development of mathematical competencies in higher ... · aimed at creating northern lithuania...

Project: “Cross-border network for adapting mathematical competences in the socio-economical development” (MATNET) LLIII-122

DEVELOPMENT OF MATHEMATICAL COMPETENCIES

IN HIGHER EDUCATION INSTITUTIONS

WITHIN SOCIO-ECONOMICAL CONTEXT

Methodical recommendations for lecturers

Sigitas Balčiūnas, Renata Macaitienė, Eglė Virgailaitė-Mečkauskaitė Siauliai University

Anna Vintere, Anda Zeidmane, Nauris Pauliņš Latvia University of Agriculture

Latvia–Lithuania, 2011

Project: “Cross-border network for adapting mathematical competences in

the socio-economical development” (MATNET) LLIII-122 Reviewers:

Assoc. Prof. Dr. Darius Šiaučiūnas, SU, Department of Mathematics, Assoc. Prof. Dr. Artūras Blinstrubas, ŠU, Social Research Centre, Assoc. Prof. Dr. Natālija Sergejeva, LUA, Department of Mathematics.

ISBN 978-609-430-104-9

© Sigitas Balčiūnas, 2011 © Renata Macaitienė, 2011 © Eglė Virgailaitė-Mečkauskaitė, 2011 © Anna Vintere, 2011 © Anda Zeidmane, 2011 © Nauris Pauliņš, 2011 © Siauliai University, 2011 © Latvia University of Agriculture, 2011 © Publishing House of Šiauliai University, 2011

„Cross-border network for adapting mathematical competences in the socio-economical development“

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Content PREFACE...................................................................................................................................6 INTRODUCTION......................................................................................................................7

Project information .............................................................................................................7 Problem of Research...........................................................................................................8 Research Focus ...................................................................................................................8 Main objective of Research ................................................................................................9 Structure of the research.....................................................................................................9

I. THEORETICAL BACKGROUND OF RESEARCH ..........................................................10 1.1. NEW CHALLENGES AND OPPORTUNITIES TO TEACH MATHEMATICS IN HIGHER EDUCATION TO CONTRIBUTE TO THE SOCIO-ECONOMIC DEVELOPMENT IN THE BORDER REGION .................................................................10

1.1.1. Background rationales ............................................................................................10 1.1.2. Mathematical competence and competencies.........................................................13

1.2. THEORETICAL FRAMEWORK OF EXTERNAL RESEARCH ..............................15 1.3.THEORETICAL FRAMEWORK OF INTERNAL RESEARCH.................................17

1.3.1. Learning outcomes and competencies....................................................................17 1.3.2. Module system........................................................................................................19 1.3.3. Collaborative learning in mathematics ...................................................................20 1.3.4. ICT usage in study process.....................................................................................23

II. METHODOLOGICAL BACKGROUND OF RESEARCH...............................................26 2.1. THE DESIGN OF THE RESEARCH...........................................................................26 2.2. METHODOLOGY OF EXTERNAL RESEARCH......................................................27

2.2.1. The Instrument of the Research..............................................................................28 2.2.2. The Sample of the Research ...................................................................................29 2.2.3. Organising of the Research ....................................................................................29 2.2.4. The Analysis of the Research Results ....................................................................30

2.3. METHODOLOGY OF INTERNAL RESEARCH .......................................................32 2.3.1. The Instruments of the Research ............................................................................32 2.3.2. The Analysis of the Research Results ....................................................................34 2.3.3. Instruments for comparison of study program .......................................................35

III. THE RESULTS OF EXTERNAL RESEARCH................................................................37 3.1. THE RESULTS OF THE OPINIONS OF THE POPULATION OF ŠIAULIAI REGION ...............................................................................................................................37

3.1.1. The Research Sample Characteristics.....................................................................37 3.1.2. Teaching of Mathematics and Attitudes towards Mathematics..............................37 3.1.3. Mathematics in the Area of Professional Activities ...............................................41 3.1.4. Directions of Improvement of Teaching of Mathematics. Respondents’ Opinion43

3.2. THE RESULTS OF OPINIONS OF ZEMGALE REGION POPULATION...............45 3.2.1. Analysis of the strategic planning documents ........................................................45 3.2.2. The Research Sample Characteristics.....................................................................48 3.2.3. Self-assessment of mathematical competences ......................................................49 3.2.4. Conformity of mathematics at higher education school to a student’s needs ........51 3.2.5. Mathematics in professional activities ...................................................................52 3.2.6. Assessment of the practical potential of mathematics............................................54 3.2.7. A need for the improvement of mathematical knowledge .....................................55

3.3. COMPARATIVE ANALYSIS OF THE OPINIONS OF LITHUANIAN AND LATVIAN POPULATION ABOUT MATHEMATICS .....................................................56 3.4. GENERALISATION AND RECOMMENDATIONS .................................................58

IV. THE RESULTS OF INTERNAL RESEARCH.................................................................60


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4.1. COMPARISON OF SIMILAR STUDY PROGRAMS ................................................60 4.1.1. Agricultural Mechanisation (LUA) and Mechanical Engineering (ŠU) ................60 4.1.2. Computer Control and Computer Science (LUA) and Informatics Engineering (ŠU) ..................................................................................................................................66 4.1.3. Environmental Science (LUA) and Ecology and Environmental Sciences (ŠU)...72 4.1.4. Organisations and Public Administration (LUA) and Public Administration (ŠU)..........................................................................................................................................78 4.1.5. General conclusions in LUA and SU study programmes.......................................83

4.2. ANALYSIS OF MATH COURSES IN ŠU (Internal + External results)....................85 4.2.1. Ecology and Environmental Sciences (1 subject) ..................................................85 4.2.2. Electrical Engineering (4 subjects).........................................................................88 4.2.3. Electronical Engineering (4 subjects).....................................................................91 4.2.4. Informatics Engineering (4 subjects)......................................................................95 4.2.5. Mechanical Engineering (4 subjects) .....................................................................99 4.2.6. Physics (5 subjects) ..............................................................................................102 4.2.7. Public Administration, Busines Aministration (1 subject)...................................105

4.3. ANALYSIS OF MATH COURSES IN LUA (Internal + External results)................110 4.3.1. Agricultural Engineering ......................................................................................111 4.3.2. Machine Design and Manufacturing ....................................................................115 4.3.3. Agricultural Power Engineering...........................................................................119 4.3.4. Computer Control and Computer Science............................................................123 4.3.5. Programming ........................................................................................................127 4.3.6. Civil Engineering..................................................................................................130 4.3.7. Landscape Architecture ........................................................................................134 4.3.8. Enironmental Science ...........................................................................................138 4.3.9. Landscape Architecture and Planning ..................................................................142 4.3.10. Food Science ......................................................................................................145 4.3.11. Food Technology................................................................................................149 4.3.12. Catering and Hotel Management........................................................................153 4.3.13. Wood Processing ................................................................................................156 4.3.14. Forest Engineering..............................................................................................160 4.3.15. Forestry Science .................................................................................................164 4.3.16. General conclusions for LUA mathematics study programs readjusting ...........168

References ..............................................................................................................................170 ANNEX No.1. ........................................................................................................................171

1.1. THE QUESTIONNAIRE FOR DIRECTORS OF STUDY PROGRAMS, DEPARTMENT CHAIR AND ACADEMIC PERSONNEL........................................171 1.2. QUESTIONNAIRE FOR EXTERNAL RESEARCH ............................................174

ANNEX No.2. ........................................................................................................................178 COMPARISON THE NUMBER OF HOURS OF THEORY, PRACTICAL AND INDEPENDENT WORK ASSIGNED FOR THE TOPICS ..........................................178 Table 2.1. Mathematics programs for Agricultural Mechanisation (LUA) and Mechanical Engineering (ŠU) (content and volume) .....................................................178 Table 2.2. Mathematics programs for Computer Control and Computer Science (LUA) and Informatics Engineering (ŠU) (content and volume) ..............................................182 Table 2.3. Mathematics programs for Environmental Science (LUA) and Ecology and Environmental Sciences (ŠU) (content and volume) .....................................................185 Table 2.4. Mathematics programs for Sociology of Organisations and Public Administration (LUA) and Public Administration (ŠU) (content and volume).............187

ANNEX No.3. ........................................................................................................................189 ANALYSIS OF CURRENT CONTENT OF MATHEMATICS SUBJECTS IN SU AND ITS EVALUATION (ASSESSMENT) ..........................................................................189


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Table 3.1. Ecology and Environmental Sciences ...........................................................189 Table 3.2. Electrical Engineering ...................................................................................190 Table 3.3. Electronical Engineering ...............................................................................194 Table 3.4. Informatics Engineering ................................................................................198 Table 3.5. Mechanical Engineering................................................................................202 Table 3.6. Physics...........................................................................................................205 Table 3.7. Public Administration, Busines Administration............................................208

ANNEX No.4. ........................................................................................................................210 COPY OF THE CONTENT OF RECOMMENDED BOOK FOR PHYSICISTS........210

ANNEX No.5. ........................................................................................................................214 COPY OF THE CONTENT OF RECOMMENDED THEMES FOR ELECTRICAL ENGINEERING .............................................................................................................214


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PREFACE

It is our pleasure to present the research report “Development of mathematical

competencies in higher education institutions within socio-economical context“. The research

was prepared in the framework of Latvia-Lithuania Cross Border Cooperation 2007-2013

Programme project “Cross-border network for adapting mathematical competences in the

socio-economical development” (MatNet). Researchers from two universities (Latvia

University of Agriculture and Siauliai University) working together has shared their research

and practical experience in the field of mathematic competence development in the border

region. The main objective of the research was to identify the needs of labour market for

socio-economic development and integration of professional competences of mathematics in

border regions, and evaluate mathematics subjects’ programs in SU and LUA compare them

and prepare recommendations and networking/collaboration strategy plan for those

improvements. The first part of the research consists on demands of labour market,

employees, employers, analysis of regional strategic plans and other national documents. The

second part consists on inner evaluation of the programs in two regional universities,

analyzing SU and LUA mathematic study programs.

We hope that the research problems addressed in this report and their solutions will

interest not only teachers, lecturers and researchers who work with mathematic subjects, but

also practitioners, who use mathematic knowledge in their work environment improving the

social reality and contribute to the socio-economical development of the region.


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INTRODUCTION

Rapid technological development is changing society and its attitudes towards education. This process is caused urgent needs to change the education environment. The Mathematics is very much touched by this process. There is a gap between Mathematics offered by universities and Mathematics needed to educate new specialist. The following challenging questions about teaching Mathematics are being arises: What is Mathematics? Why is it an essential learning area? What is the purpose of teaching Mathematics? What do we want students to understand? What do we want students to do with their understanding? What is the purpose of teaching? What are the goals for the students? What are the goals of the students? (Mustoe, 2004).

The main reason for the work of both universities Latvian University of Agriculture (LUA) and the Šiauliai University (ŠU) is to make the studies and research more pragmatic and applicable for today’s life (Kačinskaite, Vintere, 2009). Thus the cross-border cooperation between these universities between has been developed. The Latvian-Lithuanian cross-border cooperation program‘s project „Cross-border network for adapting mathematical competences in the socio-economical development” is a joint project of the LUA and the ŠU aimed at creating Northern Lithuania (Šiauliai county) and Southern Latvia (Zemgale) cross-border cooperation network, innovative educational products, new initiatives and strategies. ŠU and LUA are regional universities that actively take part in the regional development and provide intellectual services and products for the regions. Therefore, the goals of their activities are coincident and oriented to the needs of their regions.

Project information The main objective of this cooperation is to contribute to socio-economical

development in preparing high quality competitive specialists for the labour market in the field of mathematics: to proper conditions for border region specialists to develop mathematical skills; to prepare specialists who will be able to apply ICT and integrate mathematic competences for problem solving, data analysis; to raise awareness, competitiveness and qualification for users, beneficiaries and stakeholders as well as to improve the quality and enhance the accessibility of mathematical competencies across border by creating, testing and integrating innovative ICT based educational products.

Seeking to identify the needs of the labour market and integration of professional competences of mathematics in border regions, research are being carried out on the external demands of the labour market and employers who represent the need of qualified specialists with mathematic knowledge and skills. Research on internal evaluation consists of the analysis of ŠU and LUA mathematics study programmes It reveals to which extent the existing study programs correspond to the needs of the regional labour market and prepare recommendations and networking/ collaboration strategy plan for those improvements.

To improve the quality and enhance the accessibility of mathematical competencies across border, innovative ICT based educational products (three types of preparatory courses for students from high schools, first year students) have been created, tested and integrated, which contribute to the needs of the labour market and involve application of mathematics.

To raise awareness, competitiveness, qualification for users, beneficiaries and stakeholders involved in the cooperation network innovative, new-knowledge based perception of mathematical skills in their professional sectors are being provided by organizing training courses for small and medium enterprises in the regions.

The study process of ŠU and LUA are being modernized by readjusting, creating and testing mathematic study courses. The readjusted courses will be oriented to the needs and


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specifics of particular specialties for the purpose of practical knowledge application. Common trainings and constant consultations was organized for preparing ŠU and LUA lecturers to use Moodle as open source system for integrating new, interactive courses in the study process. LUA are implementing Moodle and videoconference systems in educational environment to modernize the study process and prepare more qualified specialist to the labour market.

In the frame of this cooperation, it is planned to encourage the motivation of the talented pupils and the best students that study mathematics for further carrier in this field and adaptation of knowledge and skills by gathering them together and setting the conditions for sharing scientific and cultural experiences: High School Pupil Scientific Mathematic Olympiad and International Student Scientific Mathematic Olympiad will be organised. As a stimulus for promoting fundamental sciences needed for sustainable development of Lithuania and Latvia, cooperation of school teachers and university lecturers will be enhanced to share the good experience and innovative teaching methods that are being used.

Problem of Research It is necessary to prepare specialists with abilities to use ICT, and transfer knowledge

and technology into the economy. Mathematics is a discipline which is required as a background for specialists who work in environmental protection, engineering, construction, business, telecommunication, textile, new energy sources, etc. EU directives distinguish 8 key competencies that should be developed for lifelong learning. One of them is mathematical literacy and competences in science and technology. Mathematical knowledge and competences become essential in the lifelong learning process. Mathematics as a discipline has been taught in schools, colleges, vocational training and universities. However, regional institutions have difficulties in preparation of qualified and competitive specialists of mathematics for the main economic sectors across borders. It has to be acknowledged that the quality of mathematics study process is declining and the quality level of student preparation is getting worse, thus diminishing students’ motivation and achievements of the results. According to statistical data, the highest student dropout rates are in the exact sciences: mathematics (25%), physics, technical sciences, chemistry, and only then in the humanities.

What is the problem? Mathematics must appear understandable and relevant and be of practical use in the adults’ living world. However, the subject of Mathematics is often represented as a long succession of facts to be memorized and reproduced. Many research findings show that the Mathematics in schools and universities of today does not necessarily provide a sound and unassailable foundation. The “school mathematics” and the mathematics that actually used or needed in a range of life situations are not related.

Research Focus The challenge is to find synthesis all view points and reject the attitude towards

mathematics. The mathematics must to be integrated into the world it describes. In order to achieve all goals set before and integrate the mathematics connections with daily world in mathematics syllabus the existing study programs have to be estimated and the needs of labour market to be determined. To start this process there is a need for dialogue between mathematicians and all participants of study process (directors of study programs, department chair, academic personnel, and social partners). Thus focus of this study is the interviews with the directors of study programs, chairs of departments and academic personnel as well as employer and the employee survey and its results.


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Main objective of Research To identify the needs of labour market for socio-economic development and

integration of professional competences of mathematics in border regions, and evaluate bachelor, master study programs in ŠU and LUA compare them and prepare recommendations and networking/ collaboration strategy plan for those improvements.

Structure of the research

The research manuscript consists of an introduction and four section: Theoretical background of research, Methodological background of research, Results of external research and Results of internal research. Each section is divided into chapters, which in turn contains some sub-chapter.

The first section describes new challenges and opportunities to the teaching mathematics in higher education institutions to contribute to the socio-economic development in the border region as well as given the theoretical background for external and internal research. The second section contains the research design and methodology of external and internal research.

The third section contains four chapters: The results of the opinions of the population of Šiauliai region and of Zemgale region; Comparative analysis of the opinions of lithuanian and latvian population about mathematics and Generalisation and recommendations.

Fourth section describes the results of internal research which consists of comparison of similar study programs as well of analysis of mathematics courses in Siauliai University and Latvia University of Agriculture.

Tables and figures are double numbering: the first number indicates the chapter number but the second - a table or a figure sequence number in the relevant chapter.

The research manuscript is accompanied by four annexes. The first annex contains the Questionnaire for directors of study programs, department chair and academic personnel and Questionnaire for external research. Annex No.2. contains the tables of Comparison the number of hours of theory, practical and independent work assigned for the topics; annex No.3. – seven tables with Analysis of current content of mathematics subjects and its evaluation (assesement) in Siauliai university; annex No.4. - Copy of the content of recommended book for physicists.


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I. THEORETICAL BACKGROUND OF RESEARCH

1.1. NEW CHALLENGES AND OPPORTUNITIES TO TEACH MATHEMATICS IN HIGHER EDUCATION TO CONTRIBUTE TO THE

SOCIO-ECONOMIC DEVELOPMENT IN THE BORDER REGION

1.1.1. Background rationales

In this new era of knowledge and creative society, there is a need and importance to prepare intellectual professionals and specialists for the labour market. Because of changes in the social, technological, educational and other environments, the information, knowledge and skills gets older very quickly, so there is constant need for life long learning and permanent renewing of acquired skills. Higher education institutions have new approaches toward teaching and learning paradigms, which has nothing to do with organised and system based education. Learning becomes a process, in which people develop their knowledge, understanding, skills, values, attitudes and experience. These ideas are connected with different learning environments, which have to supply various effective and interactive learning means for the students, which could be used in the education process.

It is obvious, that changes which are seen in society requires changes in higher education institutions too. Then raises the questions: what technological and educational innovative conditions would be necessary in the process of change? What educational environments it is necessary to create in order to motivate the students successfully and effectively act in the professional sphere? Teacher oriented lectures, with clear content and static information is no more effective way to transfer the knowledge. Europe 2020 puts forward three mutually reinforcing priorities: 1) Smart growth: developing an economy based on knowledge and innovation; 2) Sustainable growth: promoting a more resource efficient, greener and more competitive economy; 3) Inclusive growth: fostering a high-employment economy delivering social and territorial cohesion.

The most important priority of Lifelong Learning programme is to increase the input in educational sphere by contributing to one of the priority initiatives in the „Europe 2020“1 - "an agenda for new skills and jobs" to modernise labour markets and empower people by developing their of skills throughout the lifecycle with a view to increase labour participation and better match labour supply and demand, including through labour mobility.

EU directives distinguish 8 key competencies, which should be developed for lifelong Learning, one of them is for mathematical literacy and competences in science and technology. Mathematics knowledge and competences becomes essential in lifelong learning process. Analysing Lithuanian and Latvian socio-economic situation, it is also obvious, that mathematic competences required in the labour market are not developed enough in the region.

There are several problems and obstacles that influence slow socio-economic development of the Northern Lithuania and Southern Latvia regions:

1 Komisijos komunikatas „2020 m. Europa. Pažangaus, tvaraus ir integracinio augimo strategija“. COM(2010) 2020 (http://ec.europa.eu/eu2020/pdf/1_LT_ACT_part1_v1.pdf) ir Europos Vadovų Tarybos išvados, 2010 m. kovo 25–26 d., EUCO 7/10, CONCL 1: „Europa 2020“ – nauja Europos darbo vietų kūrimo irekonomikos augimo strategija (http://www.consilium.europa.eu/uedocs/cms_data/docs/pressdata/LT/ec/113617.pdf).


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1) according to the strategic planning documents (Siauliai Region Development 2007-2013 plan, Zemgale Region Development Programme 2008 to 2014) there is insufficient transfer of knowledge and technology to enterprises;

2) regional universities (SU, LUA) use traditional teaching/learning methods which are insufficient according to demands of labour market and knowledge society to develop mathematic competences required in technological age;

3) investments in human resource development of institutions which provide intellectual services and products for the regions for various reasons is insufficient. So it is necessary to prepare specialists with abilities to use ICT and transfer knowledge and technology to the economy.

Mathematics is a discipline, which is a background for specialists who works in environmental protection, engineering, construction, business, telecommunication, textile, new energy sources, ect. in the regions. EU directives distinguish 8 key competencies should be developed for lifelong learning, one of them is for mathematical literacy and competences in science and technology. The definition of “mathematical competence” is based on the ability to solve problems in everyday contexts, and places emphasis on aspects of the process and the habit of using models of thinking (logical and spatial) and presentation (formulas, constructs, graphs, charts, etc.). It consists in the ability to identify structures and connections, repetitions and systematicity. Moreover, a positive attitude in mathematics is based on the respect of truth and willingness to looks for reasons and so assess their validity (European Recommendation, 2006)2.

There are several definitions defining what Mathematics is. Mathematics is a multi-faceted subject, and a unique construction of human thought. Despite its high degree of abstraction, the subject has many deep and living connections with our daily world, in both simple everyday events and advanced scientific matters. Mathematics is both a basic scientific discipline with” a life of its own” and a powerful tool that is applied in numerous other disciplines. Increasingly mathematical models are being applied in dealing with both economic and social conditions. Mathematical models are embedded in technical and social artefacts and are thus generally invisible to ordinary people. Mathematics is also a domain for a particular kind of aesthetic experience, it provides moments of clarity and beautiful patterns that can create highly euphoric feelings of unexpected insight and overall understanding3

Mathematics could be explained by following aphorisms. No-one could tell anything against such truths or theses: art is any(-) thing what looks (and is) nice, and new and attractive; art is any(-) thing what is difficult to do; art is any(-) thing that stimulates our creativity, challenges our fantasy, widens our views; art is any(-)thing that is highly unusual and not standard. Replacing the word “art” by the word “Mathematics” and it is seems that all these statements are true just as they were in these sentences4. The substance of mathematics for universities is determined by the SEFI5:

1. Mathematics as a subject sees mathematics as part of the degree programme, to be studied via various teaching and learning techniques;

2 “Key competences for lifelong learning”. European Recommendation 2006-2006/962/EC. (http://www.indire.it/lucabas/lookmyweb_2_file/etwinning/eTwinning-pubblicazioni/e_twinning_volume_01ing.pdf) 3 Gustafsson, L., Ouwitz, L. (2004) Adults and Mathematics – a vital subject. ISSN 1650 -335X, NCM, 2004. 4 Kašuba, R. (2006) About so-called democratic problems proposed at international mathematical olympiads. Teaching Mathematics: Retrospective and perspectives, Proceedings of the 7thInternational conference, tartu, University of Tartu, May 12-13, 2006., 96 p. 5 Booth, S. (2004) Learning and teaching for undestanding Mathematics. 12th SEFI Maths Working Group Seminar, Proceedings, Vienna University of echnology, 2004, pp 12-25.


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2. Mathematics as the basis of other subjects, both for study and in world at large, see mathematics something existing in its own right, something to be tackled (learned and understood) for future appropriate use;

3. Mathematics as a tool for analysing problems that occur in the world at large and hence solving them. Sees mathematics as something which co-exists with other areas of knowledge and supports the study and development of that knowledge. Lithuanian authors such as Morkūnienė (2010), Dudaitė (2007), Kaminskienė,

Rimkuvienė, Laurinavičius (2010) and Latvian researchers Garleja, Kangro (2005, 2007, 2011), Ģingulis (2009, 2011)) analyse mathematic competences and didactics problems in Lithuanian higher education institutions. The innovative processes, which have started were connected with the internal condition - low student motivation to study mathematics and external conditions, connected with the change of the study regulation documents and changed socio-economical situation, which requires practical knowledge, which could be easily adapted in the labour market. It is obvious that mathematic knowledge and competences have a great input in everyday life and in the workplace.

Mathematics knowledge and competences becomes essential in lifelong learning process. What is the problem? Mathematics as discipline is been taught in schools, colleges, vocational training and universities. However, regional institutions have difficulties in preparation of qualified and competitive specialists of mathematics for the main economic sectors across borders. It has to be acknowledged that the quality of mathematics study process is declining, and preparation quality level of students is getting worse. Also diminishing students’ motivation and achievements of results. For instance, number of first year students entering Siauliai University in mathematics has declined three and a half times comparing 2005 and 2010 study years. According to the data of Statistical Department, the highest students drop out rates is in the exact sciences. In four years of studies only half of students graduate from Physics and Technology Sciences. The statistical information of drop out from Siauliai university show, that biggest drop out rates is visible in the exact sciences study programs (25 percent): mathematics, physics, technical sciences, chemistry, then – humanities sciences. Only by implementing cooperated activities it will be possible to develop holistic cross-border cooperation in Northern Lithuania and Southern Latvia. Cooperation network, educational products, new initiatives and strategies raising the awareness and creating new knowledge and education methods for mathematic study programs, will contribute to the socio-economic development and increase competitiveness in the regional level.

Mathematics must appear understandable and relevant and be of practical use in the adults’ living world. However, the subject of Mathematics is often represented as a long succession of facts to be memorised and reproduced. Many research findings show that the Mathematics in schools and universities of today does not necessarily provide a sound and unassailable foundation. The “school mathematics” and the mathematics that actually used or needed in a range of life situations are not related.

There are most important factors that affect the development of Mathematics’ education in universities: the demand for highly qualified employees; the fixed volume of the study content of mathematic studies has not changed for a few years in comparison with the time that, on the contrary has been reduced, students with poor knowledge of natural sciences and poorly developed exact– technical thinking, reasoning and world perception abilities. Some more factors are also the structure of study programs, their capability and the out-of-date study infrastructure. The essential aspect is the curriculum development. The problem is that the curriculum has evolved through addition not redesign.

The main assignment of mathematics of late years was reviewed to give knowledge for the best learning of technical subjects. But nowadays the students must be prepared for applications as well, but they will also need a basis of theoretical knowledge in their future work as well as to understand the available literature and to use it creatively.


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Basically all students spend only a year to acquire the basic course in mathematics. The questions stands as follows: how to organize the mathematical education in the first or second studies’ year in such a manner which would be appropriate at the same time both the learning technical subjects and for the working life?

The questions had been already answered by a Bologna documents6. The aim of the Bologna Process is the improvement of the mobility of students and teaching staff as well as the strengthening of the competition of the European universities in a global education market. As the consequence of the Bologna declaration is the promotion of European co-operation in higher education, particularly with regards to curricular development, inter-institutional co-operation, mobility schemes and integrated programs of study, training and research as well as exchange of experience between institutions and countries.

1.1.2. Mathematical competence and competencies

What does it mean to possess mathematical competence? According to Niss, 1999, 2003 to possess a competence (to be competent) in some domain of personal, Professional or social life is to master essential aspects of life in that domain. Mathematical competence then means 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. Necessary, but certainly not sufficient, prerequisites for mathematical competence are lots of factual knowledge and technical skills. According to Niss, 1999, there are eight competencies which can be said to form two groups. The first group of competencies are to do with the ability to ask and answer questions in and with mathematics: 1. Thinking mathematically (mastering mathematical modes of thought)

• posing questions that are characteristic of mathematics, and knowing the kinds of answers (not necessarily the answers themselves or how to obtain them) that mathematics may offer; • understanding and handling the scope and limitations of a given concept. • extending the scope of a concept by abstracting some of its properties; generalising results to larger classes of objects; • distinguishing between different kinds of mathematical statements (including conditioned assertions (‘if-then’), quantifier laden statements, assumptions, definitions, theorems, conjectures, cases):

2. Posing and solving mathematical problems

• identifying, posing, and specifying different kinds of mathematical problems – pure or applied; open-ended or closed; • solving different kinds of mathematical problems (pure or applied, open-ended or closed), whether posed by others or by oneself, and, if appropriate, in different ways.

3. Modelling mathematically (i.e. analysing and building models)

• analysing foundations and properties of existing models, including assessing their range and validity • decoding existing models, i.e. translating and interpreting model elements in terms of the ‘reality’ modelled • performing active modelling in a given context - structuring the field - mathematising - working with(in) the model, including solving the problems it gives rise to - validating the model, internally and externally - analysing and criticising the model, in itself and vis-à-vis possible alternatives - communicating about the model and its results - monitoring and controlling the entire modelling process.

6 EU, “Bolonga Declaration” (http://ec.europa.eu/education/policies/educ/bologna/bologna_en.html )


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4. Reasoning mathematically

• following and assessing chains of arguments, put forward by others • knowing what a mathematical proof is (not), ands how it differs from other kinds of mathematical reasoning, e.g. heuristics • uncovering the basic ideas in a given line of argument (especially a proof), including distinguishing main lines from details, ideas from technicalities; • devising formal and informal mathematical arguments, and transforming heuristic arguments to valid proofs, i.e. proving statements.

The other group of competencies are to do with the ability to deal with and manage mathematical language and tools: 5. Representing mathematical entities (objects and situations)

• understanding and utilising (decoding, interpreting, distinguishing between) different sorts of representations of mathematical objects, phenomena and situations; • understanding and utilising the relations between different representations of the same entity, including knowing about their relative strengths and limitations; • choosing and switching between representations.

6. Handling mathematical symbols and formalisms

• decoding and interpreting symbolic and formal mathematical language, and understanding its relations to natural language; • understanding the nature and rules of formal mathematical systems (both syntax and semantics); • translating from natural language to formal/symbolic language • handling and manipulating statements and expressions containing symbols and formulae.

7. Communicating in, with, and about mathematics

• understanding others’ written, visual or oral ‘texts’, in a variety of linguistic registers, about matters having a mathematical content; • expressing oneself, at different levels of theoretical and technical precision, in oral, visual or written form, about such matters.

8. Making use of aids and tools (IT included)

• knowing the existence and properties of various tools and aids for mathematical activity, and their range and limitations; • being able to reflectively use such aids and tools.

All these eight competencies are to do with mental or physical processes, activities,

and behaviours. In other words, the focus is on what individuals can do. This makes the competencies behavioural (not to mistake for behavioristic). The competencies are closely related - they form a continuum of overlapping clusters. All competencies have a dual nature, as they have an analytical and a productive aspect. The analytical aspect of a competency focuses on understanding, interpreting, examing, and assessing mathematical phenomena and processes, such as, for instance, following an controlling a chain of mathematical arguments or understanding the nature and use of some mathematical representation, whereas the productive aspect focuses on the active construction or carrying out of processes, such as inventing a chain of arguments or activating and employing some mathematical representation in a given situation.

Furthermore, although the competencies are formulated in terms that may apply to other subjects as well, these terms are here to be understood in a strict mathematical sense. Thus we are talking about mathematical representations, not representations in general. Similarly, we are talking about mathematical reasoning, including proof and proving, not about reasoning in general like in general logic or in a court room, and we are talking about mathematical symbols, not other kinds of symbols such as icons or chemical symbols, let alone religious or literary symbols. In other words the competencies are specific to mathematics.

A particularly important comment is to do with the relationship between the competencies and mathematical subject matter. A mathematical competency can only be developed and exercised in dealing with such subject matter.

The competencies can be used in different ways in mathematics education.


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Firstly, they can be employed for normative purposes, e.g. with respect to specification of a curriculum or of desired outcomes of student learning. In other words, they provide a tool for clarifying, in a non-circular way, how we want mathematical education to function.

Secondly, they can be used for descriptive purposes. More specifically, they can be used to describe and characterise actual teaching practice, what happens in classrooms, what is being pursued in testing and examinations, and the actual outcomes of students’ learning. They can also be used to compare different mathematics curricula and different kinds of mathematics education at different levels or in different places, and so forth.

Finally, by being explicit instruments of characterisation they can also be used as meta-cognitive support for teachers and students by assisting them to clarify, monitor and control their teaching and learning, respectively.

Another classification of mathematic competence was proposed by Wedege, 2000. There was created a Model of three levels of peoples‘ experience with mathematics7: The level of skills

Specific skills in arithmetics and mathematics which are a visible part of the work process

The level of understanding

General mathematics knowledge, e.g. understanding and dealing with the theory-practice relation in the working situation

The level of identity A judicious mixture of incorporated skills and understanding (mathematics thinking, tacit knowledge) and attitudes, feelings and motives.

In this model, qualifications are distributed on three levels of subjectivity: a basic

level, a comprehensive level and a specific level. There are also three different types of mathematics knowledge (mathematical, practical and reflective). Mathematical knowledge as such is embedded at the two levels of skills and understanding; practical (mathematics) knowledge is embedded at all three levels, while we find reflective (mathematics) knowledge at the level of understanding.

1.2. THEORETICAL FRAMEWORK OF EXTERNAL RESEARCH

Background rationales of the external research based on the need to understand and ability to use mathematics in everyday life and in the workplace has never been greater and will continue to increase. For example: • Mathematics for life. Knowing mathematics can be personally satisfying and empowering. The underpinnings of everyday life are increasingly mathematical and technological. For instance, making purchasing decisions, choosing insurance or health plans, and voting knowledgeably all call for quantitative sophistication. • Mathematics as a part of cultural heritage. Mathematics is one of the greatest cultural and intellectual achievements of human-kind, and citizens should develop an appreciation and understanding of that achievement, including its aesthetic and even recreational aspects. • Mathematics for the workplace. Just as the level of mathematics needed for intelligent citizenship has increased dramatically, so too has the level of mathematical thinking and problem solving needed in the workplace, in professional areas ranging from health care to graphic design. • Mathematics for the scientific and technical community. Although all careers require a foundation of mathematical knowledge, some are mathematics intensive. More students must pursue an educational path that will prepare them for lifelong work as mathematicians,

7 Tine Wedege, Mathematic knowledge as a vocational qualification, 2000, Bessot and Ridgway (eds.). Education for Mathematics in the Workplace, 127-136. Kluwer Academic Publishers. Printed in the Netherlands.


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statisticians, engineers, and scientists. (From the National Council of Teachers of Mathematics (NCTM)).

Therefore the theoretical basis of the external research is the researches carried out by Swedish researchers’. They findings show that in so called school mathematics an encounter takes place between the subject of mathematics and people’s attitudes, experiences, feelings and thoughts, which sometimes creates special complexes of problems in education. Attitudes toward mathematics can be twofold. On the one side mathematics is a domain for a particular kind of aesthetic experience, it provides moments of clarity and beautiful patterns that can create highly euphoric feelings of unexpected insight and overall understanding8. Mathematics is similar to the hidden land high in the mountain of mind and fantasy. All persons who happened to visit that land claim that it is impressive and at any rate worth seeing9.

Regrettably, many people’s experiences of mathematics are quite the reverse: they associate mathematics with feelings of failure, anxiety, humiliation, suspicion and disassociation. The experience of school mathematics thus becomes a life-inhibiting stigma even creates learning blockades.

The research made in Sweden shows that mathematics is to be found everywhere, but to the individual it appears to be almost nowhere, a situation usually referred to as the relevance paradox of mathematics. An adult who feels anxiety and suffers learning blockages when faced with this subject is therefore likely to conclude that the subject is meaningless; it neither improves understanding of the environment nor adds to practical knowledge.

By tradition, the subject of Mathematics has a high status. It is considered difficult to learn and yet, often without any detailed justification, it has a high value. Few people are indifferent to the subject - they either found it easy or have a good appreciation of its content, or they have feelings of anxiety, and possibly disassociation, which may be attributed to the failures and learning blockages from their school years. Therefore since the subject of Mathematics increasingly becomes an instrument for promoting broad all-round education and personal development, many education courses are mathematics-intensive, students need a high level of competence in the subject. Here, school mathematics has a specific role to play as a critical filter, a sorting instrument for admissions to many programmes in higher education. The knowledge and skills in mathematics of a student entering tertiary education is not easily predicted from the qualifications achieved prior to entry. Thus the support may well be needed throughout the first study year.

Describing learning as lifelong may be perceived as prolonging the period of youth. Furthermore, the society of tomorrow is often presented as if it existed today, which is in conflict with the idea that man creates his future through his own activities. Lifelong learning is also described as an individual life project, and one may ask if learning as a collective project does not have at least as much value. Knowledge is often described in general terms as ”a perishable item” that ages quickly, but is there not a certain type of knowledge, in mathematics for example, that is both vigorous and durable10.

8 Gustafsson, L., Ouwitz, L. (2004) Adults and Mathematics – a vital subject. ISSN 1650 -335X, NCM, 2004. 9 Kašuba, R. (2006) About so-called democratic problems proposed at international mathematical olympiads. Teaching Mathematics: Retrospective and perspectives, Proceedings of the 7thInternational conference, tartu, University of Tartu, May 12-13, 2006., 96 p. 10 Gustafsson, L., Ouwitz, L. (2004) Adults and Mathematics – a vital subject. ISSN 1650 -335X, NCM, 2004.


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1.3.THEORETICAL FRAMEWORK OF INTERNAL RESEARCH

The theoretical framework of the internal research is based on two pillars on curriculum development: learning outcomes and competencies, curriculum development in compliance with the credit – module system, as well as two pillars for study process organization: Collaborative learning in mathematics and information and communication technologies (ICT) usage in study process.

1.3.1. Learning outcomes and competencies The issue of qualifications has pushed the need for the Bologna Process participating

countries to develop qualifications framework, based on learning outcomes. If qualification is characterized by learning outcomes and competencies (in contrast with current policy - the acquired object descriptions), then it is probably much better to compare them between countries. Learning (study) results are formulated by students’ knowledge and insight, skills and ability to use at the end of particular study course and they are phrased by terms – knowledge, skills and competence.

Knowledge is described as theoretical or practical in European qualification framework - is a result of assimilation of information during learning/studying. Knowledge is the aggregate of facts, principles, theories and practice relevant to areas of work or studying/learning. Skills are described as cognitive (using logical, intuitive and creative thinking) or practical (including prestidigitation and methods, materials and tools) in European qualification framework. Skills are the application/selection/ascription and usage (know-how) of knowledge in order to fulfil practical and theoretical tasks. Competence is described in relationship with responsibility and autonomy in European qualification framework - competence is a proven capability to use knowledge and skills, including personal, social and/or methodological in situations of work and learning and personal or professional development11.

A competence is a quality, ability, capacity or skills that is developed by and that is belongs to the student, but a learning outcome is a measurable result of a learning experience which allows to ascertain to which extent/level/standard a competence has been formed or enhanced12. Learning outcomes are not properties unique to each student, but statements which allow higher education institutions to measure whether students have developed their competences to the required level. In the framework of the Tuning Process the competences are define as a dynamic combination of cognitive and metacognitive skills, demonstration of knowledge and understanding, interpersonal and practical skills, and ethical values. Fostering these is the object of all educational programmes.

Competences are developed in all course units and assessed at different stages of a programme. Some competences are subject-area related (specific to a field of studies), while others are generic (common to any degree programme). The competences differ in different degree programmes, even in academic or professional area simultaneously. The key programme competences should be the most important ones that the graduate will have achieved as a result of the specific programme. Developing the key competences is the main objective of any programme. Their achievement is verified through reference to learning outcomes.

11 http://www.nki-latvija.lv/wp-content/uploads/2011/10/Rauhvargers.pdf 12 Lokhoff, J., Wegewijs, B. Et al. (2010) A Tuning Guide to Formulating Degree Programme Profiles Including Programme Competences and programme Learning Outcomes. Bilbao, Groningen and The hague, 2010.


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As means for the purpose of development of professional competence in the process of mathematical studies can be used the development of mathematical thinking. Mathematical thinking can be defined as a fixed knowledge of mathematics and the process of cognitive activity for obtaining knowledge and applying them in practice13:

1) Mathematical thinking as the knowledge of mathematics: a link with sciences about the surrounding world – physics, chemistry, biology, economics, etc; a set of images representing definite objects – real object images, mental images;

2) Mathematical thinking as a tool of cognitive activity: actions with images; formation of knowledge (reasoning, problem-solving, decision making, creation of mental models); evaluation of the obtained knowledge, opinions, conviction, views and a practical application of the knowledge. Learning outcomes are a specification of the direct result and outcomes of a learning

process. Programmes should be designed and delivered in reference to overarching qualification framework of learning levels and qualifications. In order to design a programme in a context understandable to others, reference has to be made to general descriptors. Dublin Descriptors14 has been developed to outline the essential components of any degree programme. Bachelor’s degrees are awarded to students who:

a) have demonstrated knowledge and understanding in a field of study that builds upon and supersedes their general secondary education, and is typically at a level that, whilst supported by advanced textbooks, includes some aspects that will be informed by knowledge of the forefront of their field of study;

b) can apply their knowledge and understanding in a manner that indicates a professional approach to their work or vocation, and have competences typically demonstrated through devising and sustaining arguments and solving problems within their field of study;

c) have the ability to gather and interpret relevant data (usually within their field of study) to inform judgements that include reflection on relevant social, scientific or ethical issues;

d) can communicate information, ideas, problems and solutions to both specialist and non-specialist audiences;

e) have developed those learning skills that are necessary for them to continue to undertake further study with a high degree of autonomy. Within each area, discipline or professional sector descriptors (Dublin or other) could

be applied and adapted according to the specific way that learning is acquired in that sector. While there is a variety to describe the learning outcomes, each one contains five key components15: active verb form; indication of the type of learning outcomes (knowledge, cognitive processes, skills or other components); the topic area of learning outcomes (can be specific or general and refers to the subject matter, field of knowledge or a particular skill); indication of the standard or the level that is intended/achieved by the learning outcomes; scope and/or context of the learning outcomes. In the end of study course (module, practice, and program) the learning outcomes have been described using verbs from Bloom’s taxonomy with reference to cognitive levels16: •Evaluation - reasoning about quality (evaluation): defend, confirm regularities, conclude, substantiate, criticize..

13 Garleja, R., Kangro, I. (2011) Possibilities of Development of Students’ Mathematical Thinking in the Process of Building Their professional Competence. The 12thInternational conference Teaching Mathematics: Retrospective and perspectives, Šiauliai University, 5-6 May 2011, 27 p. 14 Descriptors. http://www.jointquality.org/ 15 Lokhoff, J., Wegewijs, B. Et al. (2010) A Tuning Guide to Formulating Degree Programme Profiles Including Programme Competences and programme Learning Outcomes. Bilbao, Groningen and The hague, 2010. 16 Reece I., Walker S. (1997) Teaching, Training and Learning - a Practical Guide// Third edition. GB. Sunderland: Business Education Publishers Ltd.


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•Synthesis - associate in common coherences: connect, improve, combine, project, mode, formulate.. •Analysis - logical dividing into parts: classify, divide, criticize, compare, polarize.. •Application - using new knowledge in new situations: control, foresee, solve, model, experiment, develop, analyse.. •Insight - sees insight in coherences: explain, polarize, illustrate, classify, interpret, reason, summarize... •Knowledge - can remember facts, recognize things: show, recollect, name, make registers, label, retell...

Several criteria for improvement of learning (study) results17 could be offered: •Student – oriented: results define what student gets (helps to coordinate things learned in formal or informal education, motivates for lifelong education); • System approach: results of higher education establishment strategy, results of study program etc., that follow the results of study program not exceeding them but not reducing them either; •Accuracy: results are formulated clearly and don’t overlap with results of different study courses; • Establishment: you can establish and evaluate the level of results achieved; •Interrelation: making a new quality, skills are based on knowledge; competence is based on skills and knowledge; •Cycle: the results are evaluated after a certain amount of time in study course; results are evaluated and corrected based on results of study program and coherent normative documents (standards, guidelines etc).

1.3.2. Module system In accordance with the requirements of the Bologna process, the mathematics syllabus

must be developed in compliance with the credit – module system. Modularisation is as known the summary of analogue fields to a thematic and temporal rounded in closed itself and with performance points provided in testable units18. ECTS (ETCS-Points = European Transfer Credit System-Points) give an average workload for the complete course of studies. It is necessary to develop the study modules. Term ″module″ in the Longman Dictionary of Contemporary English is explained as one of the units that a course of study has been divided into, each of which can be studied separately.

Study modules objective is to provide definite knowledge’s and competences which are realized integrating separately differentiated study modules. Study modules: contain certain information; are mutually coordinate (have entrance and exit); are coordinated with professional and didactic tasks. International practice shows that there are various approaches to forming modules19:

a) Modules can include various parts, united according to subject (lectures, exercises, practice);

17 http://www.nki-latvija.lv/wp-content/uploads/2011/10/Rauhvargers.pdf 18 Schlattmann J. (2007). Curriculum-Module Description on the Working with Projects in the Mechanical Design Development. Proceedings of SEFI and IGIP 2007 Joint Annual Conference, Miskolc, Hungary. Pp. 371 – 372 19 Artjukh S. (2007) Some Principles for Forming Modules in the Curriculum Development of the Bachelor’s Program of Electrical Engineering with the Scientific Direction Electric Power. Proceedings of SEFI and IGIP 2007 Joint Annual Conference, Miskolc, Hungary, 2007, pp. 291 – 292.


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b) Modules can include disciplines/courses of several semesters and have the exam after completing the module;

c) Every module requires knowledge, skills and abilities acquired in previous modules; d) Modules are defined by the number of credits and hours in semester week.

An optimal way - to form the modules for separate study subjects by elements all approaches. As the mathematics taught in various semesters the course of mathematics definitely should be designed according to study module principle20. For realizing separate subject syllabuses development the study process could be divided into study forms modules, for realizing cognitive development - the study process into study content modules. The content module must form knowledge in the concrete field. That’s why the course should be divided into separate themes, the sequence of which is appropriate to the logic of the given science. Because mathematic is the basic subject for many faculties, but every speciality have different amount of mathematics (credit points), it is necessary divide each study content module into levels. There could be pointed the advantages of the study process module method in comparison with other study process models are:

1. Systematic approach to course design and its contents; 2. Coordination of all study forms in every module and among modules; 3. Structure flexibility of study process; 4. Opportunity to find solution of the theme, to be able to see the essence and to be aware

of application of the results; 5. Effective assessment system.

1.3.3. Collaborative learning in mathematics

The tailoring of the structure of different university studies to the European Higher Education Area (EHEA), an objective forthcoming from the Bologna Declaration, is the major task outstanding in Europe’s different university systems. The reform affects not only the structure of university studies, but furthermore leads to European-wide reflection on the suitability of the syllabuses in mathematics subjects and, of greater importance, on the manner in which mathematics is taught. It is now no longer possible to uphold the same approach to teaching mathematics as fifty years ago and classrooms should reflect the technological revolution that has occurred in recent decades.

Taking into account this new frame in which we must develop our teaching activity, it is necessary to reconsider the role of mathematics. It is evident that mathematics will continue fulfilling a double objective in the new university frame. In one hand they will continue being a powerful formative tool and on the other hand they will continue being the support and background of other academic disciplines. Therefore a mathematical basic background will be essential in the new frame of acquisition of competencies.

A lot of analysed researches (Swan, 2005; Swain & Swan, 2007; Swan & Wall, 2007; Swan, 2008; Alonso, Rodriguez, Villa, 2007; Rodriguez, Villa, 2005) shows, that many students view mathematics as a series of unrelated procedures and techniques that have to be committed to memory. Instead, teachers want them to engage in discussing on explaining ideas, challenging and teaching one another, creating and solving each other’s questions and working collaboratively to share methods and results. The second aim is to develop more challenging, connected, collaborative orientations towards their teaching (Swan, 2005):

20 Zeidmane A., Vintere A. (2009) Method of Study Modules in Higher Mathematics Studies. Journal of the Korea Society of Mathematical Education Series D: Research in Mathematical Education, September 2009, Vol. 13 No. 3 (ISSUE 39), p.251-266


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A “Transmition” orientation A “Collaborative” orientation

Traditional, 'transmission' methods in which explanations, examples and exercises

dominate do not promote robust, transferrable learning that endures over time or that may be used in non-routine situations. They also demotivate students and undermine confidence. In contrast, the model of teaching that was adopted by Swan, 2006, emphasises the interconnected nature of the subject and it confronts common conceptual difficulties through discussion. Authors also reverse traditional practices by allowing students opportunities to tackle problems before offering them guidance and support. This encourages students to apply pre-existing knowledge and allows us to assess and then help them build on that knowledge. This approach has a thorough empirically tested research base (Swan, 2006). The main principles are summarised below. Teaching is more effective when it ... • builds on the knowledge students already have;

This means developing formative assessment techniques and adapting our teaching to accommodate individual learning needs (Black & Wiliam, 1998)

• exposes and discusses common misconceptions

Learning activities should exposing current thinking, create ‘tensions’ by confronting students with inconsistencies, and allow opportunities for resolution through discussion (Askew & Wiliam, 1995).

• uses higher-order questions

Questioning is more effective when it promotes explanation, application and synthesis rather than mere recall (Askew & Wiliam, 1995).

• uses cooperative small group work

Activities are more effective when they encourage critical, constructive discussion, rather than argumentation or uncritical acceptance (Mercer, 2000). Shared goals and group accountability are important (Askew & Wiliam, 1995).

encourages reasoning rather than ‘answer getting’

Often, students are more concerned with what they have ‘done’ than with what they have learned. It is better to aim for depth than for superficial ‘coverage’.

• uses rich, collaborative tasks

The tasks we use should be accessible, extendable, encourage decisionmaking, promote discussion, encourage creativity, encourage ‘what if’ and ‘what if not?’ questions (Ahmed, 1987).

• creates connections between topics

Students often find it difficult to generalise and transfer their learning to other topics and contexts. Related concepts (such as division, fraction and ratio) remain unconnected. Effective teachers build bridges between ideas (Askew et al., 1997).

• uses technology in appropriate ways

Computers and interactive whiteboards allow us to present concepts in visual dynamic and exciting ways that motivate students.

Mathematics is...

Learning is...

Teaching is...

An individual activity based on watching, listening and imitating until fluency is attained.

A collaborative activity in which learners are challenged and arrive at understanding through discussion.

Structuring a linear curriculum for learners. Giving explanations before problems. Checking that these have been understood through practice exercises. Correcting misunderstandings.

An interconnected body of ideas and reasoning processes.

Exploring meanings and connections through non-linear dialogue between teacher and learners. Presenting problems before offering explanations. Making misunderstandings explicit and learning from them.

A given body of knowledge and standard procedures that has to be ‘covered’.


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Acoording to different authors, there is need to move from an eminently face to face teaching to a mixed teaching, in which the student, under the supervision of the instructor, must acquire new knowledge across his/her own activity. With this panorama, it is necessary to highlight the importance of e-learning and b-learning techniques in the new academic configuration. In order to find a global understanding of this new paradigm of European university teaching means, we must first outline certain features of the European area that are currently being designed: · A competency-based teaching offer. Then we can plan and select the mathematical contents for specific students, engineers for example. · A diversified teaching offer (theoretical and practical teaching, supervised academic activities, independent individual work and so on). · Concern for the student’s overall work, as opposed to the current system in which the only measurement in certain European countries, is the lecture’s hours given in the classroom. This work should have a maximum annual volume estimated in 60 European credits (European Credit Transfer System, ECTS). An initial estimate suggests that one ECTS credit is equal to 25-30 hours of student work, including class attendance, laboratories, workshops, individual and group tutorials, individual or group work and assessments.

· Prevalence of student’s learning over the lectures provided by teachers. · Duration of studies that is more adapted to reality.

Before addressing the change in contents and the methodological change, we must ask ourselves again about the role to be played by mathematics in the training of various specialists (eg zample engineers, economists, etc.).

Figure 1.1. Complex phenomena of concern to the mathematics teacher (According to Davis and Simmt 2006).

The principles referred to above mark the route to be followed in University

teaching, particularly in the field of mathematics. Thus, mathematics should not be isolated from the rest of the sciences but should form an integrating system of several different systems. Only if this aim is achieved it will be possible to consolidate the mission of the University in society in general; that is, as an innovator of knowledge, thereby preventing


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scientific and technical stagnation. To achieve this objective the following guidelines should be followed:

1) The level of knowledge received through University teaching should allow students to adopt a critical and creative stance when they face with a given problem; it should offer them the necessary reasoning tools (skills) for the solution to such problems. A narrowness of view in the transmission of knowledge could become a severe disease in University teaching.

2) Learning the answer to a problem does not provide an intelligent idea of the process of solution since some step will always be overlooked and this will lead to the generation of several questions. The student identifies this step when reflecting on how the solution to the problem is arrived at. Thus, teaching should promote student’s criteria, should foster their ability to reason, and should lead them to be able to handle different theories and methods, motivating them to accept or reject them on the basis of such reasoning.

3) Scientific advances demand of students the capacity to recycle their knowledge as time progresses and at the same time demand a solid basic training. This should be an objective to be sought. This very dynamic scheme of continuing education means that students must have a sound training in their respective disciplines. Thus, computers may be of great help in this recycling process.

Finally, according to Alonso, Rodriguez, Villa (2007) there is needed to change the assessment process, seeking mechanisms that assess the whole learning process. New technologies are once again the key instrument, allowing the introduction of well-designed self-assessment processes, guided practical sessions, etc. To implement all this methodological changes it becomes indispensable to produce new didactic material to be used by the students in different situations of learning. New material is not reduced to upload the traditional materials and nice power-point presentations on the net. We have to include electronic material, to promote the use of Computer Algebra Systems, on line tutorials, etc. This material will be used according the student’s personal needs, it must enhance the mathematical competences and to promote the collaborative work.

1.3.4. ICT usage in study process

The other pillar is the implementation and use of ICT in mathematics studies. Hawkridge et al (1990) summarizes these reasons why use of ICT should be implemented within education into six arguments21: •Social argument: prepare young people for the future where knowledge of technology and basic computer skills will be needed. That means everybody should take a course in ICT; •Professional argument is to develop a database of people who acquired ICT skills. ICT based education should be connected with the future profession or work and people should be prepared to operate professionally in the community of technologies; •Pedagogic argument: use of latest technology in order to supplement the current training program ad improve the study process and achievement; •Catalytic argument: it is aimed to facilitate the change of education by improving training programs and providing better education opportunities to a larger number of people; •Technology argument: it is aimed to stimulate the national ICT industry by national investments to introducing nationally produced computers at schools; •Cost efficiency argument which gives proof that use of computers may reduce the overall costs of education.

21 Hargreaves, A. (1995), School culture, school effectiveness, and school improvement: School Effectiveness and School Improvement, vol. 6, no.1.


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The biggest benefit from the usage of information communication technologies (ICT) in education system is that the results of learning will increase, which would provide the society with the necessary workforce and at the same time increasing the expenses/benefit proportion. It is also important to increase the speed of learning and the provision of equal possibilities for everyone. Despite the fact that a qualitative learning requires making big effort from the students and the lecturers by using ICT the process itself could become more enjoyable and easier22. So ICT could be seen as a facilitator tool for the intensification of study process in the studies of mathematics development (Figure 2).

According to Pale in order to intensify the study process we have to get rid of the so called “non-educational” performances such as: administration, support, organization etc. by minimizing or automatizing the previously mentioned performances. The support functions are those that are needed in the study process but on their own do not add up to the knowledge or skills of the students. For these processes the students and lecturers use a remarkable amount of time.

At this moment an essential question is to free the student from a physical presence. Of course it is possible if students have access to all the necessary resources, as well as training and the support for the knowledgeable usage and the professionals who could assist if the student feels doubt or is in need of consultations.

Figure 1.2. The role of ICT in the mathematics studies process development (Adapted from Pale, 2005)

These aims could be reached if all the present teaching/learning materials are

digitalized and at the same time new e-materials are developed. The materials should be available on-line. A digital library is also a must. It has to be mentioned that the student’s marks in previous exams as well as the questions and answers are of great importance for the study processes.

22 Pale P. (2005) Objectives of ICT in Education. University of Zagreb.


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Communication with e-mails among students, virtual work-groups, distant synchronized (performed in real time with the lecturer who is connected to the system, a direct communication with the lecturer and with each other) and non-synchronized (communication is possible via e-mails, discussion groups etc.) learning, summaries of lectures, tasks and other processes make the study process more qualitative and easy.

A very important aspect in the development of the future ICT competence is in e-studies. Meaning the usage of internet-technologies for the provision of solutions, which increase the knowledge and the performance? The following education support services must be completed: e-studies take place online, they are accessible in computers using standard software, the student interacts not only with the material but also with the lecturers and other students and the focus is on the wide education vision23. E-studies are also defined as the usage of the technologies for the selection, development, registration, provision, management, teaching and the support24. Thus, e-studies are the process where the person learns on his own by using technologies.

It is a mistake that e-learning is suitable only for distance learning. E-learning could also be used in classrooms. If the classical teaching (face to face) to be aided with teaching using elements of e-learning it is defined as a blended learning. The term itself is quite difficult to define since it is used in diverse ways by different people. As Bonk and Graham (2005) claim, blended learning is part of the ongoing convergence of two archetypal learning environments. On the one hand, there is the traditional face-to-face learning environment that has been around for centuries. On the other hand, there are distributed learning environments that have begun to grow and expand in exponential ways as new technologies have expanded the possibilities for distributed communication and interaction.

By analysing the demographical situation in cross-border region the possible demand for the higher education in the near future as well as education as export, the authors see the development of the study process of mathematics as the creation of integrated study course.

Especially emphasized should be the studies of Mathematics and modern-business languages in the higher education programs and the studies could be carried out through e-studies, thus providing a possibility of interconnected higher school study programs not only in one country but also internationally. In the higher mathematics the publicly accessible data bases are also used.

23 Rosenberg M. (2001) E-learning. McGraw-Hill. 24 Masie E., The Masie Centre. What Is E-Learning? http://www.academyinternet.com/elearning/index.html


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II. METHODOLOGICAL BACKGROUND OF RESEARCH

The theoretical framework of the internal research is based on two pillars on curriculum development: learning outcomes and competencies, curriculum development in compliance with the credit – module system, as well as two pillars for study process organization: Collaborative learning in mathematics and information and communication technologies (ICT) usage in study process.

2.1. THE DESIGN OF THE RESEARCH Research consists of two parts (Figure 2.1.):

1. A research on external demands of labour market and employers, who represent the need of qualified specialists with mathematic knowledge and skills, who contribute to socio economical development of border regions. Also, analysis of the regional strategic plans and other national documents.

2. A research on internal evaluation consists of: analysis of ŠU and LUA mathematic study programs that shows to which level existing study programs correspond to the needs of the regional labour market; discussions with the directors of study programs, department chairs, academic personnel and social partners about the possibilities to improve the quality of study programs.

Figure 2.1. The design of the research. The sequence of activities was as follows:

• Analysis of regional strategic plans and other national documents. • Preparation of research methods and methodologies.

Research

.

Interviews with st. progr. directors, chairs,

acad. personell and soc.partn.

Internal Research Experts –

st . progr. directors, teaching staff External Research

Experts of the professional field

Analysis of regional strategic plans and other national documents

Questionnaire- “MATHEMATICS IN

PROFESSIONAL ACTIVITIES”

Questionnaire- “MATHEMATICS IN

PROFESSIONAL STUDIES”

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Analysis of the need of labour market and

employers

Analysis and comparison of current contents of math. subjects at ŠU and LUA


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• Preparation of the questionnaires “MATHEMATICS IN PROFESSIONAL STUDIES” for analysis of study programs (Internal research) and “MATHEMATICS IN PROFESSIONAL ACTIVITIES” for analysis of labour market and employers needs (External research).

• Discussions with the directors of study programs, department chairs, academic personnel (Acad. Experts' assessment) and social partners (Experts’-professionals assessment) about the possibilities to improve the quality of study programs, course materials, strengths and weaknesses, drop outs.

• Creating of the electronic questionnaires in EN, LT, LV languages and placing in the webpage http://www.matnetlatlit.eu/.

• Analysis of results from External (the needs of employees and employers) and Internal (the opinion and requirements of academic personnel, who teach the professional subjects) research, and discussions (the needs of all representatives).

• Evaluation and comparison (Experts from each institution: 3-ŠU, 3-LUA) of mathematics study programs in both partner universities. Analysis includes the body of mathematical knowledge, courses, relative strengths, weaknesses, opportunities, threats and correspondence to the needs of the regional labour market.

• Preparing recommendations (for quality improvements) and network/cooperation strategy plan.

The Research will become a knowledge background for further researches and development of educational products.

2.2. METHODOLOGY OF EXTERNAL RESEARCH

A questionnaire based research “Mathematics in Professional Activities“ is the part of an external research foreseen in the project. The questionnaire-based research (together with the analysis of documents defining the strategic development of the region as well as the survey of the regional managers) is designated to establish the requirements, which are raised by a labour market to the knowledge and skills of mathematics specialists.

Technological sciences and mathematics are specified in the strategies of European economics and higher education development as priorities developing the social-economical potential of the states. The communicate of the European Commission on Putting Knowledge into Practice25 emphasises the role of mathematics developing new products and services and generally in the process of innovative economics. Mathematics is indicated as one of the most important competences, which are necessary living and working in the modern society pursuing for novelties next to the entrepreneurship, natural science knowledge, languages and communications, social and cultural skills. The relevance of technologies and mathematics is emphasised also in strategic documents establishing the social-economical development of Lithuania and Latvia. The question arises, what are the requirements of the labour market to the knowledge and skills of mathematics of highly qualified specialists, how such requirements are understood in different areas of professional activities? One of the sources giving the answer to these questions is the questionnaire survey of highly qualified specialists.

The objective of the questionnaire-based research is to reveal the opinion of highly qualified specialists with non-mathematical higher education in Šiauliai and Zemgale regions and their attitudes towards using of mathematics in their professional activities. It shall be noticed that the questionnaire-based research highlights only the need of the labour

25Putting knowledge into practice: A broad-based innovation strategy for the EU


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market for the situation today, however it does not foresee any requirements, which are raised by the rapid development of social area and technologies.

The studies of adult mathematical competencies demonstrate that employed people meet difficulties recognising mathematical concepts, although they usually use immediately the elements of mathematics in their professional activities. Usually various standardised, algorism working instruments and procedures are based on the knowledge of mathematics. Such situation determines the following problem questions elaborating the objective of the research:

1. How do employed people with higher non-mathematical education assess their mathematical competences?

2. How do employed people with non-mathematical education assess teaching/learning of mathematics at higher education school?

3. How is mathematics applied in the professional environment of a person with higher education, which mathematical knowledge and competencies are necessary to him/her?

4. What is the image of mathematics as a real instrument of practical activities? The research of the need for mathematical competences in the labour market of two countries – Latvian Zemgala Region and Lithuanian Šiauliai Region reveals the situation and combining with the results of other studies preformed within the project enables foreseeing the areas subject to improving educating a specialist in the systems of higher education and professional development.

2.2.1. The Instrument of the Research

Answering these questions, the questionnaire designated to employees with higher education was developed. The purpose was to reveal the opinion of employed people about various aspects of the application of mathematics in professional activities (refer to Annex No.1, 1.2. “The questionnaire for external resesarch”). The questionnaire form includes 5 diagnostic blocks: Assessment of Mathematical Competences, Teaching/Learning of Mathematics at Higher Education School, Mathematics in Professional Activities, Image of Mathematics as a Real Practical Instrument, Knowledge and Competences of Mathematics in Professional Activities. In the diagnostic part of the questionnaire form, there are 23 questions of Likert Scale type and 15 selection type questions presented to the respondents. The questionnaire is presented in an electronic form, the webpage address is: www.matnetlatlit.lt

Table 2.1 Specification of the Instrument of the Questionnaire-based Research Diagnostic Block Content of Statements Assessment of Mathematical Competences (N=3)

Presented statements about the success of learning of mathematics. The presumption is made that the success learning mathematics and positive attitudes towards learning express higher mathematical competences.

Teaching/Learning of Mathematics at Higher Education School (N=6)

Presented statements cover the meanings, which are attributed to learning of mathematics at higher education school: integration with other subjects, meaningfulness, connection with the practice, teaching style.

Mathematics in Professional Activities (N=7)

Statements cover two aspects of mathematics in professional activities: mathematics in a particular professional environment and the interest to deepen mathematical knowledge in the professional area.

Image of Mathematics as a Real Practical Instrument (N=7)

Statements cover several potential values of mathematics: a tool for solving of problems, mean for thinking education, meaningless occupation, subject revealing the potential of a human being in his/ her working activities.

Mathematical Knowledge and Competences in Professional Activities (N=15)

The topics of mathematics supplemented with the description of practical tasks are presented to the respondents. The respondents are asked which area of mathematical knowledge is required in order to ensure successful performance of professional tasks by the specialists in your area, to enable analysing


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professional literature by them. Professional, Educational, Demographic Characteristics of the Respondent

Characteristics of studies (time of studies, education, area of studies), characteristics of professional activities (the area of professional activities, position), gender.

Three types of the questions are used in the questionnaire form. The questions

designated to investigate the attitudes and opinions of the respondents are presented using the modified Likert Scale of 4 scores (diagnostic blocks 1-4): statements and four optional answers expressing the level of a person’s agreement/disagreement with the statement are formulated: Strongly disagree, Disagree, Agree, Strongly agree. Other group of questions is intended to assess the expressions of the knowledge of various topics in the professional activities of population (diagnostic block 5): the respondents have to mark such topics of mathematics, which knowledge and competences are required in his/her professional activities. One open question is given. Its purpose is to reveal the opinion of population with higher education and professional experience about the directions of teaching of mathematics at higher education school.

The survey was set up using programming languages PHP, HTML and CSS: PHP and HTML provides structure and functionality, but CSS provides homepage style and visual coverage. Home page located under domain http://matnetlatlit.eu, which is like introduction for survey in all three languages – Latvian, Lithuanian and English. Depending of the language chosen by respondent can be pressed button „complete a survey”.

2.2.2. The Sample of the Research

The population of the research includes the graduates of the Agricultural University of Latvia and Šiauliai University, whose study programs included such subject as Mathematics (except for the graduates with informatics and mathematics bachelor degrees). Selecting the respondents, graduates of Šiauliai University, the databases of the university graduates were used. The sample was formed using the random probability sampling approach: from the lists of e-addresses. Thus it was ensured that the graduates of all study programs would fall into the sample proportionally to the numbers of students, who studied the programs. The sample of the research was 307 citizens of the Republic of Latvia with higher education (Zemgala Region) and 185 citizens of the Republic of Lithuania with higher education (Šiauliai Region). The total sample of the research includes 492 cases.

2.2.3. Organising of the Research

Employers and employees in Zemgale region and Siauliai county, as experts of the professional field, were asked to take part in the survey via Latvia - Lithuania cross-border cooperation programme 2007 - 2013 home page

http://www.latlit.eu/eng/events/all_events/mathematics_in_professional_ac. Latvia University of Agriculture sent written request with the address of the

webpage, where the electronic version of the questionnaire was placed (and the questionnaire), to all enterprises and to all performers of entrepreneurship, registered in Zemgale region. Invitation to participate in the survey were placed in several public information websites in Zemgale as well: http://www.zz.lv/portals/vietejas/raksts.html?xml_id=31402, http://www.zemgale.lv/index.php/izgltba/publikcijas/1411-matemtika-profesionlaj-darbb, http://www.esfondi.zemgale.lv/lv/jaunumi/matematika_profesionalaja_darbiba/,


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http://www.jelgavasvestnesis.lv/page/1?id=53&news_id=12720, http://www.koknese.lv/?o=1327, http://www.llu.lv/?mi=304&op=raksts&id=6159.

In Lithuania, the research was performed in July and August. An electronic message with the abstract of the research and invitation to participate in the research was sent to all selected respondents together with the address of the webpage, where the electronic version of the questionnaire was placed.

2.2.4. The Analysis of the Research Results

Data matrix was developed in the format compatible with the SPSS software, the answers of the respondents were coded in digits from 1 to 4 so, that an agreement to a statement would correspond a higher value of the code. The research results were systematised and analysed using statistical methods.

1. The tables of frequencies describing the distribution of answers on the basis of the respondents from one of the countries.

2. Scales were formed (using the joint data matrix of Lithuanian and Latvian respondents. In the table, there are presented possible scales, the numbers of statements included in a scale indicate the number of a diagnostic block and the number of the statement in this diagnostic block. An asterisk means that the statement indicates the opposite manifestation of the quality then the name of the scale, therefore, calculating the values of the scale, the statement is recoded. A question-mark means that the scale requires checking both with this statement and without it.

Table 2.2. Scales and Statements Included Therein

Scale (aggregated variable) Statements included into the scale (Items) Self-assessment of mathematical competences 1.1, 1.2, 1.3 Conformity of mathematics at higher education school to a student’s needs

2.1, 2.2*, 2.3, 2.4*?, 2.5*, 2.6

Mathematics in professional activities 3.1,3.2, 3.3, 3.4*, 3.5, 3.6 Assessment of the practical potential of mathematics

3.7,4.1, 4.3*,4.4, 4.5

A need for the improvement of mathematical knowledge

3.9

Validating the scales, the factorial analysis was performed. The reliability of the scales was assessed calculating Cronbach’s alpha coefficient. The values of the scales were calculated as the average of the assessment of the statements included in the scale, the weighting method was not applied.

3. The differences between the groups of the respondents were assessed in consideration of professional, educational and demographical characteristics using both individual statements and the scales formed to each country separately. A chi-squared test was applied to assess the significance of the difference between the groups.

4. A correlation analysis was performed highlighting the correlations of mathematical competences, learning of mathematics at higher education school, the application of mathematics in professional activities and assessment of the potential of mathematics. Spearman’s rank correlation coefficient was calculated.

5. The results of the diagnostic block 5 are not included into the scale. The joint rating of the topics and differences depending on the areas of professional activities. The statistical significance of the differences of the groups of the areas of activities was established applying chi-squared criterion.


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6. Establishing differences between the countries, both the individual statements (initial data) and formed scales were used. In the first case, the chi-squared test was used to assess the statistical significance, in the second - Kruskal-Wallis test. The questions of the fifth diagnostic block were compared between the countries only in the groups of respective professional activities.

7. In order to reflect more reliably the opinion of population on the matters of learning of mathematics and the application thereof in professional activities, the method of factors (scales) was applied. The essence of this method is the presumption that the answers to the questions express deeper, latent qualities – factors. The method of the alpha factor analysis Varimax rotation of factor axes was used to group the statements. The joint research data matrix of Latvia and Lithuania was used for the factor analysis. The data (the correlation matrix of qualities) was in conformity with the requirements of the method: KMO = 9.0, Bartlett's test of sphericity: Chi sq. =2575, df=171, p< 0.001), MSAi>0,5. The factor analysis revealed that the expression of 5 factors interpreted meaningfully was possible to envisage. Hereinafter we present the factor matrix:

Table 2.3. Results of Factor Analysis. Factor Loads Factor loadings Statements

1 2 3 4 5 Studying mathematics develops logical thinking, accuracy and concreteness of future specialists.

0.73

A person, who understands mathematics, will easily master most jobs that require thinking.

0,64

Mathematics develops thinking, helps to make a decision in a particular situation, find new ideas.

0,61 0,43

Mathematical thinking helps to solve real world/ professional problems. 0,51 0,32 0,45 Mathematics is a meaningless game with numbers which is played according to the rules created by scientists.

-0,78

Mathematics is only formulas that are needed to remember. -0,72

0,33

Mathematics helps to model and analyze the problems of the real world. 0,62 Mathematics gives an insight into the world we live. 0,38 0,51 I understand mathematical symbols and a formal mathematical language which is used in my professional literature.

0,36 0,33

Mathematics and the subjects, which require mathematical knowledge, have always been my favourite.

0,76

I think mathematics, which I studied at high school (university, college), could have been more complicated.

0,76

The knowledge and abilities of mathematics, mathematical thinking helped me to achieve more in my life

0,41 0,57

Mathematics was an interesting and meaningful subject. 0,35 0,49 Mathematics knowledge helped me to understand other study subjects. 0,43 0,48 Mathematics is widely used in my professional activities. 0,77 My occupation does not require deeper knowledge of mathematics: it is enough to do arithmetical calculations and count percentage.

-0,65

0,33

I have a lot of opportunities to apply my knowledge of mathematics in professional activities.

0,40 0,60

For my occupation studying mathematics at high school (university, college) is wasting of time; knowledge got in secondary school is enough.

-0,38

-0,53

0,37

Mathematics was taught matter-of-factly and boringly. 0,71I did not understand most mathematical concepts that I studied at high school (university, college).

0,63

Most of the students did not understand mathematics, tried to learn rules by heart.

0,62

Percent of variance 12 12 12 11 10Factor loads lower than 0.3 are not presented in the table.


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On the basis of the factor analysis, five scales were formed. The reliability thereof was checked calculating Cronbach’s alpha coefficient. Cronbach’ alpha coefficient to all scales was >0.5, therefore it was possible to state that the statements included into the scale were sufficiently homogenous. The values of the factors were saved using the method of a regressive analysis. Hereby we present the list of factors (a number corresponds with the number of the factor in the table):

1. Assessment of the joint educational potential of mathematics; 2. Assessment of the practical relevance of mathematics; 3. Positive attitude towards learning of mathematics; 4. Application of mathematics in professional activities; 5. Unfavourable assessment of learning of mathematics at higher education school.

2.3. METHODOLOGY OF INTERNAL RESEARCH

The internal research consists of analysis of Šiauliai University (ŠU) and Latvia University of Agriculture (LUA) mathematic study programs that will show to which level existing study programs correspond to the needs of the regional labour market. The research methodology builds on the findings of research from different theoretical approaches, disciplines and traditions, and on policy documents concerning a mathematics education and professional competence, as well as many years of personal experience in mathematics teaching.

2.3.1. The Instruments of the Research Two main research methods were used in internal research: interviews with the

directors of study programs, chairs of departments and academic personnel, and survey (using questionnaire) for curriculum analysis and comparison taking measurements at indicators created by the current researchers group.

The aim of the interviews with the directors of study programs, department chair and academic personnel to show existing mathematics courses contribution for development competences for future professionals (specialists), to show the wide range of mathematics usage in any profession in comparison with going program as well as ask to remark in what fields of mathematics oriented to the needs and specifics of particular specialties for the purpose of practical knowledge applying. The following interview areas:

1. How much and where does a specialist who has finished your programme may apply mathematical competencies in his practical activities? What mathematical knowledge, skills and attitudes may be required from him when performing the tasks of his profession?

2. Which topics of mathematics are most important in teaching mathematics to a student? What should attention be drawn to, what should be emphasized, what teaching methods should be used? The role of mathematical software in teaching mathematics.

3. What general educational value of mathematics? To what extent is mathematics necessary at university for general mathematical education?

4. What should be the conception of teaching mathematics as a subject for the students of corresponding speciality:

• emphasis on the structure, strictness of mathematics, simplest proofs are to be provided, and application in the speciality is only an illustration;

• the course is introduced through the tasks of applied nature; only several proofs and groundings are to be provided in order for the students to understand the structure of mathematics;


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• only conclusions-algorithms of applied nature are provided; the entire attention is devoted to the solution of practical problems;

• other versions. The questionnaire for directors of study programs, department chair and academic

personnel consists of 4 main parts (see Annex No.1): 1. Introduction for the teachers. 2. Currently taught content of mathematics specifying the number of hours assigned for

each topic. For each topic, the teachers will be asked to indicate if this topic is necessary for future specialists; if yes, they will be asked to name subjects/topics or specialist activities where this mathematical knowledge and skills are used/necessary (Table 2.4.);

Table 2.4. Assess the current content of mathematics subject HOURS: T-theory,

P-practical w., L – laboratory w, I- individual

w.

Subject name

…. semester

T P L I

Is it necessary?

0 – not necessary, 1 – might (should)

be taught, 2 - necessary.

Please name, if possible, the subjects/topics or specialist

activities where this mathematical knowledge and skills are

used/necessary.

LINEAR ALGEBRA Matrices, determinants, the equation system

ANALYTICAL GEOMETRY Line equations for the Cartesian and polar coordinate systems.

…………

Plane equations. Line equations in space

3. The list of mathematical areas/topics not included or only partially included into the

list of taught topics which could be useful for the future specialists. In this part, the teachers will also be asked to name if those topics will be necessary and what students should know of these topics (Table 2.5.).

Table 2.5. We are providing several areas/topics of mathematics that are not included or

are only partially included into the list of taught topics. Do you think these topics might be necessary in your study programme?

What issues of these areas might be necessary for your study programme (you may specify both the topic and concrete tasks which should be solved)?

Mathematical area

Is it necessary? 0 – not necessary,

1 – might (should) be taught, 2 - necessary.

In your opinion, what should a student know/be

able to do?

Statistical deductions (application of the sampling method, computing of confidence intervals, testing of the statistical hypotheses and etc.).

Linear programming (the description of a situation by the equations and inequalities systems and analysis of them, the tasks of productivity, recourse administration, logistics, transport and etc).

Others. Fill in……...........................................


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4. An open question on what should attention be drawn to in teaching mathematics

(content, task topics, methods). The questionnaire was presented to Dean's of Faculties in the meetings. Respondents

selected the “snowball” method - a questionnaire sent to all program managers, who in turn sent out a questionnaire at least three discipline-specific trainers.

2.3.2. The Analysis of the Research Results The questionnaire results for each program consists of five parts and collected in

tabular forms: 1. Analysis of current content of mathematics subjects (Table 2.6.).

Table 2.6. The current content of mathematics subjects

Experts' assessment (programme director + at least 3 teaching staff) Content Average

ball* MODE** comments (subject themes (topics) where these mathematics knowledge are useful / necessary)

LINEAR ALGEBRA Matrices, determinants, the equation system

*The average ball is calculated from numbers: 0,1, 2. Where 0 – not necessary, 1 –should be taught, 2 - necessary.

**If the multiple mode exists, we write all. 2. Results from internal and external research – which topics-themes could be important

for corresponding programme (Table 2.7.).

Table 2.7. Accordance of mathematical and professional competences

Themes (topics) Experts' assessment

(programme director + at least 3 teaching staff)

Experts' - professionals

assessment

Average ball

MODE*

comments (What do you think, the student should

know..?) Freq

uenc

y Range (0-25% - 0;

26%-50%-1; 51%-100%-2)

Comments for the

preferred study

program content

Statistical deductions (appli-cation of the sampling method, computing of confidence intervals, testing of the statistical hypotheses and etc.).


3. Questionnaires results:

• Experts' (programme directors, teaching staff) suggestions on the current content of mathematics subjects and teaching methods;


LLIII-122 MATNET 35

• Experts'-professionals suggestions on the content of mathematics subjects. 4. Results from discussions in the meetings. 5. Specialist’s recommendations.

2.3.3. Instruments for comparison of study program

Comparison of study program consists of three parts: 1. Comparison the number of hours of theory, practical and independent work

assigned for the same topics (Tables 2.8.-2.9.).

Table 2.8. Content and volume LUA LATVIA ŠU LITHUANIA

…. KP (=…. ECTS) ALL HOURS:

T-theory, P-practical w., L – laboratory w, I- individual w.

…. K (= …. ECTS) ALL HOURS:

T-theory, P-practical w., L – laboratory w I-individual w.

Content

T-... P-... L-... I-... T-... P-... L-... I-... 1st semester

Name of subject in LUA .... KP= .... ECTS / Name of subject in SU .... KP = .... ECTS

T-... P-... L-... I-... T-... P-... L-... I-...

1. LINEAR ALGEBRA Matrixes. Operations with matrixes.

Determinants of the second and third order, theirs properties. Adjunct and minor. Extension of determinants by rows and columns. Determinants of higher order. Reciprocal/Inverse matrix.

………………….. 2st semester

Name of subject in LUA .... KP= .... ECTS / Name of subject in SU .... KP = .... ECTS

T-... P-... L-... I-... T-... P-... L-... I-...

6. DIFFERENTIAL CALCULATION OF

FUNCTIONS IN SEVERAL VARIABLES

Notion of functions in several variables. Limit. Continuity.

Table 2.9. Comparison of the content of mathematics at the LUA and ŠU

Content

Contact classes at LUA

Com-pare

Contact classes at ŠU

Recommenda-tion for

LUA

Recommen-dation for

ŠU under the program

taking an average of 1 KP 1. Linear algebra 2. Vector geometry


LLIII-122 MATNET 36

2. Comparison study process organization by several indicators (Table 2.10.).

Table 2.10. Comparison of study process organization

Indicators groups Indicator LUA –LATVIA

(Yes/no/ partly; data, webpage adress etc)

SU – LITHUANIA (Yes/no/ partly; data, webpage address etc.)

Only for whole study program Only for math course

Described the learning outcome

(please specify descriptors to be

used)

Math outcomes are the part of study program

Content modules Levels’ modules Introduced a

module system Form modules Mandatory - in the program is allocated the ECT

Usage of ICT By the teachers initiative

In print form (material prepared by lector who teach)

In internet (only for reading) Methodical materials An interactive materials in

internet

In lectures In practical tasks

The average number of

students per faculty

Laboratory work

Exam /test Accumulative exam / test Measurement of

learning outcomes „Semester work” + exam / test Lecture Independent work tutoring by teacher

Independent work Project work Group work

Teaching methods

Collaborative learning

3. In order to evaluate of mathematics study programs in both partner universities, analysis of the body of mathematical knowledge, courses, relative strengths, weaknesses and correspondence to the needs of the regional labour market, based on indicators, will be done by SWOT analysis (Strengths, Weaknesses, Opportunities, Threats) for each program, which compares.

Recommendations to improve Mathematics programms (content and studies process organization) will done based on SWOT analysis, i.e. recomendations for program‘s: outcomes, content, study process and for study materials.

Study programs for comparison

LUA ŠU Agricultural Mechanisation ↔ Mechanical Engineering Computer Control and Computer Science

↔ Informatics Engineering

Environmental Science ↔ Ecology and Environmental sciences Sociology of Organisations and Public Administration

↔ Public Administration


LLIII-122 MATNET 37

III. THE RESULTS OF EXTERNAL RESEARCH

3.1. THE RESULTS OF THE OPINIONS OF THE POPULATION OF ŠIAULIAI REGION

3.1.1. The Research Sample Characteristics

186 persons with higher education employed in Šiauliai Region participated in the survey. The characteristics of the research sample is presented in Table 1.

Table 3.1. Characteristics of the Research Sample (N =181)

Quality Category Percent 0-4 years or less ago 86 How many years ago did you

graduate your higher education school?

5 years ago 14

Female 54 Gender Male 46 Company/department manager 16 Employee 75

Position

Other 9 Profession bachelor 16 Bachelor 69

Education

Master 14 Technological physical sciences 48 Social sciences 46

Area of studies

Other 6 Public administration 22 Services and business 21 Construction and production 16 Electronics and information technologies

19

Employment

Other 22

In the sample, the respondents, who graduated recently, prevail. The distribution of men and women is almost equal, however there are more men between the graduates of technological sciences, and more women between the graduates of social sciences. The numbers of respondents, who graduated study programs in the area of technological and physical sciences, and the respondents, who graduated study programs in the area of social sciences are almost the same.

3.1.2. Teaching of Mathematics and Attitudes towards Mathematics Mathematics and subjects, which need mathematical knowledge, are liked by more

than one half of the respondents. However, it is necessary to achieve the study results foreseen in the study program independently of whether students like mathematics or not. It shall be emphasised that according to the subjective assessment of students, the following objectives were not fully achieved: a bit more than 40 % of the respondents did not understand the mathematics, which they studied at higher education school. The tendency was noticed that students, who did not like mathematics, usually did not understand it, however it was not expressed very clearly (Spearman’s rank correlation coefficient was 0.52). On the other hand, a bit more than 20 % of the respondents thought that the mathematics, which they studied,


LLIII-122 MATNET 38

could be more complicated. The research results may be doubly interpreted: as the indicator of the condition of learning at higher education school or as the reference to organising of the educational process.

Figure 3.1. Assessment of the Complexity of Mathematics Learning Content at Higher Education School. (N =181)

We may state that the respondents express the need for organising of the differentiated teaching of mathematics, whereas some part of students achieved the defined results of Mathematics as a subject only partially, and the expectations of others about the higher level of teaching of mathematics were not realised.

What are other obstacles impeding learning of mathematics, preventing students

from pursuing better study results?

Figure 3.2. Assessment of Teaching/ Learning of Mathematics at Higher education School. (N =186)

The most of the respondents understood the relevance of learning of mathematics in

correlation with the profession (70 % stated that the level of the knowledge of mathematics in a secondary school was not sufficient to the representatives of their profession), other teaching subjects (65 %), emphasised the general educational level of learning of mathematics (80 % thought that teaching of mathematics improves thinking of future specialists). However the orientation of the process of learning of mathematics towards the result, as not only learning of mathematical rules but also as understanding, was sceptically assessed


LLIII-122 MATNET 39

(approximately 80 % of the respondents stated that it was more important to remember mechanically the rules, then to understand them). About 60 % of the respondents stated that mathematics was taught “drily and tediously”.

Interpreting these data within the context of teaching/learning of mathematics, we may envisage three factors, which probably would allow pursuing better results. The first one is learning how the learning result could be more correlated not only with repeating of a rule, algorithm solving examples, however it could be correlated with deeper understanding of mathematics; The second factor is such that pursuing of the conformity of the methods of teaching/learning of mathematics with a student’s expectations, the process of learning could be more involving, interesting; the third factor is to develop positive attitudes of students towards the relevance of mathematics in professional activities and to enhance learning motivation on this basis.

Figure 3.3. Respondents’ Attitude towards Mathematics and the Relevance Thereof (N =186)

Teaching of mathematics at higher education school is usually criticised for

insufficient correlation between the theory and practice. However, the survey demonstrated that over 80 % of the respondents agreed with the statements expressing the relevance of mathematics analysing the problems of the real world. A similar part of the respondents believed that mathematics had a general educational potential, emphasised education of thinking as one of the most important functions of teaching of mathematics at higher education school. Approximately 60 % of the respondents thought that employers treat more favourably people, who know mathematics. It shall be emphasised that managers agreed more often than other employees that mathematics was important solving professional problems in the real world, it allowed better understanding of the world, where we lived, thought that mathematical knowledge and mathematical thinking helped them to achieve more in their life.

The Aspect of a Gender. The research did not reveal any statistically significant differences between who liked mathematics more – men or women. A gender also did not


LLIII-122 MATNET 40

have any statistically significant meaning to the responses to the question about understanding of the ideas of mathematics taught at higher education school. Both men and women assessed similarly the statements describing the quality of teaching. However, men noticed interdisciplinary relationships and used mathematical knowledge studying other subjects more often than women (agreed 74 % and 58 % respectively); men stated that they understood the formal language of mathematics used in professional literature more often (95 and 68 % respectively). It shall be emphasised that women assessed the general educational value of mathematics more sceptically: 94 % of men and 77 % of women agreed that mathematics developed thinking and helped taking decisions.

The Aspect of the Study Area. Analyzing the dependence of the respondents’ responses on the study program, which they have graduated, the following two areas were distinguished: technological and physical sciences as well as social sciences. The research data do not allow stating that the population, who graduated natural and technological sciences, liked mathematics more than people, who studied social sciences. Furthermore, any differences between the groups responding to the questions about understanding of mathematics studying at higher education school and about their wish to have the more complicated course of mathematics were not established. The respondents in the abovementioned groups assessed similarly also the help of mathematics studying other subjects. Assessing the statements expressing the quality of the lectures of mathematics at higher education school, any statistically significant differences between the groups were also not noticed. The relevance of mathematics in professional activities was similarly assessed. Statistically significant differences were established in the area of study results assessing the understanding of the mathematical language and symbols used in professional literature (29 % of the representatives of technological and physical sciences as well as 5 % of the representatives of social sciences strongly agreed.). The population, who has graduated the studies of technology and physical sciences, assessed the educational value of mathematics a bit higher than the graduates of social sciences (92 % and 76 % agreed respectively).

Table 3.2. The opinion of population working in different areas about mathematics

and its teaching at higher education school (the percentage of agreement statements is presented)

Statements Public administration

Services and business

Construction and production

Electronics and information technologies

Mathematics and the subjects, which require mathematical knowledge, have always been my favourite.

49% 44% 69% 69%

Mathematics develops thinking, helps to make a decision in a particular situation, find new ideas.

67% 88% 81% 94%

I understand mathematical symbols and a formal mathematical language which is used in my professional literature.

68% 68% 96% 84%

*only the statements, which assessment differences were statistically significant, are presented. (chi squared test, p<0.1).

The Aspect of Professional Activities. Mathematics and the subjects, which require mathematical knowledge, were more liked by persons working in the area of construction, production as well as business, services than by those, who were working in the area of public administration. However assessing the aspects of teaching of mathematics, the research did not reveal any statistically significant differences. Furthermore, any statistically significant differences between the mentioned groups did not exist also assessing the statements describing the relevance of mathematics.


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3.1.3. Mathematics in the Area of Professional Activities

Mathematics, exact sciences are the grounds of technologies, therefore the studies thereof should be understood as the significant factor of the development of the competitive ability of the region. It is important developing new products and services and in the proces of the development of the innovative economics generally.

Figure 3.4. Mathematics in the Area of Professional Activities (N =186)

Approximately one half of the respondents stated that they had to apply their knowledge of mathematics directly in their professional activities, the other half stated they did not need any deeper knowledge of mathematics in their profession, it was sufficient to perform elementary calculations. However, even if the most of the respondents did not need very much mathematical knowledge obtained at higher education school in their direct activities, the practical relevance of mathematics was emphasised by the most of them. (75 % indicated that mathematical thinking helped to solve the problems of the real world and professional problems).

The need to study the applications of mathematics in relation with solving of tasks in the area of their professional activities was specified by approximately 60 % of the respondents. It shall be noticed that this need was strongly related to the situation of the application of mathematics in their professional activities (Spearman’s rank correlation coefficient was 0.38). Hence, we may state that the applications of mathematics should be an integral part of the content of professional development both to the specialists, who apply mathematics directly in their work, and the ones, who see a possibility to apply it.

The Aspect of a Gender. The research did not reveal any statistically considerable differences assessing the applications of mathematics in the professional environment, however the different need for training of the applications of mathematics in professional activities became clear: men expressed their wish to participate in such trainings more often than women (71 abd 52 % agreed respectively).


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The Aspect of the Study Area. Analysing the dependence of the respondents’ responses on the study program, which they had graduated, the following two areas were distinguished: technological and physical sciences as well as social sciences. It was natural that the statistically significant difference was observed responding the question about the applications of mathematics in professional activities, however it was expected that this difference would be greater. (refer to the picture). Such situation may be explained by the fact that graduates not necessarily worked in the same area, which they graduated.

Figure 3.5. Distribution of the answers to the question “Mathematics is widely used in my professional environment” in consideration of the area of studies (N =186)

The Aspect of Professional Activities. It shall be noticed that employed people

answered differently the question about the application of the knowledge of mathematics in the area of their professional activities. The employees working in the areas of production, construction, electronics and information technologies, who could notice more often than others that there were lots of opportunities to apply mathematics in the areas of their activities, were distinguished. Furthermore, the knowledge of people working in these areas meet better the requirements of the area: the bigger majority of the respondents understand the formal language mathematics used in professional literature. The employees working in the areas of electronics and information technologies lack the most the application of mathematics in the area of their activities; they would like the most to participate in training programs.

Table 3.3. The opinion of population working in different areas about mathematics in

professional activities (the percentage of agreement statements is presented)

Statements

Public administration

Services and business

Construction and production

Electronics and Information Technologies

Mathematics is widely used in my professional activities.

32% 36% 48% 61%

I would like to attend the training that deals with mathematics application to solve the practical problems of my professional field.

51% 47% 69% 81%

For my occupation studying mathematics at high school (university, college) is wasting of time; knowledge got in secondary school is enough.

38% 38% 11% 6%

* only statements, which assessment demonstrated significant differences between the groups (chi squared test, p<0.1.


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3.1.4. Directions of Improvement of Teaching of Mathematics. Respondents’ Opinion The respondents were asked to express their opinion, how to change teaching of

mathematics in order to ensure better conformity of the studies with the needs of students. 28 respondents answered this question. The answers of the respondents were grouped, notional elements were distinguished – the directions of improvement were suggested. Interpreting the population’s opinion, hereby we present some examples of original responses.

The most of the respondents emphasised that the major change in teaching of mathematics at higher education school would have to be related to the enhancement of the link between teaching of mathematics and practice: “In the process of teaching, it is necessary o correlate the theory with practice, to illustrate using particular examples from the real world, tasks should be related with topicalities in the life”, “Solving of true-life /professional problems invoking mathematics, but not only rough numbers/equations/formulas.”

Emphasising of the links with practice could increase students’ motivation. Respondents stated: “Today I would dare to state that mathematics is taught very theoretically. Therefore it is probable that lots of students do not understand, why should he or she study one or another theoretical subject.“, “To learn what you really need in your future life but not to drill the same with everybody knowledge, although this knowledge later becomes anything worthy.””Why is it possible to bring a famous businessman to the faculty of law or mathematics, and you cannot invite to the lectures of mathematics any mathematicians from research laboratories, who would present data from a different point of view, I think every student would be much more interested and would better learn the refinements of mathematics.”

The respondents suggested to lecturers to be more interested in the application of mathematics in the particular area of science, which would include implementation of the study program, to transform the module of mathematics in such a way, that it would better meet the speciality: “Before starting to learn a particular branch of mathematics, first of all it would be useful how it can be applied in practice in a particular specialisation. It would encourage students’ interest and would make easier the process of learning.“, “Perhaps it would be desirable to pay more attention to the professional trends of the study program and to narrow the program of mathematical subjects harmonising it with maximum requirements of the mathematical calculations of the particular teaching program”. “To teach more in consideration of the speciality, whereas the mathematics, which was taught in the first years of studies, was not required anywhere, although it was easier than learning level A at secondary school, however not only the mathematics, as a teaching subject, was not required.”

Furthermore, they emphasised that learning should be directed towards understanding, revealing of correlations within the subject: “It is necessary to strive to teach a person to understand mathematics, but not teach him/her only in order to make him know formulas without an opportunity to learn how to apply them in the life. “, “Its is important to know, how to use , apply formulas, which would make solutions easier. I think that learning of definitions by heart is not necessary anymore.” „the essential thing about mathematics is the adaptation thereof in some certain situations of the speciality.”

Links between mathematics and other subjects were missing: “Only as much, as mathematics is required studying other subjects and to tell to a student that he or she would need this and that to calculate in this or that lecture.“”Perhaps the professional trends of study programs should be more considered and the program of mathematical subjects should be narrowed adjusting it to maximum requirements to the mathematical calculations of a particular study program”.


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It was required to pay more attention to the methods of applied statistics: “We should work harder with the analysis of collection of statistical data, whereas it is required in the Bachelor’s Thesis.“

The respondents were asked to answer, which field of mathematical knowledge was the most required in order to ensure successful performance of professional tasks, ability to analyse professional literature by the specialists of their area. Hereby we present the results by the areas of professional activities.

Table 3.4. Which field of mathematical knowledge is required in order to ensure

successful performance of professional tasks, ability to analyse professional literature by the specialists of the respective area? The respondents’ choices.

those fields of the deeper knowledge of mathematics that are needed for the specialists of your field to accomplish their professional activities successfully and analyze professional literature

Topics of Mathematics Public Administration

Services and Business

Construction and Production

Electronics and Information Technologies

Descriptive statistics (grouping of the data, the tasks on the calculation of percentages, averages, and errors, estimation of statistical relations, graphical representation of the data and etc ).

86% 57% 74% 81%


73% 43% 33% 28%

More complicated statistical methods of the data analysis (market analysis, the mathematical modelling of cause-and-effect of the economic object, the use of dynamic lines and etc.)

38% 29% 37% 22%

The solving of equation systems, the operations with data matrixes (computing a demand and supply balance, making a balanced production plan, identifying the productivity of the economic system and etc.).

30% 23% 30% 47%

Geometry (the calculation of area and capacity, the determination of the power, that affects the solid, direction, computing the performed work of the power and etc.).

19% 14% 52% 56%

Derivatives and differential calculation (approximate calculation, computing of minimum and maximum value, analysis of process fluctuation: computing of speed, acceleration, gradient and the rate of the developing business profitability at a particular time and etc.)

22% 23% 33% 38%

Integral calculations (the calculation of the length of curve arc, surface area, spin capacity, work of variable power, the coordinates of heterogeneous beam mass centre, flat figure at a static and moment of inertia, the power of liquid pressure, the body at the moment of inertia and etc).

8% 9% 26% 28%

Linear programming (the description of a situation by the equations and inequalities systems and analysis of them, the tasks of productivity, recourse

11% 26% 19% 41%


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administration, logistics, transport and etc). Net planning (the tasks solutions of the

integrated jobs planning and etc.). 24% 29% 30% 41%

Discrete mathematics (combinatorics, the tasks solutions of algorithmics, graph theory, cryptography and etc.)

5% 11% 15% 31%

Mathematical logic (operations with predicates, Boole‘s algebra, predicate logic and etc.).

16% 23% 37% 63%

The application of mass service theories (the mathematical modeling and optimization of administration of clients flow service and etc.)

35% 40% 22% 16%

Decision trees (the modeling of the choice of decision alternatives by the mathematical evaluation of the conditions and etc.)

35% 20% 30% 34%

Probability theory (computing of the most presumptive events, doing the tasks on insurance, mass service, quality and control, automated management systems and etc.).

38% 54% 30% 50%

The elements of betting theory. (the mathematical modelling of decision making when the acting persons/group have conflicting aims and etc.)

16%

26%

19%

25%

* p<0.05, ** p<0.01, Chi squared test

Employees, the graduates of higher education schools, needed mostly the firm

knowledge of the application of statistics: descriptive statistics was required to the most of employees working in various fields, the applications of the theory of statistical conclusions were required more to the employees working in the area of public administration as well as in the sector of services and business. Over one third of the respondents independently of the field of their activities stated that the applications of the probability theory were required. In other areas, the need for mathematical knowledge depended on the nature of the work of specialists. Specialists in the area of public administration, service and business emphasised the higher need for mathematical theories in relation with managerial aspects. Specialists employed in the area of construction and production specified the applications of geometry, differential calculation and mathematics in production management, such as network planning, decision trees.

3.2. THE RESULTS OF OPINIONS OF ZEMGALE REGION POPULATION

3.2.1. Analysis of the strategic planning documents The European Community indicated eight competences to sustain lifelong learning.

The focus of attention here is mainly on the third competence “Mathematical competence and basic competences in science and technology”26.

The same idea can be found in national strategic planning documents. For example, Regulations issued in September 27th, 2006 by the Cabinet No.742 Approach of education

26 “Key competences for lifelong learning”. European Recommendation 2006-2006/962/EC.


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development 2007-201327 establish the fact that there are not enough students studying natural sciences, engineering and technology. Mastering professions in engineering and technological sciences is prevented by insufficient knowledge and skills of mathematics and natural sciences learned in general and professional education programs. Professional and higher education programs are not modernized enough to offer competences and skills that are required by areas of economical development. This document sees the necessity of a research studying demands of labor market, developing study programs of 1st level higher education, some higher education programs develop in modules for those who study full time and those who return to university during lifelong studying. Implementing actions in this direction also involves elaboration of profession standards study programs of 4th and 5th level.

Latvian National development plan 2007-201328 affirms that knowledge society is based on higher education. That’s why it is crucial to provide possibilities for all those who wish to gain higher education. Special attention is needed to raise the rate of students in natural sciences, medicine and engineering and increasing the number of higher qualification specialists (masters and doctorates). This document gives tasks to achieve general knowledge and skills of high quality in subjects including mathematics; and achieve harmonized rate of students of higher and professional education studying engineering, natural sciences, information technology, health care and environmental sciences.

Graduates of higher education establishments say that they’re dissatisfied with the quality of their education. Latvian National development plan 2007-2013 states the need to improve the supply of higher education based on requirements of labor market. It plans to establish interdisciplinary and inter-university study programs, evaluate and improve contents of studies regularly by involving social partners developing initiatives in state-private partnership in order to get harmonization between education supply and demand of labor market. This document also affirms the need to implement modern ITC infrastructure in education establishments of all levels and kinds. Special attention needed to improve the appeal of study programs in engineering and natural sciences.

Latvian Strategy for sustainable development till year 203029 stresses lifelong learning: development of lifelong learning, education of adults within current education system and raise educational capital gained in work environment. Latvian competitiveness will always be dependent from the connection of educational system and changes in labor market and its ability to prepare person for lifelong work in changing environment. Moreover, the document stresses that information technologies have become everyday issue and thus they have to be included in learning process. By integrating distance education elements in learning process and using advantages of information technologies there is a possibility not only to provide students with attractive, interesting and qualitative mastering of study contents in virtual environment but also get new possibilities for diversification of learning process and new organization forms. That will attract interest of youth and improve overall level of information technology level in Latvia.

Latvian Strategy for sustainable development till year 203030 determines the need for interaction between education establishments and regional entrepreneurs, the need for education establishments to follow local and global changes in economics in order to supply education of from and contents that will improve regional development and competitiveness of individuals and organizations in future economy.

27 http://izm.izm.gov.lv/normativie-akti/politikas-planosana/1016.html 28 http://www.jelgava.lv/pasvaldiba/dokumenti/dokumenti0/attistibas-planosana/valsts-attistibas-planosanas-dokumenti/ 29 http://www.latvija2030.lv/page/1 30 http://www.latvija2030.lv/page/1


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In accordance with Integrated development program for Jelgava city year 2007-201331, one of priorities of Jelgava’s development is educated, competitive, healthy, socially active and creative individual.

Development program of Zemgale planning region years 2008-2014 emphasizes qualitative and fast knowledge mastering, accumulation and application as the main factor of competitiveness of region or state. The usage of ITC and successful innovation development in all areas is seen as an instrument to develop regional economy.

Innovative development program of Zemgale’s development priority is establishing innovatively thinking labor capital. This document defines disadvantages of the region: quality of labor education, knowledge and skills are not coherent to requirements of labor market. The difficulties that graduates of university face starting work are usually connected to disadvantages of general and higher education system – insufficient knowledge in nature and technology areas and insufficient skills to work with ITC.

Development program of Zemgale planning region years 2008-201432 brings forward several tasks for solving problems mentioned. It states that employers should be involved in all of education processes – planning, realization and evaluation, especially in professional and higher education. This should lead to improvement of education quality in all the levels of education and promote individuals with high levels of competence and creative thinking.

Latvian classifier of professions33 describes necessary competences (education, knowledge, experience and skills) for completing tasks of different professions – that is dome to ensure labor record keeping and comparison appropriate to international practice. Profession standard34 gives levels of theoretical and practical background – minimal education, knowledge and skills level. Including in mathematics: insight, comprehension, application (Table 3.5.)

Table 3.5. Level of mathematics’ knowledge. Qualification Insight Comprehension Application Programmer + + Engineer of forestry + Engineer of wood-working + Engineer of power system in agriculture + Engineer of mechanics + Building engineer + Landscape architect + Engineer of land survey + Surveyor + Engineer of environment + Catering organizer + Hotel service organizer + Manager of enterprise or organization +

Siauliai University and Latvia University of Agriculture prepare primarily engineering professionals. Therefore the designing mathematics syllabus for engineers in universities across the Europe should be done in accordance with SEFI Core Curriculum35. According to the SEFI Mathematics Working group the mathematical topic of particular importance include: fluency and confidence with numbers; fluency and confidence with algebra;

31 http://www.jelgava.lv/pasvaldiba/dokumenti/dokumenti0/attistibas-planosana/jelgavas-pilsetas-attistibas-planosanas-8/ 32 http://www.jelgava.lv/pasvaldiba/dokumenti/dokumenti0/attistibas-planosana/zemgales-attistibas-planosanas-dokumenti/ 33 http://www.lm.gov.lv/upload/darba_devejiem/profesiju_klasifikators_0811.pdf 34 http://www.lm.gov.lv/upload/darba_devejiem/profesiju_standarti_0811.pdf 35 SEFI Mathematics Working Group. http://learn.lboro.ac.uk/mwg/


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knowledge of trigonometric functions; understanding of basic calculus and its application to real-world situations; proficiency with the collection, management and interpretation of data.

The pyramid of knowledge principle and others principles are considered as an obligatory methodical base of mathematics teaching however the curriculum offers many opportunities for teaching the materials in innovative ways36.

Core Curriculum has been arranged in a structure which has four levels. These levels represent an attempt to reflect the hierarchical structure of mathematics and the way in which mathematics can be linked to real applications and as the student progresses through the engineering degree programme:

a) The material in Core Zero level section is the material which ideally should have been studied before entry to an undergraduate engineering degree programme;

b) Core level 1 comprises the knowledge and skills which are necessary in order to underpin the general Engineering Science that is assumed to be essential for most engineering graduates. Items of basic knowledge will be linked together and simple illustrative examples will be used;

c) Level 2 comprises specialist or advanced knowledge and skills which are considered essential for individual engineering disciplines. Synoptic elements will link together items of knowledge and the use of simple illustrative examples from real-life engineering;

a) Level 3 comprises highly specialist knowledge and skills which are associated with advanced levels of study and incorporates synoptic mathematical theory and its integration with real-life engineering examples.

Students would progress from the core in mathematics by studying more subject-specific compulsory modules (electives). These would normally build upon the core modules and be expected to correspond to the outcomes associated with level 2 material. Typically level 2 modules would be distributed within the second or third year of an Engineering course due to the logistics of level 1 pre-requisites.

Students within the more numerate Engineering disciplines might be expected to take further more specialised modules incorporating mathematics on an optional basis, aimed to help match their career aspirations with appropriate theoretical formation. These modules will be at an advanced level, making use of appropriate technology, and heavily influenced with examples from engineering. Teaching of these level 3 modules would be most appropriate in year 3 or 4 of a degree course. It is likely that many of these topics already exist within specialist engineering courses and typically the mathematics is embedded and taught by engineers, mathematicians or both.

3.2.2. The Research Sample Characteristics

The sample of the research consists of 307 citizens of the Zemgala Region (Latvia) which characteristics given in the Table 3.6.

36 Mathematics for the European Engineer. A Curriculum for the twenty-first century. A report by the SEFI Mathematics Working Group, SEFI HQ, Brussels, Belgium, 2002. (ISBN 2-87352-045-0).


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Table 3.6. Characteristics of the Research Sample in Zemgale Region (N =294) Quality Category Percent

1-5 years ago 161 5-15 years ago 72 More then 15 years ago 34

How many years ago did you graduate your higher education school?

No answer 27 Female 145 Gender Male 149 Company/department manager 53 Employee 191

Position

Other 50 Higher professional 36 Profession bachelor 59 Bachelor 90 Master 72 Doctor 5

Education

No Answer 32 Biomedical Science 8 Engineering 111 Humanities 42 Arts 7 Technological Science 41 Social Science 52

Area of studies

No Answer 33 Agriculture 12 Forestry 11 Wood processing 19 Constructions 27 Informational technologies 31 Manufacturing 18 Electronics 13 Mechanical engineering 14 Economics, Banking 30 Services, sales, business 46 Public administration 23 Environment 4 Food Industry 12 Medicine 3

Employment

other 31

Similarly, the Siauliai region in the Zemgale sample is dominated by those respondents who have completed university only 1-5 years ago. Also the distribution between the genders are similar. Almost one-fourth of respondents have a master's degree graduates, but the third part of the respondents has a higher vocational education. The survey involved more than one-third who studied engineering at university but one sixth of the respondents have studied technological sciences. The number of respondents who have studied humanities and arts or social sciences is similar (nearly one-sixth).

3.2.3. Self-assessment of mathematical competences

The first diagnostic block “Self-assessment of mathematical abilities” are representing by three statements: Mathematics and the subjects, which require mathematical


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knowledge, have always been my favourite; I think mathematics, which I studied at high school (university, college), could have been more complicated and I did not understand most mathematical concepts that I studied at high school (university, college). Results in Figure 3.6. show that mathematics and the subjects, which require mathematical knowledge, have always been favourite for approximately 65% of respondents, but 30.1% did not understand most mathematical concepts. It could be mentioned that 82% of respondents who graduated from the last education institution more than 15 years ago answered with agree or strongly agree the question about mathematics as favourite subject.

Figure 3.6. Respondents’ self-assessment of mathematical competences (N =294)

In general 38.7% of respondents wanted that mathematics at universities or colleges could have been more complicated, but read this question in cross-section of the respondents professional fields, we see that the greatest dissatisfaction with mathematics courses are forestry workers and those whose work is related to information technology or electronics (Figure 3.7.).

Figure 3.7. Respondents' attitude towards complexity of mathematics studies (N =294)


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The other hand, Figure 3.8. illustrates the respondents' answers to the question about understanding mathematics concepts that were taught in the university. The results show that the greatest difficulty with mathematics understanding has been had Biomedical sciences and Arts students, but students of technological science has been the least of which did not understand math concepts.

Figure 3.8. Assessment of understanding of the mathematical concepts (N =294)

The main conclusion from the first diagnostic block is that people who found mathematics easy and have a good appreciation of its content are twice more than people who did not understand mathematics during studies. However it express the need for improving the studies process organization, implement new teaching methods and differentiated learning opportunities.

3.2.4. Conformity of mathematics at higher education school to a student’s needs

In diagnostic block “Conformity of mathematics at higher education school with a student’s needs” respondents should remember the mathematics lectures at university or college and evaluate them through the time perspectives. This block consists from six statements (Figure 3.9.).

The positive side of this block that 61.8% of respondents disagree or strongly disagree with statement that studying mathematics at university or college was wasting of time and that knowledge got in secondary school are not enough.

The second diagnostic block presented statements include the meanings, which are attributed by employed adults to learning of mathematics at universities and colleges: integration with other subjects, meaningfulness and connection with practice. 63% of respondents pointed out the mathematical relationships to other subjects - math knowledge helped them to understand other study subjects.

Most respondents appreciated the importance of mathematics in promoting professional competence: 71.8% of respondents indicate that studying mathematics develops logical thinking, accuracy and concreteness of future specialists, but 58% of surveyed claim that knowledge got in secondary school is not enough for their profession.

Mathematics as an interesting and meaningful subject was found by 48.9% of the respondents. To this question men positively respond three times more often than women. Despite it, 44.2% of respondents (of them almost twice more women than men) think that


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mathematics was taught matter-of-factly and boringly. Women three times often than men to claimed that most students did not understand the math and tried to learn rules by heart.

Figure 3.9. Assessment of mathematics teaching at higher schools (N =294)

This suggests that the need to seek out and implement new teaching methods in higher mathematics studies. Teaching methods should not include the repetition of the law or the training of the tasks solving techniques but raising the mathematics awareness, making learning interesting and developing link between course material with the profession.

It could be mentioned that the questionnaire have a free response question "In your opinion, what should be taught at mathematics lectures and how should it be taught at high school (university, college) to make acquirement useful in professional activity?". Most answers show that mathematics at universities and colleges should be taught in solving real life problems with the help of mathematics. The lecturers should explain examples of real life where is used particular teaching substance. It makes easier to perceive and understand the mathematics concepts differently, the question arises whether is it necessary.

3.2.5. Mathematics in professional activities

The third diagnostic block “Mathematics in professional practice” consists of six statements (Figure 3.10.). The key conclusion drawn from this block that mathematics in a particular professional environment is quite important. 68.2% of respondents answers are agree or strongly agree that the knowledge and abilities of mathematics, mathematical thinking helped them to achieve more in their life. To another two questions: "A person, who understands mathematics, will easily master most jobs that require thinking" and "People, who understand mathematics well, are highly assessed by employers" positive responses are similar (64.2%, 59.7%).

The third diagnostic block show that mathematical knowledge are required in order to perform successfully professional tasks by the specialists of their area as well as to be able to analyse professional literature (69.2% of the respondents), although the distribution is similar among respondents' positive and negative responses to the question about requirement


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for deeper mathematics knowledge. 43.6% of respondents charge that their professional activities connect with doing arithmetical calculations and count percentage. They occupation does not require deeper knowledge of mathematics.

Figure 3.10. Assessment of mathematics in the professional activities (N =294)

Viewpoint of the opportunities to use mathematical knowledge is similar for both

women and men. But the results indicate that the widest possible to use of mathematical knowledge in their professional activities was to economists as well as to workers in banks and in the information technology area but least the opportunity to apply mathematical knowledge was to workers in the mechanical engineering (Figure 3.11.). The general, the respondents think that math is the most widely used in construction, but most - agriculture.

Figure 3.11. Assessment of opportunities to apply their knowledge of mathematics in

respondents professional activities


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3.2.6. Assessment of the practical potential of mathematics

The results of the fourth diagnostic block show that the mathematics has a high status and include several potential values of mathematics: problem-solving and thinking developing means; tool for describing the real world and revealing human potential in working activities (Figure 3.12.).

Figure 3.12. Assessment of the practical potential of mathematics (N=294)

The fourth diagnostic block highlights the essence and importance of mathematics by

the impact of mathematics to the professional competence - 69.8% of the respondents emphasided that mathematics develops thinking, helps to make a decision in a particular situation, find new ideas. In figure 3.13. are given the respondents’ answers dependence on their professional field of occupation.

Figure 3.13. Assessment of mathematics impact to the professional competence (N=294)


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Other aspect is a correlation between mathematics and professional activities characterized by answers to the question: mathematics helps to model and analyze the problems of the real world (74.4%) and math gives an insight into the world we live (62.6%). The advantage of mathematics knowledge on the labor market is characterized by the answer to the question - a person who understands math will easily master most jobs that require thinking (64.2%). Only a small number of respondents (19%) think that mathematics is only formulas that are needed to remember or mathematics is a meaningless game with numbers which is played according to the rules created by scientists. It should be noted that men noted the essence and importance of mathematics twice more often than women.

3.2.7. A need for the improvement of mathematical knowledge The fifth diagnostic block includes statement about the interest for deepening of

mathematical knowledge in the professional area. 55.8% of respondents would like to attend the training that deals with mathematics application to solve the practical problems of their professional field. When analyzing the responses by gender of the respondents, was concluded that women expressed need to improve mathematics knowledge twice as likely than men. Analyzing the answers by the studies field what respondents graduated visible that the most need to improve math skills are to those who have been had studied engineering (Figure 3.14.).

To employees of biomedical sciences mathematical competence is almost sufficient. The need to improve the mathematics knowledge is similar to workers in technological field and humanities, but in the field of social sciences it is somewhat higher.

Public administration employees have a need for deeper knowledge on descriptive statistics and statistical deductions as well as on decision making instruments.

Figure 3.14. Assesment of a need for the improvement of mathematical knowledge

(N=294)

Whereas a very large number of respondents have the interest for deepening of mathematical knowledge, universities and colleges need to offer a variety of lifelong learning programs in mathematics.


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3.3. COMPARATIVE ANALYSIS OF THE OPINIONS OF LITHUANIAN AND LATVIAN POPULATION ABOUT MATHEMATICS

Latvian respondents like mathematics more than Lithuanian. The interviewed

Latvian graduates more often than Lithuanian ones indicated that mathematics and subjects, which required mathematical knowledge, were always the favourite ones.

Lithuanian respondents considered that mathematical courses taught at universities were more complicated. In Lithuania, the bigger part of them, in comparison with Latvia, did not understand the ideas of mathematics, which were taught at higher education school. Furthermore, more of the graduates of Latvian higher education school thought that mathematics could be more complicated.

The respondents thought that it was more interesting to study mathematics and the meaning of such studies was better understood in Latvia than in Lithuania. Lithuanian graduates thought more often than Latvian graduates that mathematics was taught dry and boring. Furthermore, more people between Latvian respondents agreed that mathematics was an interesting and meaningful subject.

The achieved study results were better assessed by Latvian respondents than by Lithuanian. Jei If in Lithuanian 81 % of the respondents agreed with the statement „The most of students did not understand mathematics, tried to remember mechanically the rules, which they were learning", in Latvia such percentage was only 63%.

Lithuanians appreciated the general educational potential of mathematics more than the interviewed Latvian residents. It shall be noticed that notwithstanding that Lithuanian respondents liked mathematics less, rather considerable part of the respondents thought that it was complicated at higher education school, it was understood by a small part of students, however they were more often sure than their neighbours that mathematical thinking helped to solve the real world problems, that a person knowing mathematics would easily master the most of works requiring thinking and learning of mathematics at higher education school develops thinking.

Table 3.7. Comparative Analysis of the Opinions of Lithuanian and Latvian Population

about Mathematics Country

Lithuania Latvia

Statement

Agree Disagree Agree Disagree

Mathematics and the subjects, which require mathematical knowledge, have always been my favourite.*

57% 43% 67% 33%

I think mathematics, which I studied at high school (university, college), could have been more complicated.**

22% 78% 39% 61%

I did not understand most mathematical concepts that I studied at high school (university, college).*

43% 57% 32% 68%

Mathematics knowledge helped me to understand other study subjects.

66% 34% 66% 34%

Mathematics was taught matter-of-factly and boringly.** 57% 43% 47% 53% Mathematics was an interesting and meaningful subjetc.* 46% 54% 52% 48% For my occupation studying mathematics at high school (university, college) is wasting of time; knowledge got in secondary school is enough.

30% 70% 39% 61%

Most of the students did not understand mathematics, tried to learn rules by heart.**

81% 19% 63% 37%

Studying mathematics develops logical thinking, accuracy and concreteness of future specialists.

81% 19% 76% 24%

The knowledge and abilities of mathematics, mathematical thinking helped me to achieve more in my life*

57% 43% 71% 29%


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People, who understand mathematics well, are highly assessed by employers.

57% 43% 62% 38%

I have a lot of opportunities to apply my knowledge of mathematics in professional activities.

48% 52% 51% 49%

My occupation does not require deeper knowledge of mathematics: it is enough to do arithmetical calculations and count percentage.

51% 49% 46% 54%

Mathematics is widely used in my professional activities.* 43% 57% 54% 46% Mathematical thinking helps to solve real world/ professional problems.**

76% 24% 59% 41%

A person, who understands mathematics, will easily master most jobs that require thinking.**

80% 20% 67% 33%

I understand mathematical symbols and a formal mathematical language which is used in my professional literature.

80% 20% 74% 26%

I would like to attend the training that deals with mathematics application to solve the practical problems of my professional field.

60% 40% 59% 41%

Mathematics helps to model and analyze the problems of the real world.

80% 20% 79% 21%

Mathematics is only formulas that are needed to remember. 16% 84% 23% 77% Mathematics is a meaningless game with numbers which is played according to the rules created by scientists.

14% 86% 20% 80%

Mathematics develops thinking, helps to make a decision in a particular situation, find new ideas.**

84% 16% 73% 27%

Mathematics gives an insight into the world we live. 69% 31% 66% 34% Chi squared test, *p<0.05, **p<0.01

Statements revealing various aspects of the practical relevance of mathematics received similar agreement in the both countries. The research did not reveal any statistically significant differences assessing the statements: “Mathematics helps to model and analyse the real problems of the world”, “Mathematics is only formulas that are needed to remember.", " Mathematics is a meaningless game with numbers which is played according to the rules created by scientists".

Assessing mathematics in the area of professional activities, the research did not reveal any statistically significant differences between the countries. The respondents of the both countries agreed on the following questions as the opportunities to apply mathematical knowledge in their professional activities, professional requirements to the knowledge and competences of mathematics (it is sufficient to know arithmetic actions and calculate percents) and etc.

The need for trainings in the area of mathematics was similar in the both countries. Notwithstanding the different assessment of learning of mathematics at higher education school, in presence of the similar level of mathematics in professional activities, similar assessment of own mathematical knowledge, the need for training is also similar.


LLIII-122 MATNET 58

Table 3.8. Differences of the attitudes towards mathematics of Latvian and Lithuanian population, the graduates of higher education schools. Kruskal Wallis Test results

Country N Average

rank Chi squared

value p value

Assessment of the general educational potential of mathematics

Lithuania Latvia

169 294

194,80 253,38

20,7 <0.001

Assessment of the practical relevance of mathematics

Lithuania Latvia

169 294

221,25 238,18

1,7 0.19

Positive attitude towards learning of mathematics

Lithuania Latvia

169 294

271,49 209,30

23,6 <0.001

Application of mathematics in professional activities

Lithuania Latvia

169 294

234,21 230,73

0,072 0.79

Unfavourable assessment of learning of mathematics at higher education school

Lithuania Latvia

169 294

202,14 249,16

13,3 <0.001

In the table, there are presented the average ranks of the respondents belonging to the

both groups. Whereas the following coding was used: from 1 – “Strongly agree“ to 4 – “strongly disagree”, the lower assessment implied the stronger expression of a quality. Lithuanian population assessed better the value of the educational potential of mathematics than Latvian population. Meanwhile, more residents of Latvia than of Lithuania emphasised a positive attitude towards mathematics, assessed more favourably organising of studying of mathematics at higher education school.

3.4. GENERALISATION AND RECOMMENDATIONS

The research revealed the need for organising of differentiated teaching of mathematics, whereas some part of students achieved the defined results of teaching of the subject of mathematics only partially and the expectations of others regarding the higher level of teaching of mathematics were not realised.

Students understood the general educational relevance of mathematics, however one half of them thing that mathematics was taught drily and drag. On the basis of the research results, we may foresee the following directions of teaching and improvement of mathematics: first, learning as the result of teaching could be more related not only with repeating of a rule , algorithm, solving examples, but also with deeper understanding of mathematics; second, striving for the conformity of the methods of teaching/learning of mathematics with the expectations of a student, the process of studies could be more involving, interesting; thir, to develop the positively expressed attitudes of students towards the relevance of mathematics in professional activities and to enhance their learning motivation on this basis.

Approximately one half of the respondents stated that they were applying the knowledge of mathematics directly in their professional activities, the other half stated that they did not need any deeper knowledge of mathematics in their profession, it was sufficient to know how to perform elementary calculations, however the practical relevance of mathematical thinking was raised by the majority of the respondents. About 60 % of the respondents specified the need to learn the applications of mathematics in relation with solving of tasks in their professional field.

In consideration of their experience in the field of professional activities, the respondents suggested the following methods of the improvement of teaching of mathematics at higher education school: to correlate learning of mathematics with the major subject of


LLIII-122 MATNET 59

studies, to reveal the links of mathematics with other subjects, to pay more attention to the practical applications of mathematics, especially the methods of applied statistics, it was suggested to the lecturers of mathematics to take more interest to the applications of mathematics in the particular area of science, on the basis of which the study program was developed.

Comparing the responses of Latvian and Lithuanian respondents, it was established that:

• Latvian respondents liked mathematics more than Lithuanian respondents. • Lithuanian respondents considered that the courses of mathematics taught at

universities were more complicated. • In Latvia, in the opinion of the respondents, it was more interesting to study

mathematics, the relevance of learning thereof was better perceived; • Latvian respondents also assessed better the achieved results of learning mathematics

than Lithuanian respondents. • Lithuanians appreciated the general educational potential of mathematics more than

the residents of Latvia. • Assessing mathematics in the environment of professional activities, the research did

not reveal any statistically significant differences between the countries.


LLIII-122 MATNET 60

IV. THE RESULTS OF INTERNAL RESEARCH

4.1. COMPARISON OF SIMILAR STUDY PROGRAMS For comparison of similar study programs were chosen 8 relative study programs (4 – LUA; 4 – SU):

• Agricultural Mechanisation (LUA) and Mechanical Engineering (ŠU); • Computer Control and Computer Science (LUA) and Informatics Engineering (ŠU); • Environmental Science (LUA) and Ecology and Environmental Sciences (ŠU); • Organisations and Public Administration (LUA) and Public Administration (ŠU).

The content of the mathematical programs were analysed through chosen indicators: 1. Comparing the number of hours of theory, practical and self-supporting work assigned

for the topics; 2. Comparison study process organization; 3. Analysing of the body of mathematical knowledge, courses, relative strengths,

weaknesses and correspondence to the needs of the regional labour market, based on indicators - SWOT analysis ( for LUA and ŠU);

4. The recomendations for improving these study programmes.

4.1.1. Agricultural Mechanisation (LUA) and Mechanical Engineering (ŠU)

1. Comparison the number of hours of theory, practical and independent work assigned for the topics (Tables 4.1.-4.4.). For detailed layout see ANNEX No.2.


Content

Con-tact

classes at LUA

Com-pare

Con-tact

classes at ŠU

Recommendation for LUA

Recommendation for ŠU

under the program 176 256 taking an average of 1 KP 16 18,3 Increase the

number of contact lessons

1. Linear algebra 15 > 10 Necessary a laboratory work.

2. Vector geometry 7 < 14 Necessary a laboratory work

3. Analytic geometry of plane and space

10 ~ 11 Necessary a laboratory work

4. Limit and continuity 7 < 10 5. Derivative of a function of one variable and its applications

20 > 16 Necessary a laboratory work

6. Differential calculation of functions in several variables

6 << 19

7. Indefinite integral 15 << 27 Necessary a laboratory work.

8. Definite integral and its applications

16 >> 9 Necessary a laboratory work Acad. hours cant be increased

9. Multiple and curvilinear integrals

0 << 18 Will be implemented

Necessary a laboratory work.

10. Differential equations 34 >> 26 Necessary a


LLIII-122 MATNET 61

laboratory work. Acad. hours cant be increased.

11. Numerical and functional series

21 < 28 Necessary a laboratory work.

12. Functions of complex variable

0 << 16 Will be implemented

13. Laplace transformation and its application

0 << 26 Necessary a laboratory work.

14. Random events 0 ~ 0 15. Probability 6 = 6 16. Random variables 9 ~ 8 Necessary a

laboratory work. 17. Descriptive statistic 1,5 ~ 2 Necessary a

laboratory work. 18. Statistical inferences 7,5 << 16 Necessary a

laboratory work.

2. Comparison study process organization (Table 4.2.).





Only for whole study program No No Only for math course Partly Partly



used)


No

No

Content modules Partly No Levels’ modules No No Introduced a

module system Form modules Partly No Mandatory - in the program is allocated the ECT

No

No

Usage of ICT By the teachers initiative Partly Partly In print form (material prepared by lector who teach)

Yes 1) Garbaliauskienė V., Laurutis A., Tiesinė algebra ir diferencinis skaičiavimas. Uždavinynas, Šiauliai, ŠU, 2008. 2) Macaitienė R., Steponavičienė V., Morkevičienė I., MS Excel taikymas: duomenų analizės ir verslo modeliavimo pagrindai. Šiauliai, 2010.

Methodical materials

In internet (only for reading) www.itf.llu.estudijas.lv

http://estudijas.llu.lv/course/view.php?id=296

1) http://su.lt/fic-matematikos-ir-informatikos/struktura/katedros/matematikos-katedra/5009-mok-medz

2) http://techno.su.lt/ ~laurutis/turinys.pdf

3)http://techno.su.lt/~laurutis/keleto_kint_funkcijos.pdf


LLIII-122 MATNET 62

4)http://techno.su.lt/~laurutis/kart_kreiv_ir_pav_integralai.pdf

An interactive materials in internet

No No

In lectures ~80 ~40 In practical tasks ~28 ~20

The average number of students

per faculty Laboratory work ~15 and ~15 - Exam /test No No Accumulative exam / test No No Measurement of

learning outcomes „Semester work” + exam / test Yes Yes Lecture Yes Yes Independent work tutoring by teacher

Yes Yes

Independent work Yes Yes Project work No No Group work No No

Teaching methods

Collaborative learning Partly Partly

3. Analysis of the body of mathematical knowledge, courses, relative strengths, weaknesses and correspondence to the needs of the regional labour market, based on indicators - SWOT analysis

3.1. SWOT analysis (Table 4.3.) and recomendations at LUA

Table 4.3. SWOT analysis for Agricultural Mechanisation at LUA Strengths

• Contents modules partially developed, two form modules harmonized: lectures and practical lessons.

• Methodological materials developed: available both printed and electronically for reading purposes.

• Several study methods are used – combined: lectures, where new material and approach to mathematic problems are being explained; Practical lessons where students do their regular work under the guidance of lecturer; Individual work of students is provided by homework and testing.

• Optimal measurement of learning outcomes is approved in practice, no accumulating exam needed – 30% of final grade consists of work during the semester, 70% performance in the exam.

• The laboratory works are integrated in Mathematics study course – “using software MathCad in study process”. For students - it allows to get know one of the Math software ; for teachers – decrease the time for checking the home works

Weaknesses • Has not been developed a distribution of hours of

mathematical topics for student's individual work • Modules of content levels have not been

implemented in Mathematics’ programmes. • No interactive methodological materials for

Mathematics have been developed for using them in internet.

• During Mathematics course less attention is focused on teaching methods like group work and Collaborative learning

• The number of students in lectures is a bit too large (~80), so that all students are able to acquire the substances discussed in lectures.

Opportunities • To define the outcomes of Mathematics study

course, harmonizing them with general approaches in Latvian legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe, since research of these documents will be fulfilled during this project.

• Revise content modules and implement level modules in programmes.

Threats • Heads of study programmes are not interested in

increasing ECTS for general subjects, including Mathematics.

• A lot of topics of Mathematics e.g. Statistics is lectured by special subject lecturers, not mathematicians.

• Most of subjects that are related to using mathematics in specific field are lectured by special subject lecturers, not mathematicians.

• In order to fulfill students’ need to use mathematics in their specific specialties, there is


LLIII-122 MATNET 63

need for special knowledge, but freshmen students don’t have that sort of knowledge. Lecturer of Mathematics needs to explain special terms that are hard to understand without context.

RECOMMENDATIONS to improve Mathematics program for Agricultural Mechanisation at LUA

1. Recommended for program’s outcomes

• To define the outcomes of Mathematics study programme, harmonizing them with general approaches in Latvian legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe; • Revise the aim of study Mathematics’ study course in study programme. The aim should be to gain the necessary knowledge, skills and competences in order to both master special study subjects and rise competitiveness of the young specialist in the work market;

2. Recommended for program’s content • Revise the contents of Mathematics and the depth of problems and to divide the

contents of topics into level modules – based on the inner research on mathematics’ study course in LUA and on interviews with heads of programmes and lecturers of special subjects (The 3rd level referable to engineering programs). • Course of study, supplemented by topics “Definite integral and its applications”

and “Functions of complex variable”. • Mathematical applications included in curricula.

3. Recommended for the study process • Based on SU (Lithuania) experience, there is a need to schedule individual work of

students for every topic of Mathematics and correlate the amount of individual work accordingly.

• Increase number of contact hours in course of Mathematics while keeping existing ECTS – the changes will need to be coordinated with heads of programmmes and Methodological commissions of corresponding faculties.

• Implement group work in practical lessons, think about tasks that would encourage collaborative learning

4. Recommended for study materials availability and accessibility to students • To improve study process, there is a need to implement a common electronic base

of methodological materials for department of Mathematics – students could access that base for theoretical and interactive learning materials.

• To solve problem concerning presenting usage of Mathematics’ opportunities in different specialties, there is a need to create an electronic base of special tasks – it will consist of different special problems and their solutions by using appropriate mathematical knowledge. Moreover, it will be useful to make a directory containing links to explanations of special terms.

3.2. SWOT analysis (Table 4.4.) and recomendations at ŠU


LLIII-122 MATNET 64

Table 4.4. SWOT analysis for Mechanical Engineering at ŠU Strengths

• Overall number of contact hours is sufficient and much higher comparing with LUA. Even though some topics are assigned with less contact hours, but it is compensated by high self-supporting number of hours (that is, much more self-supporting assignments are imposed in order to develop practical skills) and other related topics and themes, which requires the knowledge from previous topic.

• Specialized mathematic topics are tought: functions of complex variable, Laplase transformation and its applications.

• The amount of specialized methodical material (electronical and printed version) is sufficient, prepared by teachers who teaches Mathematics 1-2.

• The self-supporting work is planned carefully, seperated self-supporting work hours according to which amount the number of individual , self–supported works are assigned.

• The number of students in theoretical and practical lectures is optimal.

• Accumulative assessment system is implemented in University. The organizer of the programme identify all the evaluation formulas for learning outcomes (the self-supporting work number of hours, auditorials, practical assignments). Evaluation formula is offered in subject card (which is made free of use in online Information system). The evaluation formula of Engineering study programme in all 4 mathematics subject progrmames is such: 60% semester work+40% exam.

Weaknesses • There are no separation of contact hours

for laboratory works. That is a huge gap, because while working out practical, applicable assignments, it is usual to deal with a lot of results, which are complex and difficult to calculate. After the studies it is also necessary to manage mathematical package.

• The aims, objectives and learning outcomes of the mathematics subjects are not clearly defined in the subject describtions.

• Modules of content levels have not been implemented in Mathematics’ programmes.

• The amount of specialized methodical material (printed or electronic version), which was created by teachers who teaches Mathematics 3 and Probability and Mathematical Statistics, is insufficient.

• No interactive methodological materials for Mathematics.

• During Mathematics course less attention is focused on teaching methods like group work and collaborative learning.

Opportunities • To transform the subjects integrating more contact hours for

laboratory works. • To define the outcomes of Mathematics study course,

harmonizing them with general approaches in Lithuania legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe, since research of these documents will be fulfilled during this project.

• To revise the content of modules according to the remarks of specialists (teachers, external evaluators), integrate more specialized topics into the subject content.

• To implement the level modules in programmes. • To create purposefull specialized methodical materials for

mathematics while consulting with the lecturers who teaches speciality subjects.

• To provide additional consultations (“face to face” or Moodle) for students who have not taken the state examination of Mathematics or for students who have taken the ‘B’ level of Mathematics at school.

Threats • It can be difficult to reorganize the study

process, implementing additional hours for laboratory works.

• A part of students enter university study programmes with weaker and weaker mathematic knowledge every year. This reason is main for high drop out rates and main interfence to reach required knowledge level for higher institution students.

Analysing Mechanical Engineering study programme at SU, was found out that

teaching and assessment system is planned well in SU, the number of students learning in theoretical and practical lectures is not too high, that gives opportunity to develop better practical skills, to renew the knowledge from previuos topics and plan carefully self-supporting work and prepare specialized methodical materials. But there are some obvious weaknesses of this programme related to learning outcomes and competences. The theoretical framework of the whole internal research was based on two pillars on curriculum development: learning outcomes and competencies, curriculum development in compliance with the credit – module system, as well as two pillars for study process organization:


LLIII-122 MATNET 65

Collaborative learning in mathematics and information and communication technologies (ICT) usage in study process. After analysing the study programme of Mechanical Engineering at SU, was found out that the aims, objectives and learning outcomes of the mathematics subjects are not clearly defined in the subject describtions. Also we noticed that during Mathematics course less attention is focused on teaching methods like group work and collaborative learning and no interactive methodological materials for Mathematics. These results were not surprising, course the same problems were raised when the obstacles that influence slow socio-economic development of the Northern Lithuania and Southern Latvia regions were analysed - it was found out that: 1) regional universities (SU, LUA) use traditional teaching/learning methods which are insufficient according to demands of labour market and knowledge society to develop mathematic competences required in technological age; 2) investments in human resource development of institutions which provide intellectual services and products for the regions for various reasons is insufficient. So it is necessary to prepare specialists with abilities to use ICT and transfer knowledge and technology to the economy.

RECOMMENDATIONS

to improve Mathematics programs for Mechanical Engineering at ŠU 1. Recommended for program’s outcomes

• To define the outcomes of Mathematics study programme, harmonizing them with general approaches in Lithuania legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe.

• According to the regulations, Internal and External Researches results, comments and remarks from the head of the programme and professors who teach the subjects, it is necessary to redesign and transform the aims, objectives and learning outcomes of the mathematics subjects.

• To introduce a module system (of content levels) in Mathematics’ programmes.

2. Recommended for program’s content • To revise the contents of mathematics programms, based on the Internal and

External Researches also on the interviews with heads of programmes and lecturers of special subjects (see the specialist’s recommendations in the analysis of Internal Research). For example, to study deeper the matter: derivative of a function of one variable, definite integrals and its applications, differential equations.

• To work out and do more practical content assignments related to the speciality. 3. Recommended for the study process

• To redesign and transform the modules increasing contact hours for practical assignments (labaratory work). The systems Mathcad, Mathlab, Excel or SPSS should be integrated into the courses.

• To work on asignments which are more practical and applicable, consider the specificity of the programmes. To emphasize where mathematic subjects could be used and applied practically in concrete speciality/profession.

• Differentiate self-support tasks and assignments. • Implement group work in practical lessons, think about tasks that would

encourage collaborative learning. • To provide additional consultations for students. To implement the Moodle system

in the teaching process for self-study, self-control, consultations etc. 4. Recommended for study materials availability and accessibility to students


LLIII-122 MATNET 66

• As the amount of specialized methodical material (printed or electronic version), which would be created by teachers who teaches Mathematics 3 and Probability and Mathematical Statistics, is insufficient, so it would be the need for new publications with a content of more assignments, which could be applicable for professional purposes.

• To create the summary of basis mathematics school course in electronic version, which could be used in higher education institutions. Moreover, it will be useful to make a directory containing links to explanations of special terms.

• To create an electronic base of special tasks – it will consist of different special problems and their solutions by using appropriate mathematical knowledge.

• To implement interactive methodological materials.

4.1.2. Computer Control and Computer Science (LUA) and Informatics Engineering (ŠU) 1. Comparison the number of hours of theory, practical and independent work assigned

for the topics (see Tables 4.5.-4.8.). For detailed layout see ANNEX No. 2.


Content

Con-tact

classes at LUA

Com-pare

Con-tact

classes at ŠU



under the program 176 256 taking an average of 1 KP 16 18,3 Increase the number

of contact lessons

1. Linear algebra 11 ~ 10 Necessary a laboratory work.

2. Vector geometry 9 < 14 Necessary a laboratory work.

3. Analytic geometry of plane and space

9 ~ 11 Necessary a laboratory work.

4. Limit and continuity 10 = 10 5. Derivative of a function of one variable and its applications

19 > 16 Necessary a laboratory work. Acad. hours cant be increased

6. Differential calculation of functions in several variables

6 << 19 Increase the number of contact lessons

7. Indefinite integral 18 << 27 Necessary a laboratory work.

8. Definite integral and its applications

16 > 9 Necessary a laboratory work. Acad. hours cant be increased

9. Multiple and curvilinear integrals

0 << 18 It’s not necessary for IT

Necessary a laboratory work.

10. Differential equations 13 << 26 Necessary a laboratory work.

11. Numerical and functional series

17 << 28

12. Functions of complex variable

16 Will be implemented

13. Laplace transformation and its application

26 Necessary a laboratory work.

14. Random events Teach the

0


LLIII-122 MATNET 67

lectures 15. Probability from 6 16. Random variables pecial 8 17. Descriptive statistic subjects

departments

2 Necessary a laboratory work.

18. Statistical inferences 16 Necessary a laboratory work.

2. Comparison study process organization (Table 4.6.).








used)


No

No


module system Form modules Partly No Mandatory - in the program is allocated the ECT No No


Yes 1) Garbaliauskienė V., Laurutis A., Tiesinė algebra ir diferencinis skaičiavimas. Uždavinynas, Šiauliai, ŠU, 2008. 2) Macaitienė R., Steponavičienė V., Morkevičienė I., MS Excel taikymas: duomenų analizės ir verslo modeliavimo pagrindai. Šiauliai, 2010.

In internet (only for reading) www.itf.llu.estudija

s.lv http://estudijas.llu.lv/course/view.php?id=296

1) http://su.lt/fic-matematikos-ir-informatikos/struktura/katedros/matematikos-katedra/5009-mok-medz

2)http://techno.su.lt/~laurutis/turinys.pdf





No No



per faculty Laboratory work ~12 and ~13 - Exam /test No No Measurement of

learning outcomes Accumulative exam / test No No


LLIII-122 MATNET 68

„Semester work” + exam / test Yes Yes Lecture Yes Yes Independent work tutoring by teacher

Yes Yes


Teaching methods


3. Analysis of the body of mathematical knowledge, courses, relative strengths, weaknesses and correspondence to the needs of the regional labour market, based on indicators - SWOT analysis (see Table 4.7. (for LUA) and Table 4.8. (for (ŠU)).

3.1. SWOT analysis (see Table 4.7.) and recomendations at LUA

Table 4.7. SWOT analysis for Computer Control and Computer Science at LUA Strengths



• Several study methods are used – combined lectures, where new material and approach to mathematic problems are being explained. Practical lessons where students do their regular work under the guidance of lecturer. Individual work of students is provided by homework and testing.


• The laboratory works are integrated in Mathematics study course – “using software MathCad in study process”. For students - it allows to get know one of the Math software ; for teachers – decrease the time for checking the home works

Weaknesses • Has not been developed a distribution of hours

of mathematical topics for student's individual work


• No interactive methodological materials for Mathematics have been developed for using them in internet.









• In order to fulfill students’ need to use mathematics in their specific specialties, there is need for special knowledge, but freshmen students don’t have that sort of knowledge. Lecturer of Mathematics needs to explain special terms that are hard to understand without context.


LLIII-122 MATNET 69

RECOMMENDATIONS to improve Mathematics program for Computer Control and Computer Science at LUA


• To define the outcomes of Mathematics study programme, harmonizing them with general approaches in Latvian legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe;

• Revise the aim of study Mathematics’ study course in study programme. The aim should be to gain the necessary knowledge, skills and competences in order to both master special study subjects and rise competitiveness of the young specialist in the work market;


contents of topics into level modules – based on the inner research on mathematics’ study course in LUA and on interviews with heads of programmes and lecturers of special subjects (The 3rd level referable to engineering programs).

• Course of study, supplemented by topics “Functions of complex variable”. • Mathematical applications included in curricula.


students for every topic of Mathematics and correlate the amount of individual work accordingly;

• Increase number of contact hours in course of Mathematics while keeping existing ECTS – the changes will need to be coordinated with heads of programmmes and Methodological commissions of corresponding faculties;

• Implement group work in practical lessons, think about tasks that would encourage collaborative learning.

4. Recommended for study materials availability and accessibility to students • To improve study process, there is a need to implement a common electronic base

of methodological materials for department of Mathematics – students could access that base for theoretical and interactive learning materials.


3.2. SWOT analysis (Table 4.8.) and recomendations at ŠU


LLIII-122 MATNET 70

Table 4.8. SWOT analysis for Informatics Engineering at ŠU Strengths

• Overall number of contact hours is sufficient and much higher comparing with LUA. Even though some topics are assigned with less contact hours, but it is compensated by high self-supporting number of hours (that is, much more self-supporting assignments are imposed in order to develop practical skills) and other related topics and themes, which requires the knowledge from previous topic.

• Specialized mathematic topics are tought: functions of complex variable, Laplase transformation and its applications, probability theory and mathematical statistics.

• The amount of specialized methodical material (electronical and printed version) is sufficient, prepared by teachers who teaches Mathematics 1-2.

• The self-supporting work is planned carefully, seperated self-supporting work hours according to which amount the number of individual, self–supported works are assigned.


• Accumulative assessment system is implemented in University. The organizer of the programme identify all the evaluation formulas for learning outcomes (the self-supporting work number of hours, auditorials, practical assignments). Evaluation formula is offered in subject card (which is made free of use in online Information system). The evaluation formula of Engineering study programme in all 4 mathematics subject progrmames is such: 60% semester work+40% exam.

Weaknesses • There are no separation of contact hours for

laboratory works. That is a huge gap, because while working out practical, applicable assignments, it is usual to deal with a lot of results, which are complex and difficult to calculate. After the studies it is also necessary to manage mathematical package.

• The aims, objectives and learning outcomes of the mathematics subjects are not clearly defined in the subject describtions. .


• The amount of specialized methodical material (printed or electronic version), which was created by teachers who teaches Mathematics 3 and Probability and Mathematical Statistics, is insufficient.



Opportunities • To transform the subjects integrating more

contact hours for laboratory works. • To define the outcomes of Mathematics study

course, harmonizing them with general approaches in Lithuania legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe, since research of these documents will be fulfilled during this project.


• To implement the level modules in programmes. • To create purposefull specialized methodical

materials for mathematics while consulting with the lecturers who teaches speciality subjects.


Threats • It can be difficult to reorganize the study process,

implementing additional hours for laboratory works.

• A part of students enter university study programmes with weaker and weaker mathematic knowledge every year. This reason is main for high drop out rates and main interfence to reach required knowledge level for higher institution students.


LLIII-122 MATNET 71

Analysing Informatics Engineering study programme at SU, was found out very similar results as analysing Mechanical Engineering study programme; that is - teaching and assessment system is planned well in SU, the number of students learning in theoretical and practical lectures is not too high, that gives opportunity to develop better practical skills, to renew the knowledge from previuos topics and plan carefully self-supporting work and prepare specialized methodical materials. But there are some obvious weaknesses of this programme related to learning outcomes and competences - was found out that the aims, objectives and learning outcomes of the mathematics subjects are not clearly defined in the subject describtions. Also we noticed that during Mathematics course less attention is focused on teaching methods like group work and collaborative learning and no interactive methodological materials for Mathematics. Also we found out that there are a lot of opportunities to revise the whole study programme and integrate more specialized topics into the subject content. Also there is need to provide additional consultations (“face to face” or Moodle) for students who have not taken the state examination of Mathematics or for students who have taken the ‘B’ level of Mathematics at school. That could help to solve the drop out problem in University. A part of students enter university study programmes with weaker and weaker mathematic knowledge every year. This reason is main for high drop out rates and main interfence to reach required knowledge level for higher institution students. So it is important to implement the level modules in programmes.

RECOMMENDATIONS to improve Mathematics programs for Informatics Engineering at ŠU





2. Recommended for program’s content • To revise the contents of mathematics programms, based on the Internal and

External Researches also on the interviews with heads of programmes and lecturers of special subjects (see the recommendations of specialists in the analysis of Internal Research). For example, to include the following topics into the courses: Fourier transform, one-way functions and their applications, mathematical modeling, etc. (recommendation from Internal Research).

• To work out and do more practical content assignments related to the speciality. 3. Recommended for the study process

• To redesign and transform the modules increasing contact hours for practical assignments (labaratory work). The systems Mathcad, Mathlab, Excel or SPSS should be integrated into the courses.

• To work on asignments which are more practical and applicable, consider the specificity of the programmes. To emphasize where mathematic subjects could be used and applied practically in concrete speciality/profession.

• Differentiate self-support tasks and assignments.


LLIII-122 MATNET 72

• Implement group work in practical lessons, think about tasks that would encourage collaborative learning.

• To provide additional consultations for students. To implement the Moodle system in the teaching process for self-study, self-control, consultations etc.

4. Recommended for study materials availability and accessibility to students • As the amount of specialized methodical material (printed or electronic version),

which would be created by teachers who teaches Mathematics 3 and Probability and Mathematical Statistics, is insufficient, so it would be the need for new publications with a content of more assignments, which could be applicable for professional purposes.




4.1.3. Environmental Science (LUA) and Ecology and Environmental Sciences (ŠU) 1. Comparison the number of hours of theory, practical and independent work assigned

for the topics (see Tables 4.9.-4.12.). For detailed layout see ANNEX No. 2.


Content

Con-tact

classes at LUA

Com-pare

Con-tact

classes at ŠU



under the program 112 96 Increase the number of contact lessons

taking an average of 1 KP 16 16 1. The relationships between Biology, Ecology, Environmental and Mathematics

0

<<

6

2. Linear algebra 7 ~ 6 3. Vector geometry 7 >> 0 Will be

implemented 4. Analytic geometry of plane and space

11 >> 0 Will be implemented

5. Limit and continuity 10 > 6 6. Derivative of a function of one variable and its applications

21 >> 6 Acad. hours can be increased

7. Indefinite integral. Definite integral and its applications

29 >> 6 Increase the number of contact lessons

8. Functions in several variables 11 >> 6 9. Differential equations 16 >> 6 Increase the

number of contact lessons

10. Mathematical modelling of biological systems.

0 <<

6 is study in Master's course

11. Creating of real mathematical models

0 << 6 is study in Master's course

12. Random event and its probability.

0 6


LLIII-122 MATNET 73

13. Discrete and continuous random variables

6

14. Descriptive statistic. 6 15. Correlation and linear regression

6

16. Point and interval estimations of parameters. Confidence intervals. Normal distribution.

6

17. Hypotheses. Mistakes. 6 18. Modelling of biological systems.

Teach the

lectures from

special subjects departme

nts 6

2. Comparison study process organization (Table 4.10.)








used)


No

No



No

No


Yes Yes Alekna Z.P.,

Korsakienė D., Vieno kintamojo funkcijų integravimas. Uždavinynas. ŠU, 2004. Korsakienė D., Vieno kintamojo funkcijų diferencijavimas. ŠU, 2002. Baškienė A., Korsakienė D., Analizinės geometrijos uždaviniai.ŠU, 2003

In internet (only for reading) Yes www.itf.llu.estudijas.l

v http://estudijas.llu.lv/course/view.php?id=

296

Yes 1) http://su.lt/fic-matematikos-ir-informatikos/struktura/katedros/matematikos-katedra/5009-mok-medz






No No

The average In lectures 30 ~20


LLIII-122 MATNET 74

In practical tasks 30 ~20 number of students per faculty Laboratory work 0 ~20

Exam /test No No Accumulative exam / test No No Measurement of


Yes Yes


Teaching methods


3. Analysis of the body of mathematical knowledge, courses, relative strengths, weaknesses and correspondence to the needs of the regional labour market, based on indicators - SWOT analysis (Table 4.11. (for LUA) and Table 4.12. (for (ŠU)).

3.1. SWOT analysis (Table 4.11.) and recomendations at LUA

Table 4.11. SWOT analysis for Environmental Science at LUA Strengths.



• Passable number of students in lectures (30) in order to master topics viewed in lectures.



Weaknesses • Has not been developed a distribution of hours of

mathematical topics for student's individual work • IT have not been implemented into study process,

there are no laboratory works where corresponding problems could be solved by MathCad.







• Integrate MathCad programmes in laboratory works lessons, since two computer classes have been established in department of Mathematics.







LLIII-122 MATNET 75

RECOMMENDATIONS to improve Mathematics program for Environmental Science at LUA





contents of topics into level modules – based on the inner research on mathematics’ study course in LUA and on interviews with heads of programmes and lecturers of special subjects (The 2nd level referable to the technological study programs).

• Integrate MathCad programmes in laboratory works lessons • Mathematical applications included in curricula



• Increase number of contact hours in course of Mathematics while keeping existing ECTS and integrating practical lessons using MathCad in the study process – the changes will need to be coordinated with heads of programmmes and Methodological commissions of corresponding faculties.


4. Recommended for study materials availability and accessibility to students • To improve study process, there is a need to implement a common electronic base of

methodological materials for department of Mathematics – students could access that base for theoretical and interactive learning materials.


3.2. SWOT analysis (see Table 4.12.) and recomendations at ŠU


LLIII-122 MATNET 76

Table 4.12. SWOT analysis for Ecology and Environmental Science at ŠU Strengths

• The themes of mathematical modelling of biological systems, formal models, creating of real mathematical models, theirs reliability, probability theory and mathematical statistic are tought by mathematics specialists.

• To solidify one‘s mathematics knowledge students work on practical assignments in laboratories.



• Accumulative assessment system is implemented in University. The organizer of the programme identify all the evaluation formulas for learning outcomes (the self-supporting work number of hours, auditorials, practical assignments). Evaluation formula is offered in subject card (which is made free of use in online Information system). The evaluation formula of this programme is such: 60% semester work+40% exam.

Weaknesses • Insufficient amount of contact hours. • Geometry is not tought at all, which is

necessary for the students learning in this programme.

• Insufficient number of hours dedicated for such topics: derivatives and differential equations.

• The aims, objectives and learning outcomes of the mathematics subjects are not clearly defined in the subject describtions.


• The amount of specialized methodical material (printed or electronic version), which was created by teachers who teaches this course, is insufficient.



Opportunities • To renew the subject with such topics as vector and

analytic geometry and increase teaching hours for such topics as: derivatives and diffrential equations.

• To define the outcomes of Mathematics study course, harmonizing them with general approaches in Lithuania legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe, since research of these documents will be fulfilled during this project.


• To implement the level modules in the program. • To create purposefull specialized methodical materials

for mathematics while consulting with the lecturers who teaches speciality subjects.


Threats • A part of students enter university study

programmes with weaker and weaker mathematic knowledge every year. This reason is main for high drop out rates and main interfence to reach required knowledge level for higher institution students.

Analysing Ecology and Environmental Science study programme at SU, was found out

that the themes of mathematical modelling of biological systems, formal models, creating of real mathematical models, theirs reliability, probability theory and mathematical statistic are tought by mathematics specialists. Also it is very important that mathematics knowledge is used on practical assignments in laboratories and self-supporting work is planned carefully. But there are some obvious weaknesses of this programme related to learning outcomes and competences - was found out that the aims, objectives and learning outcomes of the mathematics subjects are not clearly defined in the subject describtions. Also we noticed that during Mathematics course less attention is focused on teaching methods like group work and collaborative learning and no interactive methodological materials for Mathematics. There is also lack of specialized methodical material (printed or electronic version), which was created by teachers who teaches this course. Also we found out that there are a lot of opportunities to


LLIII-122 MATNET 77

renew the subject with such topics as vector and analytic geometry and increase teaching hours for such topics as: derivatives and diffrential equations. Also there is opportunity to define the outcomes of Mathematics study course, harmonizing them with general approaches in Lithuania legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe, since research of these documents will be fulfilled during this project. It is important to revise the content of modules according to the remarks of specialists (teachers, external evaluators), integrate more specialized topics into the subject content.

RECOMMENDATIONS to improve Mathematics and its Applications in Ecology program

for Ecology and Environmental Science at ŠU

1. Recommended for program’s outcomes • To define the outcomes of Mathematics study programme, harmonizing them

with general approaches in Lithuania legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe.


• To introduce a module system (of content levels) in Mathematics’ programmes. 2. Recommended for program’s content

• To revise the contents of mathematics programms, based on the Internal and External Researches also on the interviews with heads of programmes and lecturers of special subjects (see the specialist’s recommendations in the analysis of Internal Research).

• To work out and do more practical content assignments related to the speciality. To pay more attention to statistics, which is one of the main subjects of mathematics when students writing a course paper.

• As it is necessary to incorporate themes of vector and analytic geometry (which are especially important in landscaping) and increase the number of teaching hours for derivatives, diffrental calculations and for mathematical statistics (this requirement arise in discussions with teachers), there is a need for additional statistics subject, leaving the teaching hours for required topics mentioned above. New subject could be as alternative or elective subject, which students could choose.

3. Recommended for the study process • Increase number of contact hours. • The system SPSS or MYSTAT should be integrated into the courses. • To work on asignments which are more practical and applicable, consider the

specificity of the programmes. To emphasize where mathematic subjects could be used and applied practically in concrete speciality/profession.

• Differentiate self-support tasks and assignments. • Implement group work in practical lessons, think about tasks that would encourage

collaborative learning. • To provide additional consultations for students. To implement the Moodle system in

the teaching process for self-study, self-control, consultations etc. 4. Recommended for study materials availability and accessibility to students

• As the amount of specialized methodical material (printed or electronic version), which would be created by teachers who teaches this course, is insufficient, so it


LLIII-122 MATNET 78

would be the need for new publications with a content of more assignments, which could be applicable for professional purposes.




4.1.4. Organisations and Public Administration (LUA) and Public Administration (ŠU) 1. Comparison the number of hours of theory, practical and independent work assigned

for the topics (Tables 4.13.-4.16.). For detailed layout see ANNEX No. 2.


Content

Con-tact

classes at LUA

Com-pare

Con-tact

classes at ŠU



under the program 32 64 taking an average of 1 KP 16 16

1. Mathematics in Economics 0 << 6 2. System of linear equations 2 < 4 3. Vectors 0 << 4 It’s not necessary

for sociology

4. Matrixes, determinants 4 << 14 We have not so much lessons

5. Analytic geometry of plane 6 It’s not necessary for Public Administrators

6. System of linear inequalities. Optimal planning.

4

7. Sets and functions 4 ~ 4 8. Elements of mathematical logic

4

9. Differentiations 6 < 10 10. Indefinite integral 2 << 6 We have not so

much lessons

11. Definite integral 4 << 8 We have not so much lessons

12. Elements of descriptive statistic

4

2. Comparison study process organization (Table 4.14.)


LLIII-122 MATNET 79



(Yes/no/ partly; data, webpage adress etc.)





used)


No

No



No

No


Yes Yes (-)

In internet (only for reading) Yes www.itf.llu.estudijas.lv

http://estudijas.llu.lv/course/view.php?id=296

Yes 1) http://su.lt/fic-matematikos-ir-informatikos/struktura/katedros/matematikos-katedra/5009-mok-medz






No No



per faculty Laboratory work - - Exam /test No No Accumulative exam / test No No Measurement of


Yes Yes


Teaching methods


3. Analysis of the body of mathematical knowledge, courses, relative strengths, weaknesses and correspondence to the needs of the regional labour market, based on indicators - SWOT analysis (Table 4.15. (for LUA) and Table 4.16. (for (ŠU)).

3.1. SWOT analysis (see Table No. 4.15) and recomendations at LUA


LLIII-122 MATNET 80

Table 4.15. SWOT analysis for Sociology of Organisations and Public Administration at LUA

Strengths • Contents modules partially developed, two form

modules harmonized: lectures and practical lessons.




Weaknesses • Too small number of ECTS. It is possible only

to touch to certain mathematical topics • Has not been developed a distribution of hours

of mathematical topics for student's individual work












RECOMMENDATIONS to improve Mathematics program for

Sociology of Organisations and Public Administration at LUA

1. Recommended for program’s outcomes • To define the outcomes of Mathematics study programme, harmonizing them with

general approaches in Latvian legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe;


2. Recommended for program’s content • Mathematical applications included in curricula • It is necessary to increase the number of ECTS for balance the study mathematics and

its application (The 1st level referable to the social study programs).


LLIII-122 MATNET 81




4. Recommended for study materials availability and accessibility to students • To improve study process, there is a need to implement a common electronic base of



3.2. SWOT analysis (see Table 4.16.) and recomendations at ŠU

Table 4.16. SWOT analysis for Public Administration at ŠU

Strengths • Overall number of contact hours is sufficient and

much higher comparing with LUA. • Specialized mathematic topics are tought: the

Leonjev model for the balance of economic system, linear inequalities in the optimal planing..



• Accumulative assessment system is implemented in University. The organizer of the programme identify all the evaluation formulas for learning outcomes (the self-supporting work number of hours, auditorials, practical assignments). Evaluation formula is offered in subject card (which is made free of use in online Information system). The evaluation formula of this subject is such: 50% semester work+50% exam.

Weaknesses • It is taught specialized mathematic topics: the

Leonjev model for the balance of economic system, linear inequalities in the optimal planing. But the amount of teaching hours is not sufficient.

• The amount of teaching hours is not sufficient for teaching professional (specialized) subjects, that‘s why the teachers requested that the programm should be linked to the subject taught, e.g. managers and administrators need linear programming and production planning, the graph theory, optimum planning, schedule making and feasibility trees.

• works. That is a huge gap, because while working out practical, applicable assignments, it is usual to deal with a lot of results, which are complex and difficult to calculate. After the studies it is also necessary to manage mathematical package.

• The aims, objectives and learning outcomes of the mathematics subjects are not clearly defined in the subject describtions. .


• The amount of specialized methodical material (printed or electronic version), which was created by teachers who teaches this course, is insufficient.



• The number of students in theoretical lectures is too high.

Opportunities • Redesign and transform math. program, increasing

contact hours for practical assignments (labaratory work).

• To define the outcomes of Mathematics study

Threats • It can be difficult to reorganize the study process

while trying to increase additional hours for practical assignments (labaratory work).

• A part of students enter university study


LLIII-122 MATNET 82

course, harmonizing them with general approaches in Lithuania legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe, since research of these documents will be fulfilled during this project.


• To implement the level modules in the program. • To create purposefull specialized methodical

materials for mathematics while consulting with the lecturers who teaches speciality subjects.


programmes with weaker and weaker mathematic knowledge every year. This reason is main for high drop out rates and main interfence to reach required knowledge level for higher institution students.

• Even if the number of students is decreasing in the university, there is no possibility to decrease the number of students in the theoretical lectures, this situation is in danger for proper development of mathematical knowledge.

• The amount of teaching hours is not sufficient for teaching professional (specialized) subjects. There is a need for additional statistics subject, but in the study programm no free hours.

Analysing Public Administration study programme at SU, was found out that there are

very similar strengths as in all the programmes analysed before, but as Social Science faculty has big groups of students, it was noticed that the number of students in theoretical lectures is too high. Even if the number of students is decreasing in the university, there is no possibility to decrease the number of students in the theoretical lectures, this situation is in danger for proper development of mathematical knowledge.The amount of teaching hours is not sufficient for teaching professional (specialized) subjects, that‘s why the teachers requested that the programm should be linked to the subject taught, e.g. managers and administrators need linear programming and production planning, the graph theory, optimum planning, schedule making and feasibility trees. The amount of teaching hours is not sufficient for teaching professional (specialized) subjects. There is a need for additional statistics subject, but in the study programm no free hours. Also we found out the same problems as in all the study programmes related to learning outcomes and competences - was found out that the aims, objectives and learning outcomes of the mathematics subjects are not clearly defined in the subject describtions. Also we noticed that during Mathematics course less attention is focused on teaching methods like group work and collaborative learning and no interactive methodological materials for Mathematics. There is also lack of specialized methodical material (printed or electronic version), which was created by teachers who teaches this course. It is important to revise the content of modules according to the remarks of specialists (teachers, external evaluators), integrate more specialized topics into the subject content.

RECOMMENDATIONS to improve Mathematics program for Public Administration at ŠU



• According to regulations, Internal and External Researches results, comments and remarks from the head of the programme and professors who teach the subjects, it is necessary to redesign and transform the aims, objectives and learning outcomes of the mathematics subjects.



LLIII-122 MATNET 83

2. Recommended for program’s content • To revise the contents of mathematics programms, based on the Internal and External

Researches also on the interviews with heads of programmes and lecturers of special subjects (see the specialist’s recommendations in the analysis of Internal Research).

• To work out and do more practical content assignments related to the speciality. Complicated mathematics is not needed; it should be shown that mathematical methods are related to the subject of studies.

• It would be purposeful to integrate the topics with more applicable content (that was request of teachers), such as: linear programming and production planning, the graph theory, optimum planning, schedule making and feasibility trees.

3. Recommended for the study process • To redesign and transform the modules increasing contact hours for practical

assignments (labaratory work). A part of the tasks should be placed on computers and computer modelling software should be used. The content not only of mathematics and economics, but also of mathematics and informatics should be coordinated. The systems Mathcad, Excel, SPSS (or other statistical programm) should be integrated into the courses.

• To work on asignments which are more practical and applicable, consider the specificity of the programmes. To emphasize where mathematic subjects could be used and applied practically in the profession of Administrators.

• Differentiate self-support tasks and assignments. • Implement group work in practical lessons, think about tasks that would encourage

collaborative learning. • To provide additional consultations for students. To implement the Moodle system in

the teaching process for self-study, self-control, consultations etc. • Optional subjects of applied mathematics are needed. Students may be motivated,

since the application of more complex mathematical structures is favourably accepted in a Bachelor’s paper. Maybe the content of mathematics should be differentiated according to the student’s specialisations. New subject could be as alternative or elective subject, which students could choose.

4. Recommended for study materials availability and accessibility to students • As the amount of specialized methodical material (printed or electronic version),

which would be created by teachers who teaches this course, is insufficient, so it would be the need for new publications with a content of more assignments, which could be applicable for professional purposes.


• To implement interactive methodological materials (for example, in Moodle system).

4.1.5. General conclusions in LUA and SU study programmes While analysing two study programs, we found out these differences:

1. According to the results of interviews and questionnaires with academical personel, was emphasysed the need for ICT use in all the study programs. Most of the study programs in SU and some of LUA don’t satisfy this need for ICT. In LUA special software in Mathematics study courses is integrated for Agricultural Mechanization, Computer Control and Computer Science study programs and in SU – only in Ecology and


LLIII-122 MATNET 84

Environmental Sciences. For Social Science study programs in both Universities the hours for laboratory works are not dedicated at all.

2. Contents modules are partially developed in LUA, however this content modules system wasn’t yet implemented in SU study programs.

3. Analysing the number of contact hours in Environmental Sciences in LUA and SU is very similar, but the number of contact hours for Engineers is less than 30 percent in LUA comparing to SU, and 50 percent less for Public Administrators in LUA comparing to SU. That allows to suppose, that LUA could increase the number of contact hours in study programs of Engineering and Social Sciences.

4. The hours needed for self-supporting (individual) work differs in LUA and SU. In SU self-supporting (individual) work is planned carefully, seperated self-supporting work hours according to which amount the number of individual , self–supported works are assigned. In LUA self-supporting (individual) work has not been developed. There is no distribution of hours of mathematical topics for student's individual work

5. LUA evaluation system is different than in SU. It was found out, that in LUA 30% of final grade consists of work during the semester, 70% performance in the exam. Accumulative assessment system is implemented in Siauliai University. The organizer of the programme identify all the evaluation formulas for learning outcomes (the self-supporting work number of hours, auditorials, practical assignments). Evaluation formula is offered in subject card (which is made free of use in online Information system). The evaluation formula of Engineering study programme in all 4 mathematics subject progrmames is such: 60% semester work+40% exam.

6. The number of studens participating in theoretical and practical lectures differs a lot. In SU - the number of students in theoretical and practical lectures is optimal for Engineering and Physical Sciences. In LUA - the number of students in lectures is a bit too large (~80), so that all students are able to acquire the substances discussed in lectures. The same situation was noticed in Social Sciences in SU, where the number of students in theoretical lectures is too high.


LLIII-122 MATNET 85

4.2. ANALYSIS OF MATH COURSES IN ŠU (Internal + External results)

4.2.1. Ecology and Environmental Sciences (1 subject) 1. Analysis of current content of mathematics subjects (ANNEX No.3, Table 3.1.) 2. Results from internal and external research – which topics-themes could be important for

corresponding programme (Table 4.17.).

Table 4.17. Accordance of mathematical and professional competences Experts' assessment



assessment

Themes (topics) Ave-rage ball

MO-DE*

Comments (what do

you think, the student

should know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the

preferred study

program content


1,75 2

To use the principal methods, to choose the best.

82,5% 2

Section of the programm remains unchanged. Additional course.


1,5 1

28,6% 1

Section of the programm remains unchanged.

More complicated statistical methods of the data analysis (market analysis, the mathematical modeling of cause-and-effect of the economic object, the use of dynamic lines and etc.)

1 0 14,3% 0


1 1

22% 0



1 ,75 2

71,4% 2 Add to

course!


1 1

32,8% 1



LLIII-122 MATNET 86


1 1

43,2% 2



1 1 30,2% 1

Additional course or themes.

Net planning (the tasks solutions of the integrated jobs planning and etc.). 1 1 35% 1



0,5 0;1

14,3% 0


0,5 0;1

22,8% 0


0,5 0;1

12,0% 0

Decision trees (the modeling of the choice of decision alternatives by the mathematical evaluation of the conditions and etc.).

0,5 0;1

14,3% 0


1 1

42,9% 1


The elements of betting theory (the mathematical modeling of decision making when the acting persons/group have conflicting aims and etc.)

1 1

22,4% 0

Questionnaires results Experts' (programme director, teaching staff) suggestions on the current content of mathematics subjects and teaching methods (ANNEX No.3, Table 3.1.)

1. The content of mathematics subjects correspond professional demands. 2. Most of unnecessary topics were mentioned in these chapters: matrixes, indefinite and

definite integrals, functions in several variable. 3. The most fundamental chapters or topics in them, which are practically applicable are:

functions, derivatives, descriptive statistic and statistical deductions. 4. As it is necessary to incorporate themes of vector and analytic geometry 5. The systems Mathcad, Mathlab (not only Excel) should be integrated into the courses.

Teaching to apply this software in mathematics should be coordinated with the teaching of Informatics.

Experts'-professionals suggestions on the content of mathematics subjects (Table 4.17.):1. Practice requires such knowledge as: descriptive statistics, geometry, integral

calcualtions, net planning, probability theory.


LLIII-122 MATNET 87

2. The topics which are not necessary and applicable in practice: more complicated statistical methods of the data analysis, the solving of equation systems, the operations with data matrixes, discrete mathematics, the application of mass service theories, decision trees, the elements of betting theory.

Results from discussions in the meetings

Experts' (programme director, teaching staff) suggestions on the current content of mathematics subjects and teaching methods

Members of the discussion: D. Korsakienė, D. Jurgaitis, R. Macaitienė, A. Klimienė, M. Kazlauskas, R. Mikaliūnaitė, L. Veriankaitė, E. Brinkytė.

1. Current content and teaching methods. Since lecturers had already familiarised with the current content of mathematics modules (when completing the questionnaires), they claimed that the content is of good quality and does not need any amendments. Nevertheless, the following requests were introduced:

1.1. To pay more attention to statistics, which is one of the main subjects of mathematics when writing a course paper. Several alternatives were proposed:

• to prepare an additional optional module in Statistics (which could be fee-paying) by coordinating the topics with the lecturers of different course specialities.

• for lecturers of mathematics, to work together with course paper supervisor when preparing a course paper. In such case, however, the question of payment arises.

1.2. To pay more attention to teaching geometry (which is especially important in landscaping).

1.3. To focus on application of differential equations. Lecturers very emphasised the general educational value of mathematics and its affect on general education.

2. Application of computer software. • To teach one packet of statistical data treatment on compulsory basis. The best

solution would be not to bind to one packet, but to teach the operation principles of several different packets.

• Before teaching the packet of mathematics, practical sessions are necessary, since initially it is more important to understand the methods of calculation rather than learn how to use concrete software.

3. Conception of teaching subjects of Mathematics. A number of possible conceptions of teaching mathematics were discussed: by emphasising the structure and proving of mathematics, and leaving applications as illustration only, by introducing the course with only a small number of proving and groundings, and learning the majority of the course through the tasks of applicable nature, providing only the tasks of applicable nature. The unanimous conclusion was reached that the best solution is to provide as much theoretical knowledge and provings as to form mathematical logics and to teach the deeper structure of mathematics, and then to solve tasks of applicable nature. The lecturers completely rejected the expedience of solving the tasks of applicable nature only.

Specialists' recommendations 1. To incorporate themes of vector and analytic geometry and increase the number of

teaching hours for derivatives, diffrental calculations and for mathematical statistics. There is a need for additional statistics subject, leaving the teaching hours for required topics mentioned above. New subject could be as alternative or elective subject, which students could choose.

2. To work out and do more practical content assignments related to the speciality. To pay more attention to statistics, which is one of the main subjects of mathematics when students writing a course paper.


LLIII-122 MATNET 88

3. Increase number of hours for laborathory works. The system SPSS or MYSTAT should be integrated into the courses.

4. To work on asignments which are more practical and applicable, consider the specificity of the programmes. To emphasize where mathematic subjects could be used and applied practically in concrete speciality.

5. Differentiate self-support tasks and assignments. 6. To provide additional consultations for students. To implement the Moodle system in

the teaching process for self-control studies, consultations etc.

4.2.2. Electrical Engineering (4 subjects) 1. Analysis of current content of mathematics subjects (ANNEX No.3, Table 3.2.) 2. Results from internal and external research – which topics-themes could be important for corresponding programme (Table 4.18.).




assessment


MO-DE*

Comments (what do


should know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the preferred

study program content


1 1

85,7% 2 Section of the

programm remains unchanged.


28,6% 1



0,5 -

It is in Master degree

programm.

14,3% 0





71,4% 2



LLIII-122 MATNET 89








0,5 0


programm.

57,2% 2


Net planning (the tasks solutions of the integrated jobs planning and etc.). 0,5 -


programm.

42,9% 1



0,5 -

14,3% 0


42,8% 1


0 0

0,0% 0


0,5 -

14,3% 0


0,5 -

42,9% 1



0,5 -

14,3% 0

Others. Fill in........................................

Z transformations (together with Laplas transform) Difference equations (together with differential equations) State variables and finite state machines. Eigen values and vectors Boole's algebra Euler method, Hevisaid theorem. + List of themes(ANNEX No.5)


LLIII-122 MATNET 90


1. The content of mathematics subjects correspond professional demands. 2. Most of unnecessary topics were mentioned in these chapters: multiple and

curvelinear integrals, numerical and functional series, random variables, descriptive statistics and statistical inferences.

3. The most fundamental chapters or topics in them, which are practically applicable are: linear algebra, vector and analytic geometry, derivatives and integrals, differential equations, functions of complex variable. Complex numbers and the subjects related to them should be on the 1nd semester, not on the 2nd semester as it is now.

4. Emphasized the topics which are very necessary for the programm: Fourier series, Z transforms, difference equations, būsenų (būvių) kintamieji ir baigtinių būvių sistemos, eigen values and vectors, Boole's algebra, exponential, sinusoidal and etc. functions, Laplase transform, discrete Fourier transform, Taylor series, Oiler method, Hevisaid theorem.

5. The practical assignments or tasks are required for the topics, which are used and applicable while studying subjects of the profession. For example emphasize of application of the superposition principle for analysis and modeling of processes in light technique systems and electric circuits.

6. The systems Mathcad, Mathlab or Excel should be integrated into the courses. Teaching to apply this software in mathematics should be coordinated with the teaching of Informatics.

Experts'-professionals suggestions on the content of mathematics subjects (Table 4.18.):1. Practice requires such knowledge as: descriptive statistics, the solving of equation

systems, the operations with data matrixes, geometry, integral calcualtions, linear programming.

2. The topics which are not necessary and applicable in practice: more complicated statistical methods of the data analysis, discrete mathematics, the application of mass service theories, decision trees, the elements of betting theory.



Members of the discussion: V. Garbaliauskienė, R. Macaitienė, A. Laurutis, S. Rimovskis, N. Ramanauskas, M. Pelikša, G. Daunys, G. Valiulis, A. Sabaliauskas, R. Lapė, K. Kasanavičius.

1. Subjects Mathematics1 and Mathematics2. Theirs contents are standard and based on the regulation. The main point of discussions was functions of complex variables. Basics of complex variable are necessary in the first, but not in the second year of studies, since it is used in the courses of automatic control and system modelling, studied in the first year. Also, to deepen the themes of limit and continuity.

2. Subject Applied Mathematics. It was proposed to rearrange the subject programme fundamentally. The following issues were discussed:

• 2 alternatives of reorganising this course were proposed: to include new topics into the course by continuing the deepening of knowledge of mathematics (to solve tasks on computer at the same time); to make this course of applied nature only by solving tasks of applied nature on computer and revising the material of semesters 1 and 2);


LLIII-122 MATNET 91

• to distribute the work so that the largest effectiveness is achieved: the tasks are provided by speciality teachers, and the students solve them with the help of the teacher of mathematics;

• since this subject will be taught until the year 2012, it was suggested to meet in September with concrete proposals regarding the concretizing of the content of this issue and to form the content of the subject then;

• to deepen the following topics into the course: Fourier series, functions of complex variable, Laplase transform;

• to name the subject Mathematics3, keeping the succession of the subject names. 3. Application of computer software - to integrate the use of computer mathematic

systems into the course of mathematics by assigning 8-10 contact hours in semesters 1, 2 and 4 and 16 contact hours in semester 3 to the computer work.

Specialists' recommendations 1. To integrate the use of computer mathematic systems (Mathcad, MathLab, Excel,

SPSS,..) into the course of mathematics by assigning 8-10 contact hours in semesters 1, 2 and 4 and 16 contact hours in semester 3 to the computer work.

2. Basics of complex variable is necessary to move in the first year of studies, since it is used in the courses of automatic control and system modelling, studied in the first year.

3. To include (deepen) the following topics into the course: Boole's algebra, Zand Fourier transformations, eigen values and vectors, solving difference equations, Euler method, Hevisaid theorem, Taylor series. These topics should be included in subjects Mathematics1-2 (for example, difference equations to deal together with differential equations), more difficult topics - in subject Applied Mathematics (for example, Z transform together with Laplas transform).

4. Boole's algebra is studied in the subject Discrete mathematics, Laplase transform - in Applied Mathematics.

5. Differentiate self-support tasks and assignments. 6. To work on asignments which are more practical and applicable, consider the

specificity of the programmes. To emphasize where mathematic subjects could be used and applied practically in conrete speciality/profession.

7. To provide additional consultations for students. To implement the Moodle system in the teaching process for self-control studies, consultations etc.

8. The results of the external research showed, that it is important enlarge some chapters in teaching/learning process: descriptive statistics, geometry, the solving of equation systems, but lecturers didn't empasized these topics as very important, in this case there is no need to enlarge it with more teaching hours, but there is need for assignments and task, which could have more applicable content. Linear programming and net planning could be tought in additional courses.

4.2.3. Electronical Engineering (4 subjects) 1. Analysis of current content of mathematics subjects (see ANNEX No.3, Table 3.3.) 2. Results from internal and external research – which topics-themes could be important



LLIII-122 MATNET 92




assessment


MO-

DE*

Comments (what do


should know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the

preferred study

program content


1,75 2

Learn to generalize research results and exclude correct conclusions.

100,0% 2



To design mathematical models of difficult systems.

25,0% 0

More complicated statistical methods of the data analysis (market analysis, the mathematical modeling of cause-and-effect of the economic object, the use of dynamic lines and etc.) 1 0;2

To lern the basis of linear programming and to use it in the tasks of processing.

25,0% 0


75,0% 2

Acad. hours cant be increased


50,0% 1


75,0% 2

Section of the programm remains unchanged. Can be inceresed acad. hours.


75,0% 2

Section of the programm remains unchanged. Can be incresedacad. hours.


LLIII-122 MATNET 93


1 0;2

75,0% 2 Additional course or themes.

Net planning (the tasks solutions of the integrated jobs planning and etc.). 0,5 0;1

50,0% 1



0,5 0;1

The combinatorics is needed for tasks of binary signal processing

25,0% 0


50,0% 1

The application of mass service theories (the mathematical modeling and optimization of administration of clients flow service and etc.) 0,25 0

The subjects of telecommunications scecializations.

0,0% 0


1 0;2

25,0% 0

Probability theory (computing of the most presumptive events, doing the tasks on insurance, mass service, quality and control, automated management systems and etc.). 1 0;2

Calculation of research data and evaluation of research results. Control system design.

50,0% 1



0,5 0;1

25,0% 0

Questionnaires results Experts' (programme director, teaching staff) suggestions on the current content of mathematics subjects and teaching methods (ANNEX No.3, Table 3.3.) 1. The content of mathematics subjects correspond professional demands. All the topics

marked as important or very important. 2. The systems Mathcad, Mathlab or Excel should be integrated into the courses. Teaching

to apply this software in mathematics should be coordinated with the teaching of Informatics.

Experts'-professionals suggestions on the content of mathematics subjects (Table 4.19.):1. The topics which are applicable in practice: descriptive statistics (but not necessary

statistical deductions), the solving of equation systems, derivatives, integral calcualtions, linear programming.


LLIII-122 MATNET 94

2. There is no need of such topics in practice: discrete mathematics (there are some doubts), the application of mass service theories, decision trees, the elements of betting theory.




1. Subjects Mathematics1 and Mathematics2. The contents are standard and based on the regulation. It is necessary for practical assignments in the topics, which are speciality subjects.

2. Subject Applied Mathematics: • 2 alternatives of reorganising this course were proposed: to include new topics

into the course by continuing the deepening of knowledge of mathematics (to solve tasks on computer at the same time); to make this course of applied nature only by solving tasks of applied nature on computer and revising the material of semesters 1 and 2).

• to work out the tasks for speciality subjects using computers: diffrenetial calculation, integrals, numerical and functional series, functions of complex variable;

• to name the subject Mathematics3, keeping the succession of the subject names. 3. Subject Probability Theory and Mathematical Statistics. Almost all themes are

necessary. A proposal was put forward to make deeper analysis during the lectures on issues of treatment, measurement, estimation of result reliability, generalisation of research results of different systems and digital signals.

4. Application of computer software: • to integrate the use of computer mathematic systems into the course of

mathematics by assigning 8-10 contact hours in semesters 1, 2 and 4 and 16 contact hours in semester 3 to the computer work.

• the teacher of Probability Theory and Mathematical Statistics has asked to increase the number of contact hours for semester 4, since at this time students must master at least one packet of statistical data treatment.

5. Differentiate self -supporting tasks, maybe according to different levels, or student preparation (stimulate motivation and self-reliance).

6. To provide additional consultations for students.



2. Exclude new topics and deepen practical knowledge in such topics as: vector geometry, derivatives, diffrenetial calculation, integrals, numerical and functional series, functions of complex variable, Laplace transfomations and its applications, statistics. To work out more practical content tasks and assignments, according to programme specificity. Emphasize the aplicability of teaching mathematic subjects in concrete speciality and programme.

3. Differentiate self-support tasks and assignments. 4. To provide additional consultations for students. To implement the Moodle system in the

teaching process for self-control studies, consultations etc.


LLIII-122 MATNET 95

5. The results of external research showed the necessity of such chapters as: descriptive statistics, the solving of equation systems, derivatives and difrential calculation, integration, linear programming. All mentioned topics are tought, but it is necessary to emphasize practical applicability in the content of the topics. Topics Linear programming and net planning are not tought, in this case, these topics could be included in the list of alternative subjects.

4.2.4. Informatics Engineering (4 subjects)

1. Analysis of current content of mathematics subjects (see ANNEX No.3, Table 3.4.). 2. Results from internal and external research – which topics-themes could be important





assessment


MO-

DE*

Comments (what do


should know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the preferred

study program content


1,25 1

77,3% 2



31,8% 1

Acad. hours can be increased


0,5 0

22,7% 0


45,5% 1



54,5% 2



45,5% 1



LLIII-122 MATNET 96


27,3% 1

Can be reduced acad. hours


0,5 0a

45,5% 1


Net planning (the tasks solutions of the integrated jobs planning and etc.). 1 1

54,5% 2



1,75 2

The task solutions of algorithmics, graph theory, kriptography.

45,5% 1

Mathematical logic (operations with predicates, Boole‘s algebra, predicate logic and etc.). 81,8

% 2



1 1 22,7% 0


1,25 1

For making variuos decisions.

36,7% 1



1 1

For calculation of probability events.

50,0% 1


The elements of betting theory (the mathematical modeling of decision making when the acting persons/group have conflicting aims and etc.) 1,25 1

The mathematical modelling of variuos decisions.

40,9% 1


Others. Fill in …………………………………..

One way functions and theirs applications. Cryptography.

Mathematical modelling using computer. Research works.

Signal processing (Fourier transformations, quanting,...) Electronics, signals theory.

Relational algebra (operations, calculations and etc.) Data bases.


LLIII-122 MATNET 97


1. The content of mathematic subjects exceed practical demands. 2. Most of the topics, which are excluded as having less practical application, are in the

chapters of: indefinite, multiple and curvilinear integras (but definite integrals - necessary), numerical and functional series, functions of complex variable, Laplace transformation.

Most of the topics, which were excluded as having less practical application were the last once, more mathematical and less applicable in practice. The opinions of respondents at this point are divergent - at one hand respondents excludes some topics as very important and core and the others sais that the same topics are not necessary (most common evaluation: 0- not necessary, 2- necessary). That was influenced by the profile of teaching subjects for the respondents (even if they work in the same study programme). 3. The most fundamental chapters or topics in them, which are practically applicable in

informatics are: linear algebra, vector geometry, derivatives and definite integrals, probability theory and statistic. One of the most core topics - Fourier series and its applications. While studying these chapters it is necessary to work out on the practical content assignments and tasks.

4. There were mentioned some additional topics, which were important: relational algebra (operations, calculations and etc.), one way functions and theirs applications; mathematicall modelling, processing of some signals.

5. The content of the tasks and assignments should be closely related with the study programme and speciality, emphasizing what concrete assignments helps to solve the topic., which is analysed.

Experts'-professionals suggestions on the content of mathematics subjects (Table 4.20.):1. To deepen the knowledge which is necessary for the practice: descriptive statistics,

net planning, discrete mathematics, mathematical logic. 2. The topics which are not necessary and applicable in practice: more complicated

statistical methods of the data analysis, integral calculations.

Results from discussions in the meetings Experts' (programme director, teaching staff) suggestions on the current content of mathematics subjects and teaching methods


1. Subjects Mathematics1 and Mathematics2 were discussed very briefly, since their content is standard and based on the regulation. Emhasized demonstration of relationships between mathematics and subjects of speciality (for egz. for Informatics Engineers, Electrical Engineers and etc.), differentiation of practical sessions, especially pointing out the solution of tasks in linear algebra (for chain analysis and analysis of models of management systems and controllability matrixes), derivatives and differential equations (for solving tasks in speed, acceleration, optimisations, system modelling and designing electronic equipment) as well as integrals (for calculating system modelling, power and energy).

2. The major discussions were regarding the subject of the 3rd semester Applied Mathematics. It was proposed to rearrange the subject programme fundamentally. The following issues were discussed:

• 2 alternatives of reorganising this course were proposed: to include new topics into the course by continuing the deepening of knowledge of mathematics (to


LLIII-122 MATNET 98

solve tasks on computer at the same time); to make this course of applied nature only by solving tasks of applied nature on computer and revising the material of semesters 1 and 2).


• to include (deepen) the following topics into the course: Laplace and Fourier transforms, solving inequality equations, one-way functions and their application, basics of theory of mass service.

• to name the subject Mathematics3, keeping the succession of the subject names. 3. The course of Probability Theory and Mathematical Statistics was mentioned as very

important, there was a suggestion to double the number of practical contact hours for this course (however, in such case significant changes in the study programmes would be required). A proposal was put forward to make deeper analysis during the lectures on issues of treatment, measurement, estimation of result reliability, generalisation of research results of different systems and digital signals.

4. Application of computer software: • the use of computer system Mathcad should be integrated into the courses. This

software would be better for students than Matlab, since it is less complicated and very powerful software. Moreover, in the very software, there is a possibility to learn, and students easily understand which operations should be performed to obtain the result. Teaching to apply this software in mathematics should be coordinated with the teaching of Informatics (because now students are taught to program, but is it necessary for a Bachelor?).

• to integrate the use of computer mathematic systems into the course of mathematics by assigning 8-10 contact hours in semesters 1, 2 and 4 and 16 contact hours in semester 3 to the computer work.

• the teacher of Probability Theory and Mathematical Statistics has asked to increase the number of contact hours for semester 4, since at this time students must master at least one packet of statistical data treatment.

Specialists' recommendations

1. Subjects Mathematics1 and Mathematics2 to remain unchanged , since theirs contents are standard and based on the regulation.

2. To integrate the use of computer mathematic systems (Mathcad, MathLab, Excel, SPSS,..) into the course of mathematics by assigning 8-10 contact hours in semesters 1, 2 and 4 and 16 contact hours in semester 3 to the computer work.

3. To include the following topics into the courses: Fourier transform, one-way functions and their applications, relation algebra, mathematical modelling using variuos software. These topics should be included in concrete tematics (for example, Fourier transformation instead Laplas transformation ).

4. To deepen the following topics into the course: linear algebra, vector geometry, derivatives and definite integrals, probability theory and statistic, solving inequality equations.

5. Differentiate self-support tasks and assignments. 6. To work on asignments which are more practical and applicable, consider the

specificity of the programmes. To emphasize where mathematc subjects could be used and applied practically in concrete speciality/profession.

7. To provide additional consultations for students. 8. To implement the Moodle system in the teaching process for self-control studies,

consultations etc.


LLIII-122 MATNET 99

9. According to the opinion of the majority of teachers, during theory lectures of mathematics, the structure of mathematics should be emphasized and theorem provings should be provided, whereas during practical sessions tasks of both theoretical and practical content should be solved.

10. The results of the external research showed, that it is important enlarge some practical topics in teaching/learning process: descriptive statistics, linear programming, net planning, discrete mathematics, mathematical logic. The themes of descriptive statistics and discrete mathematics (of course, mathematical logic) are tought in this study programme, the topics linear programming, decision trees, net planning could be included in Mathematics3 or could be created additional subjects as the alternatives.

4.2.5. Mechanical Engineering (4 subjects)

1. Analysis of current content of mathematics subjects (see ANNEX No.3, Table 3.5.)

2. Results from internal and external research – which topics-themes could be important for corresponding programme (Table 4.21.).


Experts' assessment (programme director + at

least 3 teaching staff)


assessment


MO-

DE*

Comments (what do


should know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the

preferred study

program content


0,5 0; 1

100,0% 2 Acad. hours

cant be increased


66,6% 2 Acad. hours

cant be increased


0,5 0; 1

100,0% 2 Acad. hours

cant be increased


33,3% 1



33,3% 1



LLIII-122 MATNET 100


33,3% 1



0,0% 0


0,5 0; 1

0,0% 0

Net planning (the tasks solutions of the integrated jobs planning and etc.). 0,5 0; 1 0,0% 0 Discrete mathematics (combinatorics, the tasks solutions of algorithmics, graph theory, cryptography and etc.)

0,5 0; 1

0,0% 0


33,3% 1 Additional

course. The application of mass service theories (the mathematical modeling and optimization of administration of clients flow service and etc.)

0,5 0; 1

33,3% 1 Additional

course. Decision trees (the modeling of the choice of decision alternatives by the mathematical evaluation of the conditions and etc.).

0,5 0; 1

33,3% 1 Additional

course. Probability theory (computing of the most presumptive events, doing the tasks on insurance, mass service, quality and control, automated management systems and etc.).

0,5 0; 1

66,6% 2



0,5 0; 1

66,6% 2 Additional

course.


1. The content of mathematic subjects exceed practical demands. A big part of subjects are indicated as not core/substantial or even unnecessary for mechanic engineers at all.

2. Most of the topics, which are excluded as having less practical application, are in the chapters of: linear algebra, integrals, numerical and functional series.

3. The most fundamental chapters or topics in them, which are practically applicable are: vector geometry, limit and continuity, derivatives, differential calculation..

4. The systems Mathcad, Mathlab, Excel or SPSS should be integrated into the courses.



Experts'-professionals suggestions on the content of mathematics subjects (Table 4.21.):1. Practice requires such knowledge as: descriptive statistics, statistical deductions, more

complicated statistical methods of the data analysis, probability theory, the elements of betting theory.

2. The topics which are not necessary and applicable in practice: integral calculations, linear proramming, more complicated statistical methods of the data analysis, discrete mathematics, the application of mass service theories.




1. Subjects Mathematics1 and Mathematics2. The contents are standard and based on the regulation. The practical assignments or tasks are required for the topics, which are used and applicable while studying subjects of the profession (vector geometry, limit and continuity, derivatives).

2. Subject Applied Mathematics. The following issues were discussed: • it was suggested to develop 2 different programmes for the subject of Applied

Mathematics: one for the study programmes of Electrical Engineering and Electronics and Informatics, another for the study programmes of Environmental and Professional Protection, Construction Engineering and Mechanical Engineering.


• since this subject will be taught until the year 2012, it was suggested to meet in September with concrete proposals regarding the concretizing of the content of this issue and to form the content of the subject then;

• to deepen the following topics into the course: vector geometry, differential calculation;

• to name the subject Mathematics3, keeping the succession of the subject names. 3. Subject Probability Theory and Mathematical Statistics. It was suggested to analyse

more precisely calculation questions of measurement results. 4. Application of computer software - to integrate the use of computer mathematic

systems into the course of mathematics by assigning 8-10 contact hours in semesters 1, 2 and 4 and 16 contact hours in semester 3 to the computer work.

5. To provide additional consultations for students.



2. The content of mathematic subjects exceed practical demands. A big part of subjects are indicated as not core/substantial or even unnecessary for mechanic engineers at all. But we have the regulation in LT.

3. To deepen the following topics into the courses: vector geometry, limit and continuity, derivatives, diferential calculations of functions in several variables.

4. Differentiate self-support tasks and assignments.



5. To work on asignments which are more practical and applicable, consider the specificity of the programmes. To emphasize where mathematic subjects could be used and applied practically in conrete speciality/profession.

6. To provide additional consultations for students. To implement the Moodle system in the teaching process for self-study, self-control, consultations etc.

7. The results of the external research showed, that it is important enlarge some chapters in teaching/learning process: descriptive statistics, statistical deductions, more complicated statistical methods of the data analysis, probability theory, the elements of betting theory. All of the chapters are closely related to the Probability Theory and Mathematical Statistics, in this case it is neccessary to increase teaching hours for this course.

8. Topics more complicated statistics, mathematical logic, the application of mass service theory, decision trees, the elements of betting theory could be included in Mathematics3 (eliminating some of the topics which are no longer in use) or create additional or extra subjects and provide the opportunity to choose these subjects as alternatives.

4.2.6. Physics (5 subjects)

1. Analysis of current content of mathematics subjects (see ANNEX No.3, Table 3.6.) 2. Results from internal and external research – which topics-themes could be important






assessment

Ave-rage ball

MO-

DE*

Comments (what do


should know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the

preferred study

program content


2 2

80% 2



2 2

28,6% 1



1,75 2

14,3% 0


1 1

71,4% 2





2 2

50,4% 1



2 2

42,8% 1



2 2

57,2% 2



2 2

58% 2 Additional course or themes.

Net planning (the tasks solutions of the integrated jobs planning and etc.). 1,5 2

42,9% 1



1,75 2 14,3% 0


1,5 2 42,8% 1


1,5 2 0,0% 0


1,5 2 14,3% 0


2 2 42,9% 1



1,75 2 14,3% 0

Others. Fill in ………………………………….. Book: P. Dennery, A. Krzywicki. Mathematics for physicists, New York, 1996 (ANNEX No. 4) Physisc.

Questionnaires results

Experts' (programme director, teaching staff) suggestions on the current content of mathematics subjects and teaching methods (ANNEX No.3, Table 3.6.)

The content of mathematic subjects fully satisfies all scientifis and practical demands! Experts'-professionals suggestions on the content of mathematics subjects (Table 4.22.):

1. To deepen the knowledge which is necessary for the practice: descriptive statistics, the solving of equation systems, linear programming.



2. The topics which are not necessary and applicable in practice: more complicated statistical methods of the data analysis, discrete mathematics, the application of mass service theories, decision trees, the elements of betting theory.



Members of the discussion: D. Korsakienė, D. Jurgaitis, R. Macaitienė, S. Balčiūnas, A. Janavičius, V. Šlekienė, A. Pelanskienė, S. Pelanskis, Ž. Norgėla, L. Ragulienė, J. Sitonytė.

1. Current programmes in Mathematics are good for preparing Bachelors; however the following should be taken into account:

• Solving differential equations: it is important for students not only to be able to solve them, but also to describe physical processes by differential equations; therefore a significant amount of attention should be paid to the development of these skills.

• Vector Geometry: students do not understand the concept of vector, they cannot perform vector operations. Students of Optometry do not need complicated triple integrals.

• Solving real tasks in physics during tutorials: a certain gap between the tasks in physics and mathematics is noticeable currently. Students are not able to perform operations with measures, solve equations with letter variables (parametric equations), and find it difficult to work with letter values. Relations between mathematics and physics should be shown in the Mathematics course. Tasks in mechanics are especially suitable for this.

• Revision of Mathematics: naturally, students forget a lot of subjects from school, they are “students, not computers”. A textbook was taken as an example, where necessary knowledge of mathematics is briefly revised before proceeding to the teaching of a certain topic in physics.

• Deeper studies: if a student later studies for doctoral degree, basic knowledge of mathematics is not sufficient. Specific topics of mathematics to be taught when studying physics may be found in textbooks of foreign authors.

• Cooperation of lecturers from different departments when teaching Mathematics: lecturers of Mathematics could study the main task books of Physics, discuss the content of practical tasks or concrete examples with the teachers of Physics courses.

2. Preparation of students for university studies. 2.1. Since the knowledge of mathematics of the first-year students, who have just

finished school, is of low level (students make mistakes in adding fractions or in short division), at the beginning of the mathematics course school mathematics should be revised, i.e.:

• students do not understand functions and physical meaning of derivative, they need to know the features of logarithmic, exponential and other functions. In Mechanics course, students need very simple knowledge: a student should be able to estimate the intervals of function variation, extrema, know trigonometry.

• Students find it difficult to solve geometry tasks, especially the ones which require spatial imagination. Sections of spatial objects should be illustrated by simple models. Unfortunately, the Ministry has launched the curriculum for secondary schools without including any revision.

2.2. The following possible solutions to the raised question how to implement the requirements of the curriculum if students do not possess elementary knowledge, were proposed:



• In theory lectures, to show not only how more complicated tasks should be solved, but also to remind basic things and during tutorials to teach the things that are necessary for solving physics tasks. To teach Mathematics as mathematics theory and the examples provided should be from the area of physics.

• To remind students of currently available methodical supplies (e.g. Mockus’ book) for revision of the topics of school mathematics; whereas new books for the students of Physics should not be prepared.

• To provide additional consultations for students who have not taken the state examination of Mathematics or for students who have taken the ‘B’ level of Mathematics at school (although earlier an equalizing course of physics was organised which was not effective).

3. Application of computer software. • The use of computer system Mathcad should be integrated into the course of

Mathematics. This software would be better for students studying physics than Matlab, since it is less complicated and very powerful software. Moreover, in the very software, there is a possibility to learn, and students easily understand which operations should be performed to obtain the result. Teaching to apply this software in mathematics should be coordinated with the teaching of Informatics (because now students are taught to program, but is it necessary for a Bachelor?).

• To use the software Excer which is also very powerful software (especially in statistics).

• Subject hours in Physics study programme should be distributed as follows: 32 theory lectures, 32 practical classes and 16 laboratory works.


1. The content of mathematic subjects fully satisfies all scientifis and practical demands 2. To provide additional consultations for students. 3. Differentiate self-support tasks and assignments. 4. To integrate the use of computer mathematic systems (Mathcad, MathLab, Excel,

SPSS,..) into the courses. Subject hours in Physics study programme should be distributed as follows: 32 theory lectures, 32 practical classes and 16 laboratory works.

5. Solving real tasks in physics. To work on asignments which are more practical and applicable, consider the specificity of the programmes. To emphasize where mathematic subjects could be used and applied practically in concrete speciality/profession (see Annex No.4).

6. To implement the Moodle system in the teaching process for self-control studies, consultations etc.

7. The results of the external research showed, that it is important enlarge some practical topics in teaching/learning process.

4.2.7. Public Administration, Busines Aministration (1 subject)

1. Analysis of current content of mathematics subjects (see ANNEX No.3, Table 3.7.). 2. Results from internal and external research – which topics-themes could be important for

corresponding programme (Table 4.23.).






assessment Themes (topics) Ave-

rage ball

MO-

DE*

Comments (what do you think, the student should

know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the

preferred study

program content


1 1

To make statistical decisions, to choise necessary methods.

90,0% 2

Acad. hours cant be increased. Additional course.


2 2

90% 2

Acad. hours cant be increased. Additional course.


2 2

To be able to prepare analysis of the market, to determine the relations between reasons and outcomes.

90,0% 2

Acad. hours cant be increased . Additional course.


33,3% 1

Section of the programm remains unchanged. Practical tasks.


0,7 0

33,3% 1



1 0;1;2

For approximte calculations, analysis of process change, to evaluate profitable differences while activating real actions.

33,3% 1



0,3 0

0,0% 0




60,0% 2 Additional

course.

Net planning (the tasks solutions of the integrated jobs planning and etc.). 1,3 1

To plan works and expenditures, calculating the resources of services.

60,0% 2 Additional

course.


0,3 0

20,0% 0


0,7 1

33,3% 1 Additional

course. The application of mass service theories (the mathematical modeling and optimization of administration of clients flow service and etc.)

2 2

For calculating the resources of services, giving the services.

60% 2 Additional course.


2 2

For decision-making , preparing project ptoposals.

33,3% 1 Additional

course.


2 2 Preparing strategies.

66,6% 2



1,3 1 Preparing strategies.

66,6% 2 Additional

course.

Questionnaires results

Experts' (programme director, teaching staff) suggestions on the current content of mathematics subjects and teaching methods (ANNEX No.3, Table 3.7.)

1. The content of mathematic subjects don‘t satisfied practical demands. A big part of subjects are indicated as not substantial or even unnecessary for administrators at all.

2. Most of the topics, which are excluded as having less practical application, are in the chapters of: vectors, matrixes, integrals. But some ideas disagrees with the facts from the discussions.

3. The most fundamental chapters or topics in them, which are practically applicable are: systems of linear equations, investigation of functions, minimal and maximal values, elements of descriptive statistics.

4. The program MS Excel should be integrated into the course. Experts'-professionals suggestions on the content of mathematics subjects (Table 4.23.):

1. To deepen the knowledge which is necessary for the practice: statistical methods and deductions, linear programming, net planning, mathematical logics, application of mass service theories, probability theory, the elements of betting theory.

2. The topics which are not necessary and applicable in practice: integral calculation, decisions trees.



Results from discussions in the meetings Experts' (programme director, teaching staff) suggestions on the current content of mathematics subjects and teaching methods

Members of the discussion: Sigitas Balčiūnas, Renata Macaitienė, Mindaugas Butkus, Jurgita Karalevičienė, Daiva Beržinskienė, Karolina Piaseckienė, Eugenijus Buivydas, Zita Tamašauskienė.

1. “What should be taught?” • The respondents doubt whether all the topics need to be taught; maybe some time

should be devoted for the revision of elementary mathematics. The curricula are too extensive. On the other hand, all the topics seem to be relevant, but some of them are necessary for master students, not for bachelor students (e.g. Leontjeva’s model is not taught to bachelor students). Thus in the course of mathematics the basics should be taught.

• Both economists and managers need tasks on linear programming and production planning.

• Graph theory is very important to the managers. • Optimum planning, schedule making and feasibility trees are useful. In the

respondents’ view, too much time is devoted for mathematical theory; the ratio of mathematical theory and tutorials should be changed.

2. Which topics of mathematics are applied most often? • In the lectures on econometrics, students apply derivatives and partial

derivatives. • Logarithms, exponential functions are very necessary. • It is important to teach topics that will be applied later.

3. Teaching mathematics. • Mathematics should be taught so that it is linked to other subjects. - • In the course of econometrics, derivatives should be explained as the topic that

students have not been aware of. • Computer modelling software should be used. • It has been proposed to differentiate the teaching of mathematics according to

the students’ abilities. • The programmes should be linked to the subject taught, e.g. both economists and

managers need linear programming and production planning; managers would need the graph theory, yet it should be of applied nature. Optimum planning, schedule making and feasibility trees are useful. Complicated mathematics is not needed; it should be shown that mathematical methods are related to the subject studies.

• In the respondents’ view, too much time is devoted for mathematical theory; the ratio of mathematical theory and tutorials should be changed.

• A part of mathematical topics might be taught by using a computer. A part of the tasks should be placed on computers, some laboratory works are needed. The content not only of mathematics and economics, but also of mathematics and informatics should be coordinated.

4. Problems. • It is very difficult for the students to interpret the results of calculations, for

example, they know how to calculate the average value, mode, median, but they are not able to explain what they mean.

• It is becoming clear that the students find it difficult to see the links of values expressed in formulas. This should be pursued not only in the lectures of economics, but in the lectures of mathematics as well.



• Students lack the knowledge of elementary mathematics and at university they are taught to calculate partial derivatives. Moreover, maybe it is not necessary for the students to acquire strong skills in routine mathematical operations (e.g. skills of calculating determinants).

5. Suggestions. • Optional subjects of applied mathematics are needed. Students may be

motivated, since the application of more complex mathematical structures is favourably accepted in a Bachelor’s paper. Maybe the content of mathematics should be differentiated according to the student’s specialisations.

• Teachers of mathematics and economics should coordinate the content of teaching. It would be useful to have closer cooperation; for mathematicians to review the content of textbooks and task books of economics, to choose the tasks of economical content for tutorials.

• Teachers also need some seminars on applied mathematics; maybe then they would start to apply mathematical models; then Bachelor and Master theses would not be prepared “according to one template”, more varied mathematical methods of economics would be applied.

6. Generalizations, decisions. • It is becoming clear that teachers of mathematics, when teaching mathematics

hardly take into consideration that at school some students studied mathematics at level A, and some of them studied mathematics at level B. It seems that this fact is only stated; however it has not been planned how to work with the students having different experience in mathematics (and skills of different level). There are two possible ways out of this situation. The first one: to differentiate the tasks of mathematics (as in the secondary school) according to the level chosen by a student. The second one: to provide the students who studied mathematics at school at the lower level with the conditions to acquire additional knowledge and skills.

• When teaching applied mathematics, it is useful to integrate the tasks of economical (managerial) content into the course of mathematics. This would not only motivate the students to study mathematics, but skills of interpreting digital information would be developed as well, which, according to the teachers, students lack.

• When teaching mathematics, it is reasonable to use a computer; however, here the compatibility of not only the modules of mathematics and economics, but also of mathematics and informatics modules is needed.

• It is necessary to show more extensively how the module of mathematical logics is related to the goals of the curriculum by emphasizing its general educational potential, by pursuing the development of intended skills.

• A question of educational aims should be raised with regard to each mathematical topic: which skills are developed, how are they related to the skills intended in the curriculum, why is one or another topic taught, how is it related to the profession of an economist or to the subjects taught in other modules?

• It is expedient to review the content of applied mathematics designed for economists and managers. The tasks should be differentiated in the tutorials according to the study programmes. It is recommended to supplement the programme (especially for managers) with “non-classic” topics of mathematics, such as graph theory, etc.

• Teaching of mathematics is not a unified, but an individual continual process. Therefore, when preparing the course of mathematics, it is reasonable to take into account the experience of learning mathematics at secondary school. The



teacher of mathematics should be well familiar with the contemporary programme of mathematics in the school of general secondary education in order to guide the student's further learning process of mathematics. On the other hand, learning mathematics does not end with the subject of mathematics, it continues through the entire studies. By applying the methods of mathematics in economics, students give practical content to the calculations and at the same time they extend their mathematical knowledge and improve their mathematical skills.

• It is recommended to include the modules of applied mathematics into the system of optional subjects; such modules would provide the students with the new instruments of mathematical analysis and would enable them to apply more varied mathematical economical methods in their Bachelor thesis.


1. Computer modelling software should be used. A part of mathematical topics might be taught by using a computer. A part of the tasks should be placed on computers, some laboratory works are needed. The content not only of mathematics and economics, but also of mathematics and informatics should be coordinated.

2. To include (deepen) the following topics into the course: systems of linear equations, logarithms, exponential functions, derivatives and elements of statistics.

3. It is important to add some practical topics in teaching/learning process: statistical methods and deductions, linear programming and production planning, discrete mathematics (graph theory!, mathematical logic), optimum planing, application of mass service theories, decision treas, probability theory. For these topics could be created additional subjects as the alternatives.

4. Differentiate the tasks of mathematics, self-support tasks and assignments. 5. The content of mathematics should be differentiated according to the student’s

specialisations. 6. To work on asignments which are more practical and applicable, consider the

specificity of the programmes. To emphasize where mathematc subjects could be used and applied practically in conrete speciality/profession.

7. To make special material. 8. To provide additional consultations for students.

4.3. ANALYSIS OF MATH COURSES IN LUA (Internal + External results)

The first phase of the study, questionnaires were sent to all program directors and to three special subjects’ teachers. Back was received quite incomplete and ill-considered response, which was influenced by teachers’ lax attitude towards survey as well as the large work load (preparation of the study programs accreditations materials). With regard to the special subjects teachers’ assessments of the relevance of these issue lecturers provide conflicting answers. For example, they indicate a very high need for area and volume calculation, but at the same time with zero points marked integral's concept and importance of integrating methods.

Teachers themselves incomplete responses argued with lack of knowledge of mathematics to evaluate subject matter. Specialists cannot be defined which mathematical knowledge is used, but claims that this topic is needed. Mathematics need not be denied, but the emphasis is on mathematics as a key factor in cognitive development. Specialists say that in their study subjects’ only formulas are used. In these study subjects emphasis is placed



more on understanding the process rather than on the mathematical justification. The teachers wishes will be respected, but in accordance with the internal logic of mathematics as scientific, namely, to learn a theme that requires - it is necessary to learn a series of concepts to which it is based. Thus, the summary of the 15 LUA mathematics study programs is given only in the Tables 4.24. – 4.38.

In order to evaluate the proposed mathematics study programs discussions were held with program managers and special subjects teachers.

4.3.1. Agricultural Engineering

Results from internal and external research – which topics-themes could be important for corresponding programme (Table 4.24.).





assessment

Ave-rage ball

MO-

DE*


know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the

preferred study

program content


1.33 1

Data processing of experimental measurements, calculations of measurement’s inaccuracy and evaluation of results.

41.18% 1

Remain unchanged


Application of technique in economics. Security of electrical devices in masters’ course

11.76% 0


11.76% 0


1.08 1

Evaluation of Automatic Management system’ (AMS) stability by using calculus criteria, e.g. Routh–Hurwitz stability criterion. Inertia tensor matrix.

11.76% 0

Can be reduced




1.67 1

Calculation of power and energy. Calculations of hold and storage. Product storing and cutting theories.

52.94% 2

Remain unchanged


1.27 1

Calculation of function changes’ speed. Analysis of connection of power phase and voltage. Product storing theory. AMS static inertial component in modeling and optimizing transition process. Optimal regulation of volume’s calculation for technological water.

29.41% 1

Can be reduced


1.2 1

In dynamics and kinematics course Elements of technological devices Transitions of integrating electrical and computerized chains and astatic acting mechanisms for process description and analysis.

29.41% 1

Needs to be increased, because double integrals need to be included.


2 1

Application of technique in economics.

17.65% 0

Additional course or include into program

Net planning (the tasks solutions of the integrated jobs planning and etc.).

17.65% 0


17.65% 0


17.65% 0




29.41% 1

Establish special course


29.41% 1


Probability theory (computing of the most presumptive events, doing the tasks on insurance, mass service, quality and control, automated management systems and etc.). 1.5 2

41.18% 1

Remain unchanged


23.53% 0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Agriculture engineering program prof.

K.Vārtukapteinis, director of Agriculture energy program asoc .prof. R.Selegovskis, director of Machine design and production program asoc. prof. D.Kanaška, lecturers of special courses in Technical faculty prof. I.Klegeris, prof. J.Vizbulis, prof. J.Priekulis, prof.A.Šnīders and lecturers from Department of mathematics.

1. Results of mathematics study course were discussed: the knowledge and skills need to be mastered and what competences need to be developed. We agreed that mastering knowledge and skills will be coherent to program of mathematics, but competences that are developed by mathematic studies are depending from the selection of mathematic tasks, calculation methods used and from the balance of abstract and concrete in the study process. Directors of study programs filled the survey and discussed the corrections and points of emphasis in Mathematics study course:

• Since there are changes in contents of mathematics in Latvian secondary schools, there is a need to include complex numbers in Mathematics study course;

• In order to get a clearer insight into basis of mathematical processes, there is a need to include multiple and curvilinear integrals;

• Unfortunately one cannot include operation calculation mathematics basis that can be used in modeling dynamic processes in software without additional credit points. We can make separate alternate study course for this reason.

2. In order to master and strengthen material of mathematics study course, we have to increase the number of contact hours in practical works keeping the preliminary credit points. 3. Based on wish of employers and students to study mathematics in more practical way, the present situation was analyzed:

• Studies of mathematics start in the 1st semester of 1st year when students has no idea about subjects of his/her specialty, nor has make him/her familiar with special terms and processes;

• Compiling mathematical equations, these processes must be familiar to both students both lecturers of mathematics – they have to be able to explain the terms although they are not specialists;

• There are not enough contact hours in specific topics. That’s why there is only chance to master basic concepts and rules of mathematics, but there is a lack of time for getting the insight of applications and usage.



• Compromise found: students who will be willing to get the insight of application of particular mathematics’ topic, will be able to get it via internet where tasks will be posted by lecturers of mathematics in cooperation with special subject lecturers.

4. Big problem in engineering is the great number of students that drop out in the first years in mathematics and physics subjects because a lot of students have insufficient background knowledge. To solve this, a special preliminary mathematics study course will be made in internet within the project. Experts' - professionals recommendations

• I studied engineering and work in technical service. It would be more useful to learn applications of mathematics for me and other technical people. For example, how to calculate requirements of different devices accurately and easy;

• There could be less learning of integrals. • You should study only those topics that are required by your profession, not all the

mathematics. • I would like to study more deeply. I graduated secondary school specialized in exact

sciences and didn’t learn anything new in the university. • Application of mathematics in solving problems of specific professional field. • More explanations, because students are not willing to attend additional lessons...

Conclusions:

• The results of external research state that specialists of this field need competence in Mass service theory, Decision making and Probability theory.

• Existing parts of the program: analytical geometry, statistics and probability theory – remain the same.

• All other parts of the existing program could be decreased and substituted with elements of linear programming.

• Extra emphasis should be made on data processing of experimental measurements and calculations of measurement’s inaccuracy and evaluation of results.

• Contents of Mathematics study course is not connected to particular specialty.

Recommendations for study program adjusting Recommended for program’s outcomes

• To define the outcomes of Mathematics study program, harmonizing them with general approaches in Latvian legislation on learning outcomes, profession standards and curriculum of Mathematics’ programs in Europe;

• Revise the aim of study Mathematics’ study course in study program. The aim should be to gain the necessary knowledge, skills and competences in order to both master special study subjects and rise competitiveness of the young specialist in the work market;

Recommended for program’s content

• Revise the contents of Mathematics and the depth of problems and to divide the contents of topics into level modules (The 3rd level referable to engineering programs) - based on the inner research on mathematics’ study course in LUA and on interviews with heads of programs and lecturers of special subjects.

• Course of study, supplemented by topics “Multiple and curvilinear integrals and its applications” and “Functions of complex variable”.

• Mathematical applications included in curricula



Recommended for the study process • Based on SU (Lithuania) experience, there is a need to schedule individual work of


• Increase number of contact hours in course of Mathematics while keeping existing ECTS – the changes will need to be coordinated with heads of programs and Methodological commissions of corresponding faculties.


Recommended for study materials availability and accessibility to students

• To improve study process, there is a need to implement a common electronic base of methodological materials for department of Mathematics – students could access that base for theoretical and interactive learning materials.


4.3.2. Machine Design and Manufacturing






assessment

Ave-rage ball

MO-

DE*


know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the

preferred study

program content


1.33 1

Data processing of experimental measurements, calculations of measurement’s inaccuracy and evaluation of results.

58.33%

2

Needs to be increased


Application of technique in economics. Security of electrical devices in masters’ course

25.00%

0


33.33%

1





1.08 1


16.67%

0

Can be reduced


1.67 1


66.67%

2

Remain unchanged


1.27 1


41.67%

1

Can be reduced


1.2 1


50.00%

1



2 1


8.33%

0 Additional course or include into program




8.33%

0


33.33%

1

Establish special additional course


8.33%

0


25.00%

0


8.33%

0


25.00%

0

Remain unchanged


16.67%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Agriculture engineering program prof.

K.Vārtukapteinis, director of Agriculture energy program asoc .prof. R.Selegovskis, director of Machine design and production program asoc. prof. D.Kanaška, lecturers of special courses in Technical faculty prof. I.Klegeris, prof. J.Vizbulis, prof. J.Priekulis, prof.A.Šnīders and lecturers from Department of mathematics.



• In order to get a clearer insight into basis of mathematical processes, there is a need to include multiple and curvilinear integrals

• Unfortunately one cannot include operation calculation mathematics basis that can be used in modeling dynamic processes in software without additional credit points. We can make separate alternate study course for this reason. 2. In order to master and strengthen material of mathematics study course, we have to

increase the number of contact hours in practical works keeping the preliminary credit points. 3. Based on wish of employers and students to study mathematics in more practical

way, the present situation was analyzed:





• There are not enough contact hours in specific topics. That’s why there is only chance to master basic concepts and rules of mathematics, but there is a lack of time for getting the insight of applications and usage. Compromise found: students who will be willing to get the insight of application of particular mathematics’ topic, will be able to get it via internet where tasks will be posted by lecturers of mathematics in cooperation with special subject lecturers. 4. Big problem in engineering is the great number of students that drop out in the first

years in mathematics and physics subjects because a lot of students have insufficient background knowledge. To solve this, a special preliminary mathematics study course will be made in internet within the project. Experts' - professionals recommendations

• I should be studying only those subjects that I will need in mu future profession, not whole mathematics.

• I think that one should learn those topics that I will need in the future. Can’t specify. • Mathematics should be clearer if we could see not only the theoretical part of

mathematics, but also the practical one – the part that can be used in life. • I studied engineering and work in technical service. It would be more useful to learn

applications of mathematics for me and other technical people. For example, how to calculate requirements of different devices accurately and easy

Conclusions:


• The results of external research state that specialists of this field need competence in Discreet mathematics.

• Contents of Mathematics study course is not connected to particular specialty. • Not enough learning statistics in Mathematics study program.

Recommendations for study program adjusting

Recommended for program’s outcomes




• Revise the contents of Mathematics and the depth of problems and to divide the contents of topics into level modules (The 3rd level referable to engineering programs) – based on the inner research on mathematics’ study course in LUA and on interviews with heads of programs and lecturers of special subjects.




• Mathematical applications included in curricula Recommended for the study process

• Based on SU (Lithuania) experience, there is a need to schedule individual work of students for every topic of Mathematics and correlate the amount of individual work accordingly.






4.3.3. Agricultural Power Engineering Results from internal and external research – which topics-themes could be important






assessment

Ave-rage ball

MO-

DE*


know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1;

51-100%-2)

Comments for the

preferred study

program content


1.33 1

Data processing of experimental measurements, calculations of measurement’s inaccuracy and evaluation of results. Application of technique in economics.

44.16%

1

Remain unchanged


Security of electrical devices in masters’ course

11.69%

0




16.24%

0


1.08 1


18.83%

0

Can be reduced


1.67 1


57.80%

2

Remain unchanged


1.27 1


32.47%

1

Can be reduced


1.2 1


37.01%

1





2 1


4.55%

0 Additional course or include into program


13.64

%

0


35.07%

1



25.98%

1 Establish special additional course


32.47%

1



35.07%

1



29.87%

1

Remain unchanged


4.55%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Agriculture engineering program prof. K.Vārtukapteinis, director of Agriculture energy program asoc .prof. R.Selegovskis, director of Machine design and production program asoc. prof. D.Kanaška, lecturers of special courses in Technical faculty prof. I.Klegeris, prof. J.Vizbulis, prof. J.Priekulis, prof.A.Šnīders and lecturers from Department of mathematics.



• In order to get a clearer insight into basis of mathematical processes, there is a need to include multiple and curvilinear integrals;



• Unfortunately one cannot include operation calculation mathematics basis that can be used in modeling dynamic processes in software without additional credit points. We can make separate alternate study course for this reason. 2. In order to master and strengthen material of mathematics study course, we have to


way, the present situation was analyzed: • Studies of mathematics start in the 1st semester of 1st year when students has no idea

about subjects of his/her specialty, nor has make him/her familiar with special terms and processes;



• Compromise found: students who will be willing to get the insight of application of particular mathematics’ topic, will be able to get it via internet where tasks will be posted by lecturers of mathematics in cooperation with special subject lecturers. 4. Big problem in engineering is the great number of students that drop out in the first


• You should study only those topics that are required by your profession, not all the mathematics.

• More explanations. • Every specialty has its needs; the most important thing is a creative lecturer.

Conclusions:


• Contents of Mathematics study course is not connected to particular specialty. • The results of external research state that specialists of this field need competence in

Discreet mathematics, Mathematics logics, Mass service theory, Decision making and Probability theory.

















4.3.4. Computer Control and Computer Science Results from internal and external research – which topics-themes could be important



Themes (topics) Experts' assessment (programme director + at least 3 teaching staff)

Experts' - professionals assessment

Ave-rage ball

MO-DE*

Comments (what do you think, the student should know?)

Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)

Comments for the preferred study program content


2 2

49.00%

1



2 2

43.50%

1





29.50%

1

Needs to be included into program


1 1 Physics – electric circuit calculations. Programming. Array calculations in application programming

18.50%

0

Can be reduced


1.5 1 Linear programming

27.00%

1

Remain unchanged


2 2 Process’ biochemical model calculations

24.50%

0

Can be reduced


1.5 1 Stationary analysis of biochemical models. Dynamic net analysis.

38.00%

1

Remain unchanged


36.50%

1



26.50

%

1 Establish special course


2 2 Combinatorics. The tasks solutions of algorithmics, Graph theory

30.00%

1

Remain unchanged


25.50

%

1 Needs to be increased


26.50%

1



25.50%

1





1.5 2

32.00%

1

Remain unchanged


31.50%

1


Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Computer control and computer science program asoc. prof. R.Čevare, asoc. prof. E.Stalidzāns, asoc.prof. L.Paura, asoc.prof. U.Gross and lecturers from Department of mathematics.



• In order to get a clearer insight into basis of mathematical processes, there is a need to include multiple and curvilinear integrals. 2. In order to master and strengthen material of mathematics study course, we have to






• Compromise found: students who will be willing to get the insight of application of particular mathematics’ topic, will be able to get it via internet where tasks will be posted by lecturers of mathematics in cooperation with special subject lecturers. 4. Big problem is the great number of students that drop out in the first years in

mathematics and physics subjects because a lot of students have insufficient background knowledge. To solve this, a special preliminary mathematics study course will be made in internet within the project. Experts' - professionals recommendations

• Mathematics should be taught through its application in real life problem solving (understandable to everybody).



• I think that current mathematics’ programs are god enough, but one should change the way of lecturing by rising students’ responsibility on the results, teaching students “to study”, and evaluating student’s performance constantly throughout the semester.

• Using mathematics in solving problems of specific area. • More responsive lecturers! • Every specialty needs its own mathematics’ program. Theory and practice 50-50. • Practice seminars should be harder and we could have more of those.

Conclusions:

• The results of external research state that specialists of this field need competence in Net planning, Mathematical logics, Linear programming, Decision making, Game and Probability theories.

• Contents of Mathematics study course is not connected to particular specialty. • Not enough learning statistics in Mathematics study program.














• To solve problem concerning presenting usage of Mathematics’ opportunities in different specialties, there is a need to create an electronic base of special tasks – it



will consist of different special problems and their solutions by using appropriate mathematical knowledge. Moreover, it will be useful to make a directory containing links to explanations of special terms.

4.3.5. Programming Results from internal and external research – which topics-themes could be important for corresponding programme (Table 4.28.).




Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



2 2

48.00%

1



2 2

52.00%

2

Needs to be included into program.


44.00%

1



1 1 Physics – electric circuit calculations. Programming. Array calculations in application programming

12.00%

0

Can be reduced


1.5 1 Linear programming

24.00%

0

Can be reduced


2 2 Process’ biochemical model calculations

24.00%

0

Can be reduced




1.5 1 Stationary analysis of biochemical models. Dynamic net analysis.

36.00%

1

Remain unchanged


48.00%

1



28.00%



2 2 Combinatorics. The tasks solutions of algorithmics, Graph theory

40.00%

1

Remain unchanged


36.00

%



28.00%

1



36.00%

1



1.5 2

44.00%

1

Remain unchanged or increase


48.00%

1


Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Computer control and computer science program asoc. prof. R.Čevare, asoc. prof. E.Stalidzāns, asoc.prof. L.Paura, asoc.prof. U.Gross and lecturers from Department of mathematics.

1. Results of mathematics study course were discussed: the knowledge and skills need to be mastered and what competences need to be developed. We agreed that mastering knowledge and skills will be coherent to program of mathematics, but competences that are developed by mathematic studies are depending from the selection of mathematic tasks, calculation methods used and from the balance of abstract and concrete in the study process.



Directors of study programs filled the survey and discussed the corrections and points of emphasis in Mathematics study course:


• In order to get a clearer insight into basis of mathematical processes, there is a need to include multiple and curvilinear integrals. 2. In order to master and strengthen material of mathematics study course, we have to






• Compromise found: students who will be willing to get the insight of application of particular mathematics’ topic, will be able to get it via internet where tasks will be posted by lecturers of mathematics in cooperation with special subject lecturers. 4. Big problem is the great number of students that drop out in the first years in

mathematics and physics subjects because a lot of students have insufficient background knowledge. To solve this, a special preliminary mathematics study course will be made in internet within the project. Experts' - professionals recommendations

• Mathematics should be taught through its application in real life problem solving (understandable to everybody).

• I think that current mathematics’ programs are god enough, but one should change the way of lecturing by rising students’ responsibility on the results, teaching students “to study”, and evaluating student’s performance constantly throughout the semester.

• Using mathematics in solving problems of specific area. • More responsive lecturers! • Every specialty needs its own mathematics’ program. Theory and practice 50-50. • Practice seminars should be harder and we could have more of those.

Conclusions:


Net planning, Mathematical logics, Linear programming, Decision making, Game and Probability theories.

• Not enough learning statistics in Mathematics study program.
















4.3.6. Civil Engineering Results from internal and external research – which topics-themes could be important for corresponding programme (Table 4.29.).






Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



1.25 1

Economics of application of technique

75.00%

2



31.25%

1



6.25%

0


1.75 2 Statically indefinable bar systems

25.00%

0

Can be reduced


1.75 2 Calculation of power and voltage in studying geometric and mechanical properties of models.

87.50%

2

Remain unchanged


1.75 2 Calculation of function changes’ speed. Analysis of connection of power phase and voltage. 31.25

%

1

Remain unchanged


1.75 2 Defining geometric characteristics of a crosscut.

31.25%

1

Remain unchanged




1.25 1 Borderline position method in calculating building structures. 25.00

%

0


18.75

%

0


12.50%

0


18.75

%

0


12.50%

0


1 1

50.00%

1



1 1

43.75%

1

Remain unchanged or increase


31.25%

1


Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of civil engineering program lect. R.Brencis, Lecturers of special subjects in Agriculture engineering asoc.prof. J.Kreilis, asoc.prof. S.Štrausa, aoc.prof. G. Andersons and lecturers from Mathematics department.



• In order to get a clearer insight into basis of mathematical processes, there is a need to include multiple and curvilinear integrals

• Unfortunately one cannot include operation calculation mathematics basis that can be used in modeling dynamic processes in software without additional credit points. We can make separate alternate study course for this reason.



2. In order to master and strengthen material of mathematics study course, we have to increase the number of contact hours in practical works keeping the preliminary credit points.

3. Based on wish of employers and students to study mathematics in more practical way, the present situation was analyzed:




• Compromise found: students who will be willing to get the insight of application of particular mathematics’ topic, will be able to get it via internet where tasks will be posted by lecturers of mathematics in cooperation with special subject lecturers. 4. Big problem in engineering is the great number of students that drop out in the first


• Lecturer should ask each student individually if the student understands the topic and can explain it using his/her own words.

• Everything mentioned should be learned till bachelor’s degree. After that student should choose what to study.

• Practical seminars should include tasks from real life so students can see the application of formulas and the sense of them, not that they are empty figures.

• You have to learn the connection between theory and real world where the knowledge can be used.

• Mathematics in connection with real work, calculations and making things easy. Conclusions:

• Contents of Mathematics study course is not connected to particular specialty. • The results of external research state that specialists of this field need competence

Decision making, Probability theory and Game theory. • Not enough learning statistics in Mathematics study program.







Recommended for program’s content • Revise the contents of Mathematics and the depth of problems and to divide the

contents of topics into level modules(The 3rd level referable to engineering programs) – based on the inner research on mathematics’ study course in LUA and on interviews with heads of programs and lecturers of special subjects.




• Increase number of contact hours in course of Mathematics while keeping existing ECTS and integrating practical lessons using MathCad in the study process – the changes will need to be coordinated with heads of programs and Methodological commissions of corresponding faculties.





4.3.7. Landscape Architecture Results from internal and external research – which topics-themes could be important for corresponding programme (Table 4.30.).




Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



1.75 2 Processing and accuracy evaluation of geodesic survey

92%

2





1 1 Land planning in error theory

54%

2



27%

1



1 1 Photogrammetry

8%

0

Can be reduced


2 2 Land surveying Land amelioration Geodesic networks 63%

2

Remain unchanged


1.75 2 Geodesic survey in error theory and equalization theory

41%

1

Remain unchanged


1.75 2 Engineering and geodesic work Geodesy Geodesic networks

22%

1

Remain unchanged


3%

0


1 1 Organization of geodesic works

13%



11%

0


11%

0



13%

0





21%

0


25%

0


18%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Landscape architecture program Anda Jankava, Director of Environmental science program Inga Grīngelde, lecturers of special subjects in Agriculture engineering faculty lect. I.Bīmane, lect. M.Kronbergs, lect. A.Ratkēvičs, lect V.Vircavs and and lecturers from Department of mathematics.



• For successful master’s degree studies, there is a need for basic knowledge on two argument functions, students must know how to calculate partial derivatives and extremes in multiple argument functions; operate with matrix and vectors, calculate inverse matrix, preferably – will be able to calculate linear equation systems with Gauss-Jordan method.

2. Since IT is developing constantly, there is a necessity to master IT software. Taking in account possibilities of Mathematics department (Mathematics department has two computer classes and MathCad licenses are already bought), lecturers of special courses suggested MathCad and Excel within Mathematics study course. Unfortunately the directors of programs do not see the way to increase credit point number of Mathematics study course, but the decision was made that the number of contact hours will be increased and 1 credit point added.








Experts' - professionals recommendations

• In engineering – only what matters to practical work. Maybe the most of things listed is needed, but we didn’t study analysis of data that is needed for information analysis in procurements, surveys etc.

• We need a basic course in higher mathematics where you could learn both theory and practice. Tasks could be more complicated.

• The existing model of lecturing is optimal. The only thing I could suggest is that students could be divided by their levels of knowledge, insight and wishes.

• You have to study theory and practical application for real problem solving of mathematics in university, to develop logical thinking.

Conclusions:

• Mathematics study course doesn’t include sufficient amount of statistics. • Contents of Mathematics study course is not connected to particular specialty.






• Revise the contents of Mathematics and the depth of problems and to divide the contents of topics into level modules (The 2nd level referable to the technological study programs) – based on the inner research on mathematics’ study course in LUA and on interviews with heads of programs and lecturers of special subjects;

• Integrate MathCad programs in laboratory works lessons; • Include complex numbers and multiple argument functions in Mathematics study

course • Mathematical applications included in curricula • Make changes in LAIS of program, according topics of mathematics and the

application of this knowledge in special subjects (e.g., derivatives in the 1st semester because it will be needed in Physics to calculate speed and acceleration in the 2nd semester)





• Increase number of contact hours from 112 to 128 in course of Mathematics while keeping existing ECTS and integrating practical lessons using MathCad in the study process – the changes will need to be coordinated with heads of programmmes and Methodological commissions of corresponding faculties.





4.3.8. Enironmental Science Results from internal and external research – which topics-themes could be important for corresponding programme (Table 4.31.).




Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



1.75 2 Processing and accuracy evaluation of geodesic survey

100.00%

2



1 1 Land planning in error theory

66.67%

2



33.33%

1





1 1 Photogrammetry

0.00%

0

Can be reduced


2 2 Underground water hydrology Geodesic nets 100.0

0%

2

Remain unchanged


1.75 2 Land surveying Hydraulics Pumps and pumping Canalization, Sewage treatment

66.67%

2

Remain unchanged


1.75 2 Engineering and geodesic work Hydraulics Pumps and pumping Canalization, Sewage treatment

33.33%

1

Remain unchanged


0.00%

0



0.00%



0.00%

0


0.00

%

0



0.00%

0



0.00%

0


33.33%

1





0.00%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Landscape architecture program Anda Jankava, Director of Environmental science program Inga Grīngelde, lecturers of special subjects in Agriculture engineering faculty lect. I.Bīmane, lect. M.Kronbergs, lect. A.Ratkēvičs, lect V.Vircavs and and lecturers from Department of mathematics.


• Since there are changes in contents of mathematics in Latvian secondary schools, there is a need to include complex numbers in Mathematics study course; • For successful master’s degree studies, there is a need for basic knowledge on two

argument functions, students must know how to calculate partial derivatives and extremes in multiple argument functions; operate with matrix and vectors, calculate inverse matrix, preferably – will be able to calculate linear equation systems with Gauss-Jordan method.









Experts' - professionals recommendations • You should study only those topics that are required by your profession, not all the

mathematics. • Application of mathematics in solving problems of specific professional field. • More logics • Mathematics should be lectured in more easy way, nothing really complicated but

close to everyday application. • Practical seminars should include tasks from real life so students can see the

application of formulas and the sense of them, not that they are empty figures. Conclusions:

• Mathematics study course doesn’t include sufficient amount of statistics. • Contents of Mathematics study course is not connected to particular specialty. • The results of external research state that specialists of this field need competence

in Probability theory.






• Integrate MathCad programs in laboratory works lessons; • Include complex numbers and multiple argument functions in Mathematics study



Recommended for the study process


• Increase number of contact hours from 112 to 128 in course of Mathematics while keeping existing ECTS and integrating practical lessons using MathCad in the study process – the changes will need to be coordinated with heads of programs and Methodological commissions of corresponding faculties.




Recommended for study materials availability and accessibility to students • To improve study process, there is a need to implement a common electronic base of



4.3.9. Landscape Architecture and Planning Results from internal and external research – which topics-themes could be important for corresponding programme (Table 4.32.).




Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



1 1

75.00%

2



50.84%

2



24.17%

1



12.50%

0

Can be reduced


2 2 Road route calculation, interpolation, calculations of twists in the road, parabolic lines and clothoides.

65.00%

2

Remain unchanged




1.75 2 Calculations of twists in the road, bends and curves. Light absorption description.

45.84%

1

Remain unchanged


1.25 1 Thermal conductivity process description.

36.67%

1

Remain unchanged


12.50%

0


12.50

%

0


10.00%

0


7.50

%

0


12.50%

0


7.50%

0


1.25 1 Insight of gas MCT

26.67%

1



7.50%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Landscape architecture and planning program lect. N.Nitavska, lecturers of special subjects in Agriculture engineering prof. M.Urtāne, doc.S.Rubene, lect.B.Helfriča and lecturers from Department of mathematics.




1. Directors of study programs filled the survey and discussed the corrections and points of emphasis in Mathematics study course:

• Unfortunately higher mathematics is almost not used in subjects of Landscape architecture and planning specialty. Calculations are made with ready formulas, and only knowledge of arithmetic is sufficient.

• Lecturers would like the students to have mastered some IT software, e.g. MathCad. But, since the number of credit points is small for mathematics (4), integrate laboratory works in computer classes is not possible. Lecturers of the study program will consider increasing the number of credit points of mathematics.







• Mathematics should be taught through its application in real life problem solving. • Should be thought more deeply, I studied exact subjects in secondary school and

haven’t learned anything new in the university. • Practical seminars should offer tasks with real life examples, so you can see the

sense of using those formulas – otherwise they seem as plain and empty numbers. • Maybe all the listed is necessary for work, but we didn’t study stuff like data

analysis that is useful in procurements, surveys etc. in engineering program. Conclusions:

• Contents of Mathematics study course is not connected to particular specialty. • Not enough learning statistics in Mathematics study program. • The results of external research state that specialists of this field need competence

in Probability theory.







• Mathematical applications included in curricula • It is necessary to increase the number of ECTS for balance the study mathematics and

its application • Make changes in LAIS of program, according topics of mathematics and the








4.3.10. Food Science





Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)





2 2 Entrepreneurship in food industry

100.00%

2

Remain unchanged


33.33%



33.33%

1



33.33%

1

Remain unchanged


1.75 2 Image of devices and processes

33.33%

1

Remain unchanged


2 2 Functions of economical dependences - functions of supply, demand, costs, income and profit. Technologic devices in establishing profit step.

33.33%

1

Remain unchanged


2 2 Calculations of area and volume taken by technological devices and premises

33.33%

1

Remain unchanged


33.33%

1



0.00

%

0


33.33%

1





0.00

%

0


1 1 Production organization, service organization. Logistics

0.00%

0


33.33%

1



33.33%

1



0.00%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Food science program prof. Inga Ciproviča, lecturers of special study courses in Food technology faculty prof. R.Galaburda, lect. V.Miķelsone, lect. I.Millere, and lecturers from Department of mathematics.




• In order to get a clearer insight into basis of mathematical processes, there is a need to include multiple and curvilinear integrals.

• In order to master and strengthen material of mathematics study course, we have to increase the number of contact hours in practical works keeping the preliminary credit points.








4. Big problem is the great number of students that drop out in the first years in mathematics and physics subjects because a lot of students have insufficient background knowledge. To solve this, a special preliminary mathematics study course will be made in internet within the project.

5. Another problem rose. In the special courses of Food technology (except for heat processes) there is no use for higher economics. In the process calculations, a formula is used, and simple arithmetic is sufficient. At the same time, knowledge of higher mathematics is neded in heat processes and master’s program. Experts' - professionals recommendations

• You should study only those topics that are required by your profession, not all the mathematics.

• You have to study theory and practical application for real problem solving. • One should study only what the profession needs.

Conclusions:

• Mathematics study course doesn’t include sufficient amount of statistics. • Contents of Mathematics study course is not connected to particular specialty. • The results of external research state that specialists of this field need

competence in Probability theory, Decision making theory and Discreet mathematics.








• Mathematical applications included in curricula





• Increase number of contact hours in course of Mathematics while keeping existing ECTS and integrating practical lessons using MathCad in the study process – the changes will need to be coordinated with heads of programs and Methodological commissions of corresponding faculties.





4.3.11. Food Technology





Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



1 1

79.17%

2



29.17%

1



33.33%

1





25.00%

0

Can be reduced


1.75 2 Image of devices and processes

50.00%

1

Remain unchanged


2 2 Functions of economical dependences - functions of supply, demand, costs, income and profit. Technologic devices in establishing profit step.

37.50%

1

Remain unchanged


1.75 2 Calculations of area and volume taken by technological devices and premises

41.67%

1

Remain unchanged


20.83%

0


4.17

%

0


33.33%

1



4.17

%

0


1.5 1 Production organization, service organization. Logistics

12.50%

0



20.83%

0


29.17%

1





8.34%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Food technology program prof. Inga Ciproviča, lecturers of special study courses in Food tchnology faculty prof. R.Galaburda, lect. V.Miķelsone, lect. I.Millere, and lecturers from Department of mathematics.




• For successful master’s degree studies, there is a need for basic knowledge on two argument functions, students must know how to calculate partial derivatives and extremes in multiple argument functions; operate with matrix and vectors, calculate inverse matrix, preferably – will be able to calculate linear equation systems with Gauss-Jordan method. 3. Since IT is developing constantly, there is a necessity to master IT software.

Taking in account possibilities of Mathematics department (Mathematics department has two computer classes and MathCad licenses are already bought), lecturers of special courses suggested MathCad and Excel within Mathematics study course. Unfortunately the directors of programs do not see the way to increase credit point number of Mathematics study course, but the decision was made that the number of contact hours will be increased and 1 credit point added.




• There are not enough contact hours in specific topics. That’s why there is only chance to master basic concepts and rules of mathematics, but there is a lack of time for getting the insight of applications and usage. 5. Another problem rose. In the special courses of Food technology (except for

heat processes) there is no use for higher economics. In the process calculations, a formula is used, and simple arithmetic is sufficient. Compromise found: students who will be willing to get the insight of application of particular mathematics’ topic, will be able to get it via internet where tasks will be posted by lecturers of mathematics looking for information in Internet..



Experts' - professionals recomendations • You should study only those topics that are required by your profession, not all the

mathematics. • I think that one should learn those topics that I will need in the future. Can’t specify. • You have to study theory and practical application for real problem solving of

mathematics in university, to develop logical thinking. Conclusions:

• Mathematics study course doesn’t include sufficient amount of statistics. • Contents of Mathematics study course is not connected to particular specialty. • The results of external research state that specialists of this field need competence in

Probability theory and Discreet mathematics.





• Revise the contents of Mathematics and the depth of problems and to divide the contents of topics into level modules (The 2nd level referable to the technological study programs) – based on the inner research on mathematics’ study course in LUA and on interviews with heads of programmes and lecturers of special subjects;

• Integrate MathCad programmes in laboratory works lessons; • Include complex numbers and multiple argument functions in Mathematics study





• Increase number of contact hours from 80 to 96 in course of Mathematics while keeping existing ECTS and integrating practical lessons using MathCad in the study process – the changes will need to be coordinated with heads of programmmes and Methodological commissions of corresponding faculties.




Recommended for study materials availability and accessibility to students • To improve study process, there is a need to implement a common electronic base of



4.3.12. Catering and Hotel Management





Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



75.00%

2

Needs to include statistics.


34.17%

1



24.17%

0


29.17%

1

Remains the same


2 2 Organization of production. Organization of service. Technological devices.

31.67%

1

Remains the same




1 1 Hospitality and catering management (determining profit margin)

29.17%

1

Can be decreased


1 1 Possibly, in Chemistry.

36.67%

1

Can be decreased


2 2 Organization of production. Hospitality and catering management Logistics

29.17%

1



1 1 Organization of production. Organization of service. Logistics

12.50%

0


26.67%

1



7.50

%

0


1 1 Organization of production. Organization of service. Logistics

12.50%

0

Additional course


24.17%

1

Additional course


26.67%

1

Additional course or include into program.


1 1 Hospitality service, Hospitality and catering management

7.50%

0

Additional course

Discussions results - experts' (programme director and teaching staff) recomendations

Members of discussion – survey: Director of Catering and hotel management program lect. I.Milere, lecturers of special study courses in Food technology faculty prof. I.Ciproviča, prof. Skrupskis, lect.V.Rozenbergs, asoc.prof. A.Blija, and lecturers from Department of mathematics.




• In the special courses of Catering and hotel management program there is no use for higher economics. In the process calculations, a formula is used, and simple arithmetic is sufficient. At the same time, knowledge of higher mathematics is neded in heat processes and master’s program.

• Lecturers would like the students to have mastered some IT software, e.g. MathCad. But, since the number of credit points is small for mathematics (3), integrate laboratory works in computer classes is not possible. Lecturers of the study program will consider increasing the number of credit points of mathematics. 2. Based on wish of employers and students to study mathematics in more practical







• Use the possibilities of computers, using Excel. • Every profession needs different mathematics – civil engineers need one, food

technology students – another, and machine engineers – another. It makes no sense to make everybody study everything – lectures become boring and uninteresting.

• You should learn what is really needed. Conclusions:

• Parts of linear algebra and analytical geometry remain the same. • Parts of decimal calculations and integrals could be decreased and including statistics

and linear programming into program. • The need to include linear programming results from both inner and external

researches. • Mathematics study course doesn’t include sufficient amount of statistics. • The results of external research state that specialists of this field need competence in

Probability theory, Decision making theory and Discreet mathematics and Probability theory. The internal research states that specialists in this field need competence in Mass service theory and Net planning.
















4.3.13. Wood Processing







Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



58.33%

2



33.33%

1



41.67%

1



1.75 2 Hydrothermal processing of wood Building structures

50.00%

1

Remain unchanged


1.5 1 Building structures Theoretical mechanics Resistance of materials

50.00%

1

Remain unchanged


2 2 Wood product manufacturing Production of glued materials Production flow planning

41.67%

1

Remain unchanged


1.75 2 Building structures Wood cutting processes

33.33%

1

Remain unchanged




50.00%

0


8.33

%

0


0.00%

0


8.33

%

0


25.00%

0


25.00%

0


1 1

41.67%

1



33.33%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Wood processing program asoc. prof. Uldis Spulle, Director of Forestry program Gints Priedītis, lecturers of special subjects in Forest faculty prof. H.Tuherms, prof. L.Līpiņš, Doc. E.Bukrāns, and lecturers from Department of mathematics.













• Every topic should be mastered through example of practical application. • More practical examples needed. • After completing the theoretical course of mathematics, students would have to use

mathematical software. Usually after graduating the knowledge soon gets forgotten, but if multiple argument calculations needed, it is easier to freshen up the knowledge of software used. Today it is more important what you can use, not the things you can remember.

• I’m satisfied with current situation. • More lectures! We need more examples and tasks connected to real life where these

topics of mathematics can be used. • Application of mathematics in decision making both in work and everyday life. • I needed more specific calculations for my specialty – those who are necessary! • More practice!!!

Conclusions:


Probability theory and Statistics.



• Revise the aim of study Mathematics’ study course in study program. The aim should be to gain the necessary knowledge, skills and competences in order to both



master special study subjects and rise competitiveness of the young specialist in the work market;



• Integrate MathCad in laboratory works lessons; • Include complex numbers and multiple argument functions in Mathematics study

course • Make changes in LAIS of program, according topics of mathematics and the









4.3.14. Forest Engineering Results from internal and external research – which topics-themes could be important for corresponding programme (Table 4.37.).






Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



1.25 1 Processing of taxation data

77.78%

2



44.44%

1



44.44%

1



1.25 1 Hydrothermal processing of wood

33.33%

1

Remain unchanged


2 2 Wood trunk, components of rotation body, stereo metric solids characteristic to wood trunks.

55.56%

2

Remain unchanged


1.25 1

22.22%

0

Can be reduced


1.5 1 Calculating the volume of wood trunk and calculating the volume of separate parts of the wood trunk

22.22%

0

Can be reduced




1 1 Optimization methods of forestry planning

33.33%

1



22.22

%

0


22.22%

0


11.11

%

0


33.33%

1



44.44%

1



33.33%

1



22.22%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Wood processing program asoc. prof. Uldis Spulle, Director of Forestry program Gints Priedītis, lecturers of special subjects in Forest faculty prof. H.Tuherms, prof. L.Līpiņš, Doc. E.Bukrāns, and lecturers from Department of mathematics.













• We definitely need a basic course in higher mathematics where you could learn both theory and practice. Tasks could be more complicated.

• The existing model of lecturing is optimal. The only thing I could suggest is that students could be divided by their levels of knowledge, insight and wishes.

Conclusions:


Probability theory and Statistics, Linear programming, Mass service theory, Decision making theory.







• Integrate MathCad in laboratory works lessons;



• Include complex numbers and multiple argument functions in Mathematics study course

• Mathematical applications included in curricula • Make changes in LAIS of program, according topics of mathematics and the









4.3.15. Forestry Science





Ave-rage ball

MO-DE*


Frequ-ency

Range (0-25% - 0; 26-50%-1; 51-100%-2)



1.25 1 Processing of taxation data

88.89%

2





55.56%

2



38.89%

1



1.25 1 Hydrothermal processing of wood

16.67%

0

Can be reduced


2 2 Wood trunk, components of rotation body, stereo metric solids characteristic to wood trunks.

77.78%

2

Remain unchanged


1.25 1

44.45%

1

Remain unchanged


1.5 1 Calculating the volume of wood trunk and calculating the volume of separate parts of the wood trunk

27.78%

1

Remain unchanged


1 1 Optimization methods of forestry planning

16.67%

0


11.11

%

0


11.11%

0


5.56

%

0


16.67%

0




22.22%

0


33.33%

1



11.11%

0

Discussions results - experts' (programme director and teaching staff) recomendations Members of discussion – survey: Director of Forestry science program asoc. prof. I.Staupe, lecturers of special subjects in Forest faculty prof. A.Dreimanis,lect.S.Luguza, lect.A.Veinbergs, and lecturers from Department of mathematics.


• Knowledge in higher mathematics is not used in special subjects (except common technical subjects like physics, technical graphics and forest melioration). That’s why the main aim in teaching mathematics is cognitive development: seeing casual relationships, solving problems, formal logics.

• Lecturers would like the students to have mastered some IT software, e.g. MathCad. But, since the number of credit points is small for mathematics (4), integrate laboratory works in computer classes is not possible. Lecturers of the study program will consider increasing the number of credit points of mathematics. 2. Based on wish of employers and students to study mathematics in more practical





Compromise found: students who will be willing to get the insight of application of particular mathematics’ topic, will be able to get it via internet where tasks will be posted by lecturers of mathematics in cooperation with special subject lecturers.



Experts' - professionals recommendations • We need basic study course of higher mathematics with theory and practice. Tasks

can be more difficult. • Application of mathematics in problem solving of specific field. • More explanations, students don’t want to attend additional lectures. • The current style of lecturing is optimal. The only thing I can add, that students

should be divided into groups by their skills, insight and plans in the specific area. Conclusions:


Probability theory and Statistics.















4.3.16. General conclusions for LUA mathematics study programs readjusting The research shows that 15 mathematics courses offered by Latvia University of

Agriculture is very different. It is necessary to combine them in blocks, defining three levels of content (setting up of three modules): engineering, technological and social unit. It is determined by several factors.

First, the mathematical study process organization at LUA - number of students in groups is large thus in the implementation of one and the same mathematics course are participating the several Mathematics Department faculties (lecturer, practical work of managers, heads of laboratory work), as a result difficult to achieve concerted action.

Second, mathematical studies in a similar manner are organized, not only in Siauliai University, but also elsewhere, such as the Estonian Science University, and Second, mathematical studies in a similar way it is organized, not only in Siauliai University, but also elsewhere, such as the Estonian University of Life Sciences and Alexandras Stulginskis University (Kaunas).

The third factor to be determined by mathematical studies University, are the common higher education curriculum development trends in Latvian. Parallel to research for adapting mathematical competences in the socio-economical development, in the Latvian University of Agriculture is also a study program evaluation, in which emphasis is placed on the "umbrella principle". This principle means that the same minimum basic knowledge blocs for all specialties are planned at the beginning of studies, but only the later specialization courses.

Given the above circumstances as the main direction for study programs readjusting are recomended to revise the contents of Mathematics and the depth of problems and to divide the contents of topics into level modules. Two other readjusting directions are associated with this direction - define the learning outcomes of each level and to schedule lectures, practical and laboratory works and students' self-supported (individual) work for each level.

Given that similar themes of contact hours of Latvian University of Agriculture is much less than the Siauliai University (for Engineers is less than 30 percent in LUA comparing to SU, and 50 percent less for Public Administrators in LUA comparing to SU), the LUA mathematics study programs readjusting should contain a view to improving the mathematics study process organisation and study materials availability to students that ensure successful program learning.

The curriculum development is connected with the proportion of compulsory and elective courses. The disadvantage of a great number of electives is in a greater demand for a number of teachers, lecture-rooms, laboratories etc. Among advantages the competition of different courses and teachers may be included. Since increase of actual lessons/lectures is not possible by decreasing the number of lessons of other subjects, the e-studies would be a beneficial solution for both the parties. It is distinguished two main usages of e-studies in higher mathematics basic course: additional materials for the acquisition of the study program content and practical tasks for different levels. The lecturer’s “improved” studies could be carried out through e-studies as additional courses with the focus only on practical application of mathematics and other themes. In accordance with Migdal-Mikuli, Bros & Bernald, (2008) classical mathematics education model could be modification (Figure 4.1.).



Figure 4.1. Modification the classical mathematics education model (Adapted from Migdal-Mikuli, Bros & Bernald, 2008)



REFERENCES

1. Ahmed, A. (1987). Better Mathematics: A Curriculum Development Study. London: HMSO.

2. Alonso F., Rodriguez, G., Villa, A. 2007. New challenges, new approaches: A new way to teach Mathematics in Engineering. International Conference on Engineering Education – ICEE 2007.

3. Askew, M., & Wiliam, D. (1995). Recent Research in Mathematics Education 5-16. London: HMSO.

4. Askew, M., Brown, M., Rhodes, V., Johnson, D., & Wiliam, D. (1997). Effective Teachers of Numeracy, Final Report. London: Kings College.

5. Black, P., & Wiliam, D. (1998). Inside the black box : raising standards through classroom assessment. London: King's College London School of Education 1998.

6. Davis B. and Simmt E. 2006. Educational Studies in Mathematics. 61: 293–319 DOI: 10.1007/s10649-006-2372-.

7. Dudaitė (2007). Matematinio raštingumo samprata. ISSN 1392–5016. ACTA PAEDAGOGICA VILNENSIA.

8. Kačinskaite, R., Vintere, A. (2009) „Cross-border cooperation in development of the mathematics study process”, 10th International conference “Teaching Mathematics: retrospective and perspectives”, Abstracts, Tallinn, p.43-44.

9. Kaminskienė, Rimkuvienė, Laurinavičius (2010). Matematikos studijos prasidėjus aukštojo mokslo reformai. Lietuvos matematikos rinkinys. LMD darbai ISSN 0132-2818 Volume 51, 2010, pages 1–14.

10. Mercer, N. (2000). Words and Minds. London: Routledge.

11. Morkūnienė (2010). Patobulintos matematikos programos diegimo tyrimas.

12. Mustoe, L. (2004) The Future of Mathematics in the united Kingdom. 12th SEFI Maths Working Group Seminar, Proceedings, Vienna University of echnology, 2004, pp 113-117.

13. Niss, M.: ‘Kompetencer og uddannelsesbeskrivelse’, Uddannelse 9, 21-29, 1999.Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM project. Paper presented at the Third Mediterranean conference on mathematics education, Athens.

14. Ofsted. (2006). Evaluating mathematics provision for 14-19-year-olds. London: HMSO.

15. Rodriguez, G., Villa, A. Could the computers change the trends in Mathematics learning?. A Spanish overview. Proceedings Applimat 2005. Slovak University of Technology Bratislava (Eslovaquia) February 2005.

16. Swain, J., & Swan, M. (2007). Thinking Through Mathematics research report. London: NRDC.

17. Swan, M. (2005). Improving Learning in Mathematics: Challenges and Strategies. Sheffield: Teaching and Learning Division, Department for Education and Skills Standards Unit.

18. Swan, M. (2006). Collaborative Learning in Mathematics: A Challenge to our Beliefs and Practices. London: National Institute for Advanced and Continuing Education (NIACE); National Research and Development Centre for Adult Literacy and Numeracy (NRDC).

19. Swan, M. (2008). Bowland Maths Professional development resources. [online]. http://www.bowlandmaths.org.uk: Bowland Trust/ Department for Children, Schools and Families.

20. Swan, M., & Wall, S. (2007). Thinking through Mathematics: strategies for teaching and learning. London: National Research and Development Centre for Adult literacy and Numeracy.

21. Watson, A., & Mason, J. (1998). Questions and prompts for mathematical thinking. Derby: Association of Teachers of Mathematics.


ANNEX NO.1.

1.1. THE QUESTIONNAIRE FOR DIRECTORS OF STUDY PROGRAMS, DEPARTMENT CHAIR AND ACADEMIC PERSONNEL

The teachers of the Department of Mathematics, in order to improve mathematical competencies of the specialists prepared in

................................................................. study programme, are asking for your assistance in coordinating teaching content of mathematics. We are asking you to express your opinion on what should be taught at the lectures of mathematics and how this should be done so that the acquired knowledge and skills were useful to the students both in learning other subjects and in the future professional activities.

The present survey is a part of the common project of Šiauliai University and Latvia University of Agriculture “Cross-border network for adapting mathematical competences in the socio-economical development (MatNet)”

If you have any questions, please do not hesitate to contact us by e-mail: [email protected]

The questionnaire consists of two parts. In the first part, the current content of teaching mathematics is provided; in the second part several additional topics of mathematics are included.

We are asking you: → to estimate to what extent, in your opinion, these topics are relevant in preparing a specialist or in the future career; → to name (if possible) the subjects/topics or specialist activities where this mathematical knowledge and skills are used/necessary.

Name of the subject you teach


Table 1.1. Assess the current content of mathematics subject HOURS: T-theory,

P-practical w., L – laboratory w, I- individual w.

Subject name

? semester

T P L I

Is it necessary?

0 – not necessary, 1 – could be taught,

2 - necessary.

Please name, if possible, the subjects/topics or specialist activities where this mathematical knowledge

and skills are used/necessary.

LINEAR ALGEBRA Matrices, determinants, the equation system 4 2 1 3

ANALYTICAL GEOMETRY Line equations for the Cartesian and polar coordinate systems. 5 5 2 2

…………

Plane equations. Line equations in space 1 2 0 2

Table 1.2. We are providing several areas/topics of mathematics that are not included or are only partially included into the list of taught topics. Do you think these topics might be necessary in your study programme? What issues of these areas might be necessary for your study

programme (you may specify both the topic and concrete tasks which should be solved)? Evaluate topics which could

be important to prepare the specialist for this study

programme 0 – not necessary

1 – could be tought 2 – necessary

Name your subject themes (topics) where

these mathematics knowledge are necessary



More complicated statistical methods of the data analysis (market analysis, the mathematical �odelling of cause-and-effect of the economic object, the use of dynamic lines and etc.)


Geometry (the calculation of area and capacity, the determination of the power, that affects the solid, direction,


computing the performed work of the power and etc.). Derivatives and differential calculation (approximate calculation, computing of minimum and maximum value, analysis of process fluctuation: computing of speed, acceleration, gradient and the rate of the developing business profitability at a particular time and etc.)



Net planning (the tasks solutions of the integrated jobs planning and etc.). Discrete mathematics (combinatorics, the tasks solutions of algorithmics, graph theory, cryptography and etc.) Mathematical logic (operations with predicates, Boole‘s algebra, predicate logic and etc.). The application of mass service theories (the mathematical �odelling and optimization of administration of clients flow service and etc.)

Decision trees (the modelling of the choice of decision alternatives by the mathematical evaluation of the conditions and etc.).


The elements of betting theory. (the mathematical modelling of decision making when the acting persons/group have conflicting aims and etc.)

Others. Fill in……........................................... 3. What should be more emphasized while teaching mathematics (content, thematics, methods, etc.)? ___________________________________________________________________________________________________________________

___________________________________________________________________________________________________________________

___________________________________________________________________________________________________________________

___________________________________________________________________________________________________________________

___________________________________________________________________________________________________________________

We are grateful for your collaboration!


1.2. QUESTIONNAIRE FOR EXTERNAL RESEARCH

MATHEMATICS IN PROFESSIONAL ACTIVITIES

To whom it may concern,

Teaching mathematics in high schools (all levels) has been brought up for the discussion recently. It is being sought the answers for the questions: who must be taught and how they must be taught to make knowledge of mathematics useful in real life.

You, the experts of the professional field, are welcome to take part in the survey. The survey concerns teaching of mathematics and its benefit in professional activities. The summarized results will be used for the improvement of teaching of mathematics subjects at a high school (university, college). Your opinion (as far as you are practicians) is very important!

The research is being carried out by the joint scientists’ group of Latvia University of Agriculture and Šiauliai University.

Questions must be sent by email, address: [email protected] or [email protected]

I. Attitude to the mathematics

1. Enjoyment and abilities of mathematics (Choose only one answer in each line and mark it) Strongly

agree Agree Disagree Strongly disagree

Mathematics and the subjects, which require mathematical knowledge, have always been my favourite. I think mathematics, which I studied at high school (university, college), could have been more complicated. I did not understand most mathematical concepts that I studied at high school (university, college).


2. Mathematics teaching at high schools. Remember the mathematics lectures at high school (university, college) (mathematics is all mathematics subjects: algebra, geometry, mathematical analysis, operations research, relevance theory, statistics etc.). How could you define them through the time perspectives? (Choose only one answer in each line and mark it) Strongly


Mathematics knowledge helped me to understand other study subjects. Mathematics was taught matter-of-factly and boringly. Mathematics was an interesting and meaningful subject. For my occupation studying mathematics at high school (university, college) is wasting of time; knowledge got in secondary school is enough.

Most of the students did not understand mathematics, tried to learn rules by heart. Studying mathematics develops logical thinking, accuracy and concreteness of future specialists. 3. Mathematics in professional activities (Choose only one answer in each line and mark it) Strongly


The knowledge and abilities of mathematics, mathematical thinking helped me to achieve more in my life. People, who understand mathematics well, are highly assessed by employers. I have a lot of opportunities to apply my knowledge of mathematics in professional activities. My occupation does not require deeper knowledge of mathematics: it is enough to do arithmetical calculations and count percentage.

Mathematics is widely used in my professional activities. Mathematical thinking helps to solve real world/ professional problems. A person, who understands mathematics, will easily master most jobs that require thinking. I understand mathematical symbols and a formal mathematical language which is used in my professional literature. I would like to attend the training that deals with mathematics application to solve the practical problems of my professional field.

4. The essence and importance of mathematics (Choose only one answer in each line and mark it) Strongly


Mathematics helps to model and analyze the problems of the real world. Mathematics is only formulas that are needed to remember. Mathematics is a meaningless game with numbers which is played according to the rules created by scientists. Mathematics develops thinking, helps to make a decision in a particular situation, find new ideas. Mathematics gives an insight into the world we live.


II. Accordance of mathematical and professional competence

5. Mark those fields of the deeper knowledge of mathematics that are needed for the specialists of your field to accomplish their professional activities successfully and analyze professional literature (you can mark several topics).

Field of mathematics Descriptive statistics (grouping of the data, the tasks on the calculation of percentages, averages, and errors, estimation of statistical relations, graphical representation of the data and etc ).

○

Statistical deductions (application of the sampling method, computing of confidence intervals, testing of the statistical hypotheses and etc.). ○ More complicated statistical methods of the data analysis (market analysis, the mathematical modeling of cause-and-effect of the economic object, the use of dynamic lines and etc.)

○


○


○


○


○


○

Net planning (the tasks solutions of the integrated jobs planning and etc.). ○ Discrete mathematics (combinatorics, the tasks solutions of algorithmics, graph theory, cryptography and etc.) ○ Mathematical logic (operations with predicates, Boole‘s algebra, predicate logic and etc.). ○ The application of mass service theories (the mathematical modeling and optimization of administration of clients flow service and etc.) ○ Decision trees (the modeling of the choice of decision alternatives by the mathematical evaluation of the conditions and etc.). ○ Probability theory (computing of the most presumptive events, doing the tasks on insurance, mass service, quality and control, automated management systems and etc.).

○

The elements of betting theory. (the mathematical modeling of decision making when the acting persons/group have conflicting aims and etc.) ○ Others. Fill in……........................................... ○

6. In your opinion, what should be taught at mathematics lectures and how should it be taught at high school (university, college) to make acquirement useful in professional activity? Write.


III. Information about a respondent 7. When did you graduate from the last

education institution?

o 1-5 years ago; o 5-15 years ago; o More than 15 years ago.

8. Your sex:

o Male; o Female.

12. Kind of employment

◦ company / department manager ◦ employee

9. Your education:

o Professional Bachelor Degree; o Bachelor Degree; o Master Degree; o Doctorate.

10. What is the field of the studies you

graduated? ○ Biomedical Science; o Physical Science; o Humanities; o Arts; o Social Science; ○Technological Sciencei.

11. Your professional field of occupation:

o Architecture o Biophysics, Biochemistry o Forestry; o Constructions; o Computer science, Informational

technologies; o Manufacturing; o Electronics; o Mechanical engineering; o Economics, Banking; o Services ,sales,, business; o Public administration; o Environment; o Food Industry; o Medicine; o Agriculture; o Other ……………..

We are grateful for your collaboration!


ANNEX NO.2.

COMPARISON THE NUMBER OF HOURS OF THEORY, PRACTICAL AND INDEPENDENT WORK ASSIGNED FOR THE TOPICS

Table 2.1. Mathematics programs for Agricultural Mechanisation (LUA) and Mechanical Engineering (ŠU) (content and volume) LUA LATVIA ŠU LITHUANIA

11 KP (=16.5 ECTS) HOURS: T-theory,P-practical w L – laboratory w I- individual w

14 K (= 21 ECTS) HOURS: T-theory,P-practical w., L – laboratory w, I- individual w.

Content

T-72 P-64 L-40 I-264 T-112 P-144 L-0 I-304 1st semester MATHEMATICS -1. (Mate 1001)- 3,5 KP = 5,25 ECTS /

MATHEMATICS - 1 (P120B111) 4 K =6 ECTS T-24 P-24 L-8 I-84 T-32 P-48 L-0 I-80

1. LINEAR ALGEBRA 5 5 3 4 6 10 Matrixes. Operations with matrixes. 2 2 1 1 2 3 Determinants of the second and third order, theirs properties. Adjunct and minor. Extension of determinants by rows and columns. Determinants of higher order. Reciprocal/Inverse matrix. 1 1 1 1 2 3

Systems of linear equations. Solution of non-degenerated linear systems. Cramer’s formulas. Gauss methods for linear systems. 2 2 1 2 2 4

2. VECTOR GEOMETRY 3 3 1 6 8 16 Vector notion. Linear vector’s operations. Vector’s projections. The vector’s coordinates. Radius vector, distance between two points. 0,5 0,5 2 2 4

Inner/Scalar product. Finding the work force. Perpendicular condition of vectors. Angle between two vectors. 0,5 0,5 1 1 2 4

Vector product. Area of parallelogram and triangle. 1 1 2 2 4 Parallelepipedal product. Volume of parallelepiped. 1 1 1 2 4

3. ANALYTIC GEOMETRY OF PLANE AND SPACE 4 4 (2) 5 6 8 General plane equation. Angle between two planes. Canonical, parametric and general line equations in the space. Line equation on the plane. 2 1

1 in the

2nd sem

3 4 6

Curves of the second order. Circle, ellipse, hyperbola, parabola. 1 2 1 2 2 2 Plane equations. Equations of curves in space. Second order surfaces. 1 1


4. LIMIT AND CONTINUIT 3 3 1 4 6 10 Elementary functions. Sequence and its limit. The number e. 1 1 1 1 2 Functions having limit zero. Limit of unboundedly increasing functions. Indefinite expressions. Using of equivalent decreasing functions for computation of some limits. 1 1 1 2 4 6

Continuity at the point. Discontinuity points. 1 1 1 1 2 5. DERIVATIVE OF A FUNCTION OF ONE VARIABLE AND ITS APPLICATIONS 9 9 2 6 10 16

Derivative. Mechanical and geometrical sense of the derivative: calculation of speed and acceleration. Approximate calculation. 1 2

1

2 2

Differentiation of non-explicit functions. Logarithmic differentiation. Differentiation of parametrically given functions. 2 2 1 2 4 6

Usage of derivates. Using derivates in geometry, Lopital’s rule 1 1 Derivatives of higher order. Differential of higher order. 1 1 1 2 4 Investigation of functions, minimal and maximal values. 4 3 1 2 2 4 2nd semester MATHEMATICS -2. (Mate2001)- 2,5 KP = 3,75 ECTS / T-16 P-16 L-8 I-60

6. DIFFERENTIAL CALCULATION OF FUNCTIONS IN SEVERAL VARIABLES 3 3 0 7 12 18 Notion of functions in several variables. Limit. Continuity. 1 1 0 1 1 2 Partial derivatives of functions of in several variables. Full differential. Partial derivatives of composite and non-explicit functions. 1 1 0 2 4 6

Partial derivatives and differentials of higher order. 0 1 2 4 Local and conditional extremes of functions in several variables. 1 1 0 2 4 4 Scalar field, direction derivative, gradient. 1 1 2 2nd semester

/ MATHEMATICS - 2 (P130B011) 4 K =6 ECTS T-32 P-48 L-0 I-80

7. INDEFINITE INTEGRAL 6 6 3 11 16 24 Primitive functions and indefinite integral. The table of indefinite integrals. 0,5 0,5 1 0 2 Methods of direct integration. Changing of variable in integration. 0,5 0,5 2 4 4 Partial integration. 1 1 1 1 2 4 Integration of functions, containing quadratic trinomial. 1 1 1 1 2 Integration of the simplest rational fractions. Expression of proper fractions by a sum of the simplest fractions. Method of indefinite coefficients. 1 1 2 3 4

Integration of irrational functions. 1 1 1 2 3 4 Integration of trigonometric functions. 1 1 1 2 3 4

8. DEFINITE INTEGRAL AND ITS APPLICATIONS 7 7 2 4 5 10 Area of curvilinear trapezium, definite integral and its properties. 0,5 0 1 0 2 Newton-Leibniz formula. Integration methods. 1,5 2 1 1 2 2 Area of a figure and length of a curve. Volume of a rotation solid and its surface area. 2 3 1 1 1 4 Work of varying force, mass centre coordinates of non-homogeneity stick, static and inertia moments of a flat figure, liquid pressure. 1 0 1 1 2

Improper integrals 2 2 0 1


5. MULTIPLE AND CURVILINEAR INTEGRALS 0 0 0 7 11 18 Double integral, properties, computation. 2 4 6 Triple integral, properties, computation. 2 4 6 Body mass centre coordinates inertial moments. 1 1 2 Curvilinear integrals. 2 2 4

3rd semester MATHEMATICS -3. (Mate4001) - 3,5 KP = 5,25 ECTS / T-24 P-24 L-8 I-84

10. DIFFERENTIAL EQUATIONS 15 15 4 10 16 28 Differential equations and the solutions.. Diff. equations of the first order. Problem Cauchy. 1 1 1 1 2 Separation of variables in differential equations. 1 1 1 2 2 Homogeneous differential equations of the first order. 1 1 1 1 2 4 Linear differential equations of the first order. 1 2 1 1 2 4 Bernoulli differential equations. 2 1 1 2 4 Linear and non-linear differential equations of the second order. 7 7 2 2 2 4 Differential equations of higher order. 1 1 1 2 2 Systems of differential equations. 1 2 4 Equation of mechanical oscillations, investigation of free oscillations. 1 1 1 1 2 3rd semester

/ APPLIED MATHEMATIC (P001B117) 4 K =6 ECTS T-32 P-32 L-0 I-96

11. NUMERICAL AND FUNCTIONAL SERIES 9 9 3 12 14 34 Series with positive members, convergence and divergence. Application of comparison, Dalamber, Cauchy radical and integral criteria. 2 2 2 2 6

Sign changing series, low of Leibniz. Absolute and relative convergence. 1 1 1 2 2 6 Convergence region of functional series. Expansion of functions by series and its applications in integration and for diff. equations. 4 4 1 4 4 12

Fourier series and application in spectral analysis. 2 2 1 4 6 12 12. FUNCTIONS OF COMPLEX VARIABLE 0 0 0 0 8 8 24

Operations with complex numbers. Algebraic, trigonometric and exponential forms of complex numbers, its applications.

2 2 6

The functions of complex variables, limit, continuity and differentiation. Cauchy and Riemann conditions. Laurent series.

3 3 9

Integral of complex variables functions. Integral Cauchy formula. Isolated singular points. 3 3 9

13. LAPLACE TRANSFORMATION AND ITS APPLICATION 0 0 0 16 10 36 Pre-image and image. Images of basics elementary functions. 4 2 12 Differentiation of images and images of derivatives. 4 4 12 Solving of linear differential equations by operational method. 4 4 12


4th semester MATHEMATICS -4. (Mate3001) - 1,5 KP = 2,25 ECTS /

PROBABILITY THEORY AND MATHEMATICAL STATISTICS P160B171) ) - 2 KP = 3 ECTS

T-8 P-0 L-16 I-36 T-16 P-16 L-16 I-48

14. RANDOM EVENTS 0 0 0 0 0 4 Elements of combinatorial analysis. Notion of the set. Operations. Random event. Compatible and incompatible events.

4

15. PROBABILITY 2 0 4 4 2 10 Classical definition of the probability. Axioms. Geometrical and statistical probabilities. Properties. The probabilities of union and intersection. 1 0 2 2 1 6

Dependent and independent events. Conditional probability. Complete probability. Bernoulli experiments. 1 0 2 2 1 4

16. RANDOM VARIABLES 3 0 6 4 4 12 Discrete and continuous random variables, theirs characteristics. 2 0 4 1 1 4 Binomial, Poisson and normal distributions. 1 0 2 1 1 4 Law of large numbers, central limit theorem. 2 2 4

17. DESCRIPTIVE STATISTIC 0,5 0 1 2 4 General set (population) and sample set. Data’s collection and classification. Notion of variable.

0.5 0 1 2 4

Frequencies. Data’s characteristics. Graphical presentation. 18. STATISTICAL INFERENCES 2,5 0 5 6 10 18

Normal distribution, 3-σ rule, Stjudent and χ2 distributions. 0.5 0 1 2 2 2 Point and interval estimations of parameters. Notion of confidence interval. Confidence intervals for mean (expected value) and variance of normal random variable. 1 0 2 1 2 4

Correlation and linear regression. 0.5 0 1 1 2 4 Hypothesis. Mistakes. Confidence level. 0.5 0 1 1 2 Parametrical statistical hypotheses for one and two sample sets. 1 4 6


Table 2.2. Mathematics programs for Computer Control and Computer Science (LUA) and Informatics Engineering (ŠU) (content and volume)

LUA LATVIA ŠU LITHUANIA

8 KP (=12 ECTS) HOURS: T-theory,P-practical w L – laboratory w,I- individual w


Content

T-64 P-48 L-16 I-192 T-112 P-144 L-0 I-304 1st semester MATHEMATICS -1. (Mate 2027) - 4 KP = 6 ECTS /

MATHEMATICS - 1 (P120B111) 4 K =6 ECTS T-32 P-24 L-8 I-84 T-32 P-48 L-0 I-80

1. LINEAR ALGEBRA 5 4 2 4 6 10 Matrixes. Operations with matrixes. 1 1 1 1 2 3 Determinants of the second and third order, theirs properties. Adjunct and minor. Extension of determinants by rows and columns. Determinants of higher order. Reciprocal/Inverse matrix. 2 1 1 1 2 3

Systems of linear equations. Solution of non-degenerated linear systems. Cramer’s formulas. Gauss methods for linear systems. 2 2 2 2 4

2. VECTOR GEOMETRY 4 4 1 6 8 16 Vector notion. Linear vector’s operations. Vector’s projections. The vector’s coordinates. Radius vector, distance between two points. 1 0,5 2 2 4

Inner/Scalar product. Finding the work force. Perpendicular condition of vectors. Angle between two vectors. 1 1,5 1 1 2 4

Vector product. Area of parallelogram and triangle. 1 1 2 2 4 Parallelepiped product. Volume of parallelepiped. 1 1 1 2 4

3. ANALYTIC GEOMETRY OF PLANE AND SPACE 4 4 1 5 6 8 General plane equation. Angle between two planes. Canonical, parametric and general line equations in the space. Line equation on the plane. 2 2 3 4 6

Curves of the second order. Circle, ellipse, hyperbola, parabola. 2 2 1 2 2 2 4. LIMIT AND CONTINUIT 5 4 1 4 6 10

Elementary functions. Sequence and its limit. The number e. 2 1 1 1 2 Functions having limit zero. Limit of unboundedly increasing functions. Indefinite expressions. Using of equivalent decreasing functions for computation of some limits. 2 2 1 2 4 6

Continuity at the point. Discontinuity points. 1 1 1 1 2 5. DERIVATIVE OF A FUNCTION OF ONE VARIABLE AND ITS APPLICATIONS 9 8 2 6 10 16

Derivative. Mechanical and geometrical sense of the derivative: calculation of speed and acceleration. Approximate calculation. 1 1

1

2 2

Differentiation of non-explicit functions. Logarithmic differentiation. Differentiation of 4 5 1 2 4 6


parametrically given functions.

Usage of derivates. Using derivates in geometry, Lopital’s rule 1 1 Derivatives of higher order. Differential of higher order. 1 1 1 2 4 Investigation of functions, minimal and maximal values. 2 0 1 2 2 4

6. DIFFERENTIAL CALCULATION OF FUNCTIONS IN SEVERAL VARIABLES 3 3 0 7 12 18 Notion of functions in several variables. Limit. Continuity. 1 1 0 1 1 2 Partial derivatives of functions of in several variables. Full differential. Partial derivatives of composite and non-explicit functions. 1 1 0 2 4 6

Partial derivatives and differentials of higher order. 0 1 2 4 Local and conditional extremes of functions in several variables. 1 1 0 2 4 4 Scalar field, direction derivative, gradient. 1 1 2 2nd semester MATHEMATICS -2. (Mate2028)- 4KP = 6 ECTS /

/ MATHEMATICS - 2 (P130B011) 4 K =6 ECTS T-32 P-24 L-8 I-84 T-32 P-48 L-0 I-80

7. INDEFINITE INTEGRAL 9 7 2 11 16 24 Primitive functions and indefinite integral. The table of indefinite integrals. 1 0,5 1 0 2 Methods of direct integration. Changing of variable in integration. 3 2,5 2 4 4 Partial integration. 1 1 1 1 2 4 Integration of functions, containing quadratic trinomial. 1 1 1 1 2 Integration of the simplest rational fractions. Expression of proper fractions by a sum of the simplest fractions. Method of indefinite coefficients. 1 0,5 2 3 4

Integration of irrational functions. 1 0,5 1 2 3 4 Integration of trigonometric functions. 1 1 2 3 4

8. DEFINITE INTEGRAL AND ITS APPLICATIONS 8 6 2 4 5 10 Area of curvilinear trapezium, definite integral and its properties. 1 0 1 0 2 Newton-Leibniz formula. Integration methods. 2 2 1 1 2 2 Area of a figure and length of a curve. Volume of a rotation solid and its surface area. 2 2 1 1 2 4 Work of varying force, mass centre coordinates of non-homogeneity stick, static and inertia moments of a flat figure, liquid pressure. 1 0 1 1 2

Improper integrals 2 2 0 6. MULTIPLE AND CURVILINEAR INTEGRALS 0 0 0 7 11 18

Double integral, properties, computation. 2 4 6 Triple integral, properties, computation. 2 4 6 Body mass centre coordinates inertial moments. 1 1 2 Curvilinear integrals. 2 2 4

10. DIFFERENTIAL EQUATIONS 6 5 2 10 16 28 Differential equations and the solutions.. Diff. equations of the first order. Problem Cauchy. 0,5 0 1 1 2 Separation of variables in differential equations. 0,5 1 1 2 2


Homogeneous differential equations of the first order. 1 1 0,5 1 2 4 Linear differential equations of the first order. 1 1 0,5 1 2 4 Bernoulli differential equations. 1 2 4 Linear and non-linear differential equations of the second order. 3 2 1 2 2 4 Differential equations of higher order. 1 2 2 Systems of differential equations. 1 2 4 Equation of mechanical oscillations, investigation of free oscillations. 1 1 2 3rd semester

/ APPLIED MATHEMATIC (P001B117) 4 K =6 ECTS T-32 P-32 L-0 I-96

11. NUMERICAL AND FUNCTIONAL SERIES 7 8 2 12 14 34 Series with positive members, convergence and divergence. Application of comparison, Dalamber, Cauchy radical and integral criteria. 2 2 2 2 6

Sign changing series, low of Leibniz. Absolute and relative convergence. 1 1 1 2 2 6 Convergence region of functional series. Expansion of functions by series and its applications in integration and for diff. equations. 4 4 0,5 4 4 12

Fourier series and application in spectral analysis. 2 2 0,5 4 6 12 12. FUNCTIONS OF COMPLEX VARIABLE 0 0 0 0 8 8 24

Operations with complex numbers. Algebraic, trigonometric and exponential forms of complex numbers, its applications.

2 2 6

The functions of complex variables, limit, continuity and differentiation. Cauchy and Riemann conditions. Laurent series.

3 3 9

Integral of complex variables functions. Integral Cauchy formula. Isolated singular points. 3 3 9 13. LAPLACE TRANSFORMATION AND ITS APPLICATION 0 0 0 16 10 36

Pre-image and image. Images of basics elementary functions. 4 2 12 Differentiation of images and images of derivatives. 4 4 12 Solving of linear differential equations by operational method. 4 4 12 4th semester

PROBABILITY THEORY AND MATHEMATICAL STATISTICS P160B171) ) - 2 KP = 3 ECTS

T-8 P-0 L-16 I-36 T-16 P-16 L-16 I-48

15. RANDOM EVENTS 0 0 4 Elements of combinatorial analysis. Notion of the set. Operations. Random event. Compatible and incompatible events.

4

16. PROBABILITY 4 2 Classical definition of the probability. Axioms. Geometrical and statistical probabilities. Properties.

2

2

2

The probabilities of union and intersection. 4 Dependent and independent events. Conditional probability. Complete probability. Bernoulli

experiments. 2 4

17. RANDOM VARIABLES 4 4 12


Discrete and continuous random variables, theirs characteristics. 2 2 4 Binomial, Poisson and normal distributions. 4 Law of large numbers, central limit theorem. 2 2 4

18. DESCRIPTIVE STATISTIC 2 4 General set (population) and sample set. Data’s collection and classification. Notion of variable.

2 4

Frequencies. Data’s characteristics. Graphical presentation. 18. STATISTICAL INFERENCES 6 10 18

Normal distribution, 3-σ rule, Stjudent and χ2 distributions. 2 2 2 Point and interval estimations of parameters. Notion of confidence interval. Confidence intervals for mean (expected value) and variance of normal random variable.

1 2 4

Correlation and linear regression. 1 2 4 Hypothesis. Mistakes. Confidence level. 1 2 Parametrical statistical hypotheses for one and two sample sets. 1 4 6

Table 2.3. Mathematics programs for Environmental Science (LUA) and Ecology and Environmental Sciences (ŠU) (content and volume)

LUA LATVIA ŠU LITHUANIA

Mathematics (Mate 1004, Mate 3010)

Mathematics and its applications in Ecology

(P110B001) 7 KP (=10.5 ECTS)

HOURS: T-theory,P-practical w L – laboratory w, I-individual w

6 K (=9 ECTS) HOURS: T-theory,P-practical w., L – laboratory w, I- individual w.

Content

T-48 P-64 L-0 I-168 T-32 P-32 L-32 I-64 1st semester MATHEMATICS -1. / MATHEMATICS AND ENVIRONMENT T-24 P-32 L-0 I-84 T-18 P-18 L-18 I-36

1.The relationships between Biology, Ecology, Environmental and Mathematics. Mathematical models. Simple ecological models.

2 2 2 4

2. MATRIXES. Operations with matrixes. Determinants. Systems of linear equations, solving methods. Calculations with mathematical software 3 4 0

2

2

2

4 3. VECTOR ALGEBRA. Operations with vectors. Scalar multiplication of two vectors. Dot product. Scalar triple product. 3 4 0

4. ANALYTICAL GEOMETRY. Straight line .Equations of curves in Cartesian and polar coordinate systems. Basic problems on straight line in plane. Other lines: Circle line, Ellipse, 5 6 0


Hyperbola, Parabola. 5. FUNCTIONS and theirs graphics. Limit, calculation laws. Continuity of function, discontinuity points. 4 6 0 2 2 2 4

6. DERIVATIVES. The laws and formulas. Differential. Applications. 9 12 0 2 2 2 4 Calculations of derivatives with computer. 2nd semester Mathematics -2. /

T-24 P-32 L-0 I-84

7. INDEFINITE INTEGRAL. Integration laws and methods. DEFINITE INTEGRAL. Applications.

6 7

8 8

0 2 2 2 4

Calculation of the areas and volumes. 8. FUNCTIONS IN SEVERAL VARIABLES. Limits. Partial derivatives and differentials. Partial and full differentials. Applications. 5 6 0 2 2 2 4

9. DIFFERENTIAL EQUATIONS, theirs solutions. Methods of calculation. Applications. 6 10 0 2 2 2 4 Using of mathematical software for searching the analytical solutions of differential equations. 10. Mathematical modelling of biological systems. Local management of complex systems. Creation of mathematical-ecological models and realizing them with computer software. 2 2 2 4

11. Creating of real mathematical models, theirs reliability. Formal models of self-regulation. Applications. 2 2 2 4

2nd semester / PROBABILITY THEORY

AND MATHEMATICAL STATISTICS IN ECOLOGY

14 14 14 28

12. Random event and its probability. Operations. Compatible and incompatible events. 2 2 2 4 13. Discrete and continuous random variables, functions and characteristics. Calculations with mathematical software.

2 2 2 4

14. Descriptive statistic. Population and sample. Data’s collection, classification and analysis. 2 2 2 4 15 .Correlation and linear regression. Giving the regression equations with computer. 2 2 2 4 16. Point and interval estimations of parameters. Confidence intervals. Normal distribution. 2 2 2 4 17. Hypotheses. Mistakes. 2 2 2 4 18. Modelling of biological systems. Work with large arrays. 2 2 2 4 3nd senmester MATHEMATICAL STATISTICS / T-32 P-32

Teach the lectures from special subjects departments


Table 2.4. Mathematics programs for Sociology of Organisations and Public Administration (LUA) and Public Administration (ŠU) (content and volume)

LUA LATVIA ŠU LITHUANIA Mathematics Applied Mathematics

Content 2K (= 3 ECTS) HOURS: T-theory,P-practical w L – laboratory w, I-Individual w


1st semester MATHEMATICS (Mate 1013)- 2 KP = 3 ECTS /

APPLIED MATHEMATICS (P120B011) 4 K =6 ECTS

T-16 P-16 L-0 I-48 T-32 P-32 L-0 I-96

1. MATHEMATICS IN ECONOMICS 3 3 8 The simplest financial calculations: interest, credit balance, discount. Using MS Excel. 2 2 4 Models for the tasks of optimal planning. 1 1 4

2. SYSTEMS OF LINEAR EQUATIONS 1 1 2 2 8 Definition. Compatibility and non-compatibility systems. Determinate and indeterminate systems. Equivalent systems. Elementary transformations. 1 1 1 1 4

Gauss and Gauss-Jordan methods. Diagonal system. 1 1 4

3. VECTORS 2 2 4 Vector and vector space. Operations with vectors. 2 2 4

4. MATRIXES, DETERMINANTS 2 2 7 7 20 Matrixes. Sorts of matrixes. The matrix operations. Economical interpretation of matrixes (costs, portages and etc.).

0,5 0

0,5 0 2 2 4

Determinants of the second, third and fourth orders. 0,5 0,5 1 1 4

Cramer method for solving systems of linear equations. 0,5 0,5 1 1 4

Reciprocal/Inverse matrix. Solving systems of linear equations by method of inverse matrixes. 0,5 0,5 1 1 4

The Leonjev model for the balance of economic system. 2 2 4

5. ANALYTIC GEOMETRY OF PLANE 3 3 General plane equation. Angle between two planes. Canonical, parametric and general line equations in the space. Line equation on the plane. 2 2

Curves of the second order. Circle, ellipse, hyperbola, parabola. 1 1

6. SYSTEM OF LINEAR INEQUALITIES AND ITS SOLUTIONS. LINEAR INEQUALITIES IN THE OPTIMAL PLANNING. 2 2 4

7. SETS AND FUNCTIONS 2 2 2 2 10 Notions of the set and subset. Operations. 1 1 1 1 4


Function’s definition. The definition range and the set of values. Examples. Limit of the function. 1 1 1 1 6

8. ELEMENTS OF MATHEMATICAL LOGIC 2 2 Elements of mathematical logic. Basic logic operations and their authenticity value 2 2

9. DIFFERENTIATION 3 3 5 5 12 Notions of derivative and differential. Derivatives of a functions in one variable. Derivatives and differentials of higher order.

2 0

2 0 2 2 4

Differential calculation of functions in several variables. 0 0 2 2 4 Investigation of functions, minimal and maximal values. 1 1 Application of the derivatives in economics. 0 0 1 1 4

10. INDEFINITE INTEGRAL 1 1 2 4 8 Primitive functions and indefinite integral. Sense. Integration laws and methods (by changing variable, partial integration). 1 1 1

1 2 2 4

4 11. DEFINITE INTEGRAL 2 2 5 3 18

Definite integral and its properties, sense. 0,5 0,5 1 6 Newton-Leibniz formula. Integration methods (term-by-term, by substitution and parts). 0,5 0,5 2 1 6 Applications (area of curvilinear trapezium, volume of a rotation solid). Definite integral in economics.

1 0

1 0 2 2 6

12. ELEMENTS OF DESCRIPTIVE STATISTIC General set and sample set. Frequencies. Data’s characteristics. Graphical presentation.

2 2 4


ANNEX NO.3.

ANALYSIS OF CURRENT CONTENT OF MATHEMATICS SUBJECTS IN SU AND ITS EVALUATION (ASSESSMENT)

Table 3.1. Ecology and Environmental Sciences Experts' assessment

(programme director + at least 3 teaching staff) Content Ave-

rage ball*

MO-DE**

Comments (subject themes (topics) where these mathematics knowledge are

useful / necessary) 1st semester MATHEMATICS AND ITS APPLICATION IN ECOLOGY (P110B001) , 4 K =6 ECTS

1. MATHEMATICS AND ENVIRONMENT The relationships between Biology, Ecology, Environmental and Mathematics. Mathematical models. Simple ecological models. 1,75 2 The need for calculation with Matcad

and Excel programms. 2. MATRIXES

Operations with matrixes. Determinants. Systems of linear equations, solving methods. Calculations with mathematical software. 1 0;1;2

3. FUNCTIONS Functions and theirs graphics. Limit, calculation laws. Continuity of function, discontinuity points. 2 2 Logarithmic curve.

4. DERIVATIVES The laws and formulas. Differential. Applications. Calculations of derivatives with computer. 2 2

5. INDEFINITE AND DEFINITE INTEGRALS Integration laws and methods. Applications. Calculation of the areas and volumes. 1 1

6. FUNCTIONS IN SEVERAL VARIABLES. Limits. Partial derivatives and differentials. Partial and full differentials. Applications. 1 0;1;2

7. DIFFERENTIAL EQUATIONS Differential equations theirs solutions. Methods of calculation. Applications. Using of mathematical software for searching the analytical solutions of differential equations. 1 0;1;2

8. MATHEMATICAL MODELLING OF BIOLOGICAL SYSTEMS

Local management of complex systems. Creation of mathematical-ecological models and realizing them with computer software. Creating of real mathematical models, theirs reliability. Formal models of self-regulation. Applications.

1 1 Very important for future studies.

PROBABILITY THEORY AND MATHEMATICAL STATISTICS IN ECOLOGY


9. RANDOM EVENTS AND ITS PROBABILITY

Random event. Compatible and incompatible events. Operations. Classical definition of the probability. Axioms. Geometrical and statistical probabilities. Properties. Applications. 1 1

10. RANDOM VARIABLES

Discrete and continuous random variables, functions and characteristics. Calculations with mathematical software. 1 1

11. DESCRIPTIVE STATISTIC General set (population) and sample set. Data’s collection, classification and analysis. 1,75 2

12. STATISTICAL INFERENCES Correlation and linear regression. Giving the regression equations with computer. 1,75 2 Very important laboratory works. Point and interval estimations of parameters. Confidence intervals. Normal distribution. 1,75 2 Hypothesis. Mistakes. Confidence level. Parametrical statistical hypotheses for one and two sample sets. 2 2 For future stuies. Modelling of biological systems. Work with large arrays. 2 2 To use the main methods, to chosse

necessary.

* The average ball is calculated from numbers: 0,1, 2. Where: 0 – not necessary, 1 –should be taught, 2 - necessary. **If the multiple mode exists, we write both.

Table 3.2. Electrical Engineering Experts' assessment


rage ball*

MO-DE**


useful / necessary) 1st semester MATHEMATICS - 1 (P120B111) , 4 K =6 ECTS

1. LINEAR ALGEBRA Matrixes. Operations with matrixes. 1,33 2 Determinants of the second and third order, theirs properties. Adjunct and minor. Extension of determinants by rows and columns. Determinants of higher order. Reciprocal/Inverse matrix. 2 2

Formal models of control systems (SS descriptors, system monitorability and controllability matrixes. Electric circuit calculations. Always is necessary.

Systems of linear equations. Solution of non-degenerated linear systems. Cramer’s formulas. Gauss methods for linear systems. 2 2 Circuit analysis. Electric circuit theory

and calculation. 2. VECTOR GEOMETRY

Vector notion. Linear vector’s operations. Vector’s projections. The vector’s coordinates. Radius vector, distance between two points. 2 2

System phasor diagrams, determination of reference signals (system with some different signals). Electric circuit theory


and calculation. Inner/Scalar product. Finding the work force. Perpendicular condition of vectors. Angle between two vectors. 2 2 Electric circuit theory and calculation. Vector product. Area of parallelogram and triangle. 1,67 2 Electric circuit theory and calculation. Parallelepipedal product. Volume of parallelepiped. 1,67 1 Electrical power calculation.

3. ANALYTIC GEOMETRY OF PLANE AND SPACE General plane equation. Angle between two planes. Canonical, parametric and general line equations in the space. Line equation on the plane. 1 0;1;2*

* Curves of the second order. Circle, ellipse, hyperbola, parabola. 1,67 2 Light technique. Electric circuit theory

and calculation. 4. LIMIT AND CONTINUIT

Elementary functions. Sequence and its limit. The number e. 1,33 1 Functions having limit zero. Limit of unboundedly increasing functions. Indefinite expressions. Using of equivalent decreasing functions for computation of some limits. 1,33 1

Continuity at the point. Discontinuity points. 0,67 1 5. DERIVATIVE OF A FUNCTION OF ONE VARIABLE AND ITS APPLICATIONS

Derivative. Mechanical and geometrical sense of the derivative: calculation of speed and acceleration. Approximate calculation. 1,67 2 The optimization exercises. Elementary

calculations. Differentiation of non-explicit functions. Logarithmic differentiation. Differentiation of parametrically given functions. 1 0;1;2 Derivatives of higher order. Differential of higher order. 1,33 1 Investigation of functions, minimal and maximal values. 1,67 2 The optimization exercises. Elementary

calculations. 6. DIFFERENTIAL CALCULATION OF FUNCTIONS IN SEVERAL VARIABLES

Notion of functions in several variables. Limit. Continuity. 1,67 2 The optimization exercises. Elementary calculations.

Partial derivatives of functions of in several variables. Full differential. Partial derivatives of composite and non-explicit functions. 1,67 2 Partial derivatives and differentials of higher order. 1,33 1 Local and conditional extremes of functions in several variables. 1 0;1;2 Scalar field, direction derivative, gradient. 1,33 1 2nd semester MATHEMATICS - 2 (P130B011), 4 K =6 ECTS

7. INDEFINITE INTEGRAL Primitive functions and indefinite integral. The table of indefinite integrals. 1,67 2 Modeling of systems. Methods of direct integration. Changing of variable in integration. 1,67 2 Partial integration. 1,67 2 Elementary calculations. Integration of functions, containing quadratic trinomial. 1,33 1 Integration of the simplest rational fractions. Expression of proper fractions by a sum of the simplest fractions. Method of indefinite coefficients. 1,33 1 Integration of irrational functions. 1,33 1


Integration of trigonometric functions. 1,67 2 8. DEFINITE INTEGRAL AND ITS APPLICATIONS

Area of curvilinear trapezium, definite integral and its properties. 1,33 1 Newton-Leibniz formula. Integration methods. 0,67 1 Area of a figure and length of a curve. Volume of a rotation solid and its surface area. 1 1 Work of varying force, mass centre coordinates of non-homogeneity stick, static and inertia moments of a flat figure, liquid pressure. 1 1

Improper integrals 1,33 1 9. MULTIPLE AND CURVILINEAR INTEGRALS Double integral, properties, computation. 0,67 1 Triple integral, properties, computation. 0,67 1 Body mass centre coordinates inertial moments. 0,67 1 Curvilinear integrals. 0,33 0

10. DIFFERENTIAL EQUATIONS

Differential equations and the solutions.. Diff. equations of the first order. Problem Cauchy. 1,33 2 Modeling of dynamic systems. Difference equation.

Separation of variables in differential equations. 1,33 2 Homogeneous differential equations of the first order. 1,33 2 Very important! Linear differential equations of the first order. 1,33 2 Bernoulli differential equations. 0,33 0 Linear and non-linear differential equations of the second order. 1,33 2 Differential equations of higher order. 1 0;1;2 Systems of differential equations. 1,33 2 Equation of mechanical oscillations, investigation of free oscillations. 0,67 1 3rd semester APPLIED MATHEMATIC (P001B117), 4 K =6 ECTS 11. NUMERICAL AND FUNCTIONAL SERIES 0,67 1 Series with positive members, convergence and divergence. Application of comparison, Dalamber, Cauchy radical and integral criteria. 0,67 1 Sign changing series, low of Leibniz. Absolute and relative convergence. 0,67 1 Convergence region of functional series. Expansion of functions by series and its applications in integration and for diff. equations. 0,67 1 Very important! Fourier series and application in spectral analysis. Very important! Complex numbers are

needfull in the 1th semester. 12. FUNCTIONS OF COMPLEX VARIABLE

Operations with complex numbers. Algebraic, trigonometric and exponential forms of complex numbers, its applications. 2 2

Automatic control, simulation of systems. Electric circuit theory and calculation.

The functions of complex variables, limit, continuity and differentiation. Cauchy and Riemann conditions. Laurent series. 1,67 2 Integral of complex variables functions. Integral Cauchy formula. Isolated singular points. 1,33 1 Very important (all themes)! Automatic


control, simulation of systems. 13. LAPLACE TRANSFORMATION AND ITS APPLICATIONS 1,67 2

Pre-image and image. Images of basics elementary functions. 1,67 2 Differentiation of images and images of derivatives. 1,67 2 Solving of linear differential equations by operational method. 0,67 1 4th semester PROBABILITY THEORY AND MATHEMATICAL STATISTICS (P160B171), 2 KP = 3 ECTS

14. RANDOM EVENTS Transient processes. Elements of combinatorial analysis. Notion of the set. Operations. Random event. Compatible and incompatible events. 1 1

15. PROBABILITY System reliability. Classical definition of the probability. Axioms. Geometrical and statistical probabilities. Properties. 1 1 The probabilities of union and intersection. 1 1 Dependent and independent events. Conditional probability. Complete probability. Bernoulli experiments. 1 1

16. RANDOM VARIABLES Discrete and continuous random variables, theirs characteristics. 0,67 1 Binomial, Poisson and normal distributions. 0,67 1 Law of large numbers, central limit theorem. 0,67 1 17. DESCRIPTIVE STATISTIC

General set (population) and sample set. Data’s collection and classification. Notion of variable. Frequencies. Data’s characteristics. Graphical presentation. 1 0;1;2

18. STATISTICAL INFERENCES

Normal distribution, 3-σ rule, Stjudent and χ2 distributions. 1 0;1;2 Point and interval estimations of parameters. Notion of confidence interval. Confidence intervals for mean (expected value) and variance of normal random variable. 1 0;1;2

Correlation and linear regression. 1 0;1;2 Hypothesis. Mistakes. Confidence level. 1 0;1;2 Parametrical statistical hypotheses for one and two sample sets. 1 0;1;2

Deductions of measurements rezults.



Table 3.3. Electronical Engineering Experts' assessment


rage ball*

MO-DE**



1. LINEAR ALGEBRA Matrixes. Operations with matrixes. 2 2 Circuit theory. Analysis of linear circuits.

Signals and systems. Determinants of the second and third order, theirs properties. Adjunct and minor. Extension of determinants by rows and columns. Determinants of higher order. Reciprocal/Inverse matrix. 1,75 2 Analysis of linear circuits.

Systems of linear equations. Solution of non-degenerated linear systems. Cramer’s formulas. Gauss methods for linear systems. 2 2 Analysis of linear circuits.

Circuit theory . Signals and systems. 2. VECTOR GEOMETRY

Vector notion. Linear vector’s operations. Vector’s projections. The vector’s coordinates. Radius vector, distance between two points. 2 2

Analysis of linear and non-linear circuits. Physics. Electronics. Master‘s degree studies

Inner/Scalar product. Finding the work force. Perpendicular condition of vectors. Angle between two vectors. 2 2 Vector product. Area of parallelogram and triangle.

2 2 Analysis of linear and non-linear circuits. Electromagnetic fields and waves.

Parallelepipedal product. Volume of parallelepiped. 1,5 1;2 3. ANALYTIC GEOMETRY OF PLANE AND SPACE

General plane equation. Angle between two planes. Canonical, parametric and general line equations in the space. Line equation on the plane. 1,25 1

Curves of the second order. Circle, ellipse, hyperbola, parabola. 1,25 1 Electronics.

4. LIMIT AND CONTINUITY

Elementary functions. Sequence and its limit. The number e.

1,75 2

Analysis of linear and non-linear circuits. In elclectronics and circuit theory is very important number e! Analogic Devices. Digital signal processing.

Functions having limit zero. Limit of unboundedly increasing functions. Indefinite expressions. Using of equivalent decreasing functions for computation of some limits. 1,75 2 Analysis of linear and non-linear

circuits.


Continuity at the point. Discontinuity points. 1,75 2 Analysis of linear and non-linear

circuits. Electronics. 5. DERIVATIVE OF A FUNCTION OF ONE VARIABLE AND ITS APPLICATIONS Physiscs. Electronics.

Derivative. Mechanical and geometrical sense of the derivative: calculation of speed and acceleration. Approximate calculation. 2 2 Analysis of Circuits and Signals.

Differentiation of non-explicit functions. Logarithmic differentiation. Differentiation of parametrically given functions. 1,75 2 Analysis of Circuits and Signals.

Derivatives of higher order. Differential of higher order. 2 2 Analysis of Circuits and Signals. Investigation of functions, minimal and maximal values.

2 2

Analysis of Circuits and Signals. Analogic Devices. Foundamentals of electronic equipment design.

6. DIFFERENTIAL CALCULATION OF FUNCTIONS IN SEVERAL VARIABLES Notion of functions in several variables. Limit. Continuity.

2 2

Processing and analysis of digital signals. Electromagnetic fields. Electronics. Circuit theory. Analogic Devices. Foundamentals of electronic equipment design. In Master’s degree studies.

Partial derivatives of functions of in several variables. Full differential. Partial derivatives of composite and non-explicit functions. 2 2

Processing and analysis of digital signals. Circuit theory. Electromagnetic fields and waves. In Master’s degree studies.

Partial derivatives and differentials of higher order.

2 2

Processing and analysis of digital signals. Circuit theory. Electromagnetic fields and waves. In Master’s degree studies.

Local and conditional extremes of functions in several variables.

2 2

Processing and analysis of digital signals. Electromagnetic fields. In Master’s degree studies.

Scalar field, direction derivative, gradient.

2 2

Processing and analysis of digital signals. Electromagnetic fields and waves. In Master’s degree studies.

2nd semester MATHEMATICS - 2 (P130B011), 4 K =6 ECTS 7. INDEFINITE INTEGRAL 2 2 In Electronics.

Primitive functions and indefinite integral. The table of indefinite integrals. 2 2 Processing and analysis of digital signals.

Methods of direct integration. Changing of variable in integration. 1,75 2 Processing and analysis of digital


signals. Partial integration. 1,75 2 Processing and analysis of digital

signals. Integration of functions, containing quadratic trinomial. 2 2 Processing and analysis of digital

signals. Integration of the simplest rational fractions. Expression of proper fractions by a sum of the simplest fractions. Method of indefinite coefficients. 1,25 2 Processing and analysis of digital

signals. Integration of irrational functions.

2 2 Processing and analysis of digital signals. Processing and analysis of digital signals. Signals and systems.

Integration of trigonometric functions. Processing and analysis of digital signals.

8. DEFINITE INTEGRAL AND ITS APPLICATIONS 2 2 Area of curvilinear trapezium, definite integral and its properties. 2 2 Newton-Leibniz formula. Integration methods. 2 2 Area of a figure and length of a curve. Volume of a rotation solid and its surface area. 1,25 2 Work of varying force, mass centre coordinates of non-homogeneity stick, static and inertia moments of a flat figure, liquid pressure. Improper integrals

2 2

Processing and analysis of digital signals. Calculation of power and energy. Digital signal processing.

9. MULTIPLE AND CURVILINEAR INTEGRALS Calculation of power and energy. Digital signal processing.

Double integral, properties, computation. 2 2

Processing and analysis of digital signals. Calculation of power and energy.

Triple integral, properties, computation. 1,75 2 Processing and analysis of digital signals.

Body mass centre coordinates inertial moments. 1,75 2

Curvilinear integrals. 2 1 Analysis of Circuits and Signals. Circuit theory.

10. DIFFERENTIAL EQUATIONS Differential equations and the solutions.. Diff. equations of the first order. Problem Cauchy. 2 1 Circuit theory. Separation of variables in differential equations. 2 1 Circuit theory. Homogeneous differential equations of the first order. 1,5 1;2 Circuit theory. Linear differential equations of the first order. 2 2 Circuit theory. Bernoulli differential equations. 2 2 Circuit theory. Linear and non-linear differential equations of the second order. 2 2 Circuit theory. Differential equations of higher order. 1,75 2 Elektronikos inžinerijos studentams ši


lygtis gali būti pakeista svyravimų lygtimi elektrinėje grandinėje.

Systems of differential equations. 2 2 Electronics.

Equation of mechanical oscillations, investigation of free oscillations. 2 2 Processing and analysis of digital signals.

3rd semester APPLIED MATHEMATIC (P001B117), 4 K =6 ECTS 11. NUMERICAL AND FUNCTIONAL SERIES

Series with positive members, convergence and divergence. Application of comparison, Dalamber, Cauchy radical and integral criteria. 1,25 2 Analysis of Circuits and Signals. Digital

signal processing. Sign changing series, low of Leibniz. Absolute and relative convergence. 1,5 2 Analysis of Circuits and Signals. Digital

signal processing. Convergence region of functional series. Expansion of functions by series and its applications in integration

and for diff. equations. 1,75 2 Fourier series and application in spectral analysis. 2 2 Circuit theory.

12. FUNCTIONS OF COMPLEX VARIABLE Operations with complex numbers. Algebraic, trigonometric and exponential forms of complex numbers, its applications. 2 2 Circuit theory. Signals and Systems.

Digital signal processing. The functions of complex variables, limit, continuity and differentiation. Cauchy and Riemann conditions. Laurent series. 2 2 Digital signal processing (are needed the

poles and range of convergation).

Integral of complex variables functions. Integral Cauchy formula. Isolated singular points. 2 2 Digital signal processing (themes: inverse Fourer transformation, inverse Z transformation).

13. LAPLACE TRANSFORMATION AND ITS APPLICATIONS Pre-image and image. Images of basics elementary functions. 2 2 Digital signal processing. Circuit theory. Differentiation of images and images of derivatives. 2 2 Digital signal processing. Solving of linear differential equations by operational method. 2 2 Digital signal processing. Circuit theory. 4th semester PROBABILITY THEORY AND MATHEMATICAL STATISTICS (P160B171), 2 KP = 3 ECTS

14. RANDOM EVENTS Elements of combinatorial analysis. Notion of the set. Operations. Random event. Compatible and incompatible events. 2 2 Programming. Digital signal processing.

15. PROBABILITY Classical definition of the probability. Axioms. Geometrical and statistical probabilities. Properties. 1,75 2 The probabilities of union and intersection. 1,75 2 Dependent and independent events. Conditional probability. Complete probability. Bernoulli experiments. 1,5 1; 2

Programming . Foundamentals of electronic equipment design.

16. RANDOM VARIABLES Discrete and continuous random variables, theirs characteristics. 2 2 Measurements and basics of metrology. Binomial, Poisson and normal distributions. 2 2 Measurements and basics of metrology. Law of large numbers, central limit theorem. 1,75 2 Measurements and basics of metrology.


17. DESCRIPTIVE STATISTIC General set (population) and sample set. Data’s collection and classification. Notion of variable. Frequencies. Data’s characteristics. Graphical presentation.

2 2

Calculation of research data and evaluation of research results. Measurement and basics of metrology. Bachelor final work.

18. STATISTICAL INFERENCES Normal distribution, 3-σ rule, Stjudent and χ2 distributions.

2 2

Calculation of research data and evaluation of research results Measurement and basics of metrology. Bachelor final work.

Point and interval estimations of parameters. Notion of confidence interval. Confidence intervals for mean (expected value) and variance of normal random variable. 2 2

Calculation of research data and evaluation of research results. Measurement and basics of metrology.

Correlation and linear regression. 1,75 2

Calculation of research data and evaluation of research results. Graduate studies.

Hypothesis. Mistakes. Confidence level. 1,75 2

Calculation of research data and evaluation of research results. Bachelor final work.

Parametrical statistical hypotheses for one and two sample sets. 1,75 2

Calculation of research data and evaluation of research results. Bachelor final work.


Table 3.4. Informatics Engineering Experts' assessment


rage ball*

MO-DE**



1. LINEAR ALGEBRA Matrixes. Operations with matrixes. 1,75 2 Programming. Determinants of the second and third order, theirs properties. Adjunct and minor. Extension of determinants by rows and columns. Determinants of higher order. Reciprocal/Inverse matrix. 0,75 0


Systems of linear equations. Solution of non-degenerated linear systems. Cramer’s formulas. Gauss methods for linear systems. 1,5 1;2

2. VECTOR GEOMETRY Vector notion. Linear vector’s operations. Vector’s projections. The vector’s coordinates. Radius vector, distance between two points. 1,5 1;2 Computer graphics. Inner/Scalar product. Finding the work force. Perpendicular condition of vectors. Angle between two vectors. 1,5 1;2 Vector product. Area of parallelogram and triangle. 1,5 1;2 Parallelepipedal product. Volume of parallelepiped. 0,25 0

3. ANALYTIC GEOMETRY OF PLANE AND SPACE General plane equation. Angle between two planes. Canonical, parametric and general line equations in the space. Line equation on the plane. 1,25 1 Curves of the second order. Circle, ellipse, hyperbola, parabola. 1,25 1

4. LIMIT AND CONTINUITY Elementary functions. Sequence and its limit. The number e. 1,5 1;2 Functions having limit zero. Limit of unboundedly increasing functions. Indefinite expressions. Using of equivalent decreasing functions for computation of some limits. 0,5 0;1

Continuity at the point. Discontinuity points. 1 1

Programming.

5. DERIVATIVE OF A FUNCTION OF ONE VARIABLE AND ITS APPLICATIONS Derivative. Mechanical and geometrical sense of the derivative: calculation of speed and acceleration. Approximate calculation. 1,5 1;2 Differentiation of non-explicit functions. Logarithmic differentiation. Differentiation of parametrically given functions. 1 0;2 Derivatives of higher order. Differential of higher order. 1,25 2 Investigation of functions, minimal and maximal values. 1,75 2 Programming.

6. DIFFERENTIAL CALCULATION OF FUNCTIONS IN SEVERAL VARIABLES Notion of functions in several variables. Limit. Continuity. 1 0;2 Partial derivatives of functions of in several variables. Full differential. Partial derivatives of composite and non-explicit functions. 1 0;2 Partial derivatives and differentials of higher order. 1 0;2 Local and conditional extremes of functions in several variables. 0,75 0 Scalar field, direction derivative, gradient. 0,75 0 2nd semester MATHEMATICS - 2 (P130B011), 4 K =6 ECTS

7. INDEFINITE INTEGRAL Primitive functions and indefinite integral. The table of indefinite integrals. 1 0;2 Methods of direct integration. Changing of variable in integration. 1 0;2 Partial integration. 1 0;2 Integration of functions, containing quadratic trinomial. 0,75 0 Integration of the simplest rational fractions. Expression of proper fractions by a sum of the simplest fractions. Method of indefinite coefficients. 0,75 0 Integration of irrational functions. 0,75 0


Integration of trigonometric functions. 0,75 0 8. DEFINITE INTEGRAL AND ITS APPLICATIONS

Area of curvilinear trapezium, definite integral and its properties. 1,5 1;2 Newton-Leibniz formula. Integration methods. 1 0;2 Area of a figure and length of a curve. Volume of a rotation solid and its surface area. 1 1 Work of varying force, mass centre coordinates of non-homogeneity stick, static and inertia moments of a flat figure, liquid pressure. 0,5 0

9. MULTIPLE AND CURVILINEAR INTEGRALS Double integral, properties, computation. 1 0;2 Triple integral, properties, computation. 1 0;2 Body mass centre coordinates inertial moments. 0,25 0 Curvilinear integrals. 0,25 0

10. DIFFERENTIAL EQUATIONS Differential equations and the solutions.. Diff. equations of the first order. Problem Cauchy. 1 0;2 Separation of variables in differential equations. 1 0;2 Homogeneous differential equations of the first order. 1 0;2 Linear differential equations of the first order. 1 0;2 Bernoulli differential equations. 0,5 0 Linear and non-linear differential equations of the second order. 1 0;2 Differential equations of higher order. 1 0;2 Systems of differential equations. 0,75 0 Equation of mechanical oscillations, investigation of free oscillations. 0,5 0 3nd semester APPLIED MATHEMATIC (P001B117), 4 K =6 ECTS

11. NUMERICAL AND FUNCTIONAL SERIES Series with positive members, convergence and divergence. Application of comparison, Dalamber, Cauchy radical and integral criteria. 0,5 0 Sign changing series, low of Leibniz. Absolute and relative convergence. 0,25 0 Convergence region of functional series. Expansion of functions by series and its applications in integration and for diff. equations. 0,25 0 Fourier series and application in spectral analysis. 1,5 1;2

12. FUNCTIONS OF COMPLEX VARIABLE Operations with complex numbers. Algebraic, trigonometric and exponential forms of complex numbers, its applications. 1 0;2 The functions of complex variables, limit, continuity and differentiation. Cauchy and Riemann conditions. Laurent series. 0,5 0;1 Integral of complex variables functions. Integral Cauchy formula. Isolated singular points. 0,25 0

13. LAPLACE TRANSFORMATION AND ITS APPLICATIONS Pre-image and image. Images of basics elementary functions. 1 0;2 Differentiation of images and images of derivatives. 0,5 0;1


Solving of linear differential equations by operational method. 0,5 0;1 4th semester PROBABILITY THEORY AND MATHEMATICAL STATISTICS (P160B171), 2 KP = 3 ECTS

3. RANDOM EVENTS

Elements of combinatorial analysis. Notion of the set. Operations. Random event. Compatible and incompatible events. 1,5 1;2

The elements of computers and the main principles of operations of electronic elements.

4. PROBABILITY Classical definition of the probability. Axioms. Geometrical and statistical probabilities. Properties.. 1,5 1;2 Counting of probablies events. The probabilities of union and intersection 2 2 Dependent and independent events. Conditional probability. Complete probability. Bernoulli experiments. 2 2

5. RANDOM VARIABLES Discrete and continuous random variables, theirs characteristics. 1 0;2 Binomial, Poisson and normal distributions. 1 0;2 Law of large numbers, central limit theorem. 0,75 0;1


2 2

18. STATISTICAL INFERENCES

Normal distribution, 3-σ rule, Stjudent and χ2 distributions. 1,5 1;2 Point and interval estimations of parameters. Notion of confidence interval. Confidence intervals for mean (expected value) and variance of normal random variable. 1,25 1

Correlation and linear regression. Hypothesis. Mistakes. Confidence level. 1,75 2 Parametrical statistical hypotheses for one and two sample sets. 1,25 1

Research works.



Table 3.5. Mechanical Engineering Experts' assessment


rage ball*

MO-DE**



1. LINEAR ALGEBRA Matrixes. Operations with matrixes. 0,5 0;1 Determinants of the second and third order, theirs properties. Adjunct and minor. Extension of determinants by rows and columns. Determinants of higher order. Reciprocal/Inverse matrix. 0,5 0;1

Such knowledge is not applicable in theoretical mechanics.

Systems of linear equations. Solution of non-degenerated linear systems. Cramer’s formulas. Gauss methods for linear systems. 0,5 0;1

2. VECTOR GEOMETRY Vector notion. Linear vector’s operations. Vector’s projections. The vector’s coordinates. Radius vector, distance between two points.

1,5 1;2

Vector - a very important definition, used while studying statics, kinematics and dinamics. Important topics are vector projections, vector coordinates - which are applicable in kinematics.

Inner/Scalar product. Finding the work force. Perpendicular condition of vectors. Angle between two vectors. 1 1 It is used in statics while analysing power moment vector.

Vector product. Area of parallelogram and triangle. 2 2

It is used while analysing power actions in space, in kinematics it is used analysing gyration and integrate motion.

Parallelepipedal product. Volume of parallelepiped. 1,5 1;2 3. ANALYTIC GEOMETRY OF PLANE AND SPACE

General plane equation. Angle between two planes. Canonical, parametric and general line equations in the space. Line equation on the plane. 1 0;1 Curves of the second order. Circle, ellipse, hyperbola, parabola. 2 2 It is used in Kinematics analysing

tangible dot motion. 4. LIMIT AND CONTINUITY

Elementary functions. Sequence and its limit. The number e. Functions having limit zero. Limit of unboundedly increasing functions. Indefinite expressions. Using of equivalent decreasing functions for computation of some limits. 1,5 1 Continuity at the point. Discontinuity points. 1 1

5. DERIVATIVE OF A FUNCTION OF ONE VARIABLE AND ITS APPLICATIONS 1 1 Derivative. Mechanical and geometrical sense of the derivative: calculation of speed and acceleration. Approximate calculation.


Differentiation of non-explicit functions. Logarithmic differentiation. Differentiation of parametrically given functions. 2 2 Knowledge applicable while working

out on kinematics tasks/assignments. Derivatives of higher order. Differential of higher order. 1,5 1;2 Investigation of functions, minimal and maximal values. 1,5 1;2

6. DIFFERENTIAL CALCULATION OF FUNCTIONS IN SEVERAL VARIABLES Notion of functions in several variables. Limit. Continuity. 1 1 Partial derivatives of functions of in several variables. Full differential. Partial derivatives of composite and non-explicit functions. 1 1 Partial derivatives and differentials of higher order. 1 1 Local and conditional extremes of functions in several variables. 1 1 Scalar field, direction derivative, gradient. 1 1 2nd semester MATHEMATICS - 2 (P130B011), 4 K =6 ECTS

7. INDEFINITE INTEGRAL Primitive functions and indefinite integral. The table of indefinite integrals. 1,5 1;2 Knowledge applicable while working

out on dynamics tasks/assignments.

Methods of direct integration. Changing of variable in integration. 1,5 1;2 Knowledge applicable while working out on gynamics tasks/assignments.

Partial integration. 1 0;1 Integration of functions, containing quadratic trinomial. 0,5 0;1 Integration of the simplest rational fractions. Expression of proper fractions by a sum of the simplest fractions. Method of indefinite coefficients. 0,5 0;1 Integration of irrational functions. 0,5 0;1 Integration of trigonometric functions. 0,5 0;1

8. DEFINITE INTEGRAL AND ITS APPLICATIONS Area of curvilinear trapezium, definite integral and its properties. 0,5 0;1 Newton-Leibniz formula. Integration methods. 0,5 0;1 Area of a figure and length of a curve. Volume of a rotation solid and its surface area. 0,5 0;1 Work of varying force, mass centre coordinates of non-homogeneity stick, static and inertia moments of a flat figure, liquid pressure. 0,5 0;1

9. MULTIPLE AND CURVILINEAR INTEGRALS Double integral, properties, computation. 0,5 0;1 Triple integral, properties, computation. 0,5 0;1 Body mass centre coordinates inertial moments. 0,5 0;1 Curvilinear integrals. 0,5 0;1

10. DIFFERENTIAL EQUATIONS Differential equations and the solutions.. Diff. equations of the first order. Problem Cauchy. 0,5 0;1 Separation of variables in differential equations. 0,5 0;1 Homogeneous differential equations of the first order. 0,5 0;1 Linear differential equations of the first order. 0,5 0;1


Bernoulli differential equations. 0,5 0;1 Linear and non-linear differential equations of the second order. 0,5 0;1 Differential equations of higher order. 0,5 0;1 Systems of differential equations. 0,5 0;1 Equation of mechanical oscillations, investigation of free oscillations. 0,5 0;1 3nd semester APPLIED MATHEMATIC (P001B117), 4 K =6 ECTS

11. NUMERICAL AND FUNCTIONAL SERIES 0,5 0;1 Series with positive members, convergence and divergence. Application of comparison, Dalamber, Cauchy radical and integral criteria. 0,5 0;1 Sign changing series, low of Leibniz. Absolute and relative convergence. 0,5 0;1 Convergence region of functional series. Expansion of functions by series and its applications in integration and for diff. equations. 0,5 0;1 Fourier series and application in spectral analysis.

12. FUNCTIONS OF COMPLEX VARIABLE 0,5 0;1 Operations with complex numbers. Algebraic, trigonometric and exponential forms of complex numbers, its applications. 0,5 0;1 The functions of complex variables, limit, continuity and differentiation. Cauchy and Riemann conditions. Laurent series. 0,5 0;1 Integral of complex variables functions. Integral Cauchy formula. Isolated singular points.

13. LAPLACE TRANSFORMATION AND ITS APPLICATIONS 0,5 0;1 Pre-image and image. Images of basics elementary functions. 0,5 0;1 Differentiation of images and images of derivatives. 0,5 0;1 Solving of linear differential equations by operational method. 0,5 0;1 4th semester PROBABILITY THEORY AND MATHEMATICAL STATISTICS (P160B171), 2 KP = 3 ECTS

14. RANDOM EVENTS Elements of combinatorial analysis. Notion of the set. Operations. Random event. Compatible and incompatible events. 1 1

15. PROBABILITY Classical definition of the probability. Axioms. Geometrical and statistical probabilities. Properties.. 1 1 The probabilities of union and intersection 1 1 Dependent and independent events. Conditional probability. Complete probability. Bernoulli experiments. 1 1

16. RANDOM VARIABLES Discrete and continuous random variables, theirs characteristics. 1 1 Binomial, Poisson and normal distributions. 1 1 Law of large numbers, central limit theorem. 1 1

17. DESCRIPTIVE STATISTIC General set (population) and sample set. Data’s collection and classification. Notion of variable. Frequencies. Data’s characteristics. Graphical presentation. 1 1


18. STATISTICAL INFERENCES Normal distribution, 3-σ rule, Stjudent and χ2 distributions. 1 1 Point and interval estimations of parameters. Notion of confidence interval. Confidence intervals for mean (expected value) and variance of normal random variable. 1 1

Correlation and linear regression. 1 1 Hypothesis. Mistakes. Confidence level. 1 1 Parametrical statistical hypotheses for one and two sample sets. 1 1


Table 3.6. Physics Experts' assessment


Content Ave-rage ball*

MO-DE**


useful / necessary)

1st semester HIGHER MATHEMATICS - 1 (P130B074) , 4 K =6 ECTS 1. COMPLEX NUMBERS

Definitions. Graphical Representation. Operations with complex numbers. Algebraic, trigonometric and exponential forms of complex numbers. Polar form. Products and Quotients. Powers and roots. Applications. 2 2

2. MATRIXES. DETERMINANTS. SYSTEMS OF LINEAR EQUATIONS Determinants and matrixes of the second and third order, theirs properties, calculation. Operations with matrixes. Determinants of higher order. Reciprocal/Inverse matrix. 2 2

Systems of linear equations. Solution of non-degenerated linear systems. Kronecker Capelli theorem. Cramer’s formulas. Gauss methods for linear systems. 2 2

3. VECTORS. OPERATIONS

Vector notion. Linear vector’s operations. Vector’s projections. The vector’s coordinates. Radius vector, distance between two points. Inner/Scalar product. Finding the work force. Perpendicular condition of vectors. Angle between two vectors. Vector product. Area of parallelogram and triangle. Parallelepipedal product. Volume of parallelepiped. A basis and rank of vector system.

2 2


4. ANALYTIC GEOMETRY OF PLANE AND SPACE General plane equation. Angle between two planes. Canonical, parametric and general line equations in the space. Line equation on the plane. Curves of the second order. Circle, ellipse, hyperbola, parabola. Polar coordinate system.

2 2

5. FUNCTIONS. LIMIT AND CONTINUITY. DERIVATIVE Elementary functions. Sequence and its limit. The number e. Functions having limit zero. Limit of unboundedly increasing functions. Indefinite expressions. Using of equivalent decreasing functions for computation of some limits. Continuity at the point. Discontinuity points.

2 2

Derivative. Mechanical and geometrical sense of the derivative: calculation of speed and acceleration. Approximate calculation. Derivatives of higher order. Differentiation of non-explicit functions. Logarithmic differentiation. Differentiation of parametrically given functions. Differential of higher order. Investigation of functions, minimal and maximal values.

2 2

1st semester DISCRETE MATHEMATICS (P001B010) , 2 K =3 ECTS

1. ELEMENTS OF SET THEORY Definition of a Set. Elements. Some special sets. Operations. Examples and applications. 1,75 2

2. COMBINATORICS General laws. Arrangements without and with repetitions. Combination. 1,75 2

3. MATHEMATICAL LOGIC Propositions. Oparations. Counting Truth tables. Predicates. Quantors. Operations. Formulation of Physical laws. Binary relations. Properties. 1,75 2

4. THEORY OF GRAPHS AND ALGORITHMS Definitions of graph. Directions, Weights and Flows. Special Graphs. Some Terms. Subgraphs. Trees and Bipartite Graphs. Hamiltonian and Eulerian Paths. Planar Graphs. Algorithms. Ways of outlines of algorithms. Substitution method. Problems.

2 2

5. ELEMENTS OF PROBABILITY THEORY Random event. Compatible and incompatible events. Classical definition of the probability. Axioms. Geometrical and statistical probabilities. Properties. The probabilities of union and intersection. Dependent and independent events. Conditional probability. Complete probability. Discrete and continuous random variables, theirs characteristics. Physical processes. Applicaions.

2 2

2nd semester HIGHER MATHEMATICS - 2 (P130B075) , 4 K =6 ECTS

1. INDEFINITE INTEGRAL Primitive functions and indefinite integral. The table of indefinite integrals. Methods of direct integration. Changing of variable in integration. Partial integration. Integration of functions, containing quadratic trinomial. Integration of the simplest rational fractions. Expression of proper fractions by a sum of the simplest fractions. Method of indefinite coefficients. Integration of irrational functions. Integration of trigonometric functions.

2 2


2. DEFINITE INTEGRAL AND ITS APPLICATIONS. IMPROPER INTEGRALS. Area of curvilinear trapezium, definite integral and its properties. Newton-Leibniz formula. Integration methods. Area of a figure and length of a curve. Volume of a rotation solid and its surface area. Work of varying force, mass centre coordinates of non-homogeneity stick, static and inertia moments of a flat figure, liquid pressure. Improper integrals. Application of Newton-Leibnic formula.

2 2

3. FUNCTIONS IN SEVERAL VARIABLES. LIMIT, CONTINUITY. DIFFERENTIAL CALCULATION.

Notion of functions in several variables. Limit. Continuity. Partial derivatives of functions of in several variables. Full differential. Partial derivatives of composite and non-explicit functions. Partial derivatives and differentials of higher order. Local and conditional extremes of functions in several variables. Scalar field, direction derivative, gradient.

2 2 Very important themes.

4-6. MULTIPLE, CURVILINEAR AND SURFACE INTEGRALS Double integral, properties, computation. Triple integral, properties, computation. Body mass centre coordinates inertial moments. Curvilinear integrals. Surface integrals. 1,75 2 2th semester PROBABILITY THEORY AND MATHEMATICAL STATISTICS (P160B147), 2 KP = 3 ECTS

1. RANDOM EVENTS AND PROBABILITY Sets. Elements of combinatorial analysis. Notion of the set. Operations. Random event. Compatible and incompatible events. Classical definition of the probability. Axioms. Geometrical and statistical probabilities. Properties. The probabilities of union and intersection. Dependent and independent events. Conditional probability. Complete probability. Bernoulli experiments.

2 2

2. RANDOM VARIABLES

Discrete and continuous random variables, theirs characteristics. Binomial, Poisson and normal distributions. Law of large numbers, central limit theorem. 2 2


2

2

4. STATISTICAL INFERENCES Normal distribution, 3-σ rule, Stjudent and χ2 distributions. Point and interval estimations of parameters. Notion of confidence interval. Confidence intervals for mean (expected value) and variance of normal random variable. Correlation and linear regression. Hypothesis. Mistakes. Confidence level. Parametrical statistical hypotheses for one and two sample sets.

2 2

5. STATISTIC IN PHYSICS 2 2

2th semester DIFFERENTIAL EQUATIONS (P130B168), 4 KP = 6 ECTS 1. SIMPLE DIFFERENTIAL EQUATIONS


Differential equations and the solutions.. Diff. equations of the first order. Problem Cauchy. Separation of variables in differential equations. Homogeneous differential equations of the first order. Linear differential equations of the first order. Bernoulli differential equations. Linear and non-linear differential equations of the second order. Differential equations of higher order. Systems of differential equations.

2 2 In general physiscs.

2. DIFFERENTIAL EQUATIONS OF PARTIAL DERIVATIVES Liner and quasi-linear differential equations of partial derivatives. Solving methods. Geometrical interpretation. Applications in physics. 1,75 2 Very important themes.

3. DIFFERENTIAL MODELS Mathematical models of real processes. Non-linear differential equations. Applications in physics and optometry. 2 2


Table 3.7. Public Administration, Busines Administration Experts' assessment


rage ball*

MO-DE**


useful / necessary) APPLIED MATHEMATICS (P120B011), 4 KP = 6 ECTS

1. MATHEMATICS IN ECONOMICS The simplest financial calculations: interest, credit balance, discount. Using MS Excel. 2 2 Models for the tasks of optimal planning. 2 2

The specialists of Public procurement. For preparing projects, counting the organizational budget.

2. SYSTEMS OF LINEAR EQUATIONS Definition. Compatibility and non-compatibility systems. Determinate and indeterminate systems. Equivalent systems. Elementary transformations. 1,67 2

For economists, finance managers, counters, bank workers, employees, counting he interests, searching for optimal decisions.

Gauss and Gauss-Jordan methods. Diagonal system. 1,33 2 3. VECTORS

Vector and vector space. Operations with vectors. 0,67 1


4. MATRIXES, DETERMINANTS Matrixes. Sorts of matrixes. The matrix operations. 1 1 Economical interpretation of matrixes (costs, portages and etc.). 1,67 2 For economists, finance managers,

counters, bank workers, employees. Determinants of the second, third and fourth orders. 0,67 1 Cramer method for solving systems of linear equations. 0.67 1 Reciprocal/Inverse matrix. Solving systems of linear equations by method of inverse matrixes. 1 1 The Leonjev model for the balance of economic system. 2 2 Very important. It is essential to

calculate with computer. 5. SYSTEM OF LINEAR INEQUALITIES AND ITS SOLUTIONS. LINEAR INEQUALITIES IN THE

OPTIMAL PLANNING. 1,33 2

6. SETS AND FUNCTIONS Notions of the set and subset. Operations. 0,33 0 Function’s definition. The definition range and the set of values. Examples. Limit of the function. 1 0; 2 Elements of mathematical logic. Basic logic operations and their authenticity value 0,33 0

7. DIFFERENTIATION Notions of derivative and differential. Derivatives of a functions in one variable. Derivatives and differentials of higher order.

1 0; 2 It is important in econometric.

Differential calculation of functions in several variables. 0,67 0 Investigation of functions, minimal and maximal values. 1,33 2 Very important. Application of the derivatives in economics. 1 0; 2

8. INDEFINITE INTEGRAL Primitive functions and indefinite integral. Sense. Integration laws and methods (by changing variable, partial integration). 0,67 1

9. DEFINITE INTEGRAL Definite integral and its properties, sense. 0,67 1 Newton-Leibniz formula. Integration methods (term-by-term, by substitution and parts). 0,67 1 Applications (area of curvilinear trapezium, volume of a rotation solid). 0,67 1 Definite integral in economics. 1 1

10. ELEMENTS OF DESCRIPTIVE STATISTIC General set and sample set. Frequencies. 1,33 2 Data’s characteristics. Graphical presentation. 1 0; 2

Very important for preparation of statisticals reports, making conclusions!

* The average ball is calculated from numbers: 0, 1, 2. Where: 0 – not necessary, 1 –should be taught, 2 - necessary. **If the multiple mode exists, we write both.


ANNEX NO.4.

COPY OF THE CONTENT OF RECOMMENDED BOOK FOR PHYSICISTS


ANNEX NO.5.

COPY OF THE CONTENT OF RECOMMENDED THEMES FOR ELECTRICAL ENGINEERING

development of mathematical competencies in higher ... · aimed at creating northern lithuania...

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