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1 Tuning Educational Structures in Europe Reference Points for the Design and Delivery of Degree Programmes in Mathematics Life Long Learning

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3 Reference Points for the Design and Delivery of Degree Programmes in Mathematics

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5 Reference Points for the Design and Delivery of Degree Programmes in Mathematics

6 Tuning Educational Structures in Europe The name Tuning was chosen for the project to reflect the idea that universities do not look for uniformity in their degree programmes or any sort of unified, prescriptive or definitive European curricula but simply for points of reference, convergence and common understanding. The protection of the rich diversity of European education has been paramount in the Tuning Project from the very start and the project in no way seeks to restrict the independence of academic and subject specialists, or undermine local and national academic authority. The Tuning Project is supported by the European Commission through the Socrates and Tempus programmes (of the Directorate Education and Culture) This publication reflects the views of the authors only, and the European Commission cannot be held responsible for any use which may be made of the information contained therein. Tuning Project Although all material that has been developed as part of the Tuning Project is owned by its formal participants, other Higher Education Institutions are free to test and use the material after publication, provided that the source is acknowledged. No part of this publication, including the cover design, may be reproduced, stored or transmitted in any form or by any means, whether electronical, chemical, mechanical, optical, by recording or photocopying, without prior permission of the publisher. Publicaciones de la Universidad de Deusto Apartado Bilbao ISBN: Depósito legal: BI Impreso en España/Printed in Spain

7 Content Introduction to the Tuning Project 9 1. Mathematics as a discipline Degree profiles Typical occupations of graduates Role of mathematics in other degree programmes Consultation process with stakeholders Towards a common framework for Mathematics degrees in Europe A common framework: what it should and should not be or do Towards a common core mathematics curriculum Some principles for a common core curriculum for the first degree (Bachelor) in mathematics The second degree (Master) in mathematics A common framework and the Bologna agreement Doctorates in Mathematics Workload and ECTS ECTS Planning form for a sample module Monitoring Learning outcomes and competences - level cycle descriptors First cycle descriptors First cycle level descriptor Second cycle descriptor Third cycle descriptor 44 7

8 9. Subject specific competences First cycle Second cycle Third cycle Commentary on some sample competences Generic competences Commonly used approaches to Learning, Teaching and Assessment in Mathematics Feedback and assessment Conclusion Quality Enhancement List of Mathematics Subject Area Group (SAG) members 55 8

9 Introduction to the Tuning project Tuning Educational Structures in Europe is a university driven project which aims to offer a universal approach to implement the Bologna Process at the level of higher education institutions and subject areas. The Tuning approach consists of a methodology to (re-) design, develop, implement and evaluate study programmes for each of the Bologna cycles. Furthermore, Tuning serves as a platform for developing reference points at subject area level. These are relevant for making programmes of studies comparable, compatible and transparent. Reference points are expressed in terms of learning outcomes and competences. Learning outcomes are statements of what a learner is expected to know, understand and be able to demonstrate after completion of a learning experience According to Tuning, learning outcomes are expressed in terms of the level of competence to be obtained by the learner. Competences represent a dynamic combination of cognitive and meta-cognitive skills, knowledge and understanding, interpersonal, intellectual and practical skills, and ethical values. Fostering these competences 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 study), others are generic (common to any degree course). It is normally the case that competence development proceeds in an integrated and cyclical manner throughout a programme. To make levels of learning comparable the subject area groups/thematic Networks have developed cycle (level) descriptors which are also expressed in terms of competences. According to Tuning, the introduction of a three cycle system implies a change from a staff centred approach to a student oriented approach. It is the student that has to be prepared as well as possible for his or her future role in society. Therefore, Tuning has organized a Europe-wide consultation process including employers, graduates and academic staff / faculty to identify the most important competences that should be formed or developed in a degree programme. The outcome of this consultation process is reflected in the set of reference points generic and subject specific competences identified by each subject area. Besides addressing the implementation of a three cycle system, Tuning has given attention to the Europe-wide use of the student workload 9

10 based European Credit Transfer and Accumulation System (ECTS). According to Tuning ECTS is not only a system for facilitating the mobility of students across Europe through credit accumulation and transfer; ECTS can also facilitate programme design and development, particularly with respect to coordinating and rationalising the demands made on students by concurrent course units. In other words, ECTS permits us to plan how best to use students time to achieve the aims of the educational process, rather than considering teachers time as a constraint and students time as basically limitless. According to the Tuning approach credits can only be awarded when the learning outcomes have been met. The use of the learning outcomes and competences approach might also imply changes regarding the teaching, learning and assessment methods which are used in a programme. Tuning has identified approaches and best practices to form specific generic and subject specific competences. Finally, Tuning has drawn attention to the role of quality in the process of (re-)designing, developing and implementing study programmes. It has developed an approach for quality enhancement which involves all elements of the learning chain. It has also developed a number of tools and has identified examples of good practice which can help institutions to boost the quality of their study programmes. Launched in 2000 and strongly supported, financially and morally, by the European Commission, the Tuning Project now includes the vast majority of the Bologna signatory countries. The work of Tuning is fully recognized by all the countries and major players involved in the Bologna Process. At the Berlin Bologna follow-up conference which took place in September 2003, degree programmes were identified as having a central role in the process. The conceptual framework on which the Berlin Communiqué is based is completely coherent with the Tuning approach. This is made evident by the language used, where the Ministers indicate that degrees should be described in terms of workload, level, learning outcomes, competences and profile. As a sequel to the Berlin conference, the Bologna follow-up group has taken the initiative of developing an overarching Framework for Qualifications of the European Higher Education Area (EQF for HE) which, in concept and language, is in full agreement with the Tuning approach. 10

11 This framework has been adopted at the Bergen Bologna follow-up conference of May The EQF for Higher Education has made use of the outcomes both of the Joint Quality Initiative (JQI) and of Tuning. The JQI, an informal group of higher education experts, produced a set of criteria to distinguish between the different cycles in a broad and general manner. These criteria are commonly known as the Dublin descriptors. From the beginning, the JQI and the Tuning Project have been considered complementary. The JQI focuses on the comparability of cycles in general terms, whereas Tuning seeks to describe cycle degree programmes at the level of subject areas. An important aim of all three initiatives (EQF, JQI and Tuning) is to make European higher education more transparent. In this respect, the EQF is a major step forward because it gives guidance for the construction of national qualification frameworks based on learning outcomes and competences as well as on credits. We may also observe that there is a parallel between the EQF and Tuning with regard to the importance of initiating and maintaining a dialogue between higher education and society and the value of consultation -- in the case of the EQF with respect to higher education in general; in that of Tuning with respect to degree profiles. In the summer of 2006 the European Commission launched a European Qualification Framework for Life Long Learning. Its objective is to encompass all types of learning in one overall framework. Although the concepts on which the EQF for Higher Education and the EQF for LLL are based differ, both are fully coherent with the Tuning approach. Like the other two, the LLL variant is based on the development of level of competences. From the Tuning perspective both initiatives have their value and their roles to play in the further development of a consistent European Education Area. This brochure reflects the outcomes of the work done by the Subject Area Group (SAG) Mathematics so far. The outcomes are presented in a template that was developed to facilitate readability and rapid comparison across the subject areas. The summary aims to provide, in a very succinct manner, the basic elements for a quick introduction into the subject area. It shows in synthesis the consensus reached by a subject area group after intense and lively discussions in the group. The more ample documents on which the template is based are also included in the brochure. They give a more detailed overview of the elaborations of the subject area group. The Tuning Management Committee 11

12 Mathematics This brochure summarises the work of the Tuning Mathematics group in the projects Tuning I to IV revised after the validation conference of Much of the material has appeared previously in the reports of Tuning I and II 1 and is also available on the Tuning project website. 2 A document which proved extremely useful in this process and which met with unanimous agreement from the group was the UK Quality Assurance Agency Benchmark document on Mathematics, Statistics and Operational Research 3. It is quoted from almost verbatim at some points in this document. 1 Final reports Tuning I and Tuning II, Julia Gonzalez and Robert Wagenaar (eds), 2002 and 2005 (resp)

13 1. Mathematics as a discipline Mathematics as an intellectual discipline traces its roots back through all the major civilisations to the earliest recorded human works. While it originated as a systematisation of the solutions of practical problems in areas such as land surveying (hence geometry), construction, war and commerce, it evolved with the realisation that abstraction of the essentials led to generalisation of the applications and hence became a science which uses rigorous deduction to arrive at solid conclusions from clearly stated assumptions. It is a discipline that requires people to think with logic, criticism, rigour and depth. Its abstraction makes it applicable to almost any discipline, since it identifies patterns that are common to many areas, such as health sciences, earth sciences, astronomy, computer sciences, etc. Mathematics is fundamental not only to much of science and technology but also to almost all situations that require an analytical modelbuilding approach, whatever the discipline. In recent decades there has been an explosive growth of the use of mathematics in areas outside the traditional base of science, technology and engineering. Statistics as a discipline within mathematics arose from probability, which seemingly originated from considerations of gambling and developed further in the nineteenth century with the growth of official statistics. It is the science of modelling random phenomena and making inferences. It uses mathematical techniques and ideas to solve problems involving randomness, chance, variability, risk and so on. Statistics plays a major and increasing role in personal and public life, particularly in medicine, quality control and management, all areas of physical and social sciences, business and economics. Programmes in mathematics can and do vary from the very pure or theory-based to the very applied or practice based. Some are broad, while others allow specialisation in particular areas, such as statistics or financial mathematics. They all share the key learning outcomes detailed in what follows. 13

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15 2. Degree profiles Typical degrees offered First cycle in Mathematics Applied Mathematics Mathematical Sciences Mathematical Physics Mathematics and Statistics Financial Mathematics Second cycle in Mathematics Statistics Financial Mathematics Biomathematics Third cycle in any specialised area of Mathematics 15

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17 3. Typical occupations of graduates First cycle Programme Profile Mathematics with Education Mathematics specialising in statistics Mathematics possibly specialising in finance, statistics or economics Mathematics with significant computer science Category / Group of professions Education Industry/Government/ Health sector Banking/Insurance/ Business Industry/Banking Example professions Secondary school teacher of Mathematics Statistician Actuary, Banker, Risk Manager, Accountant Software analyst Second cycle Programme profile Mathematics specialising in statistics at second cycle Mathematics specialising in finance, statistics or econometrics at second cycle Mathematics with specialisation at second cycle Category / Group of professions Industry/Government/ Health sector Banking/Insurance/ Business Industry/Defence Industry Statistician Example professions Actuary, Banker, Risk Manager, Accountant Researcher 17

18 Third cycle Programme profile Any specialised field of Mathematics Category / Group of professions University Example professions Researcher/Teacher Statistics Financial or actuarial mathematics Industry, in particular biotechnology and medicine Banking/Insurance/ Business Researcher Actuary, Banker, Risk Manager Algebra Government Researcher, Cryptologist 18

19 4. Role of mathematics in other degree programmes Mathematics is an essential component of all engineering and most science courses, in particular physics, but also chemistry and, increasingly, biology. Some mathematics units are included in most courses in business studies and economics; statistics has particular importance in these areas and also in programmes in the humanities where no other mathematics courses may form part of the programme. It also commonly occurs as one subject in a two subject degree, such as Mathematics and Economics, Mathematics and Computer Science, Mathematics and Biology, Mathematics and Physics, Mathematics and Economics, Mathematics and Education. This list is not exhaustive, and many variations of the listed titles exist, while more will probably emerge, reflecting Mathematics universal relevance. While no generalisations can be made, the ordering of a subject area in the title may reflect its relative weight. This document focuses mainly on degrees in Mathematics, so we merely note here that a first cycle interdisciplinary degree would need to have a significant proportion of mathematical content to enable progression to a second cycle programme in mathematics. 19

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21 5. Consultation process with stakeholders Graduates and employers were consulted about generic and some subject specific competences in a questionnaire as part of Tuning I and again as part of Tuning IV; academics were consulted also about more detailed subject specific competences. The results informed the subsequent paper Towards a common framework for Mathematics degrees in Europe, which was also published in the European Mathematical Society Newsletter in September 2002 (pp ) 4. This paper has also been widely disseminated at national levels and is substantially reproduced in section

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23 6. Towards a common framework for Mathematics degrees in Europe 6.1. A common framework: what it should and should not be or do The only possible aim in agreeing a common European framework should be to facilitate the automatic recognition of mathematics degrees in Europe in order to help mobility. By this we mean that when somebody with a degree in mathematics from country A goes to country B: a) He/she will be legally recognised as holding such a degree, and the Government of country B will not require further proof of competence. b) A potential employer in country B will be able to assume that he/ she has the general knowledge expected from somebody with a mathematics degree. Of course, neither of these guarantees employment: the mathematics graduate will still have to go through whatever procedures (competitive exams, interviews, analysis of his/her curriculum, value of the degree awarding institution in the eyes of the employer...) are used in country B to get either private or public employment One important component of a common framework for mathematics degrees in Europe is that all programmes have similar, although not necessarily identical, structures. Another component is agreeing on a basic common core curriculum while allowing for some degree of local flexibility We should emphasise that by no means do we think that agreeing on any kind of common framework can be used as a tool for automatic transfer between Universities. These will always require consideration by case, since different programmes can get students to adequate levels in different but coherent ways, but an inappropriate mixing of programmes may not. 23

24 In many European countries there exist higher education institutions that differ from universities both in the level they demand from students and in their general approach to teaching and learning. In fact, in order not to exclude a substantial number of students from higher education, it is essential that these differences be maintained. We want to make explicit that this paper refers only to universities (including technical universities), and that any proposal of a common framework designed for universities would not necessarily apply to other types of institutions Towards a common core mathematics curriculum General remarks At first sight, mathematics seems to be well suited for the definition of a core curriculum, e.g. for the first two or three years. Because of the very nature of mathematics, and its logical structure, there will be a common part in all mathematics programmes, consisting of the fundamental notions. On the other hand, there are many areas in mathematics, and many of them are linked to other fields of knowledge (computer science, physics, engineering, economics, etc.). Flexibility is of the utmost importance to keep this variety and the interrelations that enrich our science. There could possibly be an agreement on a list of subjects that must absolutely be included (linear algebra, calculus/analysis) or that should be included (probability/statistics, some familiarity with the mathematical use of a computer) in any mathematics degree. In the case of some specialised courses, such as mathematical physics, there will certainly be variations between countries and even between universities within one country, without implying any difference of quality of the programmes. Moreover, a large variety of mathematics programmes exist currently in Europe. Their entry requirements vary, as do their length and the demands on the student. It is extremely important that this variety be maintained, both for the efficiency of the education system and socially, to accommodate the possibilities of more potential students. To fix a single definition of contents, skills and level for the whole of European higher 24

25 education would exclude many students from the system, and would, in general, be counterproductive. In fact, the group is in complete agreement that programmes could diverge significantly beyond the basic common core curriculum (e.g. in the direction of pure mathematics, or probability - statistics applied to economy or finance, or mathematical physics, or the teaching of mathematics in secondary schools). The presentation and level of rigour, as well as accepting there is and must continue to be variation in emphasis and, to some extent, content, even within the first two or three years, will make all those programmes recognisable as valid mathematics programmes. As for the second cycle, not only do we think that programmes could differ, but we are convinced that, to reflect the diversity of mathematics and its relations with other fields, all kinds of different second cycles in mathematics should be developed, using in particular the specific strengths of each institution The need for accreditation The idea of a basic core curriculum must be combined with an accreditation system. If the aim is to recognise that various universities fulfil the requirement of the core curriculum, then one has to check on three aspects: a list of contents a list of skills the level of mastery of concepts These cannot be reduced to a simple scale. To give accreditation to a mathematics programme, an examination by a group of peer reviewers, mostly mathematicians, is necessary. The key aspects to be evaluated should be: a) the programme as a whole b) the units in the programme (both the contents and the level) c) the entry requirements d) the learning outcomes (skills and level attained) e) a qualitative assessment by both graduates and employers 25

26 The group does not believe that a (heavy) system of European accreditation is needed, but that universities in their quest for recognition will act at the national level. For this recognition to acquire international standing, the presence on the review panel of mathematicians from other countries seems essential Some principles for a common core curriculum for the first degree (Bachelor) in mathematics We do not feel that fixing a detailed list of topics to be covered is necessary, or even convenient. But we think that it is possible to give some guidelines as to the common contents of a European first degree in mathematics, and more important, as to the skills that all graduates should develop Contents All mathematics graduates will have knowledge and understanding of, and the ability to use, mathematical methods and techniques appropriate to their programme. Common ground for all programmes will include calculus in one and several real variables, and linear algebra Mathematics graduates must have knowledge of the basic areas of mathematics, not only those that have historically driven mathematical activity, but also others of more modern origin. Therefore graduates should normally be acquainted with most, and preferably all, of the following: language of elementary set theory, basic differential equations; basic complex functions; some probability; some statistics; some numerical methods and computer simulation; basic geometry of curves and surfaces; some algebraic structures; some discrete mathematics; and some modelling from a related discipline These need not be learned in individual modules covering each subject in depth from an abstract point of view. For example, one could learn about groups in a course on (abstract) group theory or in the framework of a course on cryptography. Geometric ideas, given their central role, could appear in a variety of courses. 26

27 Other methods and techniques will be developed according to the requirements and character of the programme, which will also largely determine the levels to which the developments are taken. In any case, all programmes should include a substantial number of courses with mathematical content In fact, broadly two kinds of mathematics curricula currently coexist in Europe, and both are useful. Let us call them, following the QAA Benchmark document, theory based and practice based programmes. The weight of each of the two kinds of programmes varies widely depending on the country, and it might be interesting to find out whether most European university programmes of mathematics are theory based or not. Graduates from theory-based programmes will have knowledge and understanding of results from a range of major areas of mathematics. Examples of possible areas are algebra, analysis, geometry, number theory, differential equations, mechanics, probability theory and statistics, but there are many others. This knowledge and understanding will support the knowledge and understanding of mathematical methods and techniques, by providing a firmly developed mathematical context. Graduates from practice-based programmes will also have knowledge of results from a range of areas of mathematics, but the knowledge will commonly be designed to support the understanding of models and how and when they can be applied. Besides those mentioned above, these areas include numerical analysis, control theory, operations research, discrete mathematics, game theory and many more. (These areas may of course also be studied in theory-based programmes.) It is necessary that all graduates will have met at least one major area of application of their subject in which it is used in a serious manner and this is considered essential for a proper appreciation of the subject. The nature of the application area and the manner in which it is studied might vary depending on whether the programme is theorybased or practice based. Possible areas of application include physics, astronomy, chemistry, biology, engineering, computer science, information and communication technology, economics, accountancy, actuarial science, finance and many others. 27

28 Skills For a standard notion like integration in one variable, the same content could imply: computing simple integrals understanding the definition of the Riemann integral proving the existence and properties of the Riemann integral for classes of functions using integrals to model and solve problems of various sciences. So, on one hand the contents must be clearly spelled out, and on the other various skills are developed by the study of the subject Students who graduate from programmes in mathematics have an extremely wide choice of career available to them. Employers greatly value the intellectual ability and rigour and the skills in reasoning that these students will have acquired, their firmly established numeracy, and the analytic approach to problem-solving that is their hallmark. Therefore, the three key skills that we consider may be expected of any mathematics graduate are: a) the ability to conceive a proof, b) the ability to model a situation mathematically, c) the ability to solve problems using mathematical tools. It is clear that, nowadays, solving problems should include their numerical and computational resolution. This requires a sound knowledge of algorithms and programming and the use of the existing software Note also that skills and level are developed progressively through the practice of many subjects. We do not start a mathematics programme with one course called how to make a proof and one called how to model a situation, with the idea that those skills will be acquired immediately. Instead, it is through practice in all courses that these develop Level All graduates will have knowledge and understanding developed to higher levels in particular areas. The higher-level content of programmes will re- 28

29 flect the title of the programme. For example, graduates from programmes with titles involving statistics will have substantial knowledge and understanding of the essential theory of statistical inference and of many applications of statistics. Programmes with titles such as mathematics might range quite widely over several branches of the subject, but nevertheless graduates from such programmes will have treated some topics in depth The second degree (Master) in mathematics We have already made explicit our belief that establishing any kind of common curriculum for second cycle studies would be a mistake. Because of the diversity of mathematics, the different programmes should be directed to a broad range of students, including in many cases those whose first degree is not in mathematics, but in more or less related fields (computer science, physics, engineering, economics, etc.). We should therefore aim for a wide variety of flavours in second cycle programmes. Rather than the contents, we think that the common denominator of all second cycles should be the level of achievement expected from students. A unifying characteristic feature could be the requirement that all second cycle students carry out a significant amount of individual work. This could be reflected in the presentation of a substantial individual project. We believe that, to be able to do real individual work in mathematics, the time required to obtain a Master s qualification should be the equivalent of at least 90 ECTS credits. Therefore, depending on the national structure of first and second cycles, a Master would typically vary between 90 and 120 ECTS credits A common framework and the Bologna agreement How various countries implement the Bologna agreement will make a difference on core curricula. In particular, 3+2 may not be equivalent to 5, because, in a 3+2 years structure, the 3 years could lead to a professional diploma, meaning that less time is spent on fundamental 29

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