107

Table 7.16 Outlines of mathematical logics and theory of algorithms courses at LETI (ML&TA)
and TUT (AM)
Course information LETI TUT
Bachelor/master level Bachelor Bachelor
Preferred year 1 2
Selective/mandatory Both Both
Number of credits 5 4
Teaching hours 40 49
Preparatory hours 65 65
Teaching assistants No 1–2
Computer labs Available Available
Average number of students on the course 400 150
Average pass % 85% 90%
% of international students 20% Less than 5%

they teach students from other faculties. International Student Office works with
foreign students, including teaching them Russian. Finally, the Faculty of Retraining
and Raising the Level of Skills does not work with students at all, it works with
university personnel.
Currently there are more than 8200 students in the university. Most of them
(more than 7500) are technical (STEM) students. Totally at five faculties there are
17 Bachelor and 13 Master STEM programs, also there is one Specialist (specific
Russian 5-year grade) program at FCTI.
Mathematical education in LETI is divided between two departments: Depart-
ment of Higher Mathematics-1 and -2.

7.2.2 Comparative Analysis of “Mathematical Logics
and Theory of Algorithms” (ML&TA)

“Mathematical Logics and Theory of Algorithms” is a more theoretical course.
There are around 400 second year students from all 6 Bachelor and 1 Specialist
FCTI programs studying this course. All of them are supposed to be applied
specializations, but, of course, each program contains a lot of theoretical courses.
The course outlines with the corresponding course from Tampere University of
Technology (TUT) “Algorithm Mathematics” are given in Table 7.16.
ML&TA is mandatory Bachelor level course of the second year. Its prerequisite
courses are Discrete Mathematics; Linear Algebra and Calculus 1–2. The follow-up
courses are Programming and various special courses from all seven FCTI Bachelor
programs.
The department responsible for the course is the Department of Higher
Mathematics-2. Now the University is in the process of restructuring and probably
the next year two departments of mathematics will be joined, but for purpose
of teaching FCTI students a new Department of Algorithm Mathematics will be
108 7 Case Studies of Math Education for STEM in Russia

founded. It will be responsible for this course and also for the prerequisite course of
Discrete Mathematics.
The overall number of credits is 5. In Russia we have 36 h in 1 credit, so the total
amount is 180 h for this course. Among them 36 h of lectures, 36 h of tutorials, 72 h
of homework and 36 h of exam preparation. There are about 400 students studying
the course every year. About 20% of them are foreign students and about 30% are
female.
The lectures are theoretically based, but applications of every theorem and
algorithm are shown. Tutorial classes are completely devoted to solving problems,
but generally with pen and paper, without computers. However, some of the
algorithms are to be implemented by students while studying this course, and almost
all of these are used later while passing the follow-up courses. We also offer some
students an alternative way to pass an exam by creating a computer program.
The course is generally oriented to individual work. However, group work can be
episodically introduced as an experiment.
Generally our students have to pass two tests and complete three individual
homeworks during the semester. When all this work is successfully done, they are
allowed to enter an exam. The examination system depends on the lecturer: oral or
written. The oral exam is a classic Russian form of examination where the student is
to answer thoroughly two questions from random topics, solve a problem and briefly
answer some additional questions.
The final mark is determined by the examiner depending on how successfully all
the parts of the exams were passed. The written exam consists of several problems
which are to be solved by students. The final mark mathematically depends on the
solved problems ratio. In both cases the final mark belongs to the classic Russian
system: from 2 (failed) to 5 (excellent).
Our course is supported by the following TEL tools: Problem generators can
create a huge amount of variants of one problem using one or few certain template(s)
and changing numbers, letters etc. Of course, this is support for teachers activity, not
for students. Google sites are used by teacher and students to exchange information.
For example, students can submit some homeworks to the teacher using this sites.
The TEL systems were not generally used in this course before the MetaMath
project. Introducing TEL systems (Moodle and our own subject manipulators) is
the main direction of our course modification.

7.2.2.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite compe-
tencies are presented in Table 7.17. Outcome competencies are given in
Tables 7.18, 7.19, 7.20, 7.21, and 7.22.
7 Case Studies of Math Education for STEM in Russia 109

Table 7.17 Prerequisite Core 1
competencies of
Competency LETI TUT
mathematical logics and
theory of algorithms courses Arithmetic of real numbers X X
at LETI (ML&TA) and TUT Algebraic expressions and formulas X X
(AM) Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their inverses X X
Sequences, series, binomial expansions X X
Logarithmic and exponential functions X X
Proof X X
Sets X X

Table 7.18 Core 1 level prerequisite competencies of mathematical logics and theory of algo-
rithms courses at LETI (ML&TA) and TUT (AM)
Core 1
Competency LETI TUT
Sets With some exceptionsa X
Mathematical induction and recursion X X
Graphs X
Matrices and determinants X X
Combinatorics X
a Excluding logical circuits

Note: sometimes graphs are completely included in Discrete Mathematics, then they should be
considered as prerequisites. In other cases they are divided between courses of DM and ML&TA,
and then they should be partially considered as outcomes

Table 7.19 Core 2 level prerequisite competencies of mathematical logics and theory of algo-
rithms courses at LETI (ML&TA) and TUT (AM)
Core 2
Competency LETI TUT
Number systems X
Algebraic operations Excluding hamming code X
Relations Excluding inverse binary relations and ternary relations X
Note: Binary relations sometimes are included in Discrete Mathematics, so they are considered as
prerequisites; in other cases they are included in ML&TA and so they are outcome competences

Table 7.20 Core 1 level outcome competencies of mathematical logics and theory of algorithms
courses at LETI (ML&TA) and TUT (AM)
Core 1
Competency LETI TUT
Mathematical logic X X
Graphs X
110 7 Case Studies of Math Education for STEM in Russia

Table 7.21 Core 2 level outcome competencies of mathematical logics and theory of algorithms
courses at LETI (ML&TA) and TUT (AM)
Core 2
Competency LETI TUT
Relations Excluding inverse binary relations and ternary relations X
Algorithms X X
Note: Binary relations sometimes are included in Discrete Mathematics, so they are considered as
prerequisites; in other cases they are included in ML&TA and so they are outcome competences

Table 7.22 Core 3 level outcome competencies of mathematical logics and theory of algorithms
courses at LETI (ML&TA) and TUT (AM)
Core 3
Competency LETI TUT
Find the distance (shortest way) between two vertices in a graph X
Find a the graph and his matrix for a relation X
Use topological sort algorithm and transitive closure algorithms X
Understand the concept of Boolean function X X
Construct a truth table for a function X X
Obtain CNF and DNF of a function X
Obtain Zhegalkin polynomial of a function X
Build a composition of two or more functions in different forms X
Recognize function membership in one of the post classes X
Use post criteria for a set of functions X
Recognize context-free grammar X
Construct context-free grammar for a simple language X
Build a parser for a grammar using Virt algorithm X
Recognize table and graph representation of final state machine X
Recognize automata language X
Carry out set operations with automata languages X
Obtain FSM for regular expression and vice versa X
Obtain determined FSM for non-determined one X
FSM minimization X
Understand the notion of a turing machine X
Run simple turing machines on paper X
Construct simple turing machine X
Run Markov algorithm X
Recognize the prenex and Skolem form of first-order formulas X
Obtain the prenex and Skolem form for a certain formula X
Unify first-order logic formulas X
Use resolution method for propositions and first-order logic X
7 Case Studies of Math Education for STEM in Russia 111

7.2.2.2 Summary of the Results

The comparison shows that both courses cover generally the same topics and
competences. However LETI ML&TA course contains more competences than the
corresponding Algorithm Mathematics course in TUT. It follows from two main
reasons: first, in LETI the course has more credits and hours; and second, LETI
course is more intensive which is good for gifted students and may be probably
be not so good for some others. This conclusion does not lead to any course
modifications just because we do not want to reduce our course.
The second conclusion is that TUT’s course is more applied and uses more
TEL systems. There is space for modifications of our course. The main idea of
modification is to introduce Moodle in our course and add there as much as possible
lectures, test and laboratory works which could help our students to get closer to
understanding of our course through their self-activity on the internet.

7.3 Analysis of Mathematical Courses in UNN

Dmitry Balandin and Oleg Kuzenkov and Vladimir Shvetsov
Lobachevsky State University of Nizhny Novgorod (UNN), Nizhny Novgorod,
Russia
e-mail: dbalandin@yandex.ru; kuzenkov_o@mail.ru; shvetsov@unn.ru

7.3.1 Lobachevsky State University of Nizhni Novgorod (UNN)

The Lobachevsky State University of Nizhni Novgorod (UNN) is one of the leading
classical research universities in Russia established in 1916. The university provides
fundamental education in accordance with the best traditions of Russian Higher
Education.
By the decision of the Russian Government, in 2009 UNN was awarded the
prestigious status of a National Research University.
Being an innovative university, the University of Nizhni Novgorod provides
high-quality research-based education in a broad range of academic disciplines
and programs. The combination of high educational quality and accessibility of
education due to a great variety of educational program types and forms of training
is a distinctive feature of the University in today’s global knowledge economy.
UNN is ranked 74 by the QS World University Ranking BRICS, and it has five
QS Stars for excellence in Teaching, Employability, Innovation, and Facilities. UNN
is one of only 15 Russian universities awarded in 2013 with a prestigious grant of
the Government of the Russian federation to implement the Leading Universities
International Competitiveness Enhancement Program.
112 7 Case Studies of Math Education for STEM in Russia

Holding leading positions among Russian universities, it is a worldwide recog-
nized institute of higher education: UNN is represented in the Executive Board of
the Deans European Academic Network (DEAN), it is a member of the European
University Association (EUA) and has direct contractual relations with more than
20 foreign educational and research centers.
The UNN has been implementing a great number of international projects,
funded by the European Commission, IREX (the International Research &
Exchanges Board), National Fund for Staff Training and other widely known
organizations. The University also closely cooperates with the institutes of the
Russian Academy of Sciences and the largest transnational high-tech companies
(Microsoft, Intel etc.).
At present about 40,000 students and post-graduate students from more than 65
countries of the world are studying at UNN, about half of which pursue STEM
courses. Training on all educational programs is conducted by a highly qualified
teaching staff, over 70% of them having a PhD or Doctor of Science degrees.
Lobachevsky State University of Nizhni Novgorod as a National Research
University has been granted the right to develop its own self-imposed educational
standards (SIES). In 2010, the first UNN standard was developed in the area
of studies “010300 Fundamental Computer Science and Information Technology
(FCSIT)” (Bachelor’s degree). In 2011, the second UNN standard was developed in
the area of studies “Applied Computer Science” (ACS) (Bachelor’s degree).
The Bachelor’s program “Information Technologies”, is aimed at training experts
in high-level programming for hi-tech companies of the information industry.
UNN is engaged in successful cooperation with major international IT companies
like Intel, Microsoft, IBM, Cisco Systems, NVIDIA that provide the University
with advanced computer equipment and software. This ensures that the educational
process is based on the latest achievements in this field of science and technology.
At the University, there are research laboratories established with the support of
Intel Corporation as well as educational centers of Microsoft and Cisco Systems.
In 2005 Bill Gates, president of Microsoft Corporation, named the University
of Nizhni Novgorod among the world’s ten leading universities in the field of
high-performance computing. In 2013 UNN built the powerful supercomputer
“Lobachevsky”, which is the second fastest supercomputer in Russia.
Teachers, working in this program, are all recognized experts in various fields of
science, Doctors and Candidates of Science. The program of studies in Information
Technologies is envisaged by Computing Curricula 2001 recommended by such
international organizations as IEEE-CS and ACM.
Graduates of the Bachelor’s program “Fundamental Computer Science and Infor-
mation Technologies”, are prepared for the following activities in their professional
sphere:
• scientific and research work in the field of theoretical computer sciences, as well
as development of new information technologies;
• design and application of new information technologies, realized in the form of
systems, products and services;
7 Case Studies of Math Education for STEM in Russia 113

• application of information technologies in project designing, management and
financial activities.
In 2014, UNN educational program in the area of studies “010300 Fundamental
Computer Science and Information Technology (FCSIT)” (Bachelor’s degree)
received the accreditation of Russian Engineer Education Association. As a rule,
UNN IT-students study in 3 groups with 20 students in each one. Mean age is 17.
Male students are twice as many as females. Moreover, there is one group with 20
foreign students (Asia, Africa), who study in English.

7.3.2 Comparative Analysis of “Mathematical Analysis”

Mathematical Analysis (Calculus) is included in curricula as one of the core subjects
of mathematics with their own distinct style of reasoning. Mathematical Analysis
is ubiquitous in natural science and engineering, so the course is valuable in
conjunction with Engineering majors. The purpose of the course is to provide a
familiarity to concepts of the real analysis, such as limit, continuity, differentiation,
connectedness, compactness, convergence etc. The number of students is 60. It is a
mandatory course in UNN that combines both theoretical and applied approaches.
Mathematics plays key role in the course and this course forms a foundation for
several other applied special disciplines in corresponding educational programs.
Mathematical Analysis contains three parts corresponding to the semesters of study:
Mathematical Analysis I, II, and III.
Mathematical Analysis I contains such topics as basic properties of inverse, expo-
nential, logarithmic and trigonometric functions; limit, continuity and derivative of
a function, evaluating rules of a derivative, function research and curve-sketching
techniques, applications of derivative in the optimization problems, L’Hôpital’s rule.
Mathematical Analysis II contains such topics as indefinite and definite integral and
their properties, rules of integration, integration of rational functions, evaluating
area between curves and surface area using integrals, integrals application in
physics, functions of several variables, implicit functions. Mathematical Analysis
III contains such topics as theory of number series and functional series, Fourier
series, double integrals and further multiple integrals, line and surface integrals.
There are three comparisons: first UNN Mathematical Analysis I (MAI) and
Tampere University of Technology (TUT) Engineering Mathematics 1 (EM1),
second UNN Mathematical Analysis II (MAII) and TUT Engineering Mathematics
3 (EM3), and third UNN Mathematical Analysis III (MAIII) and TUT Engineering
mathematics 4 (EM4). The course outlines are seen in Tables 7.23, 7.24, and 7.25,
respectively.
There are no prerequisite courses. The follow-up course for Mathematical
Analysis is Differential Equations. In UNN this course is included in the group of
core mandatory mathematical courses of the corresponding educational programs.
114 7 Case Studies of Math Education for STEM in Russia

Table 7.23 Outlines of MAI (UNN) and EM1 (TUT) courses
Course information UNN TUT
Bachelor/master level Bachelor Bachelor
Preferred year 1 1
Selective/mandatory Mandatory Mandatory
Number of credits 6 5
Teaching hours 108 57
Preparatory hours 180 76
Teaching assistants 3 1–3
Computer labs No Available
Average number of students on the course 72 200
Average pass % 90% 90%
% of international students 25% Less than 5%

Table 7.24 Outlines of MAII (UNN) and EM3 (TUT) courses
Course information UNN TUT
Bachelor/master level Bachelor Bachelor
Preferred year 1 1
Selective/mandatory Mandatory Mandatory
Number of credits 6 5
Teaching hours 108 62
Preparatory hours 180 76
Teaching assistants 3 1–3
Computer labs No Available
Average number of students on the course 72 200
Average pass % 90% 90%
% of international students 25% Less than 5%

Table 7.25 Outlines of MAIII (UNN) and EM4 (TUT) courses
Course information UNN TUT
Bachelor/master level Bachelor Bachelor
Preferred year 2 1–2
Selective/mandatory Mandatory Selective
Number of credits 8 4
Teaching hours 144 49
Preparatory hours 72 57
Teaching assistants 3 1–2
Computer labs No Available
Average number of students on the course 65 150
Average pass % 90% 85%
% of international students 25% Less than 5%
7 Case Studies of Math Education for STEM in Russia 115

Teaching the course in UNN is more theory-based and unfortunately does not
include any innovative pedagogical methods and tools such as: blended learning,
flipped classroom, MOOCs, project-based learning, inquiry-based learning, col-
laborative learning, etc. In TUT on the other hand one can find blended learning,
collaborative learning, project-based approach and active use of modern TEL tools
for administration, teaching and assessment purposes. Not all modern pedagogical
technologies are used in TUT but those of them that do exist in educational process
are applied widely and successfully.
The overall number of hours and credits in Mathematical Analysis I, II, and III is
20 cu. = 720 h. It consists of 202 h lectures, 183 h tutorials and 200 h independent
work (homework) and 135 h control of independent work (exams) There are two
types of homework assignments: these are problems which arise while lecturing,
assigned almost every class day and set of problems assigned during practical
lessons (weekly). Tests and exams are conducted some times per each term in
written, electronic, and oral forms.
There are the following types of assessment used in UNN: positive (perfect,
excellent, very good, good, satisfactory); or negative (unsatisfactory, poor). For
perfect, the student displays in-depth knowledge of the main and additional material
without any mistakes and errors, can solve non-standard problems, has acquired all
the competences (parts of competences) relating to the given subject in a compre-
hensive manner and above the required level. A stable system of competences has
been formed, interrelation with other competences is manifested.
For excellent grade, the student displays in-depth knowledge of the main
material without any mistakes and errors, has acquired all the competences (parts
of competences) relating to the given subject completely and at a high level, a
stable system of competences has been formed. For very good grade, the student
has sufficient knowledge of the main material with some minor mistakes, can
solve standard problems and has acquired completely all the competences (parts
of competences) relating to the given subject. For good grade, the student has the
knowledge of the main material with some noticeable mistakes and has acquired in
general the competences (parts of competences) relating to the given subject.
For satisfactory grade the student has the knowledge of the minimum material
required in the given subject, with a number of errors, can solve main problems,
the competences (parts of competences) relating to the subject are at the minimum
level required to achieve the main learning objectives. If the grade is unsatisfactory,
the knowledge of the material is insufficient, additional training is required, the
competences (parts of competences) relating to the subject are at a level that is
insufficient to achieve the main learning objectives. Finally, for poor grade, there is
lack of knowledge of the material, and relevant competences have not been acquired.
There are two midterm exams (tests) at the end of the first and second semesters
and there is a final exam (test) at the end of the third semester.
116 7 Case Studies of Math Education for STEM in Russia

7.3.2.1 Contents of the Course

The course comparison is based on the SEFI framework [1]. Prerequisite competen-
cies of the MAI and EM1 courses are given in Table 7.26. Core 1, core 1, core 2,
and core 3 outcome competencies for MAI and EM1, MAII and EM3, MAIII and
EM4, are given in Tables 7.26, 7.27, 7.28, 7.29, 7.30, 7.31, and 7.32.

Table 7.26 Core 0 level Core 0
prerequisite competencies of
Competency UNN TUT
the MAI (UNN) and EM1
(TUT) courses Functions and their inverses X X
Progressions, binomial expansions X X
Logarithmic and exponential functions X X
Rates of change and differentiation X X
Maximum and minimum values X X
Proof X

Table 7.27 Core 1 level Core 1
outcome competencies of the
Competency UNN TUT
MAI (UNN) and EM1 (TUT)
courses Hyperbolic functions X X
Rational functions X X
Functions X X
Differentiations X X
Sequences and series Only sequences In EM3

Table 7.28 Core 0 level Core 0
outcome competencies of
Competency UNN TUT
MAII (UNN) and EM3
(TUT) courses Indefinite integral X X
Definite integral, applications X X

Table 7.29 Core 1 level Core 1
outcome competencies of
Competency UNN TUT
MAII (UNN) and EM3
(TUT) courses Methods of integration X X
Application of integration X X

Table 7.30 Core 2 level Core 2
outcome competencies of
Competency UNN TUT
MAII (UNN) and EM3
(TUT) courses Functions of several variables X In EM4
Nonlinear optimization X
Fourier series X
7 Case Studies of Math Education for STEM in Russia 117

Table 7.31 Core 1 level Core 1
outcome competencies of
Competency UNN TUT
MAIII (UNN) and EM4
(TUT) courses Sequences and series Only series In EM3

Table 7.32 Core 2 level Core 2
outcome competencies of the
Competency UNN TUT
MAIII (UNN) and EM4
(TUT) courses Function of several variables X
Nonlinear optimization X
Fourier series X
Double integrals X X
Further multiple integrals X X
Vector calculus X
Line and surface integrals X

7.3.2.2 Summary of the Results

Comparative analysis shows that thematic content and learning outcomes for both
universities are quite close. The difference is observed in the number of hours. The
total number of hours is about 700 in UNN, whereas in TUT it is about 400. New
information technologies and, in particular, e-learning systems are actively used
in TUT. This allows TUT to take out some of the material for independent study
and focus on really difficult topics of the discipline. E-learning systems also allow
one to automate and, as a result, simplify the knowledge assessment process. In
UNN these information technologies are occasionally used some times per semester.
UNN established pre- and posttest to control incoming and outcoming students’
knowledge in mathematical analysis. The electronic course in the Moodle system
is implemented for teaching mathematical analysis for students in study programs
AMCS and FCSIT (Applied Mathematics and Computer Sciences, Fundamental
Computer Sciences and Information Technologies, respectively). All tests are based
on SEFI competences; they contain a large amount of simple tasks (during 60 min
students must fulfill 20 tasks) that allow one to control 160 SEFI competencies from
the zeroth to the second level in the areas of “Analysis and Calculus”. Authors used
Moodle system rather than Math-Bridge because it is more cross-platform and will
help the project results to be more sustainable.
The main steps of the course modernization are: including new bridging section
“Elementary Mathematics” at the beginning of Mathematical Analysis I; increasing
the number of seminars and decreasing the number of lectures; increasing the
number of consultations (from 15 to 30 h); mandatory regular testing students
during the term (includes using Math-Bridge) two tests per term; increasing the
number of engineering examples in the course; using project learning (two projects
per term at least). The topics of the projects are: “Approximate calculation of
functions: a creation of the calculator for logarithms, trigonometric and hyperbolic
functions”, “Technical and physical applications of derivatives”, “Research of the
118 7 Case Studies of Math Education for STEM in Russia

Table 7.33 Outlines of mathematical modeling courses at UNN and TUT
Course information UNN TUT
Bachelor/master level Bachelor Both
Preferred year 3 3–4
Selective/mandatory Selective Selective
Number of credits 3 5
Teaching hours 36 28
Preparatory hours 54 80
Teaching assistants – –
Computer labs Available Available
Average number of students on the course 10 60
Average pass % 100% 95%
% of international students 0% Less than 5%

normal distribution, the logistic function, the chain line”, “The calculation of the
center of gravity”, “Applications of Euler integral”, and so on.

7.3.3 Comparative Analysis of “Mathematical Modeling”

The subject of study in this course are modeling methods and relevant mathematical
model used in a variety of subject areas. As a result students should know methods
of mathematical modeling. The course helps to learn how to model situations in
order to solve problems. The course is based on the theory of differential equations
and the theory of probability. There is a final test at the end of the semester. The
“Mathematical Modeling” course at UNN is compared with the similar course
“Basic Course on Mathematical Modeling” at Tampere University of Technology
(TUT). Course outlines are given in Table 7.33.
The main goal of the course is in studying the fundamental methods of
mathematical modeling. It contains such topics as the history of modeling, classes
of models, differential equations and systems as mathematical models, dynamic
systems, mathematical models in physics, chemistry, biology, ecology, models of
a replication, the selection processes, continuous and discrete models of behaviour,
models of adaptive behaviour, models of decision making, models of a selection of
strategies, the selection and optimization, models of social and economic behaviour,
optimal control, information models, the model of transmission and storage of
information.

7.3.3.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Table 7.34.
7 Case Studies of Math Education for STEM in Russia 119

Table 7.34 Core 2 level Core 2
prerequisite competencies of
Competency UNN TUT
the mathematical modeling
courses at UNN and TUT Ordinary differential equations X X
The first-order differential equations X X
The second-order differential equations X X
Eigenvalue problems X X
Nonlinear optimization X X

7.3.3.2 Summary of the Results

The main steps of the course modernization are decreasing the number of lectures
(from 36 h to 18 h), including independent work by students (18 h), mandatory
regular testing of students (four times during the term, includes the use of Math-
Bridge), using engineering examples in the course, using method of project learning
(four projects per term). There are the following projects:
1. Introduction in mathematical modeling (simplest models: Volterra, Verhulst,
Laurence, Lotka etc.). The aim of the project is to understand principles of
mathematical modeling and the entity of the project’s work.
2. Modeling eco-systems. The aim of the project is to apply qualitative research
methods for mathematical models given in the form of systems of differential
equations.
3. Chemical kinetics modeling. The aim of the project is to apply the Lyapunov
function method.
4. The final project. The aim of the project is to apply information technologies in
mathematical modeling. The topics of the projects are “The calculation of the
index of competitiveness”, “The calculation of evolutionary stable daily vertical
migrations of aquatic organisms”, “Models of strategies for socio-economic
behaviour”, “Neural network models”, and so on.

7.4 Analysis of Mathematical Courses in OMSU

Sergey Fedosin
Ogarev Mordovia State University (OMSU), Department of Automated Systems of
Information Management and Control, Saransk, Russia
Ivan Chuchaev
Ogarev Mordovia State University (OMSU), Faculty of Mathematics and IT,
Saransk, Russia
e-mail: mathan@math.mrsu.ru
Aleksei Syromiasov
Ogarev Mordovia State University (OMSU), Department of Applied Mathematics,
Differential Equations and Theoretical Mechanics, Saransk, Russia
e-mail: syal1@yandex.ru
120 7 Case Studies of Math Education for STEM in Russia

7.4.1 Ogarev Mordovia State University (OMSU)

Ogarev Mordovia State University (hereinafter OMSU) is a classical university,
though a wide range of technical major programs are presented here. Since 2011
OMSU has the rank of National Research university, which reflects the university’s
high status in research in Russia. More than 24,000 students study in OMSU, almost
13,000 of them have intramural instruction.
There are several institutions in OMSU that may be referred to as STEM. They
are: faculty of Architecture and Civil Engineering, faculty of Biotechnology and
Biology, faculty of Mathematics and IT, institute of Mechanics and Energetics,
institute of Physics and Chemistry, institute of Electronics and Lighting Technology
and a branch in the town of Ruzaevka (institute of Machine-Building). In these
institutions there are about 5000 students.
It is difficult to determine the number of STEM disciplines because every STEM
major profile (or group of major profiles) has its special curriculum in mathematics,
its special curriculum in physics and so on. So there is variety of STEM disciplines
with the same name but with different contents. Totally, there are more than 100
different STEM disciplines in OMSU.
OMSU is the oldest and the biggest higher education institution in the Mordovian
Republic. Its main campus is situated in Saransk, which is the capital of Mordovia.
The institution was founded at the 1st of October, 1931 as Mordovian Agropedagog-
ical Institute. Next year it was transformed into Mordovian Pedagogical Institute.
Based on this institute Mordovian State University was organized in 1957. Finally,
in 1970 it was named after the poet N.P. Ogarev.
More than 150,000 people graduated from the university during its history.
OMSU graduates formed the backbone of scientific, engineering, pedagogical,
medical and administrative staff in the Mordovian Republic. Many of the graduates
work in other regions of Russia or abroad.

7.4.2 Comparative Analysis of “Algebra and Geometry”

Algebra and Geometry (AlGeo) for the major study programs Informatics and Com-
puter Science (ICS) and Software Engineering (SE) is a fundamental mathematical
course, so it is more of a theoretical than an applied course. OMSU tries to make
it more applied by giving students programming tasks (for example, students must
write a computer program which solves linear systems of equations using Gaussian
elimination). Totally, there are about 50 first year students in two study programs
(ICS and SE), and all of them must study this course.
The course information is to be compared with our European partner in the
project Université Claude Bernard Lyon 1 (UCBL). The course outlines can be seen
in Table7.35.
7 Case Studies of Math Education for STEM in Russia 121

Table 7.35 Outlines of algebra and geometry courses at OMSU and UCBL
Course information OMSU UCBL
Bachelor/master level Bachelor Bachelor
Preferred year 1 1
Selective/mandatory Mandatory Mandatory
Number of credits 6 (216 h) 6+6
Teaching hours 108 60 + 60
Preparatory hours 108 120
Teaching assistants No yes
Computer labs Several programming tasks as Gaussian elimination using
homework Sage-math
Average number of students 50 189
on the course
Average pass % 85–90% 65%
% of international students Less than 10% 14%

A prerequisite for Algebra and Geometry is mathematics as studied at secondary
school. Follow-up courses are Calculus, Discrete Mathematics, Probability theory
and Statistics. The course of Algebra and Geometry is included in the group of
mandatory mathematical courses that all the students in the ICS and IT-study
programs must study during the first years.
Though course contents for ICS and SE study programs are almost the same,
there are two departments responsible for the course of Algebra and Geometry.
The Department of Calculus is responsible for this course for ICS profile. There
are two full professors and ten associate professors working in this department.
The Department of Applied Mathematics, Differential Equations and Theoretical
Mechanics is responsible for this course for SE-study program. Four full professors,
14 associate professors and 3 teachers work in this department.
The overall number of credits for the course is 6. Russian credit is 36 h, so the
total amount is 216 h for this course. Among them one has 36 h of lectures, 72 h of
tutorials and 108 h of homework.
There is one 2-h lecture on Algebra and Geometry and two 2-h tutorials every
week. During the tutorials students solve some problems (fulfill computational
tasks) under the teacher’s direction and control. Students may be given home tasks,
which must be done during preparatory hours. We do not need computer labs for
every tutorial, but some home tasks for this course are programming tasks. Students
write simple programs on topics of the course, for example, they must write a
program that solves a linear equation system. The programming language is of the
student’s choice. Totally, there are about 25 ICS students and about 25 SE students
attending the course; for these profiles, lectures and tutorials are set separately. It
is hard to mention the number of students who finish this certain course, because
student expulsion in OMSU is the result of failing of two or more courses. But
about 10–15% students of SE and ICS profiles usually drop out after the first 1 or 2
years of study, and as a rule, these students have problems in mathematics. Usually
122 7 Case Studies of Math Education for STEM in Russia

only Russian students study on ICS and SE programs; in OMSU most popular for
international students is the Medical Institute.
The average age of students attending the course is 18, and about 75% of
the students are male. In OMSU there is no mandatory formal procedure of
course rating; unofficial feedback contains more likes than dislikes of Algebra and
Geometry.
The course of Algebra and Geometry is established for the first year students
and is quite theoretical. So the pedagogy is traditional: students listen to lectures,
fulfill some tasks during tutorials and do their homework. We think that younger
students must have less educational freedom than older ones, and the role of the
teacher in the learning process for younger students must be more explicit. That is
why we do not use project-based learning in this course. But, of course, we try to
make the learning process more interesting, sometimes funny and even competitive.
For example, from time to time a group of students in the tutorial is divided into
several subgroups and every subgroup fulfills some task. Solving linear equations
systems by Cramer’s rule is a good example of a task that can be “parallelized” so
that every student in the subgroup does his/her part of the total work, and students
in a subgroup collaborate. This kind of work in subgroups is very competitive and
students like it. Blended learning is used episodically: some teachers use Moodle
for distance learning. But for the students who have resident instruction (and here
we discuss these students) it is more the exception than the rule.
A rating system is used for assessment at OMSU. The maximum rating is 100
points; one can get 70 points during the semester and only 30 points (as a maximum)
is left for the exam procedure. In a semester students get their points for work in the
classes and for fulfilling two–three large tests. These tests include a large amount of
tasks; not only the answers, but the solutions are controlled by the teacher. An exam
is passed in oral form and it includes two theoretical questions (like a theorem with
a proof) and a computational task. A student’s final rating sums up the semester and
exam ratings. If this sum is 86 or more, the student’s knowledge of the course is
graded as excellent (ECTS grade A or B); if the sum is between 71 and 85, student
has grade “good” (approximately ECTS grade C or D); if the sum is between 51 and
70, student’s grade is “satisfactory” (ECTS grade E). Finally, a student fails (gets
grade “non-satisfactory” which is equivalent to ECTS grade F) if his/her rating is
50 or less.
As for technology, high-level programming languages (C++ or Pascal) are used
for homework. Programming (topics are harmonized with the course contents) is
a mandatory part of the tests fulfilled by the students during the semester. This
programming activity has an influence on the student’s final rating.
Until 2014, TEL systems were not used in teaching of the course for students
with resident instruction, but after participating in MetaMath project we plan to use
Math-Bridge, GeoGebra and, perhaps, Moodle in teaching Algebra and Geometry.
Now Moodle is used by students who study distantly (such distant learning is not
the part of resident instruction now and is not included in this analysis).
E-mail and social networks are used sometimes to have a closer connection with
students, to provide tasks for them and so on.
7 Case Studies of Math Education for STEM in Russia 123

7.4.2.1 Contents of the Course

The comparison is based on the SEFI framework [1], for each level from Core 0 to
Core level 3 only subareas of mathematics are listed. The symbol “X” means that
all the competencies from this subarea are prerequisite for the course; exclusions
are listed in explicit form. If few competencies (minor part) from the subarea are
prerequisite, the description begins with “Only. . . .”.
Core 1 and Core 1 level prerequisite competencies are presented in Tables 7.36
and 7.37. Outcome competencies of level zero, one, two and three, are given in
Tables 7.38, 7.39, 7.40, and 7.41.

Table 7.36 Core 0 level prerequisite competencies of algebra and geometry courses at OMSU and
UCBL
Core 0
Competency OMSU UCBL
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Linear laws With some exceptionsa X
Quadratics, cubics and polynomials X X
Proof X X
Geometry X X
Trigonometry X X
Coordinate geometry Onlyb X
Trigonometric functions and applications X X
Trigonometric identities X X
a Obtain and use the equation of a line with known gradient through a given point; obtain and use

the equation of a line through two given points; use the intercept form of the equation of a straight
line; use the general equation ax + by + c = 0; determine algebraically whether two points lie on
the same side of a straight line; recognize when two lines are perpendicular; interpret simultaneous
linear inequalities in terms of regions in the plane
b Calculate the distance between two points; give simple example of a locus; recognize and interpret

the equation of a circle in standard form and state its radius and center; convert the general equation
of a circle to standard form

Table 7.37 Core 1 level prerequisite competencies of algebra and geometry courses at OMSU
and UCBL
Core level 1
Competency OMSU UCBL
Vector arithmetic Excluding: determine the unit vector in a specified X
direction
Vector algebra and applications Excluding: all competencies due to vector product X
and scalar triple product
124 7 Case Studies of Math Education for STEM in Russia

Table 7.38 Core 0 level outcome competencies of algebra and geometry courses at OMSU and
UCBL
Core 0
Competency OMSU UCBL
Linear laws Those not in prereq. X
Functions and their inverses Onlya Onlyb
Rates of change and differentiation Only: obtain the equation of the tangent and same
normal to the graph of a function
Coordinate geometry Those not in prereq. X
a Understand how a graphical translation can alter a functional description; understand how a
reflection in either axis can alter a functional description; understand how a scaling transformation
can alter a functional description
b All competencies excluding the properties of 1/x and the concept of limit (Calculus)

Table 7.39 Core 1 level outcome competencies of algebra and geometry courses at OMSU and
UCBL
Core 1
Competency OMSU UCBL
Conic sections X X
3D coordinate geometry X X
Vector arithmetic Only: determine the unit vector in a X
specified direction
Vector algebra and applications Only: all competencies due to vector X
product and scalar triple product
Matrices and determinants Excluding: use appropriate software to Sage-math used
determine inverse matrices
Solution of simultaneous linear X X
equations
Linear spaces and With some exceptionsa X
transformations
a Define a subspace of a linear space and find a basis for it; understand the concept of norm; define

a linear transformation between two spaces; define the image space and the null space for the
transformation

7.4.2.2 Summary of the Results

The main findings that course comparison has shown are: More exact name
for OMSU course of Algebra and Geometry should be “Linear Algebra and
Analytic Geometry”. The French course is much more fundamental and much
more extensive. Though OMSU course of Algebra and Geometry is one of the
most theoretical in Software Engineering and Informatics and Computer Science
study programs, it is more adapted to the specificity of these programs. It is less
fundamental than the French one and this is the price to pay for having numerous
IT-courses in the curriculum. Though the UCBL course of Algebra and Geometry is
in total much more extensive, the amount of students’ learning load per semester is
7 Case Studies of Math Education for STEM in Russia 125

Table 7.40 Core 2 level outcome competencies of algebra and geometry courses at OMSU and
UCBL
Core 2
Competency OMSU UCBL
Linear optimization Onlya Same
Algorithms Only: understand when an algorithm
solves a problem; understand the ‘big O’
notation for functions
Helix Only: recognize the parametric equation of
a helix
Geometric spaces With some exceptionsb With some exceptionsc
and transformations
a Recognize a linear programming problem in words and formulate it mathematically; represent the

feasible region graphically; solve a maximization problem graphically by superimposing lines of
equal profit
b Understand the term ‘invariants’ and ‘invariant properties’; understand the group representation

of geometric transformations; classify specific groups of geometric transformations with respect to
invariants
c Apply the Euler transformation, cylindrical coordinates, group representation, classification of

group representation

Table 7.41 Core 3 level outcome competencies of algebra and geometry courses at OMSU and
UCBL
Core 3
Competency OMSU UCBL
Geometric core of computer graphics With some exceptionsa
Matrix decomposition Strassen’s algorithm for quick multiplying
of matrices
a Write a computer program that plots a curve which is described by explicit or parametric equations

in cartesian or polar coordinates; know Bresenham’s algorithm and Xiaolin Wu’s algorithm of
drawing lines on the display monitor

bigger in OMSU’s course (108 vs. 60 teaching hours, that is, 80% more). In OMSU
the percent of teaching hours is more than in UCBL (50% vs. 36%).
The conclusions of these findings are the measures for modernization of the
course: Compared to UCBL, there is a lack of time in OMSU (on the programs
that are involved in the MetaMath project) to study all algebra and geometry. So
it is important to define learning goals and problems of the course more precisely
and to follow them more strictly. It is necessary to collect study material according
to the learning goals and problems defined. Study material should be even more
illustrative than today and closer to IT-specificity. One of the ways to do this is to
increase the number of computer programming labs; another is to use some software
packages like GeoGebra. Students’ preparatory work should be organized in a more
effective way. The use of learning management systems (LMS) like Moodle or
Math-Bridge will be very helpful here. The advantage of Math-Bridge is that this
LMS is specially oriented to support mathematical courses.
126 7 Case Studies of Math Education for STEM in Russia

Table 7.42 Outlines of discrete mathematics (DM) and algorithm mathematics (AM) courses at
OMSU and TUT
Course information OMSU TUT
Bachelor/master level Bachelor Bachelor
Preferred year 1 or 2 2
Selective/mandatory Mandatory Mandatory
Number of credits 6 (216 h) 4 (105 h)
Teaching hours 108 49
Preparatory hours 108 65
Teaching assistants No Yes
Computer labs Several programming tasks as Available
homework
Average number of students on the 50 150
course
Average pass % 85–90% 90%
% of international students Less than 10% Less than 5%

7.4.3 Comparative Analysis of “Discrete Mathematics”

Discrete mathematics (DM) for the study programs ICS and SE is a fundamental
mathematical course, so it is more theoretical than applied. But it is obvious that
among all the theoretical courses it is the most applied and closest to the future
of IT-professions of the students. OMSU tries to make the course more applied by
giving the programming tasks to the students (for example, students must write a
computer program which returns a breadth-first search in a connected graph). There
are about 25 first year students in the SE program and 25 second year students in
the ICS program, and all of them must study this course.
The corresponding course with our European partner Tampere University of
Technology (TUT) is “Algorithm Mathematics” (AM). Course outlines can be seen
in Table 7.42.
Prerequisite courses for Discrete Mathematics are secondary school mathematics
and Algebra and Geometry. Follow-up courses are Mathematical Logic and Algo-
rithm Theory, Theory of Automata and Formal Languages, Probability Theory and
Statistics. The course of Discrete Mathematics is included in the group of mandatory
mathematical courses that must be studied by all students of ICS and IT programs
during the first years of study.
The Department of Applied Mathematics, Differential Equations and Theoretical
Mechanics is responsible for this course for both programs, SE and ICS. Four full
professors, 14 associate professors and 3 teachers work in this department.
7 Case Studies of Math Education for STEM in Russia 127

The overall number of credits is 6. As mentioned before, in Russia we have 36 h
in 1 credit, so the total amount is 216 h for this course. Among them are 36 h of
lectures, 72 h of tutorials and 108 h of homework. Finnish credits contain less hours.
As for Algebra and Geometry, there is one 2-h lecture on Discrete Mathematics
and two 2-h tutorials every week. During tutorials students solve some problems
(fulfill computational tasks) under a teacher’s direction. Students are given home
tasks which must be done during preparatory hours. Computers are used also in
controlling and grading the students’ programming homework.
Course statistics for Discrete Mathematics is similar to that of Algebra and
Geometry. There are about 25 ICS students and about 25 SE students attending
the course; for these programs, lectures and tutorials on Discrete Mathematics are
set separately. As said before, the number of students who do not finish the course is
hard to give; after failing for the first time the student has two extra tries to pass the
exam. Usually only Russian students study on ICS and SE programs. The average
age of students attending the course is 19 (because for ICS students the course is in
the second year of their study), and about 75% of the students are male. In OMSU
there is no mandatory formal procedure of course rating; but as a rule students of IT-
programs like the course because it is very “algorithmic” and close to programming.
The course of Discrete Mathematics is established for the first year or for second
year students and is more theoretical than applied. The pedagogical methods used
for this course are the same as for the course of Algebra and Geometry—and they
are traditional. Students attend to lectures, fulfill tasks during tutorials and do their
homework. Project work is not used widely, though some students interested in the
course may be given additional large tasks (doing these tasks influences their rating).
Similar to Algebra and Geometry, we do our best to make the learning process more
interesting. For example, from time to time a group of students in the tutorial is
divided into several subgroups and every subgroup fulfills some task. Searching
a path in a connected graph by various methods is an example of such work. As
mentioned before, this kind of work in subgroups is very competitive: students in
different subgroups try to fulfill their task more quickly and more correctly than
other subgroups. Blended learning is used episodically: some teachers use Moodle
for distance learning. But for the students that have resident instruction (and here
we discuss these students) it is more the exception than the rule.
Assessment and grading is similar to the course on “Algebra and Geometry”,
described above, and it is not repeated here.

7.4.3.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competencies
are presented in Table 7.43. Outcome competencies are given in Tables 7.44
and 7.45, 7.46, and 7.47.There are actually no other prerequisite competencies for
Discrete Mathematics in OMSU. The course is taught from the very beginning of
the field.
128 7 Case Studies of Math Education for STEM in Russia

Table 7.43 Prerequisite competencies of discrete mathematics and algorithm mathematics
courses at OMSU and TUT
Core 1
Competency OMSU TUT
Proof X X

Table 7.44 Core 0 level outcome competencies of discrete mathematics and algorithm mathemat-
ics courses at OMSU and TUT
Core 0
Competency OMSU TUT
Sets X X

Table 7.45 Core 1 level outcome competencies of discrete mathematics and algorithm mathemat-
ics courses at OMSU and TUT
Core 1
Competency OMSU TUT
Sets With some exceptionsa X
Mathematical induction and recursion X X
Graphs X
Combinatorics X
a Compare the algebra of switching circuits to that of set algebra and logical connectives; analyze

simple logic circuits comprising AND, OR, NAND, NOR and EXCLUSIVE OR gates

Table 7.46 Core 2 level outcome competencies of discrete mathematics and algorithm mathemat-
ics courses at OMSU and TUT
Core 2
Competency OMSU TUT
Number systems Only: use Euclid’s algorithm for finding the
greatest common divisor.
Algebraic operations X X
Recursion and difference equations Only: define a sequence by a recursive X
formula.
Relations X X
Graphs X
Algorithms Excluding: competencies due to NP and X
NP-complete problems.
Geometric spaces and transformations Only: understand the group representation
of geometric transformations.

7.4.3.2 Summary of the Results

The main findings that are consequent from the comparison made are as follows:
According to the amount of learning hours and to the list of topics covered the
OMSU course of Discrete Mathematics is more extensive than the course in TUT.
7 Case Studies of Math Education for STEM in Russia 129

Table 7.47 Core 3 level outcome competencies of discrete mathematics and algorithm mathemat-
ics courses at OMSU and TUT
Core 3
Competency OMSU TUT
Combinatorics Understanding the link between n-ary relations and relational
databases. Ability to normalize database and to convert from
1NF to 2NF.
Graph theory Write a computer program that finds the components of
connectivity, minimal spanning tree and so on.
Algebraic structures Using Shannon–Fano’s and Huffman’s methods to obtain
optimal code; the LZW zipping algorithm, the Diffie-Hellman
key exchange method; finding the RSA algorithm.

Many topics that are covered in OMSU are not covered in TUT. Teachers pay
attention to algebra within the course of Discrete Mathematics both in OMSU and
in TUT. So it is a good practice for IT-students to learn about algebraic structures in
the framework of Discrete Mathematics. Finnish colleagues widely use e-learning
(Moodle), which helps to organize students’ preparatory work more effectively. This
experience is very useful. Another useful experience is that Moodle helps teachers
to collect feedback from students after finishing the course.
These findings motivate the following modernization measures: In general,
OMSU course of Discrete Mathematics is rather good and meets the requirements
for teaching IT-students. But it will be useful to make it some more illustrative. It
will be a good practice to continue using computer programming home works. In
OMSU we should use e-learning to organize preparatory work for students in a more
effective way. Math-Bridge and Moodle will help here. Also it will be suitable to
collect some feedback from students.

7.5 Analysis of Mathematical Courses in TSU

Ilia Soldatenko and Alexander Yazenin
Tver State University (TSU), Information Technologies Department, Applied Math-
ematics and Cybernetics Faculty, Tver, Russia
e-mail: soldis@tversu.ru; Yazenin.AV@tversu.ru
Irina Zakharova
Tver State University (TSU), Mathematical Statistics and System Analysis Depart-
ment, Applied Mathematics and Cybernetics Faculty, Tver, Russia
e-mail: zakhar_iv@mail.ru
Dmitriy Nikolaev
Tver State University (TSU), International Relations Center, Tver, Russia
e-mail: Nikolaev.DS@tversu.ru
130 7 Case Studies of Math Education for STEM in Russia

7.5.1 Tver State University (TSU)

Tver State University is one of the largest scientific and educational centers in Cen-
tral Russia. Responding actively to modern-day challenges, the institution of higher
education is developing dynamically, while preserving tradition. TSU ensures the
preparation of qualified specialists in the sphere of physico-mathematical, natural,
human and social sciences, as well as of education and pedagogy, economy and
administration among other areas. Tver State University is a classical institution
with a total quantity of students equal to about 10 thousand, about half of which
pursue STEM courses.
The Tver State University has had a long and difficult developmental path. The
university’s history starts on December 1, 1870, when, in Tver, a private pedagogical
school named after P.P. Maximovich was opened. It was later on reformed in
1917 to become the Tver Teachers’ Institute, after which it became the Kalinin
Pedagogical Institute. Before the 1970s, tens of thousands of specialists graduated
at the Pedagogical Institute with university qualifications. On September 1, 1971, an
outstanding event took place in the Institute’s history; it was renamed Kalinin State
University.
In 1990, the Kalinin State University was renamed Tver State University. Its
graduates work successfully at schools, scientific institutions, as well as in economic
and social organizations. The university’s scholars have also made a considerable
contribution to making up and developing many scientific fields and research areas.
Today, our personnel consists of about 600 professors, including 100 doctors, full
professors and about 400 professors holding a PhD degree, as well as associate
professors.
TSU is comprised of 12 faculties and 2 institutes, which are the following:
• Institute of Pedagogical Education and Social Technologies,
• Institute of Economics and Management,
• Faculty of Biology,
• Faculty of History,
• Faculty of Mathematics,
• Faculty of Geography and Geo-ecology,
• Faculty of Foreign Languages and International Communication,
• Faculty of Applied Mathematics and Cybernetics,
• Faculty of Psychology,
• Faculty of Sport,
• Physico-Technical Faculty,
• Faculty of Philology,
• Faculty of Chemistry and Technology,
• Faculty of Law.
Our personnel consists of about 600 professors, including 100 doctors, full
professors and about 400 professors holding a PhD degree, as well as associate
professors.
7 Case Studies of Math Education for STEM in Russia 131

The main directions of research and development at the University are carried
out in the field of natural and exact sciences: mathematics, mechanics, physics,
chemistry, biology, geo-ecology and computer science. There is also a lot of
research in the fields of the humanities and social sciences, such as sociology,
linguistics, literature, history, economics, state and law, as well as in protection of
the environment, human ecology and demography. At the University, there are over
20 scientific schools carrying out research in relevant scientific topics of natural
sciences and humanities within 15 fields of study. Their activities are recognized
internationally, as well as domestically.
TSU maintains close ties with more than 30 universities in Europe, the USA and
the Commonwealth of Independent States, and it carries out exchange programs,
and it provides education to international students and actively participates in
various international educational programs. Tver State University’s long-standing
partners include the University of Osnabruck and the University of Freiburg
(Germany), the University of Montpellier and the University of Clermont-Ferrand
(France), the University of Turku and the University of Joensuu (Finland), the
University of Ghent (Belgium), the University of Xiamen (China), St. Cyril and
St. Methodius University of Veliko Tarnovo (Bulgaria), the University of Glasgow
(UK), etc. Thanks to many years of international cooperation, Tver State University
has established educational and cultural ties with these institutions and has increased
exchanges in the field of scientific research. TSU is one of the few higher education
institutions that develops international academic mobility.
Annually, about 100 students from TSU’s different departments attend a course
of study for one semester at universities in the Federal Republic of Germany, France,
Finland, Bulgaria, Poland, and the USA. Students attend classes according to the
profile of their learning, and they take exams. Students perfect their knowledge of
foreign languages, acquire the invaluable experience of studying at a higher learning
institution abroad, and make new and interesting friends.
The number of TSU students studying for a semester at the University of
Osnabruck at their own expense keeps increasing each year. The university’s
involvement in the “East–West” research faculty exchange program jointly financed
by DAAD and the University of Osnabruck has facilitated the creation of close-knit
research teams in the field of mathematics, geography, chemistry, botany and the
publishing of books, articles and other publications.
Since 2000, students coming from different universities in Finland have taken
part in inclusive semester courses at the Department of Russian as a Foreign
Language. The project is financed by the Ministry of Education of Finland. Since
2005 TSU has been a participant of “FIRST” program (Russia–Finland Student
Exchange Program). A similar program has been conducted with UK universities
and the cooperation of the RLUS Company.
The process of TSU integration into the global educational space is also realized
through the involvement in different international educational schemes. Each
year, more than 200 students and post-graduate students from foreign universities
(including the CIS and the Baltic nations) take a course at Tver State University.
132 7 Case Studies of Math Education for STEM in Russia

In recent years the university has significantly expanded its cooperation with the
Oxford–Russia Fund. This project is supervised by the TSU Inter-University Centre
for International Cooperation. Annually 120 TSU students are awarded scholarships
by the Fund. The University was also given access to the electronic library of the
University of Oxford. TSU was among the first 10 partners of the Foundation to
gain access to online versions of 500 British titles. The project also envisages the
TSU Library receiving books on art, languages, history, etc.
The cooperation with the Fulbright Foundation allows TSU to annually host
US guest speakers delivering lectures on international affairs, global terrorism, etc.
TSU faculty and students are actively involved in different educational and research
programs and projects financed by the European Union, the Ford Foundation,
CIMO, the DAAD, the IREX, etc.

7.5.2 Mathematics Education in Tver State University

The Faculty of Applied Mathematics and Cybernetics (AM&C) and the Mathemat-
ical Faculty are responsible for conducting mathematical courses at the university.
The Mathematical Faculty focuses on pedagogical programs; AM&C on applied
mathematics. The Faculty of Applied Mathematics and Cybernetics was founded in
1977, though the specialization “Applied Mathematics” was opened in 1974. It is in
many areas the leading educational department in the University. It includes more
than 32 full-time teachers, including 19 candidates and 10 doctors of sciences who
have major scientific achievements in their respective fields of expertise. Teaching
staff also consist of representatives of employers, who have extensive practical
experience.
Currently the Faculty has four educational programs:
• Applied Mathematics and Computer Science,
• Fundamental Computer Science and Information Technologies,
• Computer Science in Business,
• Applied Computer Science.
In addition to these programs, the Faculty provides training in mathematics
and appropriate applied disciplines in other faculties. The Faculty has four depart-
ments:
1. Information Technologies department (fields of expertise: intellectual infor-
mation systems, fuzzy systems and soft computing technologies; theory of
possibilities; probabilistic and probabilistic optimization and decision-making;
portfolio theory under conditions of hybrid uncertainty; processing and recogni-
tion of signals and images; multimedia technologies).
2. Computer Science department (fields of expertise: theoretical programming,
theory of finite models, theory of multi-agent systems, theoretical linguistics,
the development of expert systems; databases).
7 Case Studies of Math Education for STEM in Russia 133

3. Mathematical Statistics and System Analysis department (fields of expertise:
theory of sustainable and natural exponential probability distributions of data;
application of probabilistic and statistical methods in econometrics, financial and
actuarial mathematics, analysis of telecommunication networks; choice theory,
multicriteria decision making under uncertainty).
4. Mathematical Modeling and Computational Mathematics department (fields
of expertise: theoretical basis of mathematical modeling of complex systems;
development of mathematical models of critical states of nonlinear dynamical
systems, assessment of the safety performance of these systems; mathematical
models and methods for the identification of objects with large interference
and decision making under uncertainty, analysis and solution finding using
mathematical modeling applications: solution of problems of nonlinear elastic
and viscoelastic materials; solving problems of geophysical hydrodynamics
and hydrothermodynamics; solving problems of heat conduction; analysis of
the structures of insurance funds, development of recommendations on the
organization of insurance funds; development and application of optimization
methods for solving economic problems; development and implementation of a
measurement system of numerical methods for optimal digital video and audio
signals).
The faculty also has three scientific schools: Fuzzy systems and soft computing,
Mathematical modeling, Theoretical foundations of computer science.

7.5.3 Comparative Analysis of “Probability Theory
and Mathematical Statistics”

“Probability Theory and Mathematical Statistics” is a mandatory course at Tver
State University; it combines theoretical and applied approaches. Mathematics plays
a key role in the course, and it in its turn forms a foundation for several other applied
special disciplines in corresponding educational programs. The course was com-
pared with two courses at Tampere University of Technology (TUT): “Probability
Calculus” and “Statistics”. The course outlines can be seen in Table 7.48.
Prerequisite courses at TSU are Linear Algebra, Calculus and Differential Equa-
tions, and follow-up courses are Possibility Theory and Fuzzy Logic, Econometrics,
Theory of Stochastic Processes and Methods of Socio-Economic Forecasting. At
TUT, prerequisite courses are Engineering Mathematics 1–4, and follow-up courses
are all the other courses provided by the department. In Tver State University
this course is included in the group of core mandatory mathematical courses of
the corresponding study programs and is taught by the Applied Mathematics and
Cybernetics department. In TUT the course is taught by Department of Mathematics
and it is also included in the group of mandatory mathematical courses.
The teaching of the course at TSU is more theory-based and unfortunately does
not include any innovative pedagogical methods and tools, such as blended learning,
134 7 Case Studies of Math Education for STEM in Russia

Table 7.48 Outlines of the courses on probability theory and mathematical statistics at TSU and
TUT
Course information TSU TUT
Bachelor/master level Bachelor Bachelor
Preferred year 2–3 2
Selective/mandatory Mandatory Both
Number of credits 10 4+4
Teaching hours 148 42 + 42
Preparatory hours 220 132
Teaching assistants 0–1 1–4
Computer labs Practice with MATLAB, Practice in R-software
Excel, C++
Average number of students on the 85 200
course
Average pass % 90% 90%
% of international students 10% < 5%

flipped classroom, MOOCs, project-based learning, inquiry-based learning, collab-
orative learning, gamification. In TUT on the other hand one can find blended
learning, collaborative learning, project-based approach and active use of modern
TEL tools for administration, teaching and assessment purposes. Not all modern
pedagogical technologies are used in TUT but those that do exist in educational
process are applied widely and successfully.
Assessment, testing and grade computation do not differ from what is described
in Chap. 3. At TSU, as in all Russian universities, a general rating system is
implemented. Rating is the sum of points for all courses taken during the whole of 4
years of education. The rating comes into play when it is time for a student to choose
his/her major (graduating chair), which has influences on the further curriculum (set
of special professional courses) and Bachelor thesis topics. The exam for the course
is verbal; during the semester lecturer performs several written tests. A student
can receive 100 points for the discipline, 60 of which come from activities during
semester and 40 from exam work. The semester is also divided into two modules,
each of which gives a maximum of 30 points. The grading systems is the following:
less than 50 points is unsatisfactory (grade “2”), from 50 to 69 points satisfactory
(grade “3”), from 70 to 84 good (grade “4”), more than 84 excellent (grade “5”).
TUT on the other hand does not have a general rating system and uses six grades
according to European ECTS scale to assess results of education in each course:
excellent (grade A) 5 points; very good (grade B) 4 points; good (grade C) 3 points;
very satisfactory (grade D) 2 points; satisfactory (grade E) 1 point; unsatisfactory
(grade F) 0 points.
As regards educational software and TEL systems in TSU in the course:
MATLAB and Excel are used in tutorials. Lecturers teach students how to use
the tools, but these are not mandatory for solving practical and assessment tasks.
A student may also choose to write a program in any high-level multi-purpose
7 Case Studies of Math Education for STEM in Russia 135

language, which he or she knows already. In TUT, educational software and
TEL systems are used in the following ways: Moodle for file sharing and course
information, POP for course and exam enrollment and course grades, and R is
used as support for exercises. Furthermore, students in TUT form their individual
learning paths from the very beginning of their studies with the help of specialized
software. There is an electronic catalog of courses from which a student should
choose an appropriate number of courses with the needed amount of credits. This is
significantly different from what we have in Tver State University as well as in all
other Russian universities.

7.5.3.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competencies
are presented in Table 7.49. Outcome competencies are given in Tables 7.50, 7.51,
and 7.52.

7.5.3.2 Summary of the Results

Comparative analysis of the disciplines shows that thematic contents and learning
outcomes are almost identical. The difference is observed in the number of hours.
One should also note the active use of information technologies and, in particular, e-
learning systems in TUT. This allows this university to take out some of the material

Table 7.49 Prerequisite competencies of the probability theory and mathematical statistics
courses at TSU and TUT
Core 1
Competency TSU TUT
Data handling Excluding: calculate the mode,
median and mean for a set of
data items
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Functions and their inverse X X
Sequences, series, binomial X X
expansions
Logarithmic and exponential X X
functions
Indefinite integration X X
Definite integration, applications to With some exceptionsa With some exceptionsa
areas and volumes
Sets X X
a Use trapezoidal and Simpson’s rule for approximating the value of a definite integral
136 7 Case Studies of Math Education for STEM in Russia

Table 7.50 Core 0 level outcome competencies of the probability theory and mathematical
statistics courses at TSU and TUT
Core 0
Competency TSU TUT
Calculate the mode, median and mean for a set of data items X X
Define the terms ‘outcome’, ‘event’ and ‘probability’ X X
Calculate the probability of an event by counting outcomes X X
Calculate the probability of the complement of an event X X
Calculate the probability of the union of two mutually exclusive events X X
Calculate the probability of the union of two events X X
Calculate the probability of the intersection of two independent events X X

Table 7.51 Core 1 level outcome competencies of the probability theory and mathematical
statistics courses at TSU and TUT
Core 1
Competency TSU TUT
Calculate the range, inter-quartile range, variance and standard deviation for a set X X
of data items
Distinguish between a population and a sample X X
Know the difference between the characteristic values (moments) of a population X X
and of a sample
Construct a suitable frequency distribution from a data set X
Calculate relative frequencies X
Calculate measures of average and dispersion for a grouped set of data X
Use the multiplication principle for combinations X X
Interpret probability as a degree of belief X
Understand the distinction between a priori and a posteriori probabilities X X
Use a tree diagram to calculate probabilities X
Know what conditional probability is and be able to use it (Bayes’ theorem) X X
Calculate probabilities for series and parallel connections X
Define a random variable and a discrete probability distribution X X
State the criteria for a binomial model and define its parameters X X
Calculate probabilities for a binomial model X X
State the criteria for a Poisson model and define its parameters X X
Calculate probabilities for a Poisson model X X
State the expected value and variance for each of these models X X
Understand that a random variable is continuous X X
Explain the way in which probability calculations are carried out in the X
continuous case
Relate the general normal distribution to the standardized normal distribution X X
Define a random sample X X
Know what a sampling distribution is X X
Understand the term ‘mean squared error’ of an estimate X
Understand the term ‘unbiasedness’ of an estimate X
7 Case Studies of Math Education for STEM in Russia 137

Table 7.52 Core 2 level outcome competencies of the probability theory and mathematical
statistics courses at TSU and TUT
Core 2
Competency TSU TUT
Compare empirical and theoretical distributions X X
Apply the exponential distribution to simple problems X
Apply the normal distribution to simple problems X X
Apply the gamma distribution to simple problems X X
Understand the concept of a joint distribution X X
Understand the terms ‘joint density function’, ‘marginal distribution functions’ X X
Define independence of two random variables X X
Solve problems involving linear combinations of random variables X X
Determine the covariance of two random variables X X
Determine the correlation of two random variables X X
Realize what the normal distribution is not reliable when used with small samples X
Use tables of the t-distribution X X
Use tables of the F-distribution X X
Use the method of pairing where appropriate X X
Use tables for chi-squared distributions X X
Decide on the number of degrees of freedom appropriate to a particular problem X X
Use the chi-square distribution in tests of independence (contingency tables) X X
Use the chi-square distribution in tests of goodness of fit X
Set up the information for a one-way analysis of variance X X

for independent study and focus on really difficult topics of the discipline. E-
learning systems also allow one to automate and, as a result, simplify the knowledge
assessment process. This automation seems to be important for TUT, whose class
sizes substantially exceed the size of study groups in the TSU.
All this suggests the need for more active use of e-learning systems, as well as
blended learning methodology in the educational process in Tver State University.
It is also worth noting that in TUT part of the basic (input) material is moved to
a bridging course “Mathematics Basic Skills Test & Remedial Instruction”. The
material in this course is designed primarily for successful mastering of the engi-
neering mathematics courses. Nevertheless, this experience should also be useful
for Tver State University. In particular, a bridging course “Basics of Elementary
Mathematics” should be created, which will include the material from the following
topics of mathematics: Set theory, elementary functions and their graphs, series and
their properties, elements of combinatorics, equations and inequalities.
138 7 Case Studies of Math Education for STEM in Russia

Table 7.53 Outline of Course information TSU
possibility theory and fuzzy
logic course at TSU Bachelor/master level Bachelor
Preferred year 3
Selective/mandatory Selective
Number of credits 4
Teaching hours 68
Preparatory hours 76
Teaching assistants 1–2
Computer labs Not used
Average number of students on the course 50
Average pass % 95%
% of international students 17%

7.5.4 Comparative Analysis of “Possibility Theory and Fuzzy
Logic”

Because of the fact that the discipline “Possibility Theory and Fuzzy Logic” is
not widely spread among both domestic and foreign universities and is a particular
feature of Tver State University, the comparative analysis was performed with the
course “Probability Theory and Mathematical Statistics”. However, because of the
relative proximity of the disciplines all its conclusions including recommendations
on the modernization of the course are applicable to “Possibility Theory and Fuzzy
Logic”, as well. Below is the profile of the second course in terms of its structure
and prerequisite SEFI competencies. It is an elective course which combines both
theoretical and applied approaches. Mathematics plays a key role in it. The course
outline can be seen in Table 7.53.
Prerequisite courses are Probability Theory and Methods of Optimisation and
Decision Making. There are no follow-up courses for “Possibility Theory and Fuzzy
Logic”, because this course is included in the elective part of the professional
cycle of the corresponding educational programs. It is taught by lecturers from
the Applied Mathematics and Cybernetics department. The teaching of this course
is more theory-based and classical without wide support of e-learning tools and
methods.

7.5.4.1 Contents of the Course on Possibility Theory and Fuzzy Logic

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Tables 7.54, 7.55, and 7.56.
7 Case Studies of Math Education for STEM in Russia 139

Table 7.54 Core 0 level prerequisite competencies of the course on possibility theory and fuzzy
logic at TSU
Core 0
Competency TSU
Arithmetic of real numbers X
Algebraic expressions and formulas X
Linear laws X
Functions and their inverses X
Logarithmic and exponential functions X
Indefinite integration X
Proof X
Sets X
Coordinate geometry With some exceptionsa
Probability X
a Find the angle between two straight lines, recognize and interpret the equation of a circle in

standard form and state its radius and center, convert the general equation of a circle to standard
form, derive the main properties of a circle, including the equation of the tangent at a point,
recognize the parametric equations of a circle, use polar coordinates and convert to and from
Cartesian coordinates

Table 7.55 Core 1 level prerequisite competencies of the course on possibility theory and fuzzy
logic at TSU
Core 1
Competency TSU
Rational functions With some exceptionsa
Functions X
Solution of simultaneous linear equations X
Simple probability X
Probability models X
a Obtain the first partial derivatives of simple functions of several variables, use appropriate software
to produce 3D plots and/or contour maps

Table 7.56 Core 2 level Core 2
prerequisite competencies of
Competency TSU
the course on possibility
theory and fuzzy logic at TSU Linear optimization With some exceptionsa
a Understand the meaning and use of slack vari-

ables in reformulating a problem, understand
the concept of duality and be able to formulate
the dual to a given problem

Unfortunately SEFI Framework does not have learning outcomes suitable for
“Possibility Theory and Fuzzy Logic”. After successful completion of the course, a
student should have to master:
• the mathematical apparatus of the possibility theory and knowledge representa-
tion in computer science,
140 7 Case Studies of Math Education for STEM in Russia

• skills to model uncertainty of probabilistic type in decision-making problems,
• methods of knowledge representation with elements of possibilistic uncertainty,
they should be able to:
• use this mathematical apparatus in the development of fuzzy decision support
systems,
• apply mathematical apparatus of possibility theory in modern information
technologies,
• use soft computing technologies for solving applied problems,
they should know:
• elements of fuzzy sets theory, fuzzy logic theory and modern possibility theory,
soft computing technologies,
• fundamental concepts and system methodologies in the field of information
technologies based on soft computing,
• principles of construction of fuzzy decision support systems.

7.5.4.2 Summary of the Results

Because of the relative proximity of this discipline to Probability Theory all the
conclusions for the latter discipline are applicable to “Possibility Theory and Fuzzy
Logic” as well.

Reference

1. SEFI (2013), “A Framework for Mathematics Curricula in Engineering Education”. (Eds.)
Alpers, B., (Assoc. Eds) Demlova M., Fant C-H., Gustafsson T., Lawson D., Mustoe L., Olsson-
Lehtonen B., Robinson C., Velichova D. (http://www.sefi.be).

8.1 Analysis of Mathematical Courses in ATSU

Tea Kordzadze
Akaki Tsereteli State University (ATSU), Department of Mathematics, Kutaisi,
Georgia
Tamar Moseshvili
Akaki Tsereteli State University (ATSU), Department of Design and Technology,
Kutaisi, Georgia

8.1.1 Kutaisi Akaki Tsereteli State University (ATSU)

The history of Akaki Tsereteli State University (ATSU) started eight decades ago
and now it is distinguished with its traditions throughout Georgia and holds an
honorable place in the business of cultural, intellectual and moral education of the
Georgian nation. According to the Georgian government’s resolution #39, February
23, 2006 the legal entities of public law Kutaisi Akaki Tsereteli State University
and Kutaisi N. Muskhelishvili State Technical University were combined, the edu-
cational status being determined to be the university and the new entity was named
Akaki Tsereteli State University. ATSU was merged with Sukhumi Subtropical
Teaching University in 2010.
ATSU became one of the largest universities, with a wide spectrum of academic
(on BA, MA, and PhD levels), professional teaching programs and research

© The Author(s) 2018 141
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_8
142 8 Case Studies of Math Education for STEM in Georgia

fields. Today in ATSU there are about 11,000 students, 9 faculties and 11 STEM
disciplines.

8.1.2 Comparative Analysis of “Calculus 1”

Calculus 1 is a mandatory course for students of STEM specializations. This course
was developed for engineering students. It is theoretical course with practical
examples connected to real life problems. It was compared with “Engineering
Mathematics 1” (EM1), which is similar to a course at Tampere University of
Technology. The outlines of the courses are presented in Table 8.1. The Department
of Mathematics is responsible for the course at Akaki Tsereteli State University.
There are 4 full-time professors, 18 full-time associate professors, and 5 teachers.
There are no prerequisite courses. The course size is 5 credits, and it requires
on average 125 h of work (25 h per credit). The credits are divided among different
activities as follows: lectures 30 h, tutorials 30 h, homework 58 h, consultation 3 h,
and exam 4 h. On the course organizer side, there are 60 contact teaching hours.
From one to two teaching assistants work on the course to grade exams and teach
tutorials. Unfortunately, there are no computer labs available for use on the course.
There are approximately 200 students on the course. The average number of
students that finish the course is 170 (85%). The amount of international students is
less than 1%.
The pedagogical comparison of the course shows that the teaching is very much
theory based. Professors of the mathematics department are delivering the course,
and they do not use modern teaching methods such as blended learning, flipped
classroom, project or inquiry based learning etc.

Table 8.1 Outlines of Calculus 1 and EM1 course at ATSU and TUT
Course information ATSU TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 1 1
Elective/mandatory Mandatory Mandatory
Number of credits 5 5
Teaching hours 60 56
Preparatory hours 58 75
Teaching assistants 1–2 1–3
Computer labs Available Available
Average number of students on the course 200 200
Average pass% 85% 90%
% of international students Less than 1% Less than 5%
8 Case Studies of Math Education for STEM in Georgia 143

The course’s maximum evaluation equals 100 points. A student’s final grade is
obtained as a result of summing the midterm evaluation earned per semester and
final exam evaluation results.
Assessment Criteria Students are evaluated in a 100-point system in which 45
points are given from midterm assessments, 15 points from Activities and 40 points
from the final exam. Midterm assessments include the following components: Work
in group (40 points) and two midterm examinations (20 points). Within the frames of
40-point assessment for working in a group students can be given an abstract/review
work with 10 points. Students are obliged to accumulate no less than 11 points in
midterm assessments and no less than 15 points at the exams. The course will be
considered covered if students receive one of the following positive grades: (A)
Excellent: 91 points and more; (B) Very good: 81–90 points; (C) Good: 71–80
points; (D) Satisfactory: 61–70 points; (E) Sufficient: 51–60 points. (FX) No pass—
in the case of getting 41–50 points students are given the right to take the exam once
again. (F) Fail—with 40 points or less, students have to do the same course again.

8.1.2.1 Contents of the Course

The comparison is based on SEFI framework [1]. Prerequisite competencies are
presented in Table 8.2. Outcome competencies are given in Table 8.3.

Table 8.2 Core 0-level Core 0
prerequisite competencies for
Competency ATSU TUT
Calculus 1 (ATSU) and EM1
(TUT) courses Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their properties X X

Table 8.3 Core 1-level Core 1
outcome competencies for
Competency ATSU TUT
Calculus 1 (ATSU) and EM1
(TUT) courses Sets, operations on sets X X
Functions and their properties X X
Logarithmic and trigonometric functions X X
Limits and continuity of a function X X
Derivative X X
Definite integral and integration methods X
Minima and maxima X X
Indefinite integrals X
Definite integrals X
144 8 Case Studies of Math Education for STEM in Georgia

8.1.2.2 Summary of the Results

The main differences between Finnish (TUT) and Georgian (ATSU) courses that in
TUT teaching is more intense. ATSU has 30 lectures and 30 tutorials—but TUT
does all of this in 7 weeks, whereas ATSU uses 15 weeks. On the other hand
Calculus 1 covers more mathematical areas than TUT’s EM1. TUT also uses plenty
of TEL and ICT technologies to support the teaching, ICT is not used in the study
process in ATSU. Finnish students also answer in a pen and paper examination as
ATSU student, but ATSU students must pass three exams; two midterm exams and
one final exam, with no pure theoretical questions. Due to this comparison ATSU
changed syllabus in Calculus 1. To achieve SEFI competencies ATSU modernized
the syllabus by integration of Math-Bridge and GeoGebra tools in the study process.
Modernization of Syllabus was done on September 2016. In 2016 ATSU has done
pre- and post-testing in Calculus 1. During post-testing Math-Bridge tools were used
for testing and analyzing results.

8.1.3 Comparative Analysis of “Modeling and Optimization
of Technological Processes”

The course “Modeling and Optimization of Technological Processes” is offered for
Master students of technological engineering faculty in the following engineer-
ing programs: Technology of Textile Industry, Food Technology, Technology of
Medicinal Drugs, Engineering of Environmental Pollution and Ecology of Wildlife
Management. ECTS credits for the course are 5 (125 h). The teaching language
is Georgian, the average number of students is 30. It was compared with a
corresponding course “Mathematical Modelling” from TUT. The course outlines
are seen in Table 8.4.
Specific engineering departments are responsible for the course. A professor in
each department teaches this course.

Table 8.4 Outlines of modeling courses at ATSU and TUT
Course information ATSU TUT
Bachelor/Master level Master Both
Preferred year 1 1
Elective/mandatory Mandatory Elective
Number of credits 5 5
Teaching hours 48 30
Preparatory hours 77 108
Teaching assistants 1 1
Computer labs Yes Yes
Average number of students on the course 30 60
Average pass% 75% 95%
% of international students 0% Less than 5%
8 Case Studies of Math Education for STEM in Georgia 145

Prerequisite courses are the Bachelor courses of Higher Mathematics. The course
size is 5 credits, and it requires on average 125 h of work (25 h for each credit). The
credits are divided among different activities as follows: lectures 15 h, tutorials 30 h,
independent work 77 h, exam 3 h.
On the course organizer side, there are 48 contact teaching hours. There is one
teaching assistant to work on the course to grade exams and teach tutorials. There
is one computer lab available for use on the course. There are approximately 30
students on the course. The average number of students that finish the course is 20
(75%). There are no international students in the technological engineering faculty.
The pedagogical comparison of the course shows that the teaching process is not
modern. Professors delivering the course do not use modern teaching methods such
as blended learning, flipped classroom, project or inquiry based learning etc.
Grading is done on a 100 point scale with 51 points being the passing level. A
score between 41 and 50 allows the student a new attempt at the exam, and every 10
point interval offers a better grade with 91–100 being the best.
Pen and paper exams are conducted three times in a semester.
• First midterm exam comprises 1–5 weeks materials and is conducted after the
5th week in compliance with Grading Center schedule.
• Second midterm exam comprises 7–11 weeks materials and is conducted after
the 11th week in compliance with Grading Center schedule.
• Final exam is conducted after the 17–18th week.
For the final evaluation the scores of the midterm tests and independent work are
summed up.

8.1.3.1 Contents of the Course

The comparison is based on SEFI framework [1]. Outcome competencies are given
in Table 8.5.

8.1.3.2 Summary of the Results

The main differences between Finnish (TUT) and Georgian (GTU) courses are the
following: The amount of contact hours in ATSU are 48 h. This consists of lectures,

Table 8.5 Core 2-level outcome competencies of the modeling courses at ATSU and TUT
Core 2
Competency ATSU TUT
Simple linear regression X X
Multiple linear regression and design of experiments X X
Linear optimization X X
The simplex method X
Nonlinear optimization X
146 8 Case Studies of Math Education for STEM in Georgia

15 h (1 h per week), and practical work, 30 h, and midterm and final exam, 3 h. The
amount of independent work is 77 h (62%).
In TUT the course is given as a web based course. Two hours of video lectures
are implemented per week. 108 h are for laboratory work/tutorials. All of it is group
work for weekly exercises and the final project. 100% of the student’s time is for
homework, which is mandatory for the students.
There are big differences of teaching methods between our courses. In ATSU
the lectures are implemented by using verbal or oral methods: giving the lecture
materials to the students orally according to the methods of questioning and
answering, interactive work, explaining theoretical theses on the bases of practical
situation simulation.
In TUT the course is given as a web based course. Different universities around
Finland participate in the project. Each week a different university is responsible for
that week’s topic. The main coordination is done by TUT. Students form groups in
each university and work on the given tasks as teams.
Modern lecture technology is used in TUT: e-Learning, with a hint of blended
learning, Moodle, MATLAB or similar software, online lecture videos. Moodle is
used for file sharing, course information, peer assessment of tasks and MATLAB
for solving the exercises.
Video lectures online, weekly exercises are done in groups, posted online and
then reviewed and commented on by other groups. Students are awarded points for
good answers and good comments. At the end of the course a final project is given
to the students to undertake. The final project work is assessed by the other students
and by the course staff, and it is presented in a video conference.
ATSU will prepare new syllabi for modernized courses that are more in line with
European university courses. This will be done in order to better prepare the Master
students of ATSU for their future careers. The Georgian educational system teaches
less mathematics on the high school level, and thus ATSU has to design courses that
upgrade students’ knowledge in these topics as well.
As the general level of the students is quite low, ATSU considers that the increase
of the credits in mathematics on the Bachelor level is necessary. Also, the use of
different software packages to support learning (Math-Bridge, GeoGebra, etc.) will
increase the quality of knowledge of our students.

8.2 Analysis of Mathematical Courses in BSU

Vladimer Baladze, Dali Makharadze, Anzor Beridze, Lela Turmanidze, and Ruslan
Tsinaridze
Batumi Shota Rustaveli State University (BSU), Department of Mathematics,
Batumi, Georgia
e-mail: dali_makharadze@mail.ru
8 Case Studies of Math Education for STEM in Georgia 147

8.2.1 Batumi Shota Rustaveli State University (BSU)

Batumi Shota Rustaveli State University is deservedly considered to be one of the
most leading centers of education, science and culture in Georgia. Educational and
scientific activities of BSU go back to 1935. From the date of its establishment it
has been functioning as the pedagogical institute providing Western Georgia with
pedagogically educated staff for 55 years.
In 1990 the classical university with the fundamental, humanitarian and social
fields was established on the basis of the pedagogical institute. In 2006, based on
the decision of the Georgian government, scientific-research institutes and higher
education institutions of various profiles located in the territory of Autonomous
Republic of Ajara, joined Shota Rustaveli State University.
Currently 6000 students study at vocational, BA, MA and PhD educational
programs. The process of education and research is being implemented by 244
professors, 55 scientists and researchers and 387 invited specialists.
BSU offers the students a wide-range choice in the following programs at all
three levels: 43 BA, 44 MA, 28 PhD and 2 one-level programs. Based on the labor
market demand some faculties implement vocational programs as well.
Shota Rustaveli State University is located in Batumi and its outskirts area,
having six campuses. The university comprises seven faculties and three scientific-
research institutes:
• Faculties:
Faculty of Humanities;
Faculty of Education;
Faculty of Social and Political Sciences;
Faculty of Business and Economics;
Faculty of Law;
Faculty of Natural Sciences and Health Care;
Faculty of Physics-Mathematics and Computer Sciences;
Faculty of Technology;
Faculty of Tourism.
• Scientific-research institutes:
Niko Berdzenishvili Institute (Direction of Humanities and Social Studies);
Institute of Agrarian and Membrane Technologies;
Phytopathology and Biodiversity Institute.
148 8 Case Studies of Math Education for STEM in Georgia

8.2.2 Comparative Analysis of “Linear Algebra and Analytic
Geometry (Engineering Mathematics I)”

“Linear Algebra and Analytic Geometry” (or “Engineering Mathematics I)” (EMI)
is a theoretical course with approximately 50 students. The course is a first year
course for engineering students at the Faculty of Technology and is a mandatory
course of engineering programs in BSU. The course is compared with the “Engi-
neering Mathematics 1” (EM1) course at Tampere University of Technology. The
course outlines are presented in Table 8.6.
The Department of Mathematics is responsible for the course. The staff of this
department consists of three full-time professors, six full-time associate professors,
four full-time assistant professors and four teachers. The Department of Mathemat-
ics conducts the academic process within the frames of the educational programs
of the faculty Technologies, of the faculty Physics, Mathematics and Computer
Sciences as well as other faculties.
The course does not have prerequisite courses. The course size is 5 credits, and
it requires on average 125 h of student’s work (25 h for each credit). The credits are
divided among different activities as follows: lectures 15 h, tutorials 30 h, homework
80 h. Students should use about 30 h to prepare for their exam.
On the course organizer side, there are 60 contact teaching hours. From one to
three teaching assistants work on the course to grade exams and teach tutorials.
Unfortunately, there are no computer labs available for use on the course.
There are 50 students on the course. The average number of students that finish
the course is 41 (82%). The amount of international students is 0%.
The pedagogical comparison of the course shows that the teaching is very much
theory based. Professors of mathematics department are delivering the course,
and they do not use modern teaching methods such as blended learning, flipped
classroom, project or inquiry based learning etc.

Table 8.6 Outlines of EM I (BSU) and EM1 (TUT) courses
Course information BSU TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 1 1
Elective/mandatory Mandatory Mandatory
Number of credits 5 5
Teaching hours 45 57
Preparatory hours 80 76
Teaching assistants 1–3 1–3
Computer labs No Yes
Average number of students on the course 50 200
Average pass% 82% 90%
% of international students No Less than 5%
8 Case Studies of Math Education for STEM in Georgia 149

Assessment Criteria Students are evaluated in a 100-point system in which
60 points are given from midterm assessments and 40 points from the final
exam. Midterm assessments include the following components: Work in group (40
points) and two midterm examinations (20 points). Within the frames of 40-point
assessment for working in a group students can be given an abstract/review work
with 10 points. Students are obliged to accumulate no less than 11 point in midterm
assessments and no less than 21 points at the exams. The course will be considered
covered if students receive one of the following positive grades:
• (A) Excellent: 91 points and more.
• (B) Very good: 81–90 points.
• (C) Good: 71–80 points.
• (D) Satisfactory: 61–70 points.
• (E) Sufficient: 51–60 points.
(FX) No pass—in the case of getting 41–50 points students are given the right to
take the exam once again.
(F) Fail—with 40 points or less, students have to do the same course again.
“Linear Algebra and Analytical Geometry/Engineering Mathematics I” at BSU
does not have any TEL tools available on the course. Therefore, no TEL tools are
used to support learning.

8.2.2.1 Contents of the Course

The comparison is based on SEFI framework [1]. Prerequisite competencies are
presented in Table 8.7. Outcome competencies are given in Tables 8.8 and 8.9.

Table 8.7 Core 0-level Core 0
prerequisite competencies for
Competency BSU TUT
EMI (BSU) and EM1 (TUT)
courses Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their inverses X X
Sequences, series, binomial expansions Excl. series X
Logarithmic and exponential functions X X
Geometry X X
Trigonometry X X
Trigonometric identities X X
150 8 Case Studies of Math Education for STEM in Georgia

Table 8.8 Core 0-level Core 0
outcome competencies for
Competency BSU TUT
EMI (BSU) and EM1 (TUT)
courses Binomial expansions X X
Sets X X
Co-ordinate geometry X X

Table 8.9 Core 1-level Core 1
outcome competencies for
Competency BSU TUT
EMI (BSU) and EM1 (TUT)
courses Sets X X
Complex number X X
Mathematical logic X X
Mathematical induction X X
Conic sections X X
3D co-ordinate geometry X
Vector arithmetic X X
Vector algebra and applications X
Matrices and determinants X
Solution of simultaneous linear equations X

8.2.2.2 Summary of the Results

The main differences between Finnish (TUT) and Georgian (BSU) courses are the
following: in TUT, teaching is more intense and covers less topics than BSU. The
overall hours are somewhat different; TUT has 35 h of lectures and 21 h of tutorials;
BSU has 15 lectures and 30 tutorials, and TUT does all of this in 7 weeks, whereas
BSU uses 15 weeks. TUT also uses plenty of TEL and ICT technologies to support
their teaching, BSU does not. Finally, the exams are somewhat different. Finnish
students answer in a pen and paper exam, BSU students must pass three exams with
theoretical questions, but with no proofs.
The main drawbacks of the old mathematics syllabi at BSU were that mostly the
theoretical mathematical aspects were treated and the corresponding examinations
contained only purely mathematical questions.
Moreover, it should be specially mentioned that in the BSU the mathematical
syllabus “Engineering Mathematics 1” (= “Linear Algebra and Analytical Geome-
try”) in engineering BSc programs does not include the following topics: Elements
of Discrete Mathematics, Surfaces of second order. Therefore, modernization of the
syllabus of “Linear Algebra and Analytical Geometry” is very desirable.
BSU will prepare new syllabi for modernized courses that are more in line with
European technical university courses. This will be done in order to better prepare
the students of BSU for their future careers. However, modernizing courses will
not be trivial, since the university has to make up for the different levels of skills
of European and Georgian enrolling students. The Georgian educational system
teaches less mathematics on the high school level, and thus BSU has to design
courses that upgrade students’ knowledge in these topics as well.
8 Case Studies of Math Education for STEM in Georgia 151

As the overall level of students is relatively low, BSU finds that implementing
remedial mathematics courses is necessary. This could be done using the Math-
Bridge software.

8.2.3 Comparative Analysis of “Discrete Mathematics”

“Discrete Mathematics” (DM) is a theoretical course with 14 students. The course
is a second year mandatory course for BSU students in the program of Computer
Sciences at the faculty of Physics-Mathematics and Computer Sciences. It was
compared with a corresponding course “Algorithm Mathematics” (AM) from
Tampere University of Technology (TUT). Outlines of these courses are seen in
Table 8.10.
The department of Mathematics is responsible for the course. The staff of this
department consist of three full-time professors, six full-time associate professors,
four full-time assistant professors and four teachers. The Department of Mathemat-
ics conducts the academic process within the frames of the educational programs of
the faculty Physics, Mathematics and Computer Sciences as well as other faculties.
The course does not have prerequisite courses. Its size is 5 credits, and it requires
on average 125 h of work (25 h for each credit). The credits are divided among
different activities as follows: lectures 15 h, tutorials 30 h, homework 80 h. Students
should use about 30 h to prepare for their exam.
On the course organizer side, there are 60 contact teaching hours. From one to
three teaching assistants work on the course to grade exams and teach tutorials.
Unfortunately, there are no computer labs available for use on the course.
There are 14 students on the course. The average number of students that finish
the course is 11 (78%). The amount of international students is 0%.

Table 8.10 Outlines of the Discrete Mathematics (BSU) and Algorithm Mathematics (TUT)
courses
Course information BSU TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 1 2
Elective/mandatory Mandatory Elective
Number of credits 5 4
Teaching hours 45 49
Preparatory hours 80 65
Teaching assistants 1–3 1–2
Computer labs No Yes
Average number of students on the course 14 150
Average pass% 78% 90%
% of international students No Less than 5%
152 8 Case Studies of Math Education for STEM in Georgia

The pedagogical comparison of the course shows that the teaching is very much
theory based. Professors of the mathematics department are delivering the course,
and they do not use modern teaching methods such as blended learning, flipped
classroom, project or inquiry based learning etc.
Assessment criteria are as follows: Students are evaluated in a 100-point system,
in which 60 points are given on midterm assessments and 40 points on final
exams. Midterm assessments include the following components: Work in group (40
points) and two midterm examinations (20 points). Within the frames of 40-point
assessment for working in a group, students can be given an abstract/review work
for 10 points. Students are obliged to accumulate no less than 11 points in midterm
assessments and no less than 21 points at the exams.
Grading follows the same principles as with the course described above.

8.2.3.1 Contents of the Course

The comparison is based on SEFI framework [1]. Prerequisite competencies are
presented in Table 8.11. Outcome competencies are given in Table 8.12.
The main differences between Finnish (TUT) and Georgian (BSU) courses are
the following: in TUT, teaching is more intense but it covers less topics. The overall
hours are somewhat different—TUT has 35 h of lectures and 21 h of tutorials, BSU
has 15 lectures and 30 tutorials and TUT does all of this in 7 weeks, whereas BSU
uses 15 weeks. TUT also uses plenty of TEL and ICT technologies to support the
teaching, BSU does not. Finally, the exams are somewhat different. Finnish students
answer in a written exam, BSU students must pass three exams with theoretical
questions, but with no proofs.

Table 8.11 Core 0-level prerequisite competencies for DM (BSU) and AM (TUT) courses
Core 0
Competency DM (BSU) AM (TUT)
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their inverses X X
Sequences, series, binomial expansions Excl. series X
Logarithmic and exponential functions X X
Proof X X
Sets X X
Geometry X X
Data handling X x
Probability X
8 Case Studies of Math Education for STEM in Georgia 153

Table 8.12 Core 1-level outcome competencies for DM (BSU) and AM (TUT) courses
Core 1
Competency DM (BSU) AM (TUT)
Sets X X
Mathematical logic X X
Mathematical induction and recursion X X
Graphs X
Combinatorics X
Simple probability X
Probability models X

The main drawbacks of the old mathematics syllabi at BSU were that mostly
the theoretical mathematical aspects were treated and the corresponding exam lists
contained only purely mathematical questions.
Moreover, it should be specially mentioned that the BSU mathematical syllabus
“Discrete Mathematics” for Computer Sciences BS programs does not contain
the following topics: Binary relation; Boolean algebra; Groups, Rings and Fields;
Euclid’s division algorithm and Diophantine equations; Coding theory and finite
automata; Cryptography. Therefore, modernization of the syllabus “Discrete Math-
ematics” is very desirable.
BSU will prepare new syllabi for modernized courses that are more in line with
European technical university courses. This will be done in order to better prepare
the students of BSU for their future careers. However, modernizing courses will
not be trivial, since the university has to make up for the different prerequisite
skills between European and Georgian enrolling students. The Georgian educational
system teaches less mathematics on the high school level, and thus BSU has to
design courses that upgrade students’ knowledge in these topics as well.
As the overall level of students is sufficiently low, BSU finds that implementing
remedial mathematics courses is necessary. This could be done using the Math-
Bridge software.

8.3 Analysis of Mathematical Courses in GTU

8.3.1 Georgian Technical University (GTU)

David Natroshvili () and Shota Zazashvili
Georgian Technical University (GTU), Department of Mathematics, Tbilisi,
Georgia
e-mail: d.natroshvili@gtu.ge; s.zazashvili@gtu.ge
George Giorgobiani
Georgian Technical University (GTU), Department of Computational Mathematics,
Tbilisi, Georgia
e-mail: giorgobiani.g@gtu.ge
154 8 Case Studies of Math Education for STEM in Georgia

Georgian Technical University (GTU) is one of the biggest educational and
scientific institutions in Georgia. The overall number of students is approximately
20,000, and there are in total 17 STEM disciplines.
In 1917, the Russian Emperor issued an order to found a polytechnic institute in
Tbilisi, the first higher educational institute in the Caucasian region. In 1922, GTU
was originally founded as a polytechnic faculty of the Tbilisi State University. Later,
in 1928, the departments of the polytechnic faculty merged into an independent
institute called Georgian Polytechnic Institute (GPI). GTU continued under this
title until 1990, when the institute was granted university status and was renamed
Georgian Technical University.
GTU adopted the Bologna process in 2005. Today, the university hosts approxi-
mately 20,000 students, 10 faculties and 17 STEM disciplines.

8.3.2 Comparative Analysis of “Mathematics 3”

“Mathematics 3” is a theoretical course with approximately 1500 students. The
course is a second year course for engineering students at the university and is a
final mathematical mandatory course of engineering programs in GTU. This course
was compared with a corresponding course “Engineering Mathematics 4” (EM4) at
TUT. The course outlines are seen in Table 8.13.
Department of Mathematics at GTU is responsible for the course. The staff of this
department consists of 20 full-time professors, 21 full-time associate professors,
3 full-time assistant professors, 7 teachers, 16 invited professors and 5 technical
employees.
Prerequisite courses are Mathematics 1 and 2. The course is a part of the
Mathematics course cluster. The course size is 5 credits, and it requires on average
135 h of work (27 h for each credit). The credits are divided among different

Table 8.13 Outlines of Mathematics 3 (GTU) and Engineering Mathematics 4 (TUT) courses
Course information GTU TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 2 1
Elective/mandatory Mandatory Mandatory
Number of credits 5 4
Teaching hours 60 49
Preparatory hours 75 56
Teaching assistants 1–4 1–3
Computer labs No Yes
Average number of students on the course 1500 150
Average pass% 75% 85%
% of international students Less than 1% Less than 5%
8 Case Studies of Math Education for STEM in Georgia 155

activities as follows: lectures 30 h, tutorials 30 h, homework 75 h, exam 3 h. Students
should use about 10 h to prepare for their exam.
On the course organizer side, there are 60 contact teaching hours. From one
to four teaching assistants work on the course to grade exams and teach tutorials.
Unfortunately, there are no computer labs available for use on the course.
There are approximately 1500 students on the course. The average number of
students that finish the course is 1125 (75%). The amount of international students
is less than 1%. For this analysis, neither the overall student demographic nor the
average rating of the course by the students was available.
The pedagogical comparison of the course shows that the teaching is very much
theory based. Professors of the mathematics department are delivering the course,
and they do not use modern teaching methods such as blended learning, flipped
classroom, project or inquiry based learning etc.
Testing is done on a 100 point scale with 51 points being the passing level. A
score between 41 and 50 offers a new attempt at the exam, and every 10 point
interval offers a better grade with 91–100 points being the best. Exams are in
three different forms: weekly intermediate exams, two midterm exams and a final
exam. Testing methods are multiple choice answers or open ended answers done
on computers. The final grade is computed by combining the different points in the
different tests and normalizing to 100.
“Calculus 2 = Mathematics 3” at GTU does not have any TEL tools available on
the course. Therefore, no TEL tools are used to support learning.

8.3.2.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competen-
cies are presented in Tables 8.14 and 8.15. Outcome competencies are given in
Tables 8.16, 8.17, and 8.18.

8.3.2.2 Summary of the Results

The main differences between Finnish (TUT) and Georgian (GTU) courses are the
following: in TUT, teaching is more intense. The overall hours are quite similar—
TUT has 28 h of lectures and 24 h of tutorials, GTU has 30 lectures and 30
tutorials—but TUT does all of this in 7 weeks, whereas GTU uses 15 weeks. TUT
also uses plenty of TEL and ICT technologies to support their teaching, GTU does
not. Finally, the exams are quite different. Finnish students answer in a pen and
paper exam, GTU students must pass three exams (two midterm exams and one
final exam) on the computer, with no pure theoretical questions (proofs).
The main drawbacks of the old mathematics syllabi at GTU were that mostly
the theoretical mathematical aspects were treated and the corresponding exam lists
contained only purely mathematical questions. This means that the application of
156 8 Case Studies of Math Education for STEM in Georgia

Table 8.14 Core 0-level prerequisite competencies of Mathematics 3 (GTU) and EM4 (TUT)
courses
Core 0
Competency GTU TUT
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their inverses X X
Sequences, series, binomial expansions Excl. series, binomial expansion X
Logarithmic and exponential functions X X
Rates of change and differentiation X X
Stationary points, maximum and minimum values X X
Indefinite integration X X
Proof X X
Sets X X
Geometry X X
Trigonometry X X
Co-ordinate geometry X X
Trigonometric functions and applications X X
Trigonometric identities X X

Table 8.15 Core 1-level prerequisite competencies of Mathematics 3 (GTU) and EM4 (TUT)
courses
Core 1
Competency GTU TUT
Rational functions X X
Complex numbers X X
Functions X X
Differentiation X X
Sequences and series Excl. series, binomial expansion X
Vector arithmetic X X
Vector algebra and applications X X
Matrices and determinants X X
Solution of simultaneous linear equations X X
Functions of several variables X

Table 8.16 Core 0-level outcome competencies of Mathematics 3 (GTU) and EM4 (TUT) courses
Core 0
Competency GTU TUT
Sequences, series, binomial expansions X
Indefinite integration X X
Definite integration, applications to areas and volumes X X
8 Case Studies of Math Education for STEM in Georgia 157

Table 8.17 Core 1-level outcome competencies of Mathematics 3 (GTU) and EM4 (TUT) courses
Core 1
Competency GTU TUT
Sequences and series X
Methods of integration X X
Applications of integration X X

Table 8.18 Core 2-level outcome competencies of Mathematics 3 (GTU) and EM4 (TUT) courses
Core 2
Competency GTU TUT (Tampere)
Ordinary differential equations X
First order ordinary differential equations X
Second order equations—complementary function and particular X
integral
Fourier series X
Double integrals X
Further multiple integrals X
Vector calculus X
Line and surface integrals, integral theorems X

taught mathematics had almost no emphasis on the course, which led to lack of
motivation in the students.
Moreover, it should be specially mentioned that in the GTU mathematical syl-
labus “Mathematics 3” (in F inland “Calculus 2”) for engineering BSc programs,
the following topics are not included: Double integrals, Triple integrals, Curvilinear
and Surface Integrals,Vector Calculus, Divergence Theorem and Stokes’ Theorem.
These topics are widely presented in the TUT mathematical curricula.
Due to the above, it seems that an essential modernization of the syllabus
Calculus 2 (in GTU Mathematics 3) is very desirable.
At the same time it should be taken into consideration that modernization of a
particular syllabus Calculus 2 (in GTU Mathematics 3) will require modification of
the syllabuses of prerequisite courses Mathematics 1 and Mathematics 2.
GTU will prepare new syllabi for modernized courses that are more in line with
European and American technical university courses. This will be done in order to
better prepare the students of GTU for their future careers. However, modernizing
courses will not be trivial, since the university has to make up for the different
prerequisite skills of European and Georgian enrolling students. The Georgian
educational system teaches less mathematics on the high school level, and thus GTU
has to design courses that upgrade students’ knowledge in these topics as well.
As the overall level of students is sufficiently low, GTU finds that implementing
remedial mathematics courses is necessary. This could be done using the Math-
Bridge software.
158 8 Case Studies of Math Education for STEM in Georgia

8.3.3 Comparative Analysis of “Probability Theory
and Statistics”

“Probability Theory and Statistics” (PTS) is a theoretical course with approximately
200 students. The course is a second year course for engineering students of
two departments of two faculties: Faculty of Power Engineering and Telecommu-
nications and Faculty of Informatics and Control Systems. It will be compared
with a corresponding course “Probability Calculus” (PC) at Tampere University of
Technology (TUT). The outlines of the courses are presented in Table 8.19.
The department responsible for the course is the GTU Department of Math-
ematics. The staff of this department consists of 20 full-time professors, 21
full-time associate professors, 3 full-time assistant professors, 7 teachers, 16 invited
professors and 5 technical employees.
The prerequisite course is Mathematics 2. The course is a part of the Mathematics
course cluster. The course size is 5 credits, and it requires on average 135 h of work
(27 h for each credit). The credits are divided among different activities as follows:
lectures 30 h, tutorials 30 h, homework 75 h, exam 3 h. Students should use about
10 h to prepare for their exam.
On the course organizer side, there are 60 contact teaching hours. From one
to two teaching assistants work on the course to grade exams and teach tutorials.
Unfortunately, there are no computer labs available for use on the course.
There are approximately 200 students on the course. The average number of
students that finish the course is 150 (75%). The amount of international students
is less than 1%. For this analysis, neither the overall student demographic nor the
average rating of the course by students was available.
The pedagogical comparison of the course shows that the teaching is very much
theory based. Professors of mathematics department are delivering the course,
and they do not use modern teaching methods such as blended learning, flipped
classroom, project or inquiry based learning etc.

Table 8.19 Outlines of probability and statistics courses (PTS) at GTU and (PC) at TUT
Course information GTU TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 2 2
Elective/mandatory Mandatory Elective
Number of credits 5 4
Teaching hours 60 42
Preparatory hours 75 66
Teaching assistants 1–2 1–2
Computer labs No Yes
Average number of students on the course 200 200
Average pass% 75% 90%
% of international students Less than 1% Less than 5%
8 Case Studies of Math Education for STEM in Georgia 159

Testing is done on a 100 point scale with 51 points being the passing level. A
score between 41 and 50 offers a new attempt at the exam, and every 10 point
interval offers a better grade with 91–100 being the best. Exams are in three different
forms: weekly intermediate exams, two midterm exams and final exam. Testing
methods are by multiple choice answers or open ended answers done on computers.
The final grade is computed by combining the different points in the different tests
and normalizing to 100.
GTU does not have any TEL tools. Therefore, no TEL tools are used to support
teaching and learning.

8.3.3.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competen-
cies are presented in Tables 8.20 and 8.21. Outcome competencies are given in
Tables 8.22, 8.23, and 8.24.

8.3.3.2 Summary of the Results

The main differences between Finnish (TUT) and Georgian (GTU) courses are the
following: in TUT, teaching is more intense. The overall hours also are not similar—
TUT has 28 h of lectures and 14 h of tutorials for statistics (7 weeks) and 28 h of

Table 8.20 Core 0-level prerequisite competencies of the probability and statistics courses (PTS)
at GTU and (PC) at TUT
Core 0
Competency GTU TUT
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their inverses X X
Sequences, series, binomial expansions Excl. series, binomial expansion X
Logarithmic and exponential functions X X
Rates of change and differentiation X X
Stationary points, maximum and minimum values X X
Indefinite integration X X
Proof X X
Sets X X
Geometry X X
Trigonometry X X
Co-ordinate geometry X X
Trigonometric functions and applications X X
Trigonometric identities X X
160 8 Case Studies of Math Education for STEM in Georgia

Table 8.21 Core 1-level prerequisite competencies of the probability and statistics courses (PTS)
at GTU and (PC) at TUT
Core 1
Competency GTU TUT
Rational functions X X
Complex numbers X X
Functions X X
Differentiation X X
Sequences and series Excl. series X
Vector arithmetic X X
Vector algebra and applications X X
Matrices and determinants X X
Solution of simultaneous linear equations X X
Functions of several variables X X

Table 8.22 Core 0-level outcome competencies of the probability and statistics courses (PTS) at
GTU and (PC) at TUT
Core 0
Competency GTU TUT
Data handling X X
Probability X X

Table 8.23 Core 1-level outcome competencies of the probability and statistics courses (PTS) at
GTU and (PC) at TUT
Core 1
Competency GTU TUT
Data handling X X
Combinatorics X X
Simple probability X X
Probability models X X
Normal distribution X X
Sampling X
Statistical inference Excl. issues related to hypothesis testing

Table 8.24 Core 2-level outcome competencies of the probability and statistics courses (PTS) at
GTU and (PC) at TUT
Core 2
Competency GTU TUT
One-dimensional random variables Excl. Weibull and gamma distributions X
8 Case Studies of Math Education for STEM in Georgia 161

lectures and 12 h of tutorials for probability (7 weeks), GTU has 30 lectures and 30
tutorials (15 weeks). TUT also uses plenty of TEL and ICT technologies to support
their teaching, GTU does not. Finally, the exams are quite different. Finnish students
answer in a written exam, GTU students must pass three exams (two midterm exams
and one final exam) on the computer, with no pure theoretical questions (proofs).
The main drawbacks of the old mathematics syllabi at GTU were that mostly
the theoretical mathematical aspects were treated and the corresponding exam lists
contained mostly purely mathematical questions. This means that the application
of taught mathematics had almost no emphasis on the course, which led to lack of
motivation in the students.
Moreover, it should be specially mentioned that in the GTU syllabus “Probability
Theory and Statistics” for engineering BSc programs 12 lectures are devoted to
the topics of probability theory, and only the last three lectures to statistics. The
following topics are not included: Test of hypothesis; Small sample statistics issues
like t-test, F-test, chi-square tests; topics of Analysis of variance; Linear regression
etc. Due to the above, it seems that an essential modernization of the syllabus
“Probability Theory and Statistics” is very desirable.

8.4 Analysis of Mathematical Courses in UG

Kakhaber Tavzarashvili and Ketevan Kutkhashvilii
University of Georgia (UG), School of IT, Engineering and Mathematics, Tbilisi,
Georgia
e-mail: k.tavzarashvili@ug.edu.ge; k.kutkhashvili@ug.edu.ge

The University of Georgia (UG) was founded in 2004. UG is one of the largest
private universities in Georgia. Throughout past years, improvement of the quality
of education in Georgia has been one of the top priorities and, in this framework,
the aim of UG has always been to develop and accomplish high standards as
regards academic quality, and as regards student life for the benefit of both Georgian
and international students. The University of Georgia is a place that generates
and disseminates knowledge. It has created a diverse environment, which forms
open-minded and educated persons with human values and the skills necessary
to consciously and easily cope with the challenges of the modern world. The
academic faculty of the university is represented by a team of creative and enthusiast
individuals, willing to educate professionals equipped with the knowledge and skills
required in the modern world and ready to make a significant contribution to the
welfare of humanity. Today, the University of Georgia is one of the nation’s leading
universities, with the personal growth and professional development of its students
as its main goal. Graduates of the university feel confidence and are ready to enter
a competitive job market. Today, the university has the honor to offer its students
modern facilities and the learning environment in which they can gain high quality
education as well as practical experience. The knowledge and skills acquired at our
162 8 Case Studies of Math Education for STEM in Georgia

university are a guarantee of a successful career not only in Georgia, but also in the
international labor market. UG is a classical university and it offers a wide range of
specializations. From 2014 to 2016 overall numbers of UG students increased up to
50% from 4000 to 6000 students. These students are distributed in six main schools.
They are:
• School of Humanities;
• School of Law;
• School of Social Sciences;
• School of IT, Engineering and Mathematics;
• School of Health Sciences and Public Health;
• School of Business, Economics and Management.
The School of IT, Engineering and Mathematics has a STEM profile. There are
about 350 students enrolled totally in all academic level. The number of freshmen
students for 2016 was 80 students. There are seven different academic programs,
four of them on the BSc level, two on the Master level, and the seventh is a
PhD level program. For the BSc programs, Informatics has 238 students, Electrical
and Computer Engineering has 33, Engineering (for international students) has 10,
Mathematics has 7, Computer Science has 181, On the master program, Applied
Sciences has 5, and the Exact, Natural and Computer Science program, which is
PhD level, has 7 students.

8.4.1 Comparative Analysis of “Precalculus”

Precalculus is an elective course for freshmen students of STEM specializations
meant to increase their knowledge and Math skills. This course contains func-
tions, graphs, linear and quadratic functions, inverse, exponential, logarithmic and
trigonometric functions; polynomial and rational functions; solving of linear and
nonlinear systems of equations and inequalities; sequences and their properties;
combinatorial analysis and fundamentals of probability theory, binomial theorem.
Precalculus is a theoretical course but it is based on practical examples. This
course was developed for engineering students and all practical examples are
connected to real life problems. Earlier it was 26 h in semester and after course
modernization practical work hours was added and now it is totally 39 h per
semester. There have been 25 students in Precalculus in 2014. Precalculus was
compared with “Remedial Instruction” course at Tampere University of Technology
(TUT). The course is mandatory for TUT students who do not grasp mathematics
well in the beginning of their studies. All the students enrolling at TUT will
take a basic skills’ test and the weakest 20% of them are directed to “Remedial
Instruction”. This remedial course is completely done with computers with a tailored
introductory course in the Math-Bridge system at TUT. The courses’ outlines are
seen in Table 8.25.
8 Case Studies of Math Education for STEM in Georgia 163

Table 8.25 Outlines of Precalculus (UG) and Remedial Instruction (TUT) courses
Course information UG TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 1 1
Elective/mandatory Elective Mandatory
Number of credits 6 0
Teaching hours 39 0
Preparatory hours 111 0
Teaching assistants 6 0
Computer labs No Yes
Average number of students on the course 200 140
Average pass% 75% 90%
% of international students Less than 1% Less than 5%

The Precalculus course was offered for freshmen students from STEM special-
izations (Informatics, Electrical and Computer Engineering). Also it was offered to
students from Business and Economics specializations.
The Precalculus was prepared by the Department of Mathematics, which is
responsible for all mathematics courses in UG. The staff of the Department
of Mathematics consists of two full professors, three associated professors, six
assistants and invited lecturers (depending of groups). Professors and associated
professors are responsible for lectures, while assistants and invited lecturers are
responsible for tutorials.
Before course modernization “Precalculus” was mainly a theoretical course.
After modernization tutorials should be added and the course should become more
applied.
Assessment contains midterm exams (60%) and final examination (40%).
Midterm exams contain quizzes (24%) and midexams (36%). Totally, there are
eight quizzes (each for 3 points), three midexams (each for 12 points) and final
exam (40 points). The minimum number of points required for the final exam is 20.
The course is passed when student has more than 50 points and in the final exam
more than 20 points.
Each quiz (3 points) contains three tasks from the previous lecture. Each midterm
exam (12 points) consists eight tasks including two theoretical questions. The final
exam (40 points) consists of 20 tasks including six theoretical questions.
During the lectures professors use various tools for visualization, and presenta-
tion tools, in order to present some applications and some dynamic processes, in par-
ticular GeoGebra is an illustration tool. There are no mandatory parts of using TEL
systems in the exams. We used the Math-Brige system for Pre- and Post-testing.
GeoGebra was used for illustration purposes. It was used to show properties
of functions, intersection of functions by axis, finding of intersection points of
two functions and so on. The role of TEL systems in the course has been in
demonstrating basic mathematics and in visualizing different math topics.
164 8 Case Studies of Math Education for STEM in Georgia

8.4.1.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Table 8.26. Outcome competencies are given in Table 8.27.

8.4.1.2 Summary of the Results

The University of Georgia has changed the math syllabus in Precalculus. In
order to achieve SEFI competencies [1] UG have introduced modern educational
technologies in teaching methods. The aim of the modernization was an integration
of Math-Bridge and GeoGebra in the study process. UG has done the following
modification in curricula and in syllabuses.
Modernization of the syllabus was done since September of 2016 just only
in Precalculus. Comparison of syllabuses in other subjects (Precalculus) have
shown full compatibility with the subjects of TUT courses. UG have separated
Precalculus in two independent courses for STEM and for Business and Economic
specializations. In Precalculus for STEM specializations we added some subjects
according to the TUT courses content. Also we added 12 h of practical work by
using GeoGebra and MATLAB programs.
In 2016 UG has done pre- and post-testing in Precalculus and Calculus 1. Math-
Bridge was used for testing. Theoretical and practical examples were prepared in
Math-Bridge. Math-Bridge was used as a tool for analyzing the results.

Table 8.26 Core 0-level Core 0
prerequisite competencies of
Competency UG TUT
the Precalculus (UG) and
Remedial Instruction (TUT) Arithmetic of real numbers X X
courses Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their inverses X X

Table 8.27 Core 0-level Core 0
outcome competencies of the
Competency UG TUT
Precalculus (UG) and
Remedial Instruction (TUT) Graphs X
courses Linear and quadratic functions X X
Polynomial and rational functions X X
Power functions X X
Trigonometric functions X X
Sequences, arithmetic and geometric sequences X X
Combinatorics and probabilities X
8 Case Studies of Math Education for STEM in Georgia 165

8.4.2 Comparative Analysis of “Calculus 1”

Calculus 1 is a mandatory course for students of STEM specializations (Informatics
BSc, Electronic and Computer Engineering BSc, Engineering BSc). This course
contains the basic properties of inverse, exponential, logarithmic and trigonometric
functions; limit, continuity and derivative of a function, evaluating rules of a deriva-
tive, function research and curve-sketching techniques, applications of derivative
in the optimization problems, L’Hôpital’s rule, Newton’s method, indefinite and
definite integral and their properties, rules of integration, integration of rational
functions, evaluating area between curves and surface area using integrals, integrals
application in physics, numerical integration.
Calculus 1 is a theoretical course, but it is based on practical examples. This
course was developed for engineering students and all practical examples are
connected to real life problems. Before the course modernization it was 26 h in a
semester and after the course modernization practical work hours were added; now
it is totally 39 h per semester. There are 25–50 students on the course each year. It
was compared with the corresponding “Engineering Mathematics 1” (EM1) course
from Tampere University of Technology (TUT). The course outlines are seen in
Table 8.28.
This course was offered to STEM students (Informatics, Electrical and Computer
Engineering). Also it was offered to students from Business and Economics special-
izations with modified content, with more focus on real Business applications.
The “Calculus 1” was prepared by Department of Mathematics, which is respon-
sible for all mathematics courses in UG. The staff of the Department of Mathematics
consists of two full professors, three associated professors, six assistants and invited
lecturers (depending of groups).
Pedagogy and assessment are done similarly to “Precalculus”. During the
lectures the professor uses various tools for visualization, presentation tools in order
to present some applications and some dynamic processes, in particular GeoGebra

Table 8.28 Outlines of “Calculus 1” (UG) and “Engineering Mathematics 1” (TUT) courses
Course information UG TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 1 1
Elective/mandatory Mandatory Mandatory
Number of credits 6 5
Teaching hours 39 57
Preparatory hours 111 80
Teaching assistants 6 1–3
Computer labs No Yes
Average number of students on the course 25 200
Average pass% 80% 90%
% of international students Less than 1% Less than 5%
166 8 Case Studies of Math Education for STEM in Georgia

as an illustration tool. There are no mandatory parts of using TEL systems in exams.
We have used the Math-Bridge system for Pre- and Post-testing. GeoGebra was used
for illustration purposes. It was used to demonstrate the properties of functions,
derivatives of a function, applications of derivatives, integrals and applications of
integrals.

8.4.2.1 Contents of the Course

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Table 8.29. Outcome competencies are given in Table 8.30.

8.4.2.2 Summary of the Results

UG has changed the Math syllabus in “Calculus 1”. In order to achieve SEFI
competencies [1] UG has introduced modern educational technologies in teaching
methods. The aim of the modernization was an integration of Math-Bridge and
GeoGebra in the study process. UG has done the following modifications in
curricula and in syllabi.
Modernization of the syllabus was done since September of 2016 just only in
“Calculus 1”. Comparison of syllabuses in other subjects (Calculus 1 and Calculus
2) have shown full compatibility with the subjects of TUT courses. Tutorials were
added with 12 h of practical work by using GeoGebra and MATLAB programs.

Table 8.29 Core 0-level Core 0
prerequisite competencies of
Competency UG TUT
the “Calculus 1” (UG) and
“Engineering Mathematics 1” Arithmetic of real numbers X X
(TUT) courses Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their properties X X

Table 8.30 Core 0-level Core 0
outcome competencies of the
Competency UG TUT
“Calculus 1” (UG) and
“Engineering Mathematics 1” Functions and their basic properties X X
(TUT) courses Logarithmic and trigonometric functions X X
Limits and continuity of a function X X
Derivative X X
Definite integral and integration methods X
Minimum and maxima X X
Indefinite integrals X
8 Case Studies of Math Education for STEM in Georgia 167

In 2016 UG has done pre- and post-testing in “Precalculus” and “Calculus 1”.
Math-Bridge was used for testing. Theoretical and practical examples were prepared
in Math-Bridge. It was used for analyzing results as well.

Reference

1. SEFI (2013), “A Framework for Mathematics Curricula in Engineering Education” (Eds.)
Alpers, B., (Assoc. Eds) Demlova M., Fant C-H., Gustafsson T., Lawson D., Mustoe L., Olsson-
Lehtonen B., Robinson C., Velichova D. (http://www.sefi.be).

9.1 Analysis of Mathematical Courses in ASPU

Lusine Ghulghazaryan and Gurgen Asatryan
Faculty of Mathematics, Physics and Informatics, Armenian State Pedagogical
University (ASPU), Yerevan, Armenia
e-mail: lusina@mail.ru

9.1.1 Armenian State Pedagogical University (ASPU)

Armenian State Pedagogical University (ASPU) was established on November 7,
1922 and in 1948 it was named after the great Armenian Enlightener Educator
Khachatur Abovian. ASPU implements a three-level education system (Bachelor,
Master and Doctorate studies). It has ten faculties.
The Faculty of Mathematics, Physics and Informatics was formed in 2012 by
merging the two Faculties of “Mathematics and Informatics” and “Physics and
Technology”. The faculty operates under a two-level education system. It offers
Bachelor’s degree courses (4 years) and Master’s degree courses (2 years). The
faculty prepares teachers in the specializations of Mathematics, Physics, Natural
Sciences, Informatics and Technology.
The faculty has four Departments: the Department of Mathematics and its
Teaching Methodology, the Department of Physics and its Teaching Methodology,
the Department of Technological Education, and the Department of Informatics and
its Teaching Methodology.

© The Author(s) 2018 169
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_9
170 9 Case Studies of Math Education for STEM in Armenia

The Department of Mathematics and its Teaching Methodology was formed in
2016 by merging three Departments of Higher Algebra and Geometry, Mathematical
Analysis and Theory of Functions, Teaching Methodology of Mathematics.

9.1.2 Comparative Analysis of “Linear Algebra and Analytic
Geometry”

The course is given for the first year Bachelor students of specializations Informatics
and Physics. For the students specialized in Mathematics there are two separate
courses—“Linear Algebra” and “Analytic Geometry” (LA&AG). The average
number of students is 35 in each specialization. The course “Linear Algebra and
Analytic Geometry” was compared with “Engineering Mathematics 1” (EM2)
course at Tampere University of Technology; see Table 9.1 for course outlines.
The Department of Mathematics and its Teaching Methodology is responsible
for the course. The course is taught 4 h per week—2 h for lectures and 2 h for
practice. The average number of students is 35 in each specialization (Mathematics,
Physics, Informatics); most of them are female. There are no international students
in the Faculty. Students use the TEL-systems in the computer laboratory created by
Tempus MathGeAr project. In particular they use the Math-Bridge system during
their study.

9.1.2.1 Contents of the Course

The course is dedicated to working with matrices, systems of linear equations,
vector spaces and subspaces, linear mapping, coordinate method, equations of lines
and planes, second order curves and surfaces.

Table 9.1 Outlines of LA&AG (ASPU) and EM2 (TUT) courses
Course information ASPU TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 1 1
Selective/mandatory Mandatory Mandatory
Number of credits 5 5
Teaching hours 64 57
Preparatory hours 86 76
Teaching assistants 1 1–3
Computer labs Available Available
Average number of students on the course 35 200
Average pass% 70% 90%
% of international students 0% Less than 5%
9 Case Studies of Math Education for STEM in Armenia 171

The list of contents is:
1. Vector spaces, subspaces, linear mappings.
2. Complex numbers, the module and the argument of a complex number.
3. Matrix Algebra, determinant, rank, inverse of the matrix.
4. Systems of linear equations, Gauss method, Cramer’s rule, Kronecker–Capelli’s
theorem, systems of linear homogeneous equations.
5. Linear mappings, the rank and defect of a linear mapping, the kernel of a linear
mapping
6. Polynomials, the roots of a polynomial, Bézout’s theorem, the Horner scheme.
7. Coordinate method, distance of two points, equations of lines and planes,
distance of a point and a line or plane.
8. Second order curves and surfaces.
Prerequisites for the course is knowledge in elementary mathematics. Outcome
courses are General Algebra, Theory of Topology and Differential Geometry.
The objectives for the course are: To provide students with a good understanding
of the concepts and methods of linear algebra; to help the students develop the ability
to solve problems using linear algebra; to connect linear algebra to other fields of
mathematics; to develop abstract and critical reasoning by studying logical proofs
and the axiomatic method.
The assessment is based on four components, two midterm examinations, the
final examination and the attendance of the student during the semester. The points
of the students are calculated by m = (1/4)a + (1/4)b + (2/5)c + (1/10)d,
where a, b, c, d are the points of intermediate examinations, final examination and
the point of attendance, respectively (maximal point of each of a, b, c, d is 100).
The satisfactory point starts from 60.

9.1.2.2 Course Comparison Within SEFI Framework

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Table 9.2. Outcome competencies are given in Tables 9.3 and 9.4.

Table 9.2 Core 0-level prerequisite competencies of LA&AG (ASPU) and EM2 (TUT) courses
Core 0
Competency ASPU TUT
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Excl.: interpret simultaneous linear inequalities in
Linear laws terms of regions in the plane X
Quadratics, cubics, polynomials X X
Geometry X X
172 9 Case Studies of Math Education for STEM in Armenia

Table 9.3 Core 0-level Core 0
outcome competencies of
Competency ASPU TUT
LA&AG (ASPU) and EM2
(TUT) courses Linear laws X X
Coordinate geometry X X

Table 9.4 Core 1-level Core 1
outcome competencies of
Competency ASPU TUT
LA&AG (ASPU) and EM2
(TUT) courses Vector arithmetic X X
Vector algebra and applications X X
Matrices and determinants X X
Solution of simultaneous linear equations X X
Linear spaces and transformations X
Conic sections X

9.1.2.3 Summary of the Results

The course “Linear Algebra and Analytic Geometry” in ASPU has been compared
with the corresponding course “Engineering Mathematics 2” at Tampere University
of Technology (TUT).
The teaching procedure in ASPU is quite theoretical and the assessment is
based mainly on the ability of students of proving the fundamental theorems. This
theorem-to-proof method of teaching is quite theoretical, while the corresponding
courses in TUT are quite practical and applicable. In this regard, the assessment in
TUT is based on the emphasis on the ability of applying the fundamental theorems
in solving problems.
According to the analysis of the teaching methodology, the course “Linear
Algebra and Analytic Geometry” in ASPU should be reorganized so that the modern
aspects of the case can be presented more visually. In particular during the teaching
process some applications of the case should be provided. Some of the main
theorems and algorithms should be accompanied by programming in MATLAB.
The main steps of modernization to be taken are:
• Include more practical assignments.
• Demonstrate applications of algebra and geometry.
• Construct bridge between the problems of linear algebra and programming (this
would be quite important especially for students of Informatics).
• Use ICT tools for complex calculations.
• Implement algorithms for the basic problems of linear algebra. (Gauss method,
Cramer’s rule, matrix inversion. . . )
• Use Math-Bridge for more practice and for the theoretical background.
• Students should do experiments for geometric objects using ICT tools.
• Before the final examination students should prepare a paperwork with the
solution of problems assigned by the teacher and the results of their experiments.
9 Case Studies of Math Education for STEM in Armenia 173

9.1.3 Comparative Analysis of “Calculus 1”

The course is given for one and a half year Bachelor students of specializations
Informatics and Physics. For the students in the specialization Mathematics is given
2 years. Average number of students is 35 (in each specialization). The course was
compared with the corresponding course “Engineering Mathematics 1” (EM1) at
Tampere University of Technology (TUT). The course outlines are presented below;
see Table 9.5.
The Department of Mathematics and its Teaching Methodology is responsible for
the course. The course is taught 4 h per week—2 h for lectures and 2 h for practice.
The average number of students is 35 in each specialization (Mathematics, Physics,
Informatics), most of them are female. There are no international students in the
Faculty. Students use the TEL-systems in a computer laboratory created by Tempus
MathGeAr project. In particular they use the Math-Bridge system during their study.

9.1.3.1 Contents of the Course

The course is dedicated to working with the rational and real numbers, limits of
numerical sequences, limit of functions, continuity of function, monotonicity of
function, derivative, differential of function, convexity and concavity of the graph
of the function, investigation of function and plotting of graphs.
The list of contents is:
1. The infinite decimal fractions and set of real number.
2. Convergence of the numerical sequences.
3. Limit of a function and continuity of function.
4. Derivative and differential of function.
5. Derivatives and differentials of higher orders, Taylor’s formula.

Table 9.5 Outlines of Calculus 1 (ASPU) and EM1 (TUT) courses
Course information ASPU TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 1 1
Selective/mandatory Mandatory Mandatory
Number of credits 5 5
Teaching hours 64 57
Preparatory hours 86 76
Teaching assistants 1 1–3
Computer labs Available Available
Average number of students on the course 35 200
Average pass% 70% 90%
% of international students 0% Less than 5%
174 9 Case Studies of Math Education for STEM in Armenia

6. Monotonicity of a function.
7. Extremes of a function.
8. Convexity and concavity of the graph of the function.
9. Investigation of a function and plotting of graphs.
A prerequisite for the course is knowledge of elementary mathematics. Outcome
courses are Functional analysis, Differential equations and Math-Phys. equations.
Objectives of the course for the students are:
• To provide students with a good understanding of the fundamental concepts and
methods of Mathematical Analysis.
• To develop logical reasoning, provide direct proofs, proofs by contradiction and
proofs by induction.
• To teach students to use basic set theory to present formal proofs of mathematical
statements.
• To develop the ability of identifying the properties of functions and presenting
formal arguments to justify their claims.
The assessment is based on three components, two midterm examinations and
the attendance of the student during the semester. The points of the students are
calculated by m = (2/5)a +(2/5)b+(1/5)c, where a, b, c are the points of midterm
examinations and the points of attendance, respectively (maximal number of points
of each of a, b, c is 100). The satisfactory number of points starts from 60.

9.1.3.2 Course Comparison Within SEFI Framework

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Table 9.6. Outcome competencies are given in Tables 9.7 and 9.8.

Table 9.6 Core 0-level prerequisite competencies of “Calculus 1” (ASPU) and EM1 (TUT)
courses
Core 0
Competency ASPU TUT
Functions and their inverses X X
Sequences, series, binomial expansions Excl.a X
Logarithmic and exponential functions X X
Rates of change and differentiation X X
Stationary points, maximum and minimum values Excl.b X
a Obtain the binomial expansions of (a + b)2 for a rational number; use the binomial expansion to
obtain approximations to simple rational functions
b Obtain the second derived function of simple functions; classify stationary points using second

derivative
9 Case Studies of Math Education for STEM in Armenia 175

Table 9.7 Core 1-level outcome competencies of “Calculus 1” (ASPU) and EM1 (TUT) courses
Core 0
Competency ASPU TUT
Sequences, series, binomial expansions X
Stationary points, maximum and minimum values X X
Proof X X

Table 9.8 Core 1-level Core 1
outcome competencies of
Competency ASPU TUT
“Calculus 1” (ASPU) and
EM1 (TUT) courses Hyperbolic functions X X
Rational functions X X
Functions X X
Differentiation X X

9.1.3.3 Summary of the Results

The course “Calculus 1” in ASPU has been compared with the corresponding course
“Engineering Mathematics 1” at Tampere University of Technology (TUT).
The teaching procedure in ASPU is quite theoretical and the assessment is
based mainly on the ability of students of proving the fundamental theorems. This
theorem-to-proof method of teaching is quite theoretical, while the corresponding
courses in TUT are quite practical and applicable. In this, the assessment in TUT
is based on the emphasis on the ability of applying the fundamental theorems in
solving problems.
According to the analysis of the teaching methodology, the course “Calculus 1”
in ASPU should be reorganized so that the modern aspects of the case be presented
more visually. In particular during the teaching process some applications of the
case should be provided. Some of the main concepts and properties should be
accompanied by programming in Wolfram Mathematica.
The main steps of modernization are:
• Include more practical assignments.
• Demonstrate applications of calculus.
• Construct a bridge between the problems of calculus and physical phenomena
(for students of physics) and programming (for students of Informatics).
• Use ICT tools for complex calculations.
• Use Math-Bridge for more practice and for the theoretical background.
• Students should do experiments for graph of function using ICT tools.
• Before the final examination students should prepare a paperwork with the
solution of problems assigned by the teacher and the results of their experiments.
176 9 Case Studies of Math Education for STEM in Armenia

9.2 Analysis of Mathematical Courses in NPUA

Ishkhan Hovhannisyan () and Armenak Babayan
Faculty of Applied Mathematics and Physics, National Polytechnic University of
Armenia (NPUA), Yerevan, Armenia

9.2.1 National Polytechnic University of Armenia (NPUA)

National Polytechnic University of Armenia (NPUA) is the legal successor of Yere-
van Polytechnic Institute, which was founded in 1933, having only 2 departments
and 107 students. The institute grew along with the Republic’s industrialization and
in 1980–1985 reached its peak with about 25,000 students and more than 66 majors,
becoming the largest higher education institution in Armenia and one of the most
advanced engineering schools in the USSR. On November 29, 1991, the Yerevan
Polytechnic Institute was reorganized and renamed State Engineering University of
Armenia (SEUA). In 2014, by the Resolution of the Government of the Republic
of Armenia (RA) the traditional name “Polytechnic” was returned to the University
and SEUA has been reorganized and renamed to National Polytechnic University of
Armenia.
During 83 years of its existence, the University has produced nearly 120,000
graduates, who have contributed greatly to the development of industry, forming
a powerful engineering manpower and technology base for Armenia. At present
NPUA has about 9000 students. The great majority of them are STEM students.
The number of the regular academic staff of the University exceeds 800, most of
them with Degrees of Candidate or Doctor of Sciences. With its developed research
system and infrastructure the University is nationally recognized as the leading
center in technical sciences.
Today, at its central campus located in Yerevan and the Branch Campuses—in
Gyumri, Vanadzor and Kapan, the University accomplishes four study programs
of vocational, higher and post-graduate professional education, conferring the
qualification degrees of junior specialist, Bachelor, Master and researcher. Besides,
the degree programs, the University also offers extended educational courses by
means of its faculties and a network of continuing education structures. The scope of
specialization of the University includes all main areas of engineering and technolo-
gies represented by 43 Bachelor’s and 26 Master’s specializations in Engineering,
Industrial Economics, Engineering Management, Applied Mathematics, Sociology
and others, offered by 12 faculties. Totally there are more than 40 STEM disciplines
in NPUA.
Apart from the faculties, NPUA has a Foreign Students Division which organizes
the education of international students from across the Middle East, Asia and
Eastern Europe. Their overall number today is almost 200. The languages of
instruction are Armenian and English.
9 Case Studies of Math Education for STEM in Armenia 177

The University has a leading role in reforming the higher education system in
Armenia. NPUA was the first higher education institute (HEI) in RA that introduced
two- and three-level higher education systems, and it implemented the European
Credit Transfer System (ECTS) in accordance with the developments of the Bologna
Process.
During the last decade, the University has also developed an extended network of
international cooperation including many leading Universities and research centers
of the world. The University is a member of European University Association
(EUA), Mediterranean Universities Network, and Black Sea Universities Network.
It is also involved in many European and other international academic and research
programs. The University aspires to become an institution, where the education and
educational resources are accessible to diverse social and age groups of learners, to
both local and international students, as well as to become an institution which is
guided by global perspective and moves toward internationalization and European
integration of its educational and research systems.

9.2.2 Comparative Analysis of “Mathematical Analysis-1”

Mathematical Analysis-1 (MA-1) is a fundamental mathematical discipline for
major profile Informatics and Applied Mathematics (IAM). It is mainly a theoretical
course, but it also contains certain engineering applications, such as derivatives
arising from engineering and physics problems. There are about 50 first year
students (four of which are international) at IAM and they all study this course.
“Mathematical Analysis-1” was compared with a similar course “Engineering
Mathematics 1” from Tampere University of Technology (TUT). The course
outlines are seen in Table 9.9.

Table 9.9 Outlines of MA-1 (NPUA) and EM1 (TUT) courses
Course information NPUA TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 1 1
Selective/mandatory Mandatory Mandatory
Number of credits 6 5
Teaching hours 80 57
Preparatory hours 80 76
Teaching assistants 1–2 1–3
Computer labs Yes Yes
Average number of students on the course 50 200
Average pass% 75% 90%
% of international students Less than 8% Less than 5%
Description of groups 50 students in two groups, 25 in
each. Avg. age is 17 years. Male
students twice more than females
178 9 Case Studies of Math Education for STEM in Armenia

The prerequisite for Mathematical Analysis-1 is high school mathematics.
Mathematical Analysis-1 is fundamental for all Mathematical disciplines. The
course of Mathematical Analysis 1 together with Mathematical Analysis-2,-3,
Linear Algebra and Analytical Geometry, and others is included in the group of
mandatory mathematical courses. This group is a requirement for all Bachelor level
students of IAM during first year of study.
The chair of General Mathematical Education is responsible for this course for
IAM-profile. There are 2 full professors and 25 associate professors working at the
chair. The total number of credits is 6. It is an 80-h course, including 32 h of lectures
and 48 h of tutorials.

9.2.2.1 Teaching Aspects

The course of Mathematical Analysis-1 is established for the first year students
and is quite theoretical. So the pedagogy is traditional: students listen to lectures,
accomplish some tasks during tutorials and do their homework. Project-based
learning is used in this course too, which makes learning process more interesting,
sometimes funny and even competitive. Sometimes the group of students is divided
into several subgroups and every subgroup fulfills some task. This kind of work in
subgroups is very competitive and students like it. Some teachers use Moodle for
distance learning.
NPUA uses the following rating system. The maximum grade is 100 points; one
can get 50 points during the semester and another 50 points (as a maximum) is
left for the final exam. During the semester students get their 10 points for work
in the class and 20 points for each of two midterm tests. These tests allow the
teacher to assess the students’ work during the semester. Exams are either in oral
or written form and include theoretical questions (e.g. a theorem with a proof) and
computational tasks. The final grade is the sum of the semester and exam grades. A
final grade of at least 81 corresponds to “excellent” (ECTS grade A); a grade from
61 to 80 corresponds to “good” (ECTS grade B); if the sum is between 40 and 60,
the student’s grade is “satisfactory” (ECTS grade C). Finally, students fail (grade
“non-satisfactory”, equivalent to ECTS grade F), if their final grade is less than 40.
There is a 2-h lecture on Mathematical Analysis-1 and a 3-h tutorial every week.
During tutorials students solve problems (complete computational tasks) under
teacher’s direction. Students may be given home tasks, which must be done during
preparatory hours. Computer labs are not used for every tutorial, but the computers
are used to control and grade programming homework.

9.2.2.2 Use of Technology

Some programming languages (C++ or Pascal) are used for homework, writing of
programs (topics are synchronized with the course content) is a mandatory part of
the midterm tests. E-mail and social networks are sometimes used to have closer
9 Case Studies of Math Education for STEM in Armenia 179

connection with students, give assignments etc. After participation in the MathGeAr
project we use Math-Bridge and Moodle for teaching Mathematical Analysis-1.
There are about 50 IAM students attending the course; for these profiles lectures
and tutorials are set separately. Currently it is still too early to give details of the
course outcome; we will have the results after the second middle test and final
examination. But student’s unofficial feedback is very positive.
Finally, we would like to mention that quite recently we have got four foreign
students in this course, and so far, they appreciate the course Mathematical
Analysis-1.

9.2.2.3 Course Comparison Within SEFI Framework

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Table 9.10. Outcome competencies are given in Tables 9.11 and 9.12.

Table 9.10 Core 0-level prerequisite competencies for MA-1 (NPUA) and EM1 (TUT) courses
Core 0
Competency NPUA TUT
Arithmetic of real numbers X X
Algebraic expressions and formulas X X
Linear laws X X
Quadratics, cubics, polynomials Exl. derivative X
Functions and their inverses Exl. the limit of a function X
Sequences, series, binomial expansions Exl. binomial expansions X
Logarithmic and exponential functions X X
Proof X
Geometry X X
Trigonometry X X
Coordinate geometry X X
Trigonometric functions and applications X X
Trigonometric identities X X

Table 9.11 Core 0-level outcome competencies for MA-1 (NPUA) and EM1 (TUT) courses
Core 0
Competency NPUA TUT
Rates of change and differentiation X X
Stationary points, maximum and minimum values X X
Functions of one variable X X
180 9 Case Studies of Math Education for STEM in Armenia

Table 9.12 Core 1-level outcome competencies for MA-1 (NPUA) and EM1 (TUT) courses
Core 1
Competency NPUA TUT
Functions Exl. partial derivatives Exl. partial derivatives
Differentiation X X
Sequences and series Exl. series
Mathematical induction and recursion X X

9.2.2.4 Summary of the Results

One of the purposes of TEMPUS MathGeAr-project was to modernize selected
national math courses meeting SEFI criteria after comparing these with the corre-
sponding EU-courses. The SEFI framework for math curricula in STEM education
[1] provides the following list of competencies:
• thinking mathematically,
• reasoning mathematically,
• posing and solving mathematical problems,
• modeling mathematically,
• representing mathematical entities,
• handling mathematical symbols and formalism,
• communicating in with and about mathematics,
• making use of aids and tools.
After studying SEFI framework and comparing national math curricula with
those of EU considerable commonality around the aims and objectives, curriculum
content and progression, and aspirations for problem-solving are revealed. The
mathematics expectations at NPUA selected course are comparable to those at
TUT. But there are some remarkable distinctive features, discussed in comparative
analysis. The general feeling of EU experts at TUT (Tampere) and UCBL (Lyon)
is that the course in the partner universities could have a more applied nature and
in the course learning technology could be better used. In order to make the NPUA
math curricula converge to the European standards, thus ensuring transferability
of learning results and introducing best European educational technologies for
mathematics, following recommendations of EU experts, the NPUA implemented
the following to the curriculum:
• changed syllabus (contents and the way of presentation, “theorem-to-proof” style
was modified by putting more emphasis on applications);
• added more topics, applications and examples related to the engineering disci-
plines;
• started using mathematical tool programs (MATLAB, Scilab, R, etc.);
• started using Math-Bridge for the Mathematical Analysis-1 course;
• added minor student project tasks to the course, including using web resources.
9 Case Studies of Math Education for STEM in Armenia 181

9.2.3 Comparative Analysis of “Probability and Mathematical
Statistics”

This course is one of the courses in the program “Informatics and Applied Mathe-
matics”. This program is applied because it prepares specialists in IT technologies,
mainly programmers, specialists in computer sciences, financial markets experts and
so on. But for this kind of work solid mathematical knowledge is necessary, so
rigorous theoretical facts are an essential part of the course. The number of students
of our faculty is approximately 300, and every year approximately 50 students
enroll the course “Probability and Mathematical Statistics”. Mathematics is an
essential part in the study program, almost all courses of the program are connected
with mathematics, or need solid mathematical background, because the ability of
thinking mathematically and the ability to create and use mathematical models
are the most important acquirements that our graduates must have. The course
“Probability and Mathematical Statistics” was compared with a similar course on
“Probability Calculus” at Tampere University of Technology (TUT). The course
outlines are seen in Table 9.13.
This course is one of the four most important courses of the program, because
probabilistic thinking is one of the most important abilities for a modern specialist.
It starts in the fourth semester (last semester of the second year) and the duration of
this course is 1 year. Prerequisite courses are the general courses of Mathematical
Analysis, Linear Algebra and Analytic Geometry. This course was selected from
the cluster of mathematical courses mandatory for the students of the “Informatics”
specialization. Topics of the course are used in following courses: “Numerical
Methods” “Mathematical Physics Equations” and this course is the base for
the Master’s program courses “Mathematical Statistics”, “Stochastic Processes”,
“Information Theory”.
The Chair of Specialized Mathematical Education is responsible for the course.
This chair is responsible for the mathematical courses of the Master programs of
NPUA and all courses (in Bachelor and Master programs) for “Informatics and

Table 9.13 Outlines of the “Probability and Mathematical Statistics” (NPUA) and “Probability
Calculus” (TUT) courses
Course information NPUA TUT
Bachelor/Master level Bachelor Bachelor
Preferred year 2 2
Selective/mandatory Mandatory Selective
Number of credits 6 4
Teaching hours 52 42
Preparatory hours 52 66
Teaching assistants – 1–2
Computer labs 2 Available
Average number of students on the course 50 200
Average pass% 80% 90%
% of international students None Less than 5%
182 9 Case Studies of Math Education for STEM in Armenia

Applied Mathematics” speciality. There are five full professors and eight associate
professors working at the Chair.
The number of the credits for the course “Probability and Mathematical Statis-
tics” is six. Two of them are for the lectures, and four for the practical work.
Teaching hours are three hours per week. One hour for lecture and 2 h for
practical work. No preparatory hours are planned, because it is supposed that
students have enough mathematical knowledge (general courses of mathematical
analysis, linear algebra and analytic geometry). Two computer laboratories help us
organize teaching process effectively.
The average number of students in the course is 50. Approximately 80%
successfully finish it. This course is delivered in Armenian, so foreign students may
appear in the group only occasionally. But the same course may be delivered in
English for foreign students.

9.2.3.1 Course Comparison Within SEFI Framework

The comparison is based on the SEFI framework [1]. Prerequisite competencies are
presented in Tables 9.14 and 9.15. Outcome competencies are given in Tables 9.16
and 9.17.

Table 9.14 Core 0-level Core 0
prerequisite competencies of
Competency NPUA TUT
the “Probability and
Mathematical Statistics” Arithmetic of real numbers X X
(NPUA) and “Probability Algebraic expressions and formulas X X
Calculus” (TUT) courses Linear laws X X
Quadratics, cubics, polynomials X X
Functions and their inverses X X
Analysis and calculus X X
Sets X X
Geometry and trigonometry X X

Table 9.15 Core 1-level Core 1
prerequisite competencies of
Competency NPUA TUT
the “Probability and
Mathematical Statistics” Hyperbolic and rational functions X X
(NPUA) and “Probability Functions X X
Calculus” (TUT) courses Differentiation X X
Methods of integration X X
Sets X X
Mathematical induction and recursion X X
Matrices and determinants X X
Least squares curve fitting X X
Linear spaces and transformations X X
9 Case Studies of Math Education for STEM in Armenia 183

Table 9.16 Core 0-level outcome competencies of the “Probability and Mathematical Statistics”
(NPUA) and “Probability Calculus” (TUT) courses
Core 0
Competency NPUA TUT
Data handling X X
Probability X X

Table 9.17 Core 1-level outcome competencies of the “Probability and Mathematical Statistics”
(NPUA) and “Probability Calculus” (TUT) courses
Core 1
Competency NPUA TUT (Tampere)
Data handling X X
Combinatorics X X
Simple probability X X
Probability models X X
Normal distribution X X
Statistical inference Exc. advanced part of hypothesis testing X

9.2.3.2 Summary of the Results

The general feeling of EU experts at TUT and Lyon is that the course could have
a more applied nature and the learning technology could be better used. In order to
make the NPUA math curricula converge to the European standards, thus ensuring
transferability of learning results and introducing the best European educational
technologies for mathematics, following recommendations of EU experts, the
NPUA implemented the following to its curriculum: one
• changed syllabus (contents and the way of presentation, “theorem-to-proof” style
was modified by putting more emphasis on applications);
• added more topics, applications and examples related to the engineering disci-
plines;
• started using mathematical tool programs (MATLAB, Scilab, R, etc.);
• started using Math-Bridge for the course;
• added minor student project tasks to the course, including using web resources.
184 9 Case Studies of Math Education for STEM in Armenia

Reference

Sergey Sosnovsky, Christian Mercat, and Seppo Pohjolainen

10.1 Introduction

The two EU Tempus-IV projects MetaMath (www.metamath.eu) and MathGeAr
(www.mathgear.eu) have brought together mathematics educators, TEL specialists
and experts in education quality assurance from 21 organizations across six
countries. A comprehensive comparative analysis of the entire spectrum of math
courses in the EU, Russia, Georgia and Armenia has been conducted. Its results
allowed the consortium to pinpoint and introduce several curricular modifications
while preserving the overall strong state of the university math education in these
countries. The methodology, the procedure and the results of this analysis are
presented here.
During the first project year 2014 three international workshops were organized
in Tampere (TUT), Saarbrucken (DFKI & USAAR) and Lyon (UCBL), respectively.
In addition, national workshops were organized in Russia, Georgia and Armenia.
The purpose of the workshops was to get acquainted with engineering mathematics
curricula in the EU and partner countries, with teaching and learning methods used
in engineering mathematics, as well as the use of technology in instruction of
mathematics. Finally, an evaluation methodology was set up for degree and course
comparison and development.

S. Sosnovsky
Utrecht University, Utrecht, the Netherlands
e-mail: s.a.sosnovsky@uu.nl
C. Mercat
IREM Lyon, Université Claude Bernard Lyon 1 (UCBL), Villeurbanne, France
e-mail: christian.mercat@math.univ-lyon1.fr
S. Pohjolainen ()
Tampere University of Technology (TUT), Laboratory of Mathematics, Tampere, Finland
e-mail: seppo.pohjolainen@tut.fi

© The Author(s) 2018 185
S. Pohjolainen et al. (eds.), Modern Mathematics Education for Engineering
Curricula in Europe, https://doi.org/10.1007/978-3-319-71416-5_10
186 S. Sosnovsky et al.

10.2 Curricula and Course Comparison

To accomplish curricula comparison and to set up guidelines for further develop-
ment SEFI framework [1] was used from the beginning of the project. The main
message of SEFI is that in evaluating educational processes we should shift from
contents to competences. Roughly said, competences are the knowledge and skills
students have when they have passed the courses and this reflects not only on
the course contents but especially on the teaching and learning processes, use of
technology and assessment of the learning results. SEFI presents recommendations
on the topics that BSc engineering mathematics curricula in different phases or
levels of studies should contain. The levels are Core 0—prerequisite mathematics,
Core 1—common contents for most engineering curricula, Core 2—elective courses
to complete mathematics education on chosen engineering area and finally Core
3 for advanced mathematical courses. SEFI makes recommendations on assessing
students knowledge and on the use of technology. Here it is also emphasized that
learning should be meaningful from the students’ perspective. Students’ perceptions
on mathematics in each of the universities were investigated and results were
presented in Chap. 1, Sect. 1.3.
The SEFI framework and related EU-pedagogy was described in Chap. 1,
Sect. 1.1. To make a comparison of curricula and courses, a methodology was
created. This methodology was described in detail in Chap. 2. Comprehensive
information on partner countries’ curricula, courses and instruction was collected
and organized in the form of a database. For comparison similar data was collected
from participating EU-universities, Tampere University of Technology (TUT),
Finland and university of Lyon (UBCL), France. This data includes:
• University: university type, number of students, percentage engineering stu-
dents, number of engineering disciplines, degree in credits, percentage of math
in degree.
• Teaching: Teacher qualifications, delivery method, pedagogy, assessment, SEFI
depth aim, modern lecture technology, assignment types, use of third party
material, supportive teaching.
• Selected course details: BSc or MSc level, preferred year, selective/mandatory,
prerequisite courses, outcome courses, department responsible, teacher position,
content, learning outcomes, SEFI level, credits, duration, student hours (and their
division), average number of students.
• Use of ICT/TEL: Tools used, mandatory/extra credit, optional, e-learning/
blended/traditional, Math-Bridge, calculators, mobile technology.
• Resources: Teaching hours, assistants, computer labs, average amount of stu-
dents in lectures/tutorials, use of math software, amount of tutorial groups, access
to online material.
10 Overview of the Results and Recommendations 187

10.3 Comparison of Engineering Curricula

The overall evaluation of partner universities curricula shows that they aim to cover
the SEFI core content areas, especially Cores 0 and 1 of BSc-level engineering
mathematics.
Considering only coverage may, however, lead to erroneous conclusions, if the
amount of mandatory ECTS in mathematics is not considered simultaneously.
For example, one university may have one 5 ECTS mathematics course during a
semester, while the second has two 5 ECTS courses, which cover the same topics in
the same time. In this case much more time is allocated in the second university to
study the same topics. This means that the contents will be studied more thoroughly,
students can use more time for their studies, and the learning results will better.
The ECTS itself are comparable, except in Russia, where 1 Russian credit unit
corresponds to 36 student hours compared with 25–30 h per ECTS in other univer-
sities. The amount of mandatory ECTS and contact hours used in teaching depends
on the policy of the university and it varies more as is seen in Tables 10.1, 10.2,
and 10.3 below.
To compare curricula, the following information was collected from EU and
partner universities. Table 10.1 shows the amount mandatory mathematics in
Engineering BSc programs in Finland (TUT), France (UCBL) and Russia (OMSU);
the corresponding information for Armenia (ASPU), (NPUA), and Georgia (ATSU),
(BSU), (GTU), (UG), is given in Tables 10.2 and 10.3. The first column presents the
country and partner university. The second column shows one ECTS as the hours a
student should work for it. It covers both contact hours (lectures, exercise classes,
etc.) and independent work (homework, project work, preparation for exams etc.).
The third column shows one ECTS as contact hours like lectures, exercise classes
etc., where the teacher is present. The fourth column shows the mandatory amount
of mathematics in BSc programs, and the fifth column shows all (planned) hours a
student should use to study mathematics. It has been calculated as the product of
mandatory ECTS and hours/ECTS for each university/BSc program.
In the first table, figures from Finland (TUT), France (UCBL), and Russia
(OMSU) are given. As the educational policy in Russia is determined on the national
level, the numbers from other Russian partner universities are very much alike. That
is why we have only the Ogarev Mordovia State University (OMSU) representing
the Russian universities for comparison.
188

Table 10.1 Mandatory mathematics in EU- and Russian engineering BSc programs
One ECTS as One ECTS as Mathematics in Mathematics in
Country student hours contact hours BSc program ECTS student hours Notes
Finland (TUT) 26.67 11–13 All (except 27 720 Additional elective courses
natural sciences) can be chosen
Finland (TUT) 26.67 11–13 Natural sciences 60 1600 Mathematics major
France (UCBL) 25–30 10 Generic 48 1200–1440 First 2 years of study
France (GPCE) Generic 864 Higher School Preparatory
Classes
Russia (OMSU) 1 CU = 36 h ≈ 1CU = 18 contact 16 technical and 7–75 CU (9–100 252–2700 Russian CU = 36 h. Half of
1.33 ECTS hours engineering BSc ECTS) the programs have more
programs than 20 CU (27 ECTS),
half less than 20 CU (27
ECTS). Exploitation of
Transport and
Technological Machines
and Complexes BSc has
only 7 CU, Fundamental
Informatics and
Information Technologies
has 75 CU
Russia (OMSU) 1 CU = 36 h 1CU = 18 contact Informatics and 35 CU (47 ECTS) 1260 Russian CU = 36 h. An
hours Computer Science example of a BSc program
in OMSU
S. Sosnovsky et al.
Table 10.2 Mandatory mathematics in Armenian engineering BSc programs
One ECTS as One ECTS as Mathematics in Mathematics in
Country student hours contact hours BSc program ECTS student hours Notes
Armenia (ASPU) 30 13 GROUP 1 BSc 42 1260 Specialization—
Engineering programs Informatics
Armenia (ASPU) 30 13 GROUP 2 BSc 32 960 Specializations—
Engineering programs Physics and Natural
Sciences
Armenia (ASPU) 30 13 GROUP 3 BSc 9 270 Specialization—
Technology and Technology and
Entrepreneurship, Entrepreneurship,
10 Overview of the Results and Recommendations

Chemistry Chemistry
Armenia (ASPU) 30 13 GROUP 4 BSc 9 270 Specializations—
Psychology and Psychology and
Sociology Sociology
Armenia (NPUA) GROUP 1 BSc 28 Faculty of Applied
Engineering programs Mathematics and
Physics
Armenia (NPUA) GROUP 2 BSc 18 All remaining faculties
Engineering programs
189
190

Table 10.3 Mandatory mathematics in Georgian engineering BSc programs
One ECTS as One ECTS as Mathematics in Mathematics in
Country student hours contact hours BSc program ECTS student hours Notes
Georgia (ATSU) 25 12 GROUP 1 BSc 35 875 Faculty of Exact and
Engineering Natural Sciences. BSc
programs Informatics
Georgia (ATSU) 25 12 GROUP 2 BSc 32.5 812.5 Faculty of Technical
Engineering Engineering
programs
Georgia (ATSU) 25 12 GROUP 3 BSc 15 375 Faculty of Technical
Engineering Engineering
programs
Georgia (ATSU) 25 12 GROUP 4 BSc 10 250 Faculty of Technical
Engineering Engineering
programs
Georgia (BSU) 25 9 GROUP 1 BSc 10 250 Programs of Civil
Engineering Engineering; Transport;
Telecommunication;
Mining and
Geoengineering
Georgia (BSU) 25 9 GROUP 2 5 125 Program of Architecture
Architecture
Georgia (BSU) 25 9 GROUP 3 Computer 5 125 Program of Computer
Science Science
S. Sosnovsky et al.
Georgia (GTU) 27 12 GROUP 1 BSc 15 405 Faculties of Power
Engineering programs Engineering and
Telecommunications;
Civil Engineering;
Transportation and
Mechanical Engineering;
Informatics and Control
Systems; Agricultural
Sciences and Biosystems
Engineering
Georgia (GTU) 27 12 GROUP 2 BSc 15 405 Business-Engineering
Business-engineering Faculty
programs
Georgia (GTU) 27 12 GROUP 3 BSc 10 270 Faculties of Architecture;
Mining and Geology;
Chemical Technology
and Metallurgy)
10 Overview of the Results and Recommendations

Georgia (GTU) 27 12 GROUP 4 BSc 10 270 International Design
School
Georgia (UG) 25–27 6.5 GROUP 1 BSc 24 600–648 Informatics (+elective
courses)
Georgia (UG) 25–27 6.5 GROUP 2 BSc 30 750–810 Electronic and Computer
Engineering (+ elective
courses)
Georgia (UG) 25–27 6.5 GROUP 3 MSc 12 300–324 + elective courses
191
192 S. Sosnovsky et al.

Some differences may be detected from the tables. The amount of ECTS varies
between engineering BSc programs from 10 ECTS to 75 Russian CUs, which
is about 100 ECTS. The contact hours per ECTS are between 6.5 and 13. The
time students use in studying engineering mathematics is different between the
universities. If all the universities would like to fulfill the SEFI 1 Core, then some
universities are resourcing less time for teaching and learning. This is unfortunate,
as the quality of learning depends strongly on the amount of time spent on teaching
and learning. This is not the only criteria, but one of the important criteria. As
mathematics plays an essential role of engineering education it should have a
sufficient role in engineering BSc curricula and it should be resourced to be able
to reach its goals.
The major observations from the national curricula are the following:
• Russian courses cover more topics and seem to go deeper as well. The amount
of exercise hours seems to be larger than EU. The overall number of credits
is comparable, but the credits are different (1 cr = 36 h (RU), 1 ECTS = 25–30 h
(EU)) therefore per credit, more time is allotted to Russian engineering students
for studying mathematics. The amount of mandatory mathematics varies with
BSc programs between 7–75 CU (9–100 ECTS). The medium is 20 CU, which
is about 27 ECTS. This means that coverage of the SEFI topical areas varies with
BSc program. The highest exceeds well the SEFI Core 1, but the lowest lacks
some parts. The medium of mandatory mathematics among the programs (20
CU ≈ 27 ECTS) is close to European universities.
• In Armenia, the amount of engineering mathematics in engineering BSc pro-
grams varies from 42 ECTS to 18 ECTS. The contents of engineering math-
ematics is very much the same as in EU but Armenian courses must cover
more topics for 18 ECTS than the EU-universities (27–40 ECTS). The amount
of lecture/exercise hours/ECTS is about the same as in the EU and the overall
number of credits are well comparable.
• In Georgia, the amount of engineering mathematics varies from 35 to 10 ECTS
in engineering BSc programs. The minimum 10 ECTS is low compared with
the comparable EU, Russian and Armenian degree programs. The coverage of
engineering mathematics courses is still very much the same as in EU. This
means that there is not as much time for teaching/studying as in other universities.
This may reflect negatively to students’ outcome competencies. In some cases
Georgian credits seem to be higher for the same amount of teaching hours.

10.4 Course Comparison

For course comparison, each partner university selected 1–3 courses, which were
compared with similar courses from the EU. In the SEFI classification, the selected
courses are the key courses in engineering education. They are taught mostly on BSc
and partly on MSc level. In general, the BSc level engineering mathematics courses
10 Overview of the Results and Recommendations 193

should cover contents described by SEFI in Core 0 and Core 1. Core 0 contains
essentially high-school mathematics, but it is not necessarily on a strong footing or
studied at all in schools in all the countries at a level of mastery. The topical areas
of Core 1 may vary, depending on the engineering field. Engineering mathematics
curriculum may contain elective mathematics courses described in SEFI Core 2 or
Core 3. Depending on engineering curricula, these courses can be studied at the BSc
or MSc level.
Courses on the following topical areas were selected for comparison between the
EU and Russia:
• Engineering Mathematics, Mathematical Analysis
• Discrete Mathematics, Algorithm Mathematics
• Algebra and Geometry
• Probability Theory and Statistics
• Optimization
• Mathematical Modeling
Courses on the following topical areas from the Georgian and Armenian
universities were compared with EU universities:
• Engineering Mathematics, Mathematical Analysis
• Calculus
• Discrete Mathematics, Algorithm Mathematics
• Linear Algebra and Geometry
• Probability Theory and Statistics
• Mathematical Modeling
As mathematics is a universal language, the contents of the courses were always
in the SEFI core content areas. The course comparison shows that the contents of
the courses are comparable. Sometimes a direct comparison between courses was
not possible because the topics were divided in the other university between two
courses and thus single courses were not directly comparable.
In most of the courses the didactics was traditional and course delivery was
carried out in the same spirit. The teacher gives weekly lectures and assignments
related with the lectures to the students. The students try to solve the assignments
before or in the tutorials or exercise classes.
Students’ skills are assessed in exams. The typical assessment procedure may
contain midterms exams and a final exam or just a final exam. In the exams students
solve examination problems with pen and paper. The teacher reviews the exam
papers and gives students their grades. Sometimes student’s success in solving
assignments or their activity during class hours was taken into account. In some
Georgian universities there were tendencies to use multiple choice questions in the
exam.
194 S. Sosnovsky et al.

The use of technology to support learning was mainly at a developing stage,
and the way it was used depended very much on the teacher. In some universities
learning platforms or learning management systems like LMS Blackboard or
Moodle was used. Mathematical tool programs (MATLAB, R, Scilab, Geogebra)
were known and their use rested much on the teacher’s activity.

10.5 Results and Recommendations

10.5.1 Course Development

The contents of the engineering mathematics courses is very much the same in
the EU and Russian and Caucasian universities. However, in the EU engineering
mathematics is more applied. In other words Russian and Caucasian students
spend more time learning theorems and proofs, whereas European students study
mathematics more as an engineering tool. We recommend changing slightly the
syllabus and instruction from “theorem-to proof” style by putting more emphasis
on applications. Topics, applications, examples, related to engineering disciplines,
should be added to improve engineering student’s motivation to study mathematics.
Traditionally mathematics has been assessed by pen and paper types examina-
tions. Students’ assessment could be enhanced so that it covers new ways of learning
(project works, essays, peer assessment, epistemic evaluation etc.). Multiply choice
questions may be used to give feedback during the courses, but replacing final exams
by multiple choice questions cannot be recommended.

10.5.2 Use of Technology

Mathematical tool programs (Sage, Mathematica, Matlab, Scilab, R, Geogebra etc.)
are common in EU in teaching and demonstrating how mathematics is put into
practice. These programs are known in Russia, Georgia and Armenia, but their
use could be enhanced to solve modeling problems from small to large scale. The
use of e-Learning (e.g. Moodle for delivery and communication, Math-Bridge as
an intelligent platform for e-learning), could be increased in the future to support
students’ independent work and continuous formative assessment.
10 Overview of the Results and Recommendations 195

10.5.3 Bridging Courses

In the EU, the practices for bridging/remedial courses have been actively devel-
oping in the last several decades. With the shift to Unified State Exam and the
abolishment of preparatory courses for school abiturients, Russian universities lack
the mechanisms to prepare upcoming students to the requirements of university-
level math courses. There is also a lack of established practices for bridging
courses in the Georgian and Armenian universities to prepare upcoming students
to the requirements of university-level math courses. With the shift to Standardized
SAT Tests, this becomes problematic, as many students enroll in engineering
studies without even a real math test, hence with incorrect expectations and low
competencies. Moreover, the needs differ from one student to the next and bridging
courses have to be individualized and adapted to specific purposes.

10.5.4 Pretest

Many universities have level tests on Core 0 level for enrolling students to gain an
understanding of the mathematical skills new students have and do not have. This
makes it possible to detect the weakest students and their needs, providing them
further specific support from the beginning of their studies. Math-Bridge system
might be a valuable tool here.

10.5.5 Quality Assurance

Quality assurance is an important part of studies and development of education in
the EU. Student feedback from courses should be collected and analyzed, as well as
acceptance rates, distribution of course grades, and the use of resources in Russian
and Caucasian universities.
A necessary, but not sufficient, principle to guarantee the quality of mathematics
education is that mathematics is taught by professional mathematicians. This is one
of the cornerstones which, in addition to mathematics being an international science,
make degrees and courses in mathematics comparable all over the world.
196 S. Sosnovsky et al.

Reference

1. SEFI (2013), A Framework for Mathematics Curricula in Engineering Education. (Eds.) Alpers,
B., (Assoc. Eds) Demlova M., Fant C-H., Gustafsson T., Lawson D., Mustoe L., Olsson-
Lehtonen B., Robinson C., Velichova D. (http://www.sefi.be).