# India's field standards have fallen

## Concepts and studies on university didactics and teacher training in mathematics

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1 Concepts and studies on university didactics and teacher training in mathematics Isabell Bausch Rolf Biehler Regina Brother Pascal R. Fischer Reinhard Hochmuth Wolfram Koepf Stephan Schreiber Thomas Wassong Ed. Mathematical preparatory and bridging courses, concepts, problems and perspectives

2 Concepts and studies on university didactics and teacher training in mathematics

3 Edited by Prof. Dr. Rolf Biehler (managing editor), University of Paderborn Prof. Dr. Albrecht Beutelspacher, Justus Liebig University Giessen Prof. Dr. Lisa Hefendehl-Hebeker, University of Duisburg-Essen, Essen campus Prof. Dr. Reinhard Hochmuth, Leuphana University Lüneburg Prof. Dr. Jürg Kramer, Humboldt University of Berlin Prof. Dr. Susanne Prediger, Technical University of Dortmund Prof. Dr. Günter M. Ziegler, Freie Universität Berlin Teaching mathematics at all levels of the educational chain plays a key role in promoting interest and performance in the field of mathematics, science and technology. Related didactic research and development work provides theoretical and empirical foundations as well as good practical examples. The series "Concepts and studies for university didactics and teacher training in mathematics" documents scientific studies as well as theoretically sound and practically proven innovative approaches for teaching in mathematics-based courses and all phases of teacher training in mathematics.

4 Isabell Bausch Rolf Biehler ReginaBruder Pascal R. Fischer Reinhard Hochmuth Wolfram Koepf Stephan Schreiber Thomas Wassong Editor Mathematical preparatory and bridging courses Concepts, problems and perspectives

5 volume editors: Isabell Bausch Technical University Darmstadt, Germany Prof. Dr. Regina Bruder Technical University of Darmstadt, Germany Prof. Dr. Reinhard Hochmuth Leuphana University of Lüneburg, Germany r e i n h a r d. h o c h m u h l e u p h a n a.d e Dr. Stephan Schreiber Leuphana University of Lüneburg, Germany Prof. Dr. Rolf Biehler University of Paderborn, Germany Dr. Pascal R. Fischer University of Kassel, Germany Prof. Dr. Wolfram Koepf University of Kassel, Germany Thomas Wassong University of Paderborn, Germany ISBN DOI / ISBN (ebook) The German National Library lists this publication in the German National Bibliography; detailed bibliographic data are available on the Internet at. Springer Spectrum Springer Fachmedien Wiesbaden 2014 The work including all of its parts is protected by copyright. Any use that is not expressly permitted by copyright law requires the prior consent of the publisher. This applies in particular to copying, editing, translation, microfilming and saving and processing in electronic systems. The reproduction of common names, trade names, trade names etc. in this work does not justify the assumption that such names are to be regarded as free within the meaning of the trademark and trademark protection legislation and can therefore be used by everyone, even without special identification. Planning and editing: Ulrike Schmickler-Hirzebruch Barbara Gerlach Printed on acid-free and chlorine-free bleached paper. Springer Spectrum is a trademark of Springer DE. Springer DE is part of the specialist publishing group Springer Science + Business Media

6 Foreword All articles published in this conference volume are based on a lecture or a poster that was presented at the first working conference of the Competence Center for University Didactics Mathematics (. The conference took place from to in the Gießhaus of the University of Kassel with over 100 participants and was in conjunction with the associated project Virtual Introductory Mathematics for the MINT Subjects (VEMINT, formerly VEMA, which has existed since 2003, and the authors of this foreword work together on its board of directors The conference should provide an opportunity to find out about the answers and solutions found in each case and offer opportunities to establish cooperation relationships. In our opinion, the contributions in this conference proceedings show impressively that this has also been achieved gungsbandes after a first nationwide conference on preliminary and bridging courses with such a large participation as well as the very intensive and also controversial discussions is a very special challenge and has now also required some time. We would like to take this opportunity to thank all of the contributing authors and the publisher for their patience. We would like to thank the contributors, especially for their reviews, which have contributed significantly to the improvement of the submitted manuscripts. The publication of this volume is a joint effort by all eight editors, which should also be expressed in the alphabetical order of the editors. The four locations each handled a quarter of the articles and all eight editors were already actively involved in the preparation and implementation of the conference. We would like to express our special thanks to Stephan Schreiber, who took on the overall coordination of the preparation of this conference proceedings. Darmstadt, Kassel, Lüneburg, Paderborn, in August 2013 Rolf Biehler, Regina Bruder, Reinhard Hochmuth, Wolfram Koepf

7 Table of contents 1 Introduction ... 1 Rolf Biehler, Regina Bruder, Reinhard Hochmuth and Wolfram Koepf Part I: Aims, content and addressees of preliminary courses Years Esslinger Modell New students and mathematics ...

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9 Contents IX 20 A diagnostic approach to identify knowledge gaps at the beginning of mathematical preliminary courses Stefan Halverscheid, Kolja Pustelnik, Susanne Schneider and Andreas Taake 21 Math0 the introductory course for all freshmen of a technical teaching unit Maria Krüger-Basener and Dirk Rabe Part IV: Support measures in the introductory phase Understanding the mathematics better project Christoph Ableitinger and Angela Herrmann 23 Promotion of self-regulated learning for students in preliminary mathematical courses, web-based training Henrik Bellhäuser and Bernhard Schmitz 24 Self-assessment test mathematics for students of physics at the University of Vienna Franz Embacher 25 What is Mathematics? Introduction to mathematical work and study selection review for student teachers Tanja Hamann, Stephan Kreuzkam, Barbara Schmidt-Thieme and Jürgen Sander 26 Fifth semesters as mentors for freshmen Walther Paravicini 27 Do engineers need mathematics? How practical relevance changes views on the compulsory subject mathematics Aeneas Rooch, Christine Kiss and Jörg Härterich 28 Introduction to the mathematical work of the Passage-Point at the University of Vienna Roland Steinbauer, Evelyn Süss-Stepancik and Hermann Schichl

10 1 Introduction 1 Rolf Biehler (University of Paderborn), Regina Bruder (Technical University Darmstadt), Reinhard Hochmuth (Leuphana University Lüneburg) and Wolfram Koepf (University of Kassel) R. Biehler, R. Bruder, R. Hochmuth and W. Koepf Mathematical Vor - and bridging courses are now offered at almost all universities and technical colleges in Germany. In particular, they serve to facilitate the transition from school to university. Mathematical content from the school relevant to the respective degree programs is repeated and, in some cases, supplemented. In addition, there is often a requirement to introduce students to the more abstract way of speaking, writing and reasoning in mathematics courses at universities. After all, learning in school takes place under different time restrictions and in very different arrangements compared to university, which requires a different learning organization and different learning strategies from the students. While the daily routine is clearly pre-structured in school and new learning content is usually practiced immediately, the study day must be organized independently and each lecture must first be reworked independently. Recognizing this and reacting to it in a goal-oriented manner is often difficult for students. It is quite understandable that strategies that have often proven themselves in school and over many years are at least retained for the time being. In probably all preliminary and bridging courses, such breaks in the transition from school to university are addressed and made aware of. Sometimes training elements aimed directly at this are integrated into the courses or supplemented appropriately to support self-regulated learning. Preliminary and bridging courses are unquestionably faced with diverse and demanding tasks. The hurdles of the transition from school to university have generally been known for a long time and every successful or unsuccessful student of mathematics could report about them. Nevertheless, the time in which the preliminary courses essentially consisted of quickly writing down central sec-II content in a one-week course on the blackboard or showing it on slides and possibly having a few tasks added to the central content is still there not long back. Therefore the question arises what to change the relevance of these hurdles, the way around I. Bausch et al. (Ed.), Mathematical preparatory and bridging courses, concepts and studies on university didactics and teacher training in mathematics, DOI / _1, Springer Fachmedien Wiesbaden

11 2 R. Biehler, R. Bruder, R. Hochmuth and W. Koepf with these and ultimately to the current expansion of preliminary and bridging courses. In most cases, major local changes cannot usually be explained and understood solely from local developments, in this case those in schools and universities. Little has so far been written (and researched) on rather higher-level processes and relevant social developments, which also exist here, and we, too, want to limit ourselves in this volume to essential local phenomena and a few general remarks. With regard to schools, we believe that there are two main changes that should be mentioned as reasons for these developments. First of all, due to legal changes in the individual federal states, there are now numerous school-leaving certificates that enable university studies. The high school Abitur or, in relation to the universities of applied sciences, the advanced technical college entrance qualification, acquired through a high-level technical diploma, are just two options among many others. Since not only the final degrees differ, but also the ways to get there are often significantly different, the differences in the mathematical knowledge and skills that students bring to the universities not only affect the question of whether, for example, integration is known or not, But in particular the availability of knowledge areas from lower secondary level I. There now seems to be a consensus in the preparatory and bridging course community that there are considerable gaps, especially in central areas of secondary school mathematics, and that these make it more difficult to develop in advanced areas of the To acquire basic mathematics in a flexible way and, in particular, to apply them. Fractional arithmetic, term transformations and the understanding of variables also play an important role in differential and integral calculus or in not entirely trivial modeling tasks and may become an insurmountable hurdle if they are not mastered. The differences in the competencies of students, which are partly due to the many possible learning biographies on the way to university, are typically understood as an essential element of an increasing problem of heterogeneity. In addition, the mathematical content and requirements in grammar school and in the classic Abitur have changed. This is only partly due to the G8. Even before that, for example, evidence played a much smaller role than it did about 25 years ago. Instead, other competencies, such as modeling or the use of technologies, have come to the fore. Accordingly, one of the reproaches by the schools to the universities is that they should not only take note of deficits, but also recognize the new competencies and build on them more closely in their teaching. How and at which points this is actually possible, however, is not quite as obvious as is sometimes assumed or expected in discussions. The universities have, we think, that we as university lecturers should note this here with a remarkably long hold back with reactions to the changed entrance skills of students. There are many reasons for this, in part it was

12 1 Introduction 3 in our opinion on the so-called Bologna Process and the way in which it is implemented and implemented. Even if tuition fees are viewed critically overall, it is important to recognize that the flow of money generated by them to the universities, in conjunction with the political guidelines regarding the use of the funds, has stimulated processes at universities that have long been necessary: Many universities were faced with the question of how they can spend this money sensibly. The introduction or significant expansion of preliminary and bridging mathematical courses seemed to be a sensible option in any case. Among other things, because of the drop-outs in subjects that are not mathematically in the strict sense of the word, mainly due to the mathematical parts of the course, their representatives could also be quickly won over for preliminary and bridging courses. It is of course an empirically open question whether, even with optimal support for students from the perspective of mathematics education, drop-out rates in such subjects can be significantly reduced. For the image of mathematics, however, it would certainly be an advantage if it did not primarily have a certain, systemically possibly even desired readout function. As a result of the new financial opportunities, initiatives have quickly formed at practically all universities in Germany where mathematics plays a role in some way and often represents a major obstacle on the way to a successful course of study, and preparatory courses or preparatory courses before the course Introduce bridging courses during the course of studies. These initiatives were subsequently further supported by the so-called Excellence of Teaching Initiative, which brought additional funding to numerous universities to strengthen teaching. However, filling the positions was not entirely unproblematic, not only because of their large number. On the one hand, these positions require specialist expertise, which is usually only found in successful master's degree students in mathematics. On the other hand, the students should also have certain higher education didactic and subject-related didactic knowledge and related skills. Mathematics-related university didactics as a scientific discipline is still at the beginning of its development. This is where the conference started. One of its goals is to bring about an exchange between the various efforts at preliminary and bridging courses. In our estimation, the newly hired employees faced similar questions and problems in many places. The conference should therefore offer an opportunity to exchange ideas about the answers and solutions that have been found and offer opportunities to establish cooperative relationships. All contributions submitted for this conference volume are based on a lecture or a poster that was presented at the khdm conference, but they were specially prepared for this volume. All contributions have been carefully reviewed by two other authors of this volume and one person from the editorial team, and some have been revised several times. The review process we organized served not only to optimize individual contributions but also to establish certain science-oriented standards for this area. It is obvious that the so-called

13 4 R. Biehler, R. Bruder, R. Hochmuth and W. Koepf named hard criteria for empirical educational research. On the one hand, there have so far only been very few, if any, contributions that meet these criteria and, on the other hand, contributions should be published that are of direct practical relevance for the actions of the employees and those responsible for preliminary and bridging courses, i.e. of a different interest serve as a purely scientific one. We have taken into account that in addition to scientific studies, best practice examples could also be used as impulses for the university didactic discussion. Even if we are convinced that ultimately both belong together, it takes time for a development process that establishes standards in this field that adequately balances the various perspectives. If we have brought this process a small step forward with our conference, we would be satisfied. The chapters in the present conference proceedings are essentially based on the structure of the conference. As the following brief remarks on the individual sections of this volume indicate, there are many overlaps between the individual works in terms of their content and orientations. A really disjoint division therefore does not seem possible to us. Ultimately, almost all of the papers contain references to questions that are actually mainly dealt with in other chapters. That is why we refrain from quoting individual papers on special issues in the brief overview.In our opinion, the respective priorities and orientations of the individual works can be taken from their informative headings. Chapter 1 deals with the goals, content and addressees of preliminary and bridging courses. The goals vary between the compensation of mathematical deficits, the repetition of school mathematical contents up to the consolidation and the careful expansion of this mathematical basis. In addition to specific mathematical content in the narrower sense, further central elements of the courses offered are the introduction to the use of mathematical language at the university and, in addition, the development of university-related mathematical ways of thinking and working. Related goals include raising the level of knowledge and skills available from school to university, developing a meta-level from which elementary mathematical content can be rethought, and finally making subject-related learning processes more effective and, in particular, promoting reflective skills. The specific design of the respective preliminary courses is geared towards the respective local addressees, both in terms of content and level of aspiration, i.e. their (mostly assumed) previous knowledge compared to the skills they need at the beginning of their respective chosen course. The scenarios in which the preliminary and bridging courses are offered are quite different. In the second chapter, a whole series of different course scenarios and teaching-learning concepts are presented, with special emphasis on the role of e-learning elements. A pre-

14 1 Introduction 5 The course variant consists of a block course that takes place before the start of the semester, with lectures in the mornings and tutorials in the afternoons from students from higher semesters. E-learning is sometimes offered in addition to these face-to-face events, and sometimes the teaching is also offered entirely virtually. Face-to-face teaching and e-learning elements are sometimes closely interlinked, so that, for example, the students' learning essentially takes place outside the university, but this is supported and structured by presence components. Sometimes they do without it themselves and rely on a model of pure self-study. Bridging courses during the course of study only last three weeks, but can also last up to two semesters. There are also differences with regard to the degree of obligation of the preliminary or bridging course. The majority of the courses are offered on a voluntary basis. In some cases, and in our perception increasingly, there are also models in which participation in the courses is more or less mandatory. This is achieved, for example, through special examination regulations in which so-called entrance tests have to be passed by a certain point in the course, otherwise the specialist course may not be continued, or by the fact that in the regular maths introductory courses no account is actually taken of existing deficits and this is also emphatically communicated. An important element of preparatory and bridging courses that can certainly be expanded is a target and target-oriented assessment and the establishment of efficient diagnostics within and after the courses taken. Chapter 3 presents several papers in which the first approaches are described. It can be assumed that the students at different universities and especially different courses differ significantly in their entry requirements. The course design in its various elements and their weighting should of course take this into account. But also the question of the effect of preliminary and bridging courses calls for instruments that record reliable, objective and valid entry and exit skills taking into account the respective course objectives. Diagnostic elements are also of great importance for the individual students themselves. As a rule, first-year students are not able to reliably assess their mathematical skills with regard to the requirements at the beginning of their studies. And for the learning process itself, especially in the context of e-learning offers, it seems important to us that students receive feedback on their learning status as precisely as possible and constructively for the further learning process and, if possible, receive information on how to continue their learning process should design. The work in Chapter 4 in particular deals with this topic, i.e. the question of effective subject- and course-related support measures. Here, the range of offers includes complete self-assessment tests, the promotion of self-regulated learning within the framework of web-based training, or the

15 6 R. Biehler, R. Bruder, R. Hochmuth and W. Koepf Motivation for engineering students, for example, by examining the role of mathematics in their respective study program. Both the contributions in this volume and the final plenary session of the conference make it clear that preliminary and bridging courses still raise many open practical and scientifically interesting questions. For example, the efficiency of preliminary and bridging courses does not seem to have really been answered empirically, although of course the question of what would be understood here by efficiency will not be easy to answer in each specific case. As a rule, preliminary and bridging courses accompany a whole range of high expectations. Should taxonomies to be developed now be more oriented towards normative expectations of universities and their lecturers or towards the wishes and current satisfaction of future students? Certainly there is no real contradiction here, and we teachers of course assume or hope that both perspectives can be reconciled. Whether and how this can be done has not yet been discussed in detail, nor are there any concrete, practice-related and empirically proven procedures for realizing this. Open questions along the way that are of their own importance include: How can and should e-learning and face-to-face learning be designed and combined in blended learning scenarios in order to offer the best possible support for the respective learning objectives? Are e-learning elements particularly suitable for individual self-diagnoses? Which local or global possibilities actually lie in so-called adaptive learning systems? How should calculators or mathematics programs be used within the framework of preliminary and bridging courses? Should entrance tests be based primarily on technical skills? Which digitally evaluable tasks can be used to test a certain understanding of mathematics? What influence do preliminary and bridging courses have on the further course of studies? Or to put it somewhat heretically: Do deficits in school mathematics at the start of studies in MINT subjects actually lead to higher drop-out rates? What do universities have realistic expectations of schools? Some of these questions, especially those that affect the transition from school to university, were taken up and pursued again in particular at the second khdm working conference on mathematics in the transition from school to university and in the first year of study, which took place this year. This conference took place in February 2013 at the University of Paderborn in cooperation with the joint mathematics commission for transition from school to university of the DMV, GDM and MNU. A conference volume with further interesting contributions will also be published for this conference. It will be exciting to understand the further developments in it.

16 1 Introduction 7 Part I Aims, content and addressees of preliminary courses

17 2 28 years of the Esslingen Model Newcomers and Mathematics 2 Heinrich Abel (University of Esslingen, Faculty of Fundamentals) and Bruno Weber (State Institute for School Development Stuttgart) Summary At the beginning of their studies, many engineering students show serious deficiencies in basic mathematical knowledge and skills. These difficulties are particularly great at universities of applied sciences, which envisage the technical college entrance qualification as an entry qualification in addition to the general and subject-specific higher education entrance qualification (Abitur). Applicants with a technical college entrance qualification usually have an intermediate educational qualification, a completed apprenticeship and a one-year additional school education, the completion of which entitles them to be admitted to a university of applied sciences. To alleviate these difficulties, the Esslingen University of Applied Sciences (formerly FHTE) has been offering a compact course in elementary mathematics since the winter semester 83/84. In the present article, after a brief outline of the historical development, the organization, content and current development of this Esslingen model are presented. In addition, reports are made on two new preliminary course models that have arisen from the collaboration between mathematics teachers at vocational schools and universities of applied sciences in Baden-Württemberg in the COSH (Cooperation Schule Hochschule) working group: the advanced mathematics course for students at the vocational college and the refresher courses for BK- School beginners before school starts at some vocational colleges. H. Abel and B. Weber I. Bausch et al. (Ed.), Mathematical preparatory and bridging courses, concepts and studies on university didactics and teacher training in mathematics, DOI / _2, Springer Fachmedien Wiesbaden

18 10 H. Abel and B. Weber 2.1 Initial situation A constant topic at the universities is the general complaint about the inadequate ability of new students to study. The main complaints are considerable knowledge gaps in mathematics. These difficulties are particularly great at universities of applied sciences, which envisage the technical college entrance qualification as an entry qualification alongside the general and subject-specific higher education entrance qualification (Abitur). Applicants with a technical college entrance qualification usually have an intermediate educational qualification, a completed apprenticeship and a one-year additional school education, the completion of which entitles them to be admitted to a university of applied sciences. Since the winter semester 1979/80, all newcomers to the Esslingen University of Applied Sciences (formerly: University of Applied Sciences Esslingen (FHTE)) have been taking entrance knowledge tests in mathematics (31 questions in multiple-choice format). Detailed data and interpretations can be found in Brenne, Hohloch and Kurz (1981); Brenne, Hohloch and Kümmerer (1982) and Kurz (1988). The results and statements of these tests have remained relatively stable for many years. However, it should be noted that the mean values achieved have sunk considerably in the course of the last 20 years: in winter semester 92/93 the mean value of the correct answers for all 400 new students was 18.2 points or 58.7%; in WS 11/12 with 537 first-year students with 14.3 points or 46.1%. In the course of time, however, the curricula have also changed. For example, in some federal states the logarithm no longer occurs as a function, and neither does the cotangent. The entrance knowledge test has been modified accordingly; Tasks related to these terms have been removed or replaced with similar ones. The tests reveal terrifying weaknesses of the first-year students in elementary mathematics: lack of knowledge of bourgeois arithmetic, uncertainty with the simplest algebraic transformations, insufficient knowledge of elementary functions and their diagrams, insufficient skills in solving trigonometric equations, etc. The main results of some test questions are mentioned as examples (SS 12; 504 participants; no aids allowed): Only 50% recognize the question 2 3 =? the correct solution in decimal representation 0.125 (25 years ago it was just over 70%). By the way, 8 is a popular answer% think the equation is correct. a b a b Only 46% can specify the smallest value of the function f (x) 2sin (3 x). 56% cannot convert radians 2 into degrees 120. 3 Only 31% recognize the identity: sin (90) cos. In the evaluation of the entrance test and in the exercises for the compact course in elementary mathematics (see below), further deficits emerge, e. B .:

19 2 28 years of the Esslinger Modell First-year students and mathematics 11 Sin is something on a right triangle; it is not known that this can be used to describe vibrations. When solving exponential equations, it is multiplied by ln; the definition of the logarithm is largely unknown. Global qualitative properties of elementary function curves (asymptotic behavior, geometric meaning of the derivatives) are not known. Estimates of the order of magnitude cause great difficulties; on the other hand, percentage errors are given with ten decimal places. Total dependence on the pocket calculator for even the simplest of calculations. Sloppy work, careless spelling (e.g. omitting brackets). Recipe-like application of formulas and rules without understanding the context. This lack of basic mathematical knowledge and skills is, in our opinion, one of the most serious problems when starting an engineering degree. The introduction of graphical pocket calculators and / or CAS systems certainly opens up a potential for mathematics lessons that should not be underestimated; however, there is a risk of even greater neglect of elementary, manual arithmetic skills. To counteract this disastrous tendency, z. For example, at the HS Esslingen no electronic arithmetic aids are permitted for the mathematics 1 exams in most courses. In addition to the relatively low mean value of the test results, the evaluations show a second result, which is almost even more important for the design of mathematics lessons for first-year students: namely a very wide range of results in the input knowledge test. This heterogeneity of initial knowledge is the main problem for the design of lessons in the introductory phase. At universities of applied sciences, it is largely due to the different entry requirements. The main access to the second educational path in Baden-Württemberg is via the vocational college. It is noticeable that the first-year students with this access authorization score particularly poorly in our tests. The ratio of first-year students with a high school diploma to those with a technical college entrance qualification (mostly vocational college) at Esslingen University is currently 3: Compact course elementary mathematics: organization and content as of 2012 From attempts by individual lecturers, through additional exercises in the first semester, the initial difficulties of the new students in mathematics At the beginning of the 1980s, the Esslingen model for first-year students was developed, consisting of the above-mentioned knowledge test in MC form (diagnosis) and a compact course in elementary mathematics (therapy) (Hohloch and Kümmerer 1994).

20 12 H. Abel and B. Weber This course is offered by the University Didactics Working Group of the Faculty Fundamentals in cooperation with the Steinbeis Transfer Center Technical Advice to all new students (except for students in the Faculty of Social Work, Health and Nursing (SAGP)). Its content is essentially limited to mathematics at secondary level 1. Participation in the course is voluntary; the participation fee is 75 euros for 40 hours of course tuition and teaching material. All first-year students in technical and economics courses receive a registration form for the compact course with their notification of admission, which takes place in the last two weeks before the start of the lectures. The registration form contains a number of typical tasks that the new students should be able to solve. If this is not the case, we strongly recommend that you take the compact course. Participation in the compact course is based on a self-assessment of the students. On eight mornings, professors or experienced lecturers will repeat important terms from the areas of algebra (10 h), trigonometry (8 h), elementary functions (8 h) and analytical geometry (6 h) and use examples to discuss them. In additional exercises (2 hours each on four afternoons), the participants are supervised by students from higher semesters who, in addition to helping with solving mathematical tasks, also provide information about studying at our university. The groups are basically divided according to degree programs and do not consist of more than 30 participants. Volume 1 of the Bridges for Mathematics series is used as course material (Hohloch and Kümmerer 1994). A CD is enclosed with this volume, which, in addition to detailed sample solutions for all exercises, also contains several interactive tests for self-control (with correction and solution tips) (Hohloch, Kümmerer and Gilg 2006). The number of participants has risen since the first course in the winter semester 1983/84 from almost 40% to currently around 60% of all new students. In WS 11/12 there were 537 participants with 22 lecturers and 33 tutors. Overall, in the 30 years since the course was first offered, more than first-year students have taken part in the elementary mathematics compact course; they were supervised in the additional exercises by almost 300 tutors. 2.3 Effects of the compact course The entrance knowledge test is carried out in the first week of the lecture period. It contains almost the same tasks as the test at the beginning of the compact course. When evaluating the test, a distinction is also made between participation / non-participation in the compact course. This shows positive effects of the compact course: participants in the compact course achieve an average of around 10% more correct answers than students who did not take part in the compact course.

21 2 28 years of the Esslinger Modell First-year students and mathematics 13 Fig. 2.1 Before the compact course (n = 237; x = 12.4; s = 5.7).The number of points reached is plotted on the x-axis, the absolute frequency on the y-axis. The number of participants is marked with n, the mean value of the points achieved with x, the standard deviation with s Fig. 2.2 After the compact course (n = 237; x = 18.5; s = 5.9). Axes and designations as in Fig.2.1 The positive effects of participating in the compact course are even clearer if one compares the results of the test before the start of the course with the results at the start of the course: the evaluations show increases in learning in terms of size

22 14 H. Abel and B. Weber order of 20 percentage points. In the example shown, there were 40% correct solutions before the start of the compact course and 60% afterwards. However, these learning gains were measured using methods that do not stand up to modern empirical and statistical principles. It could be that the better students in particular attend the compact course and therefore particularly benefit from it. A comparison of the two groups of KK participants before KK and first-year students without KK; shows, however, that the course is mainly attended by the weaker first-year students, for whom it is ultimately organized. It would also be interesting to pair the data, i.e. to show the difference between the pre- and post-test for each participant. However, that did not happen here. It is also not certain whether this is only a short-term increase in learning or whether the compact course can permanently resolve gaps in knowledge. At irregular intervals, we examined the influence of mathematics knowledge at the beginning of the course on academic success (= success in the mathematics examination). It was shown in each case that the test results correlate significantly with the results in the math test at the beginning of the course; The results were correlation coefficients of the order of magnitude of student surveys. The statements cited below are based on surveys of the participating first-year students and the tutors employed. In these repeated surveys immediately after the compact course or after the end of the first semester of study (organized by the university didactics working group or independently from the AStA), the course met with great approval. The vast majority of students think that they have benefited from the course and will recommend it to others. Only a few participants felt they were under-challenged. The choice of material and the accompanying material, the explanations by the lecturers and the speed of their presentations are particularly positive. In addition to refreshing math skills and practicing arithmetic skills, the compact course also has a social function that should not be underestimated thanks to the cooperation of the tutors. It makes it easier to get used to the university, helps to break down inhibitions, establishes contacts with fellow students and thus makes an important contribution to student advice. The success of our Esslingen model, documented through various questionnaires and the results of knowledge tests before and after the course, has led to comparable courses, partly based on the materials developed in Esslingen, now being held at most universities of applied sciences in Baden-Württemberg (e.g. . FH Karlsruhe, FH Stuttgart).

23 2 28 years of the Esslinger Modell First-year students and mathematics Quotes from course participants The course is a very good opportunity to refresh your elementary knowledge. The ability to learn is also awakened. For people who graduated from school two years ago, the course should be a bit more detailed and slower. The best invested 75 euros last year. I expected more, but I also didn't know that the differences in knowledge when it came to math were so great. I wasn't challenged. The course should fill gaps, not reveal new ones! Quotes from tutors The confirmation letter for the math tutorial is really great for application letters. It was also fun to work as a tutor. I'll be back next time! My work as a tutor in the compact course was the decisive factor in my receiving the scholarship from the Carl Duisberg Society. 2.5 Further developments Suggestions for improving the further development of the course are often mentioned in the student surveys: the desire for a two-week course (with differential and integral calculus), the desire for a longer period between the compact course and the start of the course, a continuation as a support course to the mathematics 1 lecture during the First semester, offer course and accompanying material earlier in order to prepare for the course independently (e.g. during military service or civil service). This last wish is the reason for the further development of the mathematics knowledge test and the compact course documents into an offer of orientation aids for university applicants and first-year students. As a continuation or as a supplement to the compact course that takes place before the start of the lectures, there are now tutorials on mathematics 1 for almost all courses of study. In response to the heterogeneity of previous knowledge, primarily due to the different previous education (high school with Abitur, vocational college with technical college

24 16 H. Abel and B. Weber readiness for school), models are discussed how to distribute the lecture content and exams of the first two semesters over three semesters for weaker students. However, there are considerable formal hurdles to be overcome. As part of the cooperation between the vocational college and the university of applied sciences, additional mathematics courses are held at several locations in Baden-Württemberg, which are aimed at BK students who are willing to study. The aim is to improve the starting chances of BK graduates at the university in the subject of mathematics. The following is a report on two new preliminary course models that have arisen from the collaboration between mathematics teachers at vocational schools and universities of applied sciences in Baden-Württemberg in the COSH (Cooperation School University) working group: the advanced mathematics course for students at the vocational college and the refresher courses for BK -School beginners PISA shock and teaching style As a result of the PISA shock, a change in mathematics teaching in schools has been demanded in recent years. Lessons should focus less on imparting facts and arithmetic skills than on the learning process, reasoning, problem solving and communication. The use of graphical pocket calculators and computer algebra systems also play a role here. On the other hand, knowledge transfer at the university continues to be traditionally lecturer-oriented; In addition, in many courses, especially at universities of applied sciences, mathematics is only used as an auxiliary science. Of course, the changes in mathematics lessons in schools are at the expense of the amount of material taught so far. Some university professors, especially in technical and natural science courses, fear that the associated shortening of the exercise phases will also result in a further decline in elementary arithmetic skills of the first-year students. Unfortunately, ignorance of developments on the other side often results in mutual undifferentiated accusation between school and university: teachers complain that the universities do not take note of changes in the school sector and stick to their traditional teaching methods, university teachers complain about the lack of knowledge of school leavers and that they cannot take this into account due to the tight time schedule. The first-year students suffer. They have to cope with the changed teaching methods in schools as well as with the style of lectures at universities

25 2 28 years of the Esslinger Modell New students and mathematics 17 universities of applied sciences; almost two thirds of the new students come from this school area. It was therefore natural to strive for cooperation between these types of schools and the universities of applied sciences. Based on private contacts between teachers at vocational colleges and professors at universities of applied sciences, the working group COSH 1 (Cooperation School and University in Mathematics) was established ten years ago: Talking about one another turned into talking to one another. A catalog of requirements in mathematics for the transition from school to university was initially defined at several working group meetings and large-scale conferences, each with equal representation from teachers at vocational schools and universities of applied sciences with the participation of students and financed by both responsible ministries. This led to the idea of an advanced course for students who are willing and able to study at the one-year vocational college to acquire the advanced technical college entrance qualification (Dürrschnabel and Weber 2005; Weber 2010), primarily inspired by the students. For four years now, such courses, supervised by student tutors, have been taking place at twelve vocational colleges in Baden-Württemberg with great success. Unfortunately, the planned nationwide introduction at all vocational colleges could not be implemented. Some schools took over our idea and now offer such additional courses at their vocational colleges on their own. If this improves the starting conditions for first-year students, then of course we will also consider that a success of the efforts of COSH. The work of COSH is not limited to the development and implementation of advanced courses and the organization of cooperation meetings. There are also a number of other activities that document the trusting cooperation between vocational schools and universities of applied sciences in Baden-Württemberg. In 2001/2002, for example, the mathematics curriculum commission at vocational high schools sought the opinion of professors from the technical colleges. In 2006, three professors from different universities and different fields were officially invited to the meetings of the Mathematics Curriculum Commission for the vocational colleges to acquire the advanced technical college entrance qualification. Suggestions and requests from university representatives were carefully examined and some of them were included in the new curriculum. The cooperation of the technical college representatives can also be found in the foreword to the redesigned curriculum. During this joint work, the university representatives noticed certain discrepancies between the number of hours in the curriculum and the actual school hours and initiated a letter from the chairman of the rectors' conference to Minister Rau. After an initially somewhat disappointing answer, as a result of this letter in the following school year in the vocational colleges with the lowest mathematics offerings, the number of mathematics lessons was actually increased. Joint public appearances by schools and universities are now standard. Representatives from schools and universities reported on the day of teaching in Ulm in November 2005 about the joint efforts to improve 1

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