Case Studies of Math Education for STEM in Armenia

Abstract

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.

9.1 Analysis of Mathematical Courses in ASPU

• Lusine Ghulghazaryan &
• Gurgen Asatryan 

1. Faculty of Mathematics, Physics and Informatics, Armenian State Pedagogical University (ASPU), Yerevan, Armenia

   Lusine Ghulghazaryan & Gurgen Asatryan

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 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.

Table 9.1 Outlines of LA&AG (ASPU) and EM2 (TUT) courses

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.

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

Table 9.3 Core 0-level outcome competencies of LA&AG (ASPU) and EM2 (TUT) courses

Table 9.4 Core 1-level outcome competencies of LA&AG (ASPU) and EM2 (TUT) courses

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.

9.1.3 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.

Table 9.5 Outlines of Calculus 1 (ASPU) and EM1 (TUT) courses

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.

• 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

Table 9.7 Core 1-level outcome competencies of “Calculus 1” (ASPU) and EM1 (TUT) courses

Table 9.8 Core 1-level outcome competencies of “Calculus 1” (ASPU) and EM1 (TUT) courses

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.

9.2 Analysis of Mathematical Courses in NPUA

• Ishkhan Hovhannisyan &
• Armenak Babayan 

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

   Ishkhan Hovhannisyan & Armenak Babayan

9.2.1 National Polytechnic University of Armenia (NPUA)

National Polytechnic University of Armenia (NPUA) is the legal successor of Yerevan 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 technologies 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.

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

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 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

Table 9.11 Core 0-level outcome competencies for MA-1 (NPUA) and EM1 (TUT) courses

Table 9.12 Core 1-level outcome competencies for MA-1 (NPUA) and EM1 (TUT) courses

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 corresponding 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 disciplines;

• 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.

9.2.3 Comparative Analysis of “Probability and Mathematical Statistics”

This course is one of the courses in the program “Informatics and Applied Mathematics”. 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.

Table 9.13 Outlines of the “Probability and Mathematical Statistics” (NPUA) and “Probability Calculus” (TUT) courses

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 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 Statistics” 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 prerequisite competencies of the “Probability and Mathematical Statistics” (NPUA) and “Probability Calculus” (TUT) courses

Table 9.15 Core 1-level prerequisite competencies of the “Probability and Mathematical Statistics” (NPUA) and “Probability Calculus” (TUT) courses

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

Table 9.17 Core 1-level outcome competencies of the “Probability and Mathematical Statistics” (NPUA) and “Probability Calculus” (TUT) courses

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 disciplines;

• 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.