Transcript
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Faculteit wiskunde en natuurwetenschappen

Portfolio template

BKO-registration FWN October 2011 This template provides guidance for the composition of your education-portfolio for obtaining a university teaching qualification (BKO in Dutch). Use the order provided by the template, remove “italic type”, and fill out the open spaces. Provide clear references to relevant attachments for each of the required competences.

Name Prof. Dr. Holger Waalkens

P-number P252003

Address Boenster Str. 54

26826 Weener, Germany Telephone number

Date of birth 27 August 1969

Place of birth Weener, Germany

E-mail address [email protected]

Institute/Faculty FWN

Working at the University since:

at the RUG since 2007

Teaching certificate

BKO

Date: March 2012

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De bijlagen bij dit portfolio bevinden zich aan het eind. Als naar een bijlage verwezen wordt, staat in de marge het paginanummer van de bijlage. Elke bijlage heeft bovendien een bookmark.
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Table of contents

1� Personal information and CV 3�

3� Vision on scientific education 11�

4� Self-analysis BKO criteria 18�4.1� Criterion (re)design of education 18�4.2� Criterion teaching and supervising students 25�4.3� Criterion testing and assessment 29�4.4� Criterion Evaluation 33�

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1 Personal information and CV

Insert CV as attachment, without publications. A General information

1. Personal Information Name (title, initials, name), first name, M/F? Prof. Dr. Holger Waalkens

Year of birth 27 August 1969

Faculty FWN

Actual University Job Ranking Adjunct hoogleraar

Employment at RUG (in hours/week) 40 hours/week

Employment for teaching tasks (in fte) 0.4 fte

Course in which you are the principal lecturer this academic year: Lineaire Algebra I, Vectoranalyse, CaputMathematical Physics, Student Colloquium (the latter together with A.C.D. van Enter)

2. Experience as a lecturer

Number of years as a lectures in higher education

12 years: 3 years in Bremen, Germany, as Wissenschaftlicher Assistent (which corresponds to Junior Professorschips in the new German academic system) 5 years in Bristol, UK, as lecturer (which is similar to an Assistant Professorship with tenure) 4 years in Groningen, the Netherlands, as adjunct hoogleraar

Other experience as a teacher?

7 years: 3 years as a PhD student giving tutorials 4 years as a student giving tutorials (Analysis I and II, Linear Algebra, Algebra, Calculus, Statistical Physics, Mechanics, Quantum Mechanics)

Remarks:

3. Courses and program in which you participate Never Sometimes Regularly

Own bachelors program, including minor x

Bachelor program other courses x

Own master program x

Master programs other courses x B Experience in education

4. Experience with education Little Moderate Much Direct preparation and performing a lecture x Designing a course x (Re)designing curriculum x Coordinating or management tasks x

5. Your experience regarding teaching methods Little Moderate Much

Lecture x

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Tutorial x Guided study x Lab work x Educational Projects x Problem Based Learning x Case-studies Computer based education x Individual teaching (AIO, thesis) x Other:………………..

6. Experience using (multi)media Little Moderate Much Simple visual media x Written material and hand-outs x Digital learning environment (TeleTOP, Blackboard) x

Remarks:

7. Experience with assignments Little Moderate Much Construction of assignments x Evaluation of written materials x Evaluation of electronic results x Remarks:

8. Subjects for which you were the coordinating lecturer: Subject code Name subject Teaching method Course

load Academic program and

period

1

WILA1-06.2011-2012.1B

Linear Algebra I Lectures (by me) + tutorials + computer practicals

30+30+ 10 hours

Mathematics (+physics, astronomy, chemistry), bachelor course, Groningen, 2011, 1 time

2 WIFP1006.2010-2011.1

Mathematics&Statistics for Pharmacy

Lectures (by me) + tutorials + computer practicals

30+30+ 24

bachelor course for pharmacy students, Groningen, 2010, 1 time

3 WICMF-09.2009-2010

Caput Mathematical Physics Lectures (by me) 30 hours Mathematics, master course, Groningen, 2010, 1 time

4 WIAM-03.2007-2008.1

Algebra Lectures + tutorials (both by me)

30 + 15 hours

Mathematics, bachelor course, Groningen, 2007-2011, 4 times

5 WIIPDS-07.2009-2010

Integrerend Project Dynamische Systemen

Lectures + tutorials + office hours (all by me)

25+15+ 10 hours

Mathematics, bachelor course, Groningen, 2007-2011, 4 times

6 General Graduate Lectures Lectures (by me) 8 hours Mathematics, Graduate

course, Bristol, 2006, 1 time

7 Quantum Mechanics Lectures (by me) 30 hours Mathematics, Bristol, 2004-2007, 3 times

8 Numerical Projects Lectures +

computer practicals (both by me)

30+15 hours

Mathematics, Bristol, 2003-2007, 4 times

9 Advanced Quantum Mechanics

Lectures (by me) 30 hours Physics, Bremen, 2001, 1 time

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10 Computational Physics Lectures/Computer

practicals (both by me)

30 hours Physics, Bremen, 2001, 1 time

11 Bose-Einstein Condensation Lectures (by me) 30 hours Physics, Bremen, 2000, 1 time

9. Subects for which you were a co-lecturer Subject code Name subject Teaching method Course

load Academic program and period

1

Conservative Dynamical Systems

Lectures + Tutorials (both by me and H. Hanßmann)

30+15 hours

MasterMath course, with H. Hanßmann, Utrecht, 2010, 1 time

2

Geometric Mechanics Lectures + Tutorials (one quarter by me)

30+15 hours

MasterMath course, with H. Hanßmann, A. Kiselev and B. Rink, Utrecht, 2009, 1 time

3

WLPB1013.2007-2008.1

Biomathematics Lectures (one half by me) + Tutorials + Computer practicals (for the tutorials and the computer practicals I was presented about half the time)

30 + 30+ 24 hours

bachelor course for biology students, with F. Weissing, Groningen, 2007-2010, 3 times

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10. Training in the field of education

Name course Provider Result Course load Year Induction course for lecturers new to teaching mathematics & statistics in UK higher education in Birmingham

the Higher Education Academy; Maths, Stats & OR Network

Contents:- What makes a good mathematics lecture? - Teaching students mathematical proof - The computer environment: recent developments and systems to support learning - Examinations, writing and marking

2 days 2005

Further Higher Education Training

the higher education didactics department at Bremen University

Contents:- Basic elements of didactics and their practical realization - Simulations of learning situations - Video recording of these simulations and individual feedback - Handling difficult situations (like students with low motivation levels)

2 day course + three supervision sessions

2000

Workshop Eindhoven: Teaching Mathematical Proofs

ICAB Project Bachelor Wiskunde Nederland (BWN)

Contents: discussion among Dutch maths departments on how to teach students to do mathematical proofs; this included also the implications for assessing of maths students

1 day 2012

PhD coaching course RuG, Brigitte Hertz Contents:- situational leadership - counseling - negotiations - conflict management - influence techniques - providing feedback to presentations - feedback during the writing process

4 days 2011

Introductieprogramma hoogleraren

RuG, Ben Verheijen Contents (among others):- situational leadership - counseling - negotiations - conflict management - influence techniques

2 days 2009

11. Teaching methods which you plan to use in the near future Little Moderate Much

Lecture x Tutorial x Guided study x Lab work x Educational Projects x Problem Based Learning x

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Case-studies Computer based education x Individual teaching (AIO, thesis) x Other:………………..

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C. Evaluation didactic knowledge and skills Important: Indicate for each of the 45 statements to what extent you’ve objectively mastered the skill that is described in the statement. The way you score a given item is an indication of your experience with the described situation. Scores must be read as: 1=none; 2=some; 3=adequate; 4=good; 5=excellent; n=not applicable. Knowledge/skill 1 2 3 4 5 n

Design courses 1 2 3 4 5 n

1. Formulate learning goals for a course (part of a course) X

2. Choosing teaching methods to achieve certain learning goals X

3. Preparation of a lecture, including lecture scheme, PowerPoint, assignments, student activity

x

4. Design a course (minimum 6 meetings) including selection of course content and course planning

x

5. Design of lab work, including assignments and planning x

6. Design of project or problem based learning (PBL), including assignments, PBL task description and a study guide

x

7. Design of electronic learning environment, digital content x

8. Write study material in English x

9. Knowledge of the professions for which the degree program prepares students

x

10. Knowledge of the benchmarks for the degree program your course is part of

x

Elucidation: concerning writing study material: I almost never did this; the only exception was a manual on how to use Maple for the course Biomathematica together with Franjo Weissing

Practice as teacher 1 2 3 4 5 n

11. Presentation/teaching skills x

12. Performing a lecture, including presenting study material, involving students, posing questions to students, handling questions from students

x

13. Assist with lab-work, including organization of assistance, involving students

x

14. Monitoring of electronic learning environment, providing feedback to students

x

15. Teaching in English x

16. Teaching to student groups of diverse cultural background x

Elucidation:

Coaching students (e.g. PHD students) 1 2 3 4 5 n

17. Formulate a graduation assignment or internship assignment, including coaching plan

x

18. Coaching students during bachelor project or stage, including providing feedback

x

19. Coaching groups of students during independent study assignments or PBL, including providing feedback

x

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Elucidation:

Assessment of student progress 1 2 3 4 5 n

20. Knowledge concerning formative and summative assessments x

21. Knowledge concerning alternative forms of assessment; e.g. peer-assessment, presentation, written report

x

22. Constructing multiple choice questions to assess reproductive knowledge, including correction model

x

23. Constructing open test questions including correction model X

24. Determining norms for passing grades for an exam or test x

25. Taking verbal exams, including correction model x

26. Analyzing an exam or test x

27. Establish criteria for assessing students results on lab work, determining achieved learning goals

x

28. Establish criteria for assessing results on a bachelor project or stage, determining achieved learning goals

X

29. Establish criteria for assessing results on project or PBL, determining achieved learning goals

x

Elucidation:

Evaluation of teaching 1 2 3 4 5 n

30. Evaluation of own lectures X

31. Construct an evaluation for a course (part of a course) x

32. Provide feedback on evaluation results and formulate steps for improvement based on evaluation results

x

33. Contribute to evaluation of the degree program x

Elucidation: The evaluations of my courses were performed by the faculty/departments for which I gave the course.

The organization of education 1 2 3 4 5 n

34. Working in a team, coordinating activities with other lecturers x

35. Logistic support of teaching materials, written exams, student administration, progress reports

x

36. Understands regulations relevant for education, including exam regulations (OER), and the role of relevant bodies, including examination board, program board

x

Elucidation:

Didactic professionalism 1 2 3 4 5 n

37. Reflects on functioning as a lecturer and is able to set up a reflection report

x

38. Can formulate a personal vision on education x

39. Observes other lecturers and provides feedback x

40. Analyses video-recordings of own lectures x

41. Formulates and maintains a personal development planning x

42. Knowledge concerning developments for teaching at secondary school and at University (e.g. competency based learning)

Elucidation:

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Addition remarks

Teaching responsibilities: o Head of the teaching committee (OC) in mathematics at Groningen University (since 2010) o Head of the joint degree program Mathematics and Physics at Bristol University (2006-

2007) (including the organization of the curriculum, conducting examiners meeting etc.) o Member of the Erasmus exchange committee in the mathematics department at Bristol

University (2002-2007) (transferring grades obtained by students at European universities to the UK grading scheme)

o Coordinator of the mathematics education in the physics department at Bremen University (2000-2001)

Did you miss any didactic skills that you have not yet acquired in this self-evaluation form? o o o

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3 Vision on scientific education

My role as lecturer Responsibilities as lecturer Areas you feel responsible for in your role as lecturer: Transfer of knowledge

Acquiring competences Achieving a learning goal with students Inspire and engage students in the subject you teach

Other I have always been very engaged in teaching, and I have always spent a lot of time not only on my teaching (designing and redesigning courses; supervising undergraduate and graduate students) but also on the teaching/education in my department in general (for example, I am head of the opleidingscommissie in mathematics in Groningen and I was head of the joint Maths./Phys. programme at Bristol University). Teaching at a University in general and in mathematics in particular has many different facets. Besides acquiring knowledge the students have to develop various competences which are necessary to successfully complete their studies on the one hand, and which form part of a university degree on the other. Concerning the former the transition has to be overcome from more spoon fed school learning in the well organized environment of a school to university learning which gives more freedom to the student on the one hand and as a consequence of this freedom requires the student to develop more responsibility to find his/her way through the degree programme and complete it successfully. In mathematics the differences between school and university education are particularly significant. Whereas school learning is often more focused on developing computational skills the focus in university mathematics is another one. Mathematics is not only a language (as Johann Wolfgang von Goethe and others phrased it) but it also is a way of thinking. Teaching students to use this language and to think as a mathematician is especially challenging. This first of all applies to the bachelor level. The students need to learn to make precise statements and draw conclusions through precise chains of reasoning based on precisely defined objects with precisely defined properties. It is interesting to see that once the students are sensitized to mathematical rigour in the bachelor programme their creativity often gets blocked such that the students cannot do the next step which consists of transferring their knowledge as is required in their master and PhD studies. To stimulate and encourage the students to make this step is one of the main aspects of the master programme. In order to achieve these goals the teaching in the bachelor and master programmes should in my opinion be quite different. Whereas I prefer the courses on the bachelor level to be based on self-contained lectures, I usually try to organiye the courses on the master level more in the form of seminars. I think that the main emphasis for inspiring the students to spend their energy on learning the subject mathematics should be built on the pleasure that arises from understanding or making a connection between two stages of a reasoning. The discipline of mathematics has the advantage that often understanding can be measured in quite an objective way, namely in the form of writing up a proof. The pleasure involved in understanding then automatically leads to the desire to write this up in a mathematically appealing way. The pleasure of understanding something is most of the time a very private experience, which you enjoy when you really have a chance to think about a problem by yourself. This is not the case in a lecture and most of the time not in a tutorial either. I therefore find it very important that the students develop the culture to work a lot by themselves at home. This does not mean that they should not discuss with their fellow students. Developing an idea together with others, and convincing others of your own idea are also important. But I think that for mathematics, the independent private work is crucial. In my undergraduate courses I always tell the students that the lectures are not

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the most important part of the course, and that it is usually not possible to grasp every aspect of the material presented during the lecture. What instead is important is that the students go through their course notes at home and, using pen and paper, try to understand every single line of it, and that they very thoroughly do their homework assignments. Self-analysis What are your strengths in education? What improvements have you realized in your teaching? What improvements do you aspire to achieve? How did you become the lecturer you are today? In what way are you developing as a lecturer? Which aspects of teaching are important to you? I think that one of the strengths of my lectures is that they are well planned and clear. This is not as trivial as it might sound. I always try to cover a clearly defined subsection in the individual lectures. This helps to motivate the corresponding subject at the beginning of the lecture by defining the problem (for example, by setting a question that we are going to answer) to which one then develops the solution and ideally provide the solution/answer at the end of the lecture. Of course I do not always succeed in following this procedure. I believe that with the enthusiasm for my own research I am also able to inspire project (bachelor/master) and PhD students. Here I certainly went through a development as too much enthusiasm for the subject does not always help the students under supervision. At the beginning I was often too involved in my own research and had a too narrow view of how a project should be done. By now I learned to give the students more freedom and put the educational aspect of supervision more to the foreground. There is of course always the problem to find the right balance between bringing the research forward and the educational aspect. The time constraints that I have now are in a way also helpful in this respect. Due to the time pressure I have to make clear plans for how the project work should proceed, and the student rather than myself has to follow the plan and fill in the work. My attitude towards teaching in general (in particular undergraduate teaching) also went through a development. When I started in Germany the responsibility to succeed with the studies was very much with the student and not so much with the lecturer. When I came to the UK this has changed. In the UK one teaches with the awareness that the students pay quite a lot of money for their studies, and this way also can expect some value for the money. Also in Bristol, there were about 10 applications for each the about 200 free places in the first year of mathematics, and as a consequence each failure of students was very carefully considered (for example, there is an extensive tutoring system). I think that more generally the universities in Europe (including the Netherlands) see themselves more and more as providers of a teaching service to the students, and this is also reflected in how the university staff (including myself) define their role as teachers. As far as I understand the developments especially in the Dutch academic system it is getting closer and closer to the system in the UK where the standards and levels of the examinations are comparable at the different universities, and the quality of a university as a teaching institution is measured by its success in getting the students through the degree programme. This then also implies that the universities put a lot of effort into improving and monitoring the quality of the lecturers. Here the BKO is one example, and the increasing attention paid to the evaluation of courses (see, e.g., the new rules in the Handboek Kwaliteitszorg FWN) is another example. I think that among all members of staff there is now a high degree of awareness of the importance of high quality teaching (besides high quality research), and many lecturers (including myself) are engaged in improving the lectures and the curriculum. In my opinion the personal contact between staff and student is very important for the success of the student, and I think this should be further improved (for more on this, see Personal vision on education; in particular the last paragraph).

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Personal vision on education I have very ambiguous views on what university teaching should be like. On the one hand I have (re)developed quite a conservative opinion of how to teach mathematics. For example, I got back to using chalk and a blackboard more than I did for a while, and I think that lecturers should try to inspire the students by their enthusiasm for the subject and their competence rather than using any kinds of artificial entertainment tricks. On the other hand I think a university should try to be innovative and, e.g., more heavily exploit different media to catch the students’ interest and to also meet their lifestyle which has changed considerably in the last one or two decades. With interest I read an article about the former Stanford Professor Thrun who says that lectures should be completely abandoned, and replaced by virtual learning. I have always been skeptical whether this might work also for mathematics (Thrun, e.g, is teaching artificial intelligence). However, there are examples (like Osnabrück in Germany) where they video record the lectures in mathematics and make them available to the students, and the students like this as they can watch the lecture whenever they like and repeat parts they did not understand. Interestingly, the possibilities for video recording our MasterMaths lectures was recently also brought up by our students in an OC meeting. My own view on this is not yet conclusive. As I think that the homeworks and tutorials play a more important role in learning mathematics than lectures do, different ways to deliver lectures might be useful when they are intelligently combined with extensive homework and tutoring. There also important issues concerning the maths programme in general. Should we preferably aim for delivering high standard mathematicians, or should we gear the education more towards the needs of society? It is a fact that most of the students we are educating will not continue to work in but outside of academia, in jobs that often have relatively little to do with mathematics. My impression is that students today have a much more practical view on the role of a university degree than they used to have, e.g., at the time when I completed my degree. To attract more students in mathematics it is important to take this into account. There are efforts for doing this in the Groningen maths department (e.g., there is the applied mathematics degree besides the (pure) maths degree). But I think that this is not enough. My impression is that in the Netherlands mathematics is suffering from an image problem. Mathematics is mainly perceived as an academic discipline. In the UK, e.g., mathematics was historically always tied to engineering and the number of maths students is much higher (although the curriculum is very similar). We should put more efforts into promoting mathematics. For example, in 2000 the Isaac Newton Institute initiated a poster campaign in the London tube for promoting mathematics. It would be nice to have something of this kind also in the Netherlands. I also think that we should improve the assistance that we provide to our students for successfully completing their degree. This is in particular a must if we impose measures like the BSA. In the UK where I worked before the formal requirements the students need to fulfill are even stricter than in the Netherlands. But in the UK they have a very intensive tutoring system which helps the students to cope with the requirements. For example, in the first year there are small group tutorials: the students are divided into groups of 5 in which they have two meetings with two different members of staff each week. The staff mark their homework and discuss the homework in the tutorials. For each student, one member of staff is also his/her personal tutor who accompanies him/her through the full degree programme (this includes a discussion of the student’s performance, the choice of courses, help with administrative issues, writing reference letters after the graduation etc.). In my opinion (even though the student numbers are relatively low in mathematics) the personal contact between staff and students falls short in Groningen (this is particularly true in the first year when this help is most important). However, with the current resources that we have in the maths department in Groningen this is very difficult to improve (see also my comments at the end of Sec.3 below).

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How do you motivate students and stimulate their participation? Apart from a few courses in the bachelor programme the maths courses in our department have relatively small numbers of students, and I usually know most students by name. This private atmosphere very much stimulates the interaction during the lectures. I always encourage the students to ask questions, interfere when something is not clear, and I also ask short questions which the students are supposed to answer during the lectures. One aspect of my strategy to keep the attention of the students during the lectures in general (i.e. in small and big classes) is that I often ask rhetoric questions like “what do we need to show here?” or “what is the next step?” Even though the answer is then not supposed to be articulated by the students, the students are stimulated to think by themselves, and when the answer I provide to the question I posed is not clear to them, then at least in the small courses the students are not afraid to interrupt me. For the big course the active student participation mainly happens in the tutorials. For the tutorials, I also put special emphasis on letting the students present the results of the exercises (and not the student assistants). I always instruct my student assistance accordingly. What are the most effective teaching methods in your view? I think that at the bachelor level the most effective teaching methods in mathematics are lectures courses accompanied by extensive homework assignments and tutorials. To some extent this is how I try to organize my courses already (for more on this, see item 4 Self-analysis BKO below), but I think that more should be done in this direction. The objective should be (as mentioned above) to more strongly stimulate the students to work at home. For example, our first year lecture courses are accompanied by two times two hours of tutorials each week. The students are usually supposed to solve exercises in the tutorials rather than at home. This delimits the type of exercises that you can set since the working atmosphere of a tutorial makes it difficult to work on exercises sufficiently thoroughly. In my opinion it would be better to skip one of these tutorials each week, and instead set more extensive homework assignments which the students can work, e.g., in groups of two, and let these homework assignments count significantly for the final mark. I would also prefer to have PhD students rather than student assistants to mark the homework assignments and to give the tutorials. This is to ensure that sufficient attention is paid on the write up of the solutions. The full solution of representative exercises should then be discussed and presented in full detail by the students (usually by those who found a good solution; the tutor can provide assistance). I find it important that at the end of the tutorial every student has the complete solution written up in detail in his/her notes as a “carry home result”. On the master level the situation is different. There are course which are more part of a standard maths curriculum and which should be done in a style similar to the bachelor courses, and there are more specialized course where in my experience a combination of lectures and student presentations work very well (like in my courses Caput Mathematical Physics and Conservative Dynamical Systems). In what way do you incorporate new insights regarding the subject you teach? Concerning the lectures I give there are rarely new results which I need to incorporate in what I teach. Of course I constantly try to improve the way I teach. I carefully monitor problems occurring in the course of a lecture and I try to improve them in the course in the following year. For example, in every lecture I have a precise idea of how far I want to get with the course material. It happens that while giving a lecture I find out that more or less than the planned time is required for one aspect, or that something should be explained in more or less detail. I include this in my notes and take it into account in the following year.

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How do you incorporate (your own) scientific research in your teaching? In master courses I have incorporated my own scientific research on various occasions: The three courses Geometric Mechanics, Conservative Dynamical Systems and Caput Mathematical Physics which I taught resp. teach in Groningen are all related to the theory of Hamiltonian systems which forms a main part of my own research. In all three of these courses I spent one or two lectures on reporting on results of my own research (namely my results on phase space structures governing reaction dynamics on the one hand and Hamiltonian monodromy on the other). In Bristol I once gave a whole series of postgraduate lectures on phase space structures governing reaction dynamics. Do you consult publications on education and didactic methods? Do you check on how your subject is handled at other universities? I attended a few workshop on teaching (see item 10. Training in the field of education). For example, I recently attended a workshop in Eindhoven where lecturers from different departments in the Netherlands discussed how to teach students to do mathematical proofs. More generally, there are efforts to streamline the bachelor programmes at the different maths departments in the Netherlands. As head of the OC in the Groningen I will probably be closely involved in this. Within the Netherlands there is also the MasterMaths teaching where lecturers from different universities teach together and/or coordinate their teaching with others. Last but not least I have been teaching in three departments in three different countries which gives me quite a good idea of how maths is taught at other places. In particular in the UK system the exams at different departments are supposed to be on a similar level, and the quality of the degree programme of a university is measure by the percentage of students who successfully complete the degree programme given the standards for the exam. To ensure that the standards are met external examiners are involved in the examinations at the different departments. This way I was always up to date with the teaching at other universities in the UK for the five years I was teaching in Bristol. What new developments did you implement recently (e.g. new learning methods, Nestor, new media,..)? I have been using a broad variety of teaching methods (besides pure lecture course, I have given project based course like Integrerend Project Dynamische Systemen and Numerical Projects, seminar type courses like Caput Mathematical Physics and Conservative Dynamical Systems), and I will continue to do so. For example, I am in charge of the Studentencolloquium which is a master course where the students learn how to give mathematical presentations. This course does not involve any lectures from me at all (apart from the initial one). The course is based solely on giving feedback to students on the quality of their presentation. Similarly, I often incorporate different media (electronic presentations, showing animations/illustrations with Maple), and I always use Nestor. As indicated above my attitude on an extensive use of electronic presentations like powerpoint and latex beamer has changed, and I am more back to using chalk and the blackboard. The studies that I read on the use of electronic presentations in education (see, e.g., the studies by John Sweller from the University of New South Wales) seem to confirm that the extensive use of electronic presentations can be very counterproductive.

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Faculty vision on education Do you think that the course you teach fits in with other components of the degree program? For example, does your course contribute to acquiring academic skills? I am involved in all aspects of the teaching in our department (bachelor and master lecture courses; tutorials and computer practicals; supervision of students and PhD students; head of the OC), and in all of this the acquisition of academic skills plays a major role. The bachelor courses in mathematics are very much interlinked, and the contents of these courses are coordinated by the curriculum committee and the lecturers (this concerns for example the courses Linear Algebra I, Vectoranalyse (both first year) and Integrerend Project Dynamische Systemen (which has just been moved from the second to the third year) that I am teaching). The course Integrerend Project Dynamische Systemen is very much relevant for the track Dynamical Systems and Analysis which the students can follow in the master programme. Whereas as the full programme is devoted to developing a large spectrum of academic skills (as, e.g., mentioned in the Eindtermen van de bacheloropleiding, and other places), the courses Integrerend Project Dynamische Systemen and the Studentenkolloquium which I teach explicitly lay special stress on developing communication and presentation skills, and on the independent acquisition and processing of information and data sources. In most of the master courses this holds in a similar way. For the supervision of undergraduate and graduate students, the development of skills to independently pursue a research project are always key objectives. In what way do you take the faculty vision on education into account in your teaching? How does this relate to your personal vision on education? I mainly encounter the faculty vision on education via the OERen and the visitatie documents (in particular through my role as head of the OC). The faculty is aiming for high quality teaching, and this is what every one of us is aiming at. I very much welcome recent developments which seem to indicate that (after the master programme) we will also switch our bachelor programme to English. This is a must when you want to attract more students from abroad, and in my opinion, this development is already overdue (in particular given the good English language skills of Dutch students, and the fact that most of the textbooks we are using are in English anyway). The faculty decided to have a broad education in the first year (in the form of the flexible bachelor). Whereas I generally like this idea its present implementation has its problems in mathematics. The most significant of these problems is that in my opinion that the maths students get too late in touch with the mathematical thinking I mentioned further above. To some extent the students get spoiled in the first year because they neither get sufficient stimulation to work at home nor do they get training on writing up mathematics in the way required for the successful completion of a mathematics degree. This is because the maths courses they do together with the students from other disciplines have to meet the requirements of all students, and this implies that the needs in mathematics fall short. Also the maths students get a wrong impression of what mathematics at a university is like. For example, the course Analysis which is one of the key courses in mathematics should be taught in the first year rather than the second year. Now the subject Analysis is motivated by saying that it gives the mathematical foundation of what the students learned in Calculus in the first year. I do not think that it is a good motivation if the students have already seen what they can do with what they learned in Calculus without ever bothering about the mathematical foundation. This way mathematics comes across as nitpicking. Also starting the “genuine” maths education in the second year leads to the difficulty that it leaves too little time to prepare the students for the master programme. The students from Groningen have serious problems when they start to attend the MasterMaths courses which are taught on a national level. I think that the broad education in the first year (which I like) needs to be backed up and accompanied by more courses for maths students only to make the whole idea a real success. It seems however that unfortunately we do not have the resources (in the form of man power) to provide this breadth and depth at the same time.

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The lacking resources more generally form a serious problem in mathematics. A work load of four courses per academic year that we have to teach in mathematics is just too much, and it is the main obstacle to meet the faculties objectives to have high standard teaching in combination with high standard research. I personally will have taught 10 different courses during my first five years in Groningen. How can I keep up my research standards under such circumstances? The high teaching load also implies that the staff in mathematics has no time for “extra” activities. For example, people stay away from the maths colloquium, I personally stopped giving Honours College Tutorials due to lack of time (after offering them in the first two years after the invention of the Honours College programme), a student colloquium to promote mathematics and to stimulate the interaction between students and staff (which I coorganized with Jaap Top) stopped due to the simple fact that the staff does not have time to take part in it, etc. I very much hope that the situation will improve in the coming years.

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4 Self-analysis BKO criteria

4.1 Criterion (re)design of education 1) Design and redesign of teaching for a course unit: selecting and developing suitable learning objectives, working methods and assessment methods that contribute to the degree program’s learning outcomes.

I have been giving lecture courses (as fully responsible lecturer) for more than 12 years in three different countries: 3 years in Bremen, Germany, as Wissenschaftlicher Assistent (these positions are now called Junior Professorschips) 5 years in Bristol, UK, as lecturer (this is similar to an Assistant Professorship with tenure) 4 years in Groningen, the Netherlands, as adjunct hoogleraar In Groningen my teaching load comprises four lecture courses per year. In the UK I did two courses per year. In Germany I did about one per year. In these 12 years I have given 14 different (i.e. not counting the diverse repetitions) lecture courses. Two more new lecture courses (Vectoranalyse (bachelor) and Student Colloquium (master)) will follow this academic year. The following courses under items 8 and 9 above I designed from scratch: 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.8, 8.9, 8.10, 8.11, 9.2. The following courses from items 8 and 9 above existed before, but I developed my lecture notes for them from scratch: 8.1, 8.2, 9.1, 9.3.

I have given courses in three different languages (all courses in Germany I did in German, all courses in the UK in English, and in the Netherlands I just finished my first course in Dutch (Lineaire Algebra I); other course in the Netherlands I taught in English). The courses were in two different disciplines (I first had a position in physics and then in mathematics) plus service courses to biology and pharmacy students. The courses were on all levels (bachelor, master and PhD), see items 8 and 9 above. Sizes of the courses I have been teaching: big courses (100-200 students): 8.1, 8.2, 8.2, 9.3 medium (15-50 students): 8.4, 8.5, 8.5, 8.6, 8.7, 8.8, 8.10, 9.1 small courses (less than 10 students): 8.9, 8.11, 9.2 I did the following tutorials/problem classes (as permanent staff, PhD or undergraduate student): Analysis I and II, Linear Algebra, Algebra, Calculus, Statistical Physics, Mechanics, Quantum Mechanics Like every lecturer in Bristol, I regularly did first year small group tutorials (this comprises two groups of five students whose homework I marked, whom I met once a week to discuss their homework, and whom I helped to find their way through the administration including giving advice on their choice of courses; and for whom I also was the contact person for reference letters after graduation). I have given courses on the black/white-board only, using electronic presentations like latex beamer or powerpoint only, transparencies only, and all possible combinations of all of these depending on the subject, size and level of the course. For all my (big) courses in Groningen and Bristol, I (heavily) use Nestor/Blackboard. In the following I give an indication of my ability to design a course for one example.

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Background to the Course: Linear Algebra I, WILA1-06 (5 ECTS) I picked this course because - it involves a broad variety of teaching and assessment methods (lectures, tutorials and computer practicals; homework assignments, midterm exam, final exam) - it is a big course (about 200 students) which is quite different from most other maths courses which usually have much smaller numbers of students (typically the order of 20) which makes it very different to interact with the students - it is an especially challenging course because it is attended by students from different degree programmes with different expectations and backgrounds (mathematics, physics, chemistry and astronomy) - it is a first year course where students still need to acquire basic learning skills and mathematical techniques Moreover, this is a course which I just taught and which I will teach again in the next couple of years (which is not the case for most other bachelor courses I have been teaching at the RUG so far.)

Learning goals for the course The mathematical learning objective of the course is to give the student an introduction to Linear Algebra which is one of the main pillars of the mathematics curriculum (see attachments Appendix_Ocasys_LA1 and Appendix_course_information_LA1_Nestor). In mathematics the course forms the input for Linear Algebra II (first year), Vector Analysis (also first year), Ordinary Differential Equations (second year), Algebra (third year), and other more general courses like Dynamical Systems Theory. The course moreover provides the mathematical foundation for many concepts and theories in physics and chemistry which come up, e.g., in mechanics, quantum mechanics and special relativity. More generally, Linear Algebra I is the first course in the maths curriculum where the students get in touch with abstract mathematical notions like a vector space and a linear map. It is the first course where mathematical objects are defined in a formal way and where the students have to learn to use these formal definitions to prove properties of and relations between these mathematical objects (i.e. the students learn how to prove mathematical theorems). The learning objectives directly relate to several of the Eindtermen van de bacheloropleiding in mathematics (see the attached bijlage I OER). As Linear Algebra is a first year course it first of all concerns the acquisition of knowledge. It in particular addresses the points 1.1, 1.3, 1.6, 1.8 mentioned under 1. Kennis. The course contributes to the most of the points mentioned under 2. Attitude and also provides a first step towards the points mentioned under 3.1 Probleem-oplossen en onderzoeken. The team work in the computer practicals addresses the point 3.3 Samenwerken. This also applies to the tutorials which moreover address item 3.4 Communiceren. More precisely, in line with my teaching vision outlined in Sec. 3, I organize the tutorials in such a way that they stimulate the students to work at home. In addition to the homework assignments to be handed in for marking the students should also study the exercises set for the tutorials at home. The student assistants are instructed to assign representative exercises to different groups at the beginning of the tutorial session. The groups then work out the details of the solution in such a way that the full solution, and one group member presents the full solution on the blackboard to all students in the tutorial group so that at the end of the tutorial all students have in their notes a complete set of solutions of all exercises. The course builds on two items of preknowledge already acquired at school: vectors and systems of linear equations.

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At school vectors are introduced as “arrows” which you can add to get a new “arrow” and multiply by numbers to change the lengths of the “arrow”. These properties which can be understood geometrically and still be visualized on the blackboard are used to give a more abstract definition of a vector space as a space of objects (vectors) which you can add and multiply by numbers in such a way that certain properties hold. The concept of linear independence then leads to the notions of a basis and the dimension of a vector space. The notion of a basis is very important since it tells the student that the coordinate vectors with respect to a basis are just the vectors in the Euclidean space they are already familiar with from school. Henc also in an abstract vector space (like a space of polynomials of degree less then a certain maximal degree) the students can use their knowledge and intuition they developed for the ‘simple’ Euclidean vector space. The preknowledge from school about linear systems of equations is used to introduce matrices and discuss the properties of matrices (elementary row transformations, row echelon form, invertability of matrices, etc.). This is essential for the study of linear maps which form the second main subject of the course. Linear maps are maps which map vectors from one vector space to vectors from another (or the same) vector space in a linear fashion. Introducing bases in these vector spaces allows one to represent a linear map in terms of a matrix. So the knowledge acquired about matrices can be transferred one-to-one to the analysis of linear maps (e.g. the nullspace of a matrix to the kernel of a linear map). The contents of the course (in chronological order) is the following: - Systems of linear equations; row echelon form - Matrices and matrix algebra - Elementary matrices and block matrices - Determinant - Vector spaces - Linear independence - Basis and dimension - Basis transformations and row and column space of a matrix - Linear maps - Matrix representation of linear maps and similarity of matrices - Scalar product, orthogonality - Eigenvalues and eigenvectors - Diagonalisation of matrices The course begins with systems of linear equations. Using the student’s knowledge from school a systematic discussion of the structure of the solution set of linear systems and its computation/determination using elementary row operations, etc. is discussed. Such systems are then written in a matrix notation. Also column and row vectors are introduced at this stage (as matrices with one column or row only). The structure of the solution set of linear system and the computation of solution is then discussed in great details in terms of matrices. This in particular implies the definition and computation of the inverse of a matrix which in turn also involves the definition of the determinant of a matrix. Vectors are then discussed in a more systematic way after the axiomatic definition of a vector space. Linear independence of vectors, the basis and dimension of vector space are then all discussed with a particular emphasis on the solution set of linear systems. All of this forms then an input for the definition and discussion of linear maps. It is pointed out that matrices are example of linear maps, and more importantly, that by introducing bases a linear map can be represented in terms of matrices. Getting across that all the properties of matrices can now be used to study linear maps is one of the key points of the lecture. Finally orthogonality of vectors with the particular application of the least square fit of data sets, and eigenvalues, eigenvectors and the diagonalization of matrices are discussed.

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Teaching Method The text book for the course is Linear Algebra with Applications. 8th edition. (Pearson International Edition) by Steven J. Leon This book is chosen because - it is an American style textbook which is relatively easy to read not only for mathematics students but also for physics, astronomy and chemistry students; the book is well known and used at many places in the US - it provides a very careful introduction to abstract notions without sacrifying mathematical rigor - it discusses very many applications (partly in separate chapters) - it has many exercises - each chapter contains a separate section where an implementation on the computer using MATLAB is discussed The many applications make the book too comprehensive to be covered as a whole (even though the second half is covered in Linear Algebra II), see objectives of the lectures below. The course comprises lectures, tutorials and computer practicals. There are 4 hours of lectures per week, 4 hours of tutorials, and 2 hours of computer practicals (starting in the fourth week). The course lasts one period (i.e. 8 weeks). The main objectives of the lectures are: - provide a red line through the course - complete and self contained presentation of the required material on the blackboard (picking the bits from the textbook which are necessary to get a proper introduction to Linear Algebra) - motivate, illustrate, explain abstract notions and ideas - get the students to see the bigger picture of the subject Linear Algebra (Eindtermen 1.4) - provide the students with different techniques to prove theorems; give many examples where these techniques are used throughout the course (Eindtermen 1.1, 1.3, 1.6) For this course, I decided to do the lectures on the blackboard only (i.e. there are no electronic presentation). I have also given big courses based on electronic presentations only. But from my experience blackboard presentations for a subject like linear algebra are more suitable than electronic presentations because - they are more flexible - they enforce an appropriate pace - they are better suited for developing an idea on the spot - they better motivate students to copy the material presented - there are no animations or complex pictures or diagrams required in Linear Algebra I Special emphasis is laid on learning how to do proofs (Eindtermen 1.3). Learning to do proofs is often difficult from reading a textbook. The problem is that the proofs often appear as “magic” because the creativity involved in developing the proof is lost. Often there is a number of properties shown in an unmotivated fashion which in the end then fall into place to prove the theorem. This is very irritating for the untrained student because it suggests that only a genus could have come up with a proof like that and that he/she him/herself would have never been and will never be able to produce anything like that. For the student, it is important to understand that the proof as it can be found in a book is just the result of putting those bits together in some logical order which have led to a proof of the theorem; the unsuccessful attempts and directions possibly pursued earlier are no longer visible and the question of how the proof starts this way rather than another way is not addressed. The question then is how the student can him/herself develop a proof. In Linear Algebra I this is done by showing several little proofs in the lecture and letting the students do proofs in the homework and tutorials themselves. Besides learning different techniques of doing proofs (direct proofs; by contradiction; showing the reverse direction for the negated statements; induction) and little things like showing equality between sets by showing left and right inclusions the very basic but

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also very important question of how to start a proof is addressed over and over again. To the latter end I emphasize how important it is to clarify what exactly is to be shown. This question is then usually posed to the students in the lecture and the answer is written on the board. The students see that once it is clarified what exactly is to be shown already gives half the proof. The objectives of the tutorials are: - familiarize students with notions and concepts introduced in the lectures by doing (usually short) exercises which involves these notions and concepts - develop and train computation skills - train techniques to prove mathematical theorems - force/enable the students to stay on the ball with the material presented in the lectures (also as a preparation for the final exam) - get the students to present their results to their fellow students (further develop communication skills) The homework assignments had the following objectives: - encourage the students to continuously follow the course (also as a preparation for the final exam) - give the students continuous feedback on their progress (see assessment below) - encourage the students to also work at home (which is a goal that is only partly successfully met also with the tutorials) - give the students who have difficulties with an exam situation an opportunity to compensate their mark for the final exam (see assessment below) Finally the computer practicals have the following objectives: - learn how to use computers to solve mathematical problems in general and to use MatLab for solving linear algebra type problems in particular - study longer examples/applications; the computer assignments often involve more extensive applications for which there was no time to cover them in the lectures. An example are Markov chains which I discussed only very briefly in the lecture (in the context of eigenvalues and eigenvectors). For the third computer practical assignment, the student had to go through the corresponding section of the book essentially by themselves. The different teaching methods are also used to meet the different requirements/expectations of the students from the different degree programmes: For mathematics students, it is crucial to develop the skill to make precise mathematical statements, to get an intuition for abstract mathematical definitions, and to be able to use different techniques to prove mathematical statements. This is also true for students from the other degree programmes but the focus there is often more on the application and computational side. As the lectures are designed in such a way that they give a complete and self-contained coverage of the material (e.g. basically all theorems which come up in the lectures are also proven in the lectures) they first of all meet the requirements/expectations of mathematics students. Examples and applications are of course also given and discussed in the lectures, but the focus for these is in the tutorials and computer practicals. The latter therefore especially address the needs of physics, chemistry and astronomy students. The teaching methods address the Eindtermen van de Bacheloropleiding in the following way: Concerning 1. Knowledge: 1.1: the lectures contribute to the acquisition of knowledge of basic notions, theories and methods 1.3: the lectures, tutorials and homework assignments familiarize the students with the essence of mathematics: mathematical proofs 1.4: in the lectures linear algebra is put into perspective to other parts of mathematics by pointing out its importance for many courses that follow later in the curriculum like ODEs, Vectoranalyse

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1.5: in the lectures motivations are giving for formulating more general theories, e.g., the vector calculus known for vectors in Euclidean space give rise to the definition of an abstract vector space 1.6: in the lectures, tutorials and homework assignments the students get familiarized with various techniques to proof theorems 1.8: the students see applications of linear algebra in other disciplines like biology in tutorials, homework assignments and computer practicals Concerning 2. Attitude: 2.2: all aspects of the course contribute to developing a systematic mathematical approach to the solution of problems, which includes using different methods by, e.g., trying different approaches to prove a theorem 2.3/2.4: the more extensive assignments in the computer practicals stimulate the students to explore mathematical problems using different methods 2.5: independent, persistent and determined work is stimulated in the homework assignments and the computer practicals Concerning 3. Academic skills 3.1: all teaching methods stimulate the development of a scientific attitude and approach to solving problems (including different levels of abstraction) 3.3/3.4: communication skills and team work are stimulated in various forms: the oral communication of mathematics and teamwork are stimulated in the tutorials because the students have to work out problems in groups on the one hand and present results to their peers on the other and also in the computer practicals where the student are allowed to work in groups of two. The written communication of mathematics is mainly stimulated in the homework assignments 3.5: the feedback in the form of the various assessment methods (homework, computer practicals, midterm exam and final exam) help the students to reflect on his performance and mathematical quality, to discover his/her strengths and weaknesses (and apart from the final exam they also give the students time to work on their weaknesses) Assessment The assessment is based on - 4 homework assignments (HW) - 3 computer practical assignments - 1 mid term exam (MT) - 1 final exam (FE; see attachment Appendix_exam_LA_I_February_2012) For the computer practical assignments, the students only get a pass or fail mark. The students are allowed to work in groups of two on computer practical assignments (this is to stimulate teamwork; moreover computer practicals always have the problem that a lot of time is wasted on getting the syntax right; looking at the programs in groups of two helps to some extent to overcome this problem). The students have to pass all three computer practicals. The homework assignments are not mandatory (this way the bright students are not forced to spend more time on the course than necessary). The final grade is composed as follows: If the mark for the final exam is smaller than 4.5 then the mark for the course is the mark of the final exam (i.e. the student failed the course). If the mark of the final exam is 4.5 or higher then the final mark for the course is computed according to the formula: max(FE, (HW1+HW2+HW3+HW4+4MT+12FE)/20) Hence 40% of the final mark can be compensated by accomplishments prior to the final exam (assuming a reasonable mark for the final exam).

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Goals of midterm exam: - enforce the students to learn material covered in the first half of the course - get the students to become acquainted with style of the final exam - get the students to understand what is expected from them (in particular in the final exam) - take away some pressure from the final exam - monitor whether the students have acquired the expected knowledge as planned for the first half of the course Goals of the final exam: - test whether the students have successfully acquired the required knowledge in breadth and depth - use the results to discover deficiencies/points for improvements in the course ICT Nestor: All the information about the course can be found on Nestor: outline, organization and assessment of the course (see Appendix_course_information_LA1_Nestor); homework assignments; assignments for the computer practicals; old exam papers for preparing for the exams; contact details of the lecturer and student assistants; grades (for each individual aspect). It is of course always good to use Nestor. But for a course of this size, it is almost impossible to do it without Nestor anyway. Nestor does not only serve as a source for information about the course for the students but also for the student assistants who are in charge of the tutorials and computer practicals. Also the communication with the students and student assistants is done via Nestor’s email facility. The lecture notes are not put on Nestor. This is because there is a textbook for the course (of which the course notes essentially only form a subset), and also because the students should not be discouraged to attend the lectures. However, the information which sections from the book are covered in which week of the course can be found on Nestor. MATLAB: In the computer practicals the students use MATLAB to do assignments. MATLAB is one of the big computer maths packages (like Mathematica and Maple) which is especially geared towards linear algebra problems. Learning MATLAB is one of the Eindtermen van de bacheloropleiding. The students are provided with the assignment sheet and a formatted answer sheet on Nestor. Attachments: Appendix_Ocasys_LA1 Appendix_course_information_LA1_Nestor Appendix_exam_LA_I_February_2012

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4.2 Criterion teaching and supervising students Teaching and supervising students: being widely deployable in the main working methods of academic teaching.

Lecture In the following I describe two Quantum Mechanics lectures (see attachments Appendix_QM_Lecture_1 and Appendix_QM_Lecture_2) of a course which I held at the University of Bristol. The reason for choosing this lecture is that I still have quite a detailed written feedback report from a colleague (Dr. Nina Snaith) who peer observed these two lectures (see attachments Appendix_QM_observation_1 and Appendix_QM_observation_2, and the section Analysis below). This is a third-year course in mathematics. It was attended by 22 students. There were three hours of lectures per week over a full semester. There were homework assignments and a final exam but no problem classes/tutorials (as is usual for a third-year lecture in the UK). First lecture (50 minutes):

1. I started with a revision of the material we discussed the week before. I did this in first lecture of every new week to remind the student of where we ended in the lecture in the week before and what the important points were. I did this on transparencies to save time (the students were not supposed to copy this material since they already had it in their notes). The subject was the Fourier decomposition of quantum wavefunctions to solve the time dependent Schrödinger equation. This took about 10 minutes.

2. A Maple worksheet was shown on the computer which illustrate the Fourier decomposition of a wavepacket in a one-dimensional box potential (see attachment Appendix_QM_Maple_Fourier). It was shown how the wavepacket was better and better approximated the longer the Fourier series was. Then the time evolution of the wavepacket using its Fourier decomposition was discussed. A picture of a quantum carpet was shown and discussed which demonstrated the rich structure developing in the time evolved probability density of a wavepacket (see attachments Appendix_QM_carpet_1 and Appendix_QM_carpet_2). This took another 10 minutes. Such quantum carpets were actively discussed in the research at that time. The carpet presented was produced by a student who did a project with me. The Maple Worskheet (and also the picture of the quantum carpet) was sent to the students by email prior to the lecture so that they could also try it out/look at it by themselves.

3. I started a new subject on the white board: probability currents. I started by asking the question under what conditions the norm of an initial wavepacket is conserved under the time evolution. This eventually led to probability currents and its conservation. This took about 30 minutes.

Second lecture (50 minutes):

1. Since this was again the first lecture of a new week I started the lecture as usual with a summary of the material from the week before using transparencies (see attachment Appendix_QM_transparancies_rev_2). (time: 10 minutes)

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2. As a continuation of the material discussed the week before the lecture proceeded with

the computation of transmission and reflection probabilities of a wavepacket initiated at infinity which encountering rectangular potential barriers/wells in one dimension. I first presented the study of a potential step on the white board for the case where the energy lies above the energy of the step. This involves to write down the general solution of the stationary Schrödinger equation in the two region of the piecewise constant potential and to match the solutions at the step. The same problem for energies below the step was then left to the students to be solved by themselves. I gave them about 5 minutes to do so. During these five minutes I walked through the class and provided assistance. The solution was then presented on the white board in interaction with the students. This was then followed by me working out more involved examples where the computation for one particular example was presented on transparencies due to the length of the computations. Copies of the transparencies were handed out to the students (see attachment Appendix_QM_transparencies_finite_barriers). The planned illustration of the computed reflection and transmission using Maple had to be cancelled due to a lack of time. The Maple Worksheet was sent beforehand to the students so that they could try it out themselves (see attachment Appendix_QM_Maple_finite_barriers). (total duration: about 40 minutes)

Analysis The following is taken from the peer review by Nina Snaith (see attachments Appendix_QM_Lecture_1 and Appendix_QM_Lecture_2). The strong points of both lectures: - Clarity of the lectures - Good structure and pace of the lectures; motivation by an opening question which was answered at the end of the lecture - Revision at the beginning (always in the first lecture of a new teaching week) - Physical interpretation and motiviation of the mathematical computation - Illustration of the computations and the underlying physics using Maple - Use of different media (transparancies, computer, white board) to break up the lecture Weak points of the first lecture: The main criticism brought up by the observer was that I could have engaged the students more during the lecture. Whereas I asked a couple of questions the students were not sufficiently stimulated/encouraged to stand up and answer the question. This is a valid criticism. Most of the questions were only meant to get the students to think about the problem but they were not supposed to answer them themselves. When I asked a question that meant to be answered by the students I did not wait long enough and did not give sufficient encouragement for the students to give the answer, and instead gave the answer myself. The observer also made the suggestion that in order to take the pressure from the students I could let them discuss the question with their neighbours and/or break up the questions into smaller pieces. Improvements in the second lecture: The main issue was that in order to get the students more engaged I filled in a little exercise the students had to work out by themselves. The students had about 5 minutes to work out the problem. I walked around in the class and provided assistance. This worked to some extent. However, there was not really time to help those students who did not even have a clue on how to start.

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I have to admit, as also pointed out by the observer, that there is in general no time to do such exercises during the lectures due to the tight syllabus. In Bristol the problem is that third year courses are usually not accompanied by a problem class/tutorial. This is partially compensated by homework assignments but this is not sufficient. In Groningen this is done better because all lectures have some sort of problem class/tutorial. So this problem is not directly transferrable as the students in Groningen have to get active in the tutorials. However, the interaction with the students during the lectures remains a general problem. For the small classes I have in Groningen (order of 20 students) this is usually not a problem as I always encourage the students to ask questions, and usually the students are not afraid to do so. So during the small lectures there are often moments of little discussions between me and the students to explain something in more detail or to clarify the connections to what we discussed before. In a big lecture the situation is however quite different. Not only are the students often too afraid to ask a question (although I always encourage them to do so) but it is also often quite difficult to discuss a question because they are often not quite to the point and then still answering them just takes too much time. The way I address this problem is that I ask many little questions to which I also give the answer myself but which still create an atmosphere where students follow the thoughts. I would like to add that after all the quantum mechanics course in Bristol got very good student evaluations (see the attachment Appendix_QM_evaluation). Alternative teaching methods I have used several different teaching methods in my academic career. This concerns some of my master courses which (if appropriate) I organize in a more seminar type style. In Groningen this has been the case for the master courses Caput Mathematical Physics and Conservative Dynamical Systems. Both of these courses are essentially lecture courses where some of the lectures are taken over by the student participants. The students are assessed on the basis of the quality of their presentation. In these courses it often happens that not every single detail is represented but the emphasis is more on conveying general ideas which is very hard to assess in a written exam. For the students, the preparation for these lectures is typically quite extensive because they have to go through different sources and retrieve diverse background information to get knowledge sufficiently profound to give present the lecture. Like myself who gives most of the other lectures the student who presents a lecture has to develop the skill to distil the most important points of his/her subject and prepare it in such a way that it is comprehensible for his/her peers. After each student presentation questions can be asked on the subject, and the strong and weak points of the presentation are discussed by all participants of the course. The teaching of how to give a maths presentation is also the subject of the master course Student Colloquium which I will give for the first time in the fourth period of this academic year. As one example where I use other teaching methods than lectures in the bachelor programme I mention the course Integrerend Project Dynamische Systemen which I designed from scratch (for an evaluation see attachment Appendix_Int-Proj-Dyn-Syst-evaluation). As the name suggests this is not a pure lecture course but a course in which the emphasis is on project work. The course starts as a regular lecture course with four hours of lectures and two hours of tutorials per week. In the fourth week I introduce projects on which the students can work on in groups of two (or possibly three). From then on I only give two hours of lectures each week plus a two-hours tutorial, and I supervise the project work by meeting the groups of students individually once a week. In the last week of the course the students present their projects in a talk and they need to hand in a written report the week after. Moreover there is a short (90-minutes) exam. The objectives of theses course are twofold. On the one hand the students have to get a proper introduction to the subject dynamical systems theory (which is especially

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important for those students who want to choose the direction Dynamical Systems and Analysis in their master), and on the other hand the oral and written presentation of a mathematical subject are trained in this course. All students get a detailed feedback on their oral and written presentation which together also form 75% of their mark for the course (the remaining 25% come from the exam). Supervising individual bachelors-, master-, and PhD-students At the moment I have two PhD students. Before I had one PhD student who graduated in 2010 and I supervised two postdoctoral students. I have supervised several bachelor and master theses, and also (inofficially) co-supervised several further PhD theses. At the end of last year I attended a four-day PhD coaching course (provided by the RUG, conducted by Brigitte Hertz). We here covered - situational leadership - counseling - negotiations - conflict management - influence techniques - feedback during the writing process - providing feedback to presentations In the coaching course we extensively reflected on our role as supervisors (in particular during the different phases of a PhD), discussed our visions, strengths and weaknesses, tried communication techniques in various situations in practical sessions. I can recommend this course to anybody involved in supervising students. My role as a supervisor of PhD students has changed considerably when I came to Groningen. In Germany and the UK I still had the time to really work with my PhD students resp. the PhD students in our group in the manner of a collaboration rather than in the roles of supervisor and students. I experienced this as very productive. Most things which are important for a successful PhD then occur naturally. Due to my extensive teaching and administrative obligations in Groningen I simply do no longer have the time to follow this style of dealing with PhD students. Since due the lacking time I can basically see my students only once a week the meetings have quite an hierarchical structure where I ask the student to report on his progress in the preceding week and we together discuss the programme for the following week. There is almost no time to really discuss any ideas in detail in the way I am used to from other places. For the future, it will be important for me to also have one or two postdocs which can take over some of the supervision tasks. Attachments: Appendix_QM_Lecture_1 Appendix_QM_Lecture_2 Appendix_QM_observation_1 Appendix_QM_observation_2 Appendix_QM_Maple_Fourier Appendix_QM_Maple_finite_barriers Appendix_QM_transparencies_finite_barriers Appendix_QM_transparancies_rev_2 Appendix_QM_carpet_1 Appendix_QM_carpet_2 Appendix_QM_evaluation Appendix_Int-Proj-Dyn-Syst-evaluation

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4.3 Criterion testing and assessment Testing and assessment: compiling and using test types that are consistent with the learning objectives and working method used in content, form and assessment. -

Test In my academic career I have used several different kinds of assessments: written and oral examinations, reports, computer practicals, oral presentations and homework assignments, and all kinds of combinations of these. The way I assess a course depends on many different aspects which include the size of the course, the subject, the level, the objectives (e.g. development of communication and presentation skills in addition to the scientific content), etc. The particular difficulties involved in assessing maths students was the subject of two workshop I attended: one in Eindhoven and one in Birmingham (see item 10, Training in the field of education, above). The main difficulty here is that mathematics involves a great deal of creativity, and that this creativity is very difficult to grasp and assess. This first of all concerns the competence to prove theorems. Whereas one can assess the familiarity with various techniques to prove theorems by letting the students do straightforward proofs of simple statements by, e.g., explicitly asking for a specific technique (like proof by contradiction) in the exam question or in a hint, it is very difficult to go beyond this. The creative process in working out a proof often does not only involve the ``right idea'' of how to start the proof, but also having the ``right idea'' on which route to take in the course of a proof, and the ``right ideas'' on how to do intermediate steps in a proof. Getting the ``right ideas'' at the right time in the environment of an exam then often involves a great deal of good luck which cannot (or at least should not) be the basis of an assessment. Of course one can provide the students with some appropriate hints which guides the students through a proof. But then the creative aspects are again less pronounced. Finding the right balance here is very difficult because there is no clear definition of what the right balance is. I think that all maths lecturers struggle with this dilemma when they set exams. The problem above also arises in the marking of an exam, namely in how far do you reward right ideas and punish computational faults? If two or more people have to give individually a mark for the same project report, then usually the marks agree amazingly well. If two different examiners mark the same exercise of an exam the mark/number of points might differ quite heavily even if you provide them with a very detailed marking. At the workshop in Birmingham on teaching maths in higher education the attendees were asked to mark the same student answer to an exam question. The extent to which the marks were varying was shocking. To partly meet this problem I at least try to be fair in the sense that I mark an exam question by question and not exam paper by exam paper. If I have students assistance involved in the marking then I make sure that the marking of one question is not split up between different markers. The problem of assessing creativity is first of all a problem with the bachelor courses as I access my master courses usually by student presentations (possibly combined with written reports). The way I try to meet the problem with the bachelor courses is that I try to rely on different assessment methods in a course. For example, in my course Linear Algebra I, there are homework assignments and computer practicals besides the midterm and final exam. In the homework assignments and computer practicals one can then set more complex problems where the students can find the right idea or trick without time pressure. But the problem with the final exam remains.

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Having worked at departments in three different countries I have come across very different assessment cultures: In Germany all my courses where assessed by oral exams (in the meantime this has also changed in Germany). The advantages of oral examinations are: - There is much room for flexibility: if the student gets stuck with answering a question, one can provide assistants; one can possibly even resort to another question - One gets a good idea of whether the problems in answering questions are more of a technical or a fundamental nature - One can ask the student about the “bigger picture”, i.e., how are certain aspects of a course interconnected to others? It is very difficult (almost impossible) to retrieve such information in a written exam. Disadvantages: - Developing an idea on the spot in the exam situation is difficult which delimits the

extent to which you can assess creativity (however, I also did oral exams where the students got some of the questions before the start of the oral exam so that he/she had a few minutes to think about the questions; then the situation is as far as this aspect is concerned very similar to a written exam)

- Some students feel very much under stress in an oral examination because they are completely exposed to the examiners

- A student might have his/her personal problems with the examining lecturer which might lead to difficulties; also oral examinations require the lecturer to be quite sensitive to the communication style and skills of the student

- Oral examinations are very time consuming Despite the disadvantages I still think that oral examinations are better than a written exam. For big (first year) courses, this is however not feasible for practical reasons. Interestingly the students I have talked to about the issue of oral or written exams all mentioned that they prefer written exams. In the UK the whole examination procedure is extremely professional. All exam papers are checked by an internal and an external reviewer. The final marks are usually scaled to achieve a reasonable/desired distribution of the marks (there were about 200 students per year; so that a scaling makes sense). This scaling is discussed with the internal reviewer and it is documented in detail so that it can be discussed in the examiners meeting. The examiners meeting is a big two-day meeting once a year where all members of staff (about 60 in Bristol) take part and where the results of all courses are discussed in detail, and moreover the degree classifications of the final year students are considered. All borderline cases (students on the border between pass and fail) are looked at separately by the examiner, and possibly also by the internal and external reviewer. The decision on pass or fail is then usually done on the basis of the matheamtical quality of the exam paper, i.e. on the basis of whether the respective student gave sufficient indication that he/she achieved the learning goals of the course. The exact number of points only plays a less important role. This is a practice which I also follow with my exams in Groningen. In the Netherlands the examination is very different in the sense that the whole responsibility for the examination is with the lecturer only. Coming from the UK this was a shock and relief at the same time. It is interesting to see that with the introduction of the BSA and other measures the examination procedure in the Netherlands will also have to change. It seems that in the end we will get very close to the UK system. In the following I give some comments on my Linear Algebra I exam (see attachment Appendix_exam_LA_I_February_2012). The exam consists of 6 exercises which cover all main subjects of the course.

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Exercise 1: solving systems of linear equations. This is in a way a basic "warm up" question which, as it is based on a specific example, is very concrete and computational. On the other hand the exercise also requires some broader understanding, namely the structure of a solution set of a linear systems depending on whether it is homogenous or inhomogenous. Exercise 2: the mainly computational Excersise 1 is picked up again and discussed in a more general/abstract way. The students have essentially seen the statements and their proofs in the lectures. However, the students are not expected to know the proofs by heart. The proofs are sufficiently simple (all direct proofs) so that they can be developed in an exam. In part (b) the students have to recognize that they have to prove both directions of the if-and-only-if statement. The exercises tests familiarity with key notions like linear subspaces and key ideas namely the general structure of the solution set of homogenous and inhomogenous systems. Exercise 3: invertability of matrices, and block structure of matrices. Invertiblility of matrices and the computations of the inverse were discussed in detail in the lecture. Also block matrices and the calculus of block matrices were treated in the lectures and in the tutorials. However, block matrices were not discussed from the perspective of invertability of matrices. The exercise therefore assesses some transferable skills. For example, part (a) is slightly challenging: how do you prove invertability of the given block matrix given the invertability of its diagonal blocks? One can simply computed the inverse (as required in part (b) anyway), or one can argue on the basis of the linear independences of the row vectors of the matrices involved. This requires a good understanding of when a matrix is invertible. Exercise 4: notion of a linear map, matrix representation of a linear map, the rank and nullspace of the linear map and how it is related to the matrix representation. This exercise addresses one of the key points of the course: namely that linear maps are essentially identical with matrices, and that this identification can be obtained by introducing bases in the vector space the map is mapping between. In the present example, the domain vector space is rather abstract: a vector space of polynomials. However, the students have seen this vector space before in the lectures and in the tutorials. In this exercise the students have to show that they can compute the representing matrix of a linear map (which computationally is not at all challenging but which requires a good understanding of how to this), and that they can use this representing matrix to analyse the linear map. The fact that this exercise embodies the key point of the course is also reflected in the number of points that the students can achieve here. Exercise 5: this exercises concerns the notion of orthogonality of vectors and the application of the Rank-Nullity Theorem (which is one of the main theorems in Linear Algebra). Part (a) requires a good understanding of the outer product type definition of the matrix A. The students have briefly seen this in examples in the lecture. Even if a student does remember the example from the lecture, part (a) is straightforward for a student confident with matrix multiplications and with the notion of a symmetric matrix part (and a student should have acquired this confidence). In part (b) the connection between the orthogonal complement and the nullspace of the matrix is to be shown. This is again relatively straightforward, if the student is well familiar with the notions. Part (c) requires the student to see that the Rank-Nullity Theorem is required here. The Rank-Nullity Theorem was heavily discussed in the lectures and in the tutorials. Every student should be able to see (without a hint) that this important theorem is to be used here. However, there is the additional creative aspect that one has to use part (b) and the fact that the full space given by the direct sum of S and its orthogonal complement. Putting this together requires the students to have a good overview over these aspects of the course. Exercise 6: tests familiarity with the notions eigenvalues, eigenvectors, and diagonalization of a matrix; when the student is familiar with these notions then this exercise is mainly computational.

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Analysis Overall result: About 72% of the students passed the course (with the resit exam the percentage will even further increase). This is a very good performance of the students which lies above the results in previous years. I take this as an indication that the course was successful. In the 6 exercises the student scored as follows: In Exercise 1 the students on average reached about 80% of the maximal number of points that could be achieved. The students reached on average 50% in the 2nd Exercise, 75% in the 3rd Exercise, 65% in the 4th Exercise, 45 % in the 5th Exercise and 75% in the 6th Exercise. The results for the individual exercises are in line with expectations. The students do very well with the computational Exercises 1 and 6. It is a very pleasing result that the students did so well with Exercise 3 which also is computational but which also involves a little proof in part (a). The result of 65% in Exercise 4 is also very pleasing. From the previous lecturer I know that making the connection between linear maps and matrices used to be a problem in previous years. The exercises the students were struggling with are Exercises 2 and 5. Both of them are based on little proofs only. Since these exercises required quite high degree of creativity compared to the other exercises it was to be expected that the students scored here worse. Concerning the skill of learning to work out and write up proofs the results in the range of 50% still give an acceptable outcome for a course in which this skill is expected from the students for the first time. The number of students who improved their final mark using the results for the midterm exam and the homework assignments is almost negligible. You can interpret this in two ways: the students who where successful with the homework assignments and the midterm exam did also well in the final exam, or the students who had problems with the final exam already had problems with the midterm and the homework assignment.

Attachments: Appendix_exam_LA_I_February_2012

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4.4 Criterion Evaluation Evaluation: Evaluating the teaching using a variety of data sources in order to arrive at a well-reasoned improvement proposal. - Where do you obtain information for evaluation of the course? - A proposal to improve based on a review of different evaluation forms that were used, and

subsequent reflection.

Information In order to evaluate my course I used data from various sources: - performance of the students in the exams - course questionnaires (which accompany all our bachelor courses) - ask for feedback from students during the course simply by talking to them in the breaks between lectures; for small to medium size course I often ask for comments and suggestions for improvements in the last lecture - ask student assistants for feedback (they are in direct contact with the students in the tutorials) - if I do the course together with a colleague then we of course also critically review the course at intermediate stages and at the end

Evaluation In Groningen, I did all courses apart from the courses Algebra, Integrerend Project Dynamische Systemen and Biomathematics only once and there hence was no need for adjustments. For the courses I have been teaching more regularly, there has rarely been any criticism from student evaluations which would have required changes to my courses. One here has to point out that in mathematics the contents (at least of a bachelor course) is pretty clear cut. The teaching methods are too a large extent determined by the curriculum commissie. The contents has to fit with the overall curriculum which is worked out in great detail by the curriculum committee. There is hence not that much room for changes. Nevertheless there were a few occasions on which I made adjustments to my courses. One example is that I introduced a final exam to my course Integrerend Project Dynamische Systemen (see attachment Appendix_Int-Proj-Dyn-Syst-evaluation). Originally the course was evaluated on the basis of project work only. This however had the disadvantage that the students are not sufficiently motivated to follow all the lectures of the course and also attend the tutorials. The objectives of the lectures were however not only to provide the students with a background on dynamical systems theory sufficient to enable them to do their projects but also to give a general introduction to dynamical systems theory as is then also required to follow the direction Dynamical Systems and Analysis in the master programme. Without an exam the students would only focus on those parts of the lecture which were absolutely necessary for their particular project. The introduction of the exam to some extent helped to overcome this problem. Since the exam contributed only 25% of the final mark for the course the introduction of the exam did still not completely solve the problem. As the course has been moved from the second to the third year the course is not given this academic year. When I reshape the course for the third year I will need to review the exam issue again. As another example where I initiated some changes I mention the service course Mathematics&Statistics for Pharmacy. This course originated from the course Biomathematics for biology students (essentially set up by Franjo Weissing) from which certain sections were replaced by topics from statistics. When I took over this course I realized (too late to make adjustments) that the statistics came far too short in the course, and instead there was too much

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emphasis on ordinary differential equations. The student evaluations were quite bad for this course which is quite typical for this course as I found out later. The OC of pharmacy arranged a meeting with some students who took the course to find out what should be improved. It seemed that the main issue was that the pharmacy students just did not see the relevance of the course for their degree programme, and to some extent I agreed with them. My suggestions to bring in more statistics which is very important for the pharmacy students was taken up by the lecturer who succeeded me for this course. Finally I would like to remark that also in Bristol I consistently had very good student evaluations. This holds for the small group tutorials (see attachment Appendix_Small_Group_Tutorials_evaluations) as well as the courses I taught (see attachments Appendix_Numerical_Projects_evaluations and Appendix_QM_evaluation).

Attachments: Appendix_Int-Proj-Dyn-Syst-evaluation Appendix_Small_Group_Tutorials_evaluations Appendix_Numerical_Projects_evaluations Appendix_QM_evaluation

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3/13/12 11:27 AMOcasys: Toon vak Lineaire algebra 1

Page 1 of 3http://www.rug.nl/ocasys/fwn/vak/show?code=WILA1-06&ocasysyear=2011

Lineaire algebra 1

Faculteit Wiskunde enNatuurwetenschappen

Jaar 2011/12Vakcode WILA1-06Vaknaam Lineaire algebra 1Niveau(s) bachelor, propedeuse, facultaire

minorVoertaal NederlandsPeriode semester I bECTS 5Rooster Roosters FWN

Uitgebreidevaknaam

Lineaire algebra 1

ProgRESS naam Lineaire Algebra 1DoelstellingOmschrijving Veel natuurkundige grootheden, zoals 'kracht',

'positie', 'snelheid', en 'versnelling', hebben nietalleen een grootte maaar ook een richting.Dergelijke grootheden worden 'vectoren'genoemd. Een vector wordt vaakgerepresenteerd door een pijl waarvan de lengtede grootte, en de richting de richting van devector is. Vectoren kunnen worden opgeteld enworden vermenigvuldigd met getallen. Eenverzameling van vectoren die, met deze tweebewerkingen, aan bepaalde regels (axioma's)voldoet wordt een 'vectorruimte' genoemd. Hetblijkt dat verzamelingen van heel andereobjecten dan dergelijke drie-dimensionale pijlenook aan deze axioma's voldoen. Zo is deverzameling van alle polynomen ook eenvectorruimte. Ook de verzameling van continuefuncties op de reele getallen is een vectorruimte.Vaak wordt een vectorruimte voortgebracht dooreen eindig aantal van zijn elementen. Zo'neindige verzameling elementen heet dan een'basis'; van de vectorruimte, en het aantalelementen heet de 'dimensie' van devectorruimte. We bekijken ook (lineaire)operaties die vectoren omzetten in vectoren. Eenvoorbeeld is het spiegelen van driedimensionale'pijlen' in de oorsprong. Een ander voorbeeld isde operatie 'differentieren', die functies omzet infuncties. In het geval van vectorruimtes met een

Toon vak Lineaire algebra 1Jaar: 2011/12

Rijksuniversiteit Groningen

Zoek vakken

Zoek opleidingen

Toon opleidingen per faculteit

Over Ocasys

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eindige dimensie kan een dergelijke operatieworden voorgesteld door een 'matrix'.

Uren per weekOnderwijsvorm computerpracticum, hoorcolleges, werkcollegesToetsvorm huiswerk, opdrachten, schriftelijk tentamen

(Alle computerpractica moeten voldoende zijnafgerond om een voldoende op het vak tekunnen halen. Eind = max(ET, 0.2 HW + 0.2 MT+ 0.6 ET) mits ET >=4.5 anders Eind = ET, metHW gemiddelde van huiswerkcijfers, MTmidtoetscijfer en ET tentamencijfer. HW en MTtellen niet mee bij hertentamen.)

Vaksoort bachelorCoördinator prof. dr. H. WaalkensDocent(en) prof. dr. H. WaalkensVerplichteliteratuur

Titel Auteur ISBN Prijs

LinearAlgebra withApplications.8th edition.(PearsonInternationalEdition)

StevenJ.Leon

978-0135128671

EntreevoorwaardenOpmerkingenOpgenomen in Opleiding Jaar Periode Type

BScNatuurkundeen Wiskunde(dubbelebachelor)

1 semesterI b verplicht

FWN MinorWiskunde - semester

I b keuze

BScNatuurkunde,richtingEnergie enMilieu

1 semesterI b verplicht

BScTechnischeWiskunde

1 semesterI b verplicht

KeuzevakkenFarmacie - semester

I b keuze

BScNatuurkunde,richting LevenenGezondheid

1 semesterI b verplicht

BScScheikunde (SmartMaterials)

2 semesterI b verplicht

BSc

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Scheikunde (Chemistry ofLife)

2 semesterI b

verplicht

BScScheikunde (SustainableChemistry andEnergy)

2 semesterI b verplicht

BScNatuurkunde,richtingExperimenteleenTheoretischeNatuurkunde

1 semesterI b verplicht

BScScheikundigeTechnologie

2 semesterI b verplicht

BScSterrenkunde 1 semester

I b verplicht

BScTechnischeNatuurkunde

1 semesterI b verplicht

BSc Wiskunde (Statistiek enEconometrie)

1 semesterI b verplicht

BSc Wiskunde ( WiskundeAlgemeen)

1 semesterI b verplicht

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3/13/12 1:33 PMLinaire Algebra 1

Page 1 of 4http://www.math.rug.nl/~holger/linalg_2011.html

Lineaire Algebra 1, 2011-2012Recente Mededelingen:Op donderdag 15 december, van 14:00 tot 17:00, Tentamenhal 01 is de midtoets. De stof voor de midtoetsis: Hoofdstukken 1 en 2, en van hoofdstuk 3 alleen Sectie 3.1.Hoorcollege:Het hoorcollege Lineaire Algebra begint op maandag 14 november, en loopt tot en met vrijdag 20 januari.Wekelijks zijn er twee hoorcolleges, namelijk op maandag van 11:00 tot 12:45 in zaal 5111.0022, enwoensdag van 13:00 tot 14:45 in 5111.0022.Werkcollege en practicum:Wekelijks zijn er twee bijeenkomsten ingeroosterd waarop afwisselend een werkcollege ofcomputerpracticum plaatsvindt. Er zijn seven werkcollege/practicum-groepen, ieder met een eigen docent.De roosters voor het werkcollege/computerpracticum van de verschillende groepen kunnen wordengevonden op de website http://www.rug.nl/fwn/onderwijs/roosters/2011/vakken/WILA1-06. Verdereinformatie kan nog worden gevonden op http://www.rug.nl/ocasys/fwn/vak/show?code=WILA1-06&jaar=2009. Hieronder vind u de namen van de werkcollegedocenten met de bijbehorende groepen: Tjerk Stegink: Na A-DBart Visser: Wi1 1Lucas Stam: Ch2, ST2Jasper van Dijk: Na K-RRemko Klein: Na S-ZWiebe Kees Goodijk: Na E-JPeter Gooijert: HIO (IS of CSV), Na+Wi 1, Wi1 3, Wi1 2a Tijdens het werkcollege wordt gewerkt aan opgaven uit het boek. In het schema hieronder kunt u een lijstvinden vinden van de opgaven die per week moeten worden gemaakt. Bij het werkcollege is eenwerkcollege-docent aanwezig met wie u eventuele moeilijkheden bij het maken van de opgaven kuntbespreken. Gedurende de cursus is er vijf keer een computer-practicum ingeroosterd. In het computerpracticum wordtin (vaste) koppels van twee studenten gewerkt aan computeropdrachten. Tijdens het doen van dezeopdrachten leert u de basisbeginselen van het wiskundige software-pakket Matlab. Depracticumopdrachten en bijbehorende antwoordformulieren kunnen gedownload worden op Nestor. Ditdient voor de aanvang van elk practicum worden gedaan.BELANGRIJK: HET PRACTICUM IS VERPLICHT. Per koppel moet het bij het practicum horendeantwoordformulier worden ingevuld en aan het einde van het practicum worden ingeleverd bij dewerkcollegedocent. Van elk antwoordformulier zal worden beoordeeld of het voldoende of onvoldoendeis. Om toegelaten te worden tot het eindtentamen moeten alle drie de practica met een voldoendebeoordeeld zijn.Data voor huiswerk en midtoets:Gedurende de loop van de cursus bestaat de gelegenheid vier keer Huiswerk in te leveren en te latennakijken. De cijfers voor het huiswerk tellen mee voor het eindcijfer (zie verderop). Het Huiswerk moetingeleverd worden in apart hiervoor bestemde postvakken in de hal van Nijenborgh 4 (zelfde plek alsCalculus 1). De data waarop het huiswerk moet worden ingeleverd zijn als volgt:

Alle groepen: maandag 21 november, 5 december, 9 januari en 16 januari voor 16:30 uur.

Er is een midtoets op 15 december, van 14:00 tot 17:00, Tentamenhal 01. Het eindtentamen vind plaats op2 februari 2012, van 18:30 - 21:30, Tentamenhal 01. Verder is er een herhalingstentamen 18 april 2011van 14:00 - 17:00, Tentamenhal 01.

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3/13/12 1:33 PMLinaire Algebra 1

Page 2 of 4http://www.math.rug.nl/~holger/linalg_2011.html

Berekening eindcijfer:Het eindcijfer voor het vak Lineaire Algebra wordt als volgt bepaald: Als ET < 4.5, dan Eindcijfer = ET. Als ET > = 4.5, dan Eindcijfer = max( ET, (H1 + H2 +H3 + H4 + 4*MT + 12*ET)/20 ). In deze formule zijn H1 t/m H4 de huiswerkcijfers, MT het cijfer voor de midtoets, en ET het cijfer voorhet tentamen. Rooster per week:

1e week(46): 14 november - 18 november 2011

hoorcollege maandag

Zevende druk: Hst. 1, sectie 1.1: Stelsels lineaire vergelijkingen, sectie 1.2: Derij-echelon-vorm. Achtste druk: Hst. 1, sectie 1.1: Stelsels lineaire vergelijkingen, sectie 1.2: Derij-echelon-vorm.

woensdag Zevende druk: Hst. 1, sectie 1.3: Matrix algebra. Achtste druk: Hst. 1, secties 1.3 en 1.4 : Matrix algebra.

werkcollege1

Zevende druk: sectie1.1: opg. 1c, 3c,d, 6b,f, 9,10,11, sectie1.2: opg. 5a,c,e, 7, 13,17, 20c Achtste druk: sectie1.1: opg. 1c, 3c,d, 6b,f, 9,10,11, sectie1.2: opg. 5a,c,e, 7, 15,19, 22c

werkcollege2 Zevende druk: sectie 1.3: opg. 2, 9, 15, 16, 17, 18, 19, 22

Achtste druk: sectie 1.3: opg. 2, sectie 1.4: opg. 6, 12, 15, 16, 17, 18, 3

huiswerk 1 Zevende druk: sectie 1.3: opg. 13, 14, 26, 27, 29 Achtste druk: sectie 1.3: opg. 11, 12, sectie 1.4: opg. 28, 29, 30

2e week(47): 21 november - 25 november 2011

hoorcollege maandag

Zevende druk: Hst. 1, sectie 1.4: Elementaire matrices, sectie 1.5:Gepartitioneerde matrices. Achtste druk: Hst. 1, sectie 1.5: Elementaire matrices, sectie 1.6:Gepartitioneerde matrices.

woensdag Zevende druk: Hst. 2, sectie 2.1: Determinant van een matrix Achtste druk: Hst. 2, sectie 2.1: Determinant van een matrix

werkcollege1 Zevende druk: sectie 1.3: 32, 33 , sectie 1.4: 5, 6, 8b,d, 10b,e,h, 15, 18, 20

Achtste druk: sectie 1.4: 32, 33 , sectie 1.5: 5, 6, 8b,d, 10b,e,h, 15, 18, 20werkcollege2 Zevende druk: sectie 1.4: 22, 23, 24, 26, sectie 1.5: 1, 2, 6, 7, 10, 11

Achtste druk: sectie 1.5: 22, 23, 24, 26, sectie 1.6: 1, 2, 6, 7, 10, 113e week

(48): 28 november - 2 december 2011

hoorcollege maandag

Zevende druk: Hst. 2, sectie 2.2: Eigenschappen van de determinant, sectie 2.3:Regel van Cramer, Uitproduct Achtste druk: Hst. 2, sectie 2.2: Eigenschappen van de determinant, sectie 2.3:Regel van Cramer, Uitproduct.

woensdag

Zevende druk: Hst. 3, sectie 3.1: Definitie van vector-ruimte, sectie 3.2:Deelruimten. Achste druk: Hst. 3, sectie 3.1: Definitie van vector-ruimte, sectie 3.2:Deelruimten.Zevende druk: sectie 1.5: 15, 16, sectie 2.1: 3b,e,h, 4, 5, 6, 9,10, sectie 2.2: 5, 6,

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werkcollege1

10. Achtste druk: sectie 1.6: 16, 17, sectie 2.1, 3b, e, h, 4, 5, 6, 9, 10. sectie 2.2, 5, 6,10.

werkcollege2 Zevende en achtste druk: sectie 2.2: 11, 12, 14, 15, 16, sectie 2.3: 1a,c, 2d, 12,

sectie 3.1: 4,6, 8,13, 15huiswerk 2 Achtste druk: sectie 2.2: opg. 18, 19, sectie 2.3: opg. 8, 9

4e week(49): 5 december - 9 december 2011

hoorcollege maandag

Zevende druk: Hst. 3, sectie 3.3: Lineaire onafhankelijkheid, sectie 3.4: Basis endimensie. Achtste druk: Hst. 3, sectie 3.3: Lineaire onafhankelijkheid, sectie 3.4: Basis endimensie.

woensdag

Zevende druk: Hst. 3, sectie 3.5: Basis-transformaties, sectie 3.6: Rij-ruimte enkolom-ruimte van een matrix. Achtste druk: Hst. 3, sectie 3.5: Basis-transformaties, sectie 3.6: Rij-ruimte enkolom-ruimte van een matrix.

werkcollege Zevende druk: sectie 3.2 1a,b,d, 3a,c,f,g, 4b, 6a,b,c, 10b,d, 12, 13, 15, 18, 20. Achtste druk: sectie 3.2 1a,b,d, 3a,c,f,g, 4b, 6a,b,c, 8b,d, 10, 11, 13, 16, 18.

5e week(50): 12 december - 16 december 2011

hoorcollege maandag Hst. 4, sectie 4.1: Lineaire afbeeldingen

werkcollege maandag/dinsdag Nog overgebleven opgaven en opgaven uit midtoetsen van voorgaande jaren

Midtoets donderdag 15 december, 14:00 - 17:00, Tentamenhal 01practicum Matlab Practicum 1

6e week(51): 19 december - 23 december 2011

hoorcollege maandag Hst. 4, sectie 4.2: Matrix representaties van lineaire afbeeldingen. woensdag Hst. 4, sectie 4.3: Similariteit van matrices.werkcollege1 Zevende druk: sectie 3.3: 1e,b,d, 2d,e, 6c,d, 11, 12, sectie 3.4: 7, 8, 15.

Achtste druk: sectie 3.3: 1e,b,d, 2d,e, 8c,d, 13, 14, sectie 3.4: 7, 8, 15.werkcollege2 Achtste druk: sectie 3.6: 8, 24, 25, 28, 31, sectie 4.1: 6, 7, 9, 16, 17, 20.

huiswerk 3 Achtste druk: sectie 3.6: 22, 26, sectie 4.1: 21, 22.practicum Matlab Practicum 2

7e week(2): 9 januari - 13 januari 2012

hoorcollege maandag Hst. 5, sectie 5.1: Skalair product, orthogonaliteit, Hst. 5, sectie 5.2: Orthogonaledeelruimten.

woensdag Hst. 5, sectie 5.3: Kleinste kwadraten methode.werkcollege1 sectie 4.2: 4, 14, 15, sectie 4.3: 3, 5, 13.

werkcollege2 sectie 5.1: 3b,d, 6, 8, 10, sectie 5.2: 2, 4, 8, 9, 13.

huiswerk 4 Achtste druk: sectie 5.3: 5, 7, 8, 11

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8e week(3):

16 januari - 20 januari 2012

hoorcollege maandag Hst. 6, sectie 6.1: Eigenwaarden en eigenvectoren. woensdag Hst. 6, sectie 6.3: Diagonalisatie van matrices, Markov ketens.werkcollege sectie 6.1: 1a,b,c,f,g,,j, 2, 3, 4, 9, 15, sectie 6.3: 1b,d, 2b,d, 6, 13practicum Matlab Practicum 3

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Lineaire Algebra I

2 Februari 2012

Het tentamen bestaat uit 6 vraagstukkken. U krijgt 180 minuten om dezevraagstukken te beantwoorden. De puntenwaardering kunt u vinden aan hetbegin van de vraagstukken. Het totale aantal punten die u kunt bereiken is100. U krijgt 8 punten gratis. Each question is also translated into English.You may answer in Dutch or English.

1. (Nederlands) [4+4+4+4+4 Punten.]

Gegeven is de matrix

A =

1 2 02 3 11 1 β

met β ∈ R.

(a) Bestaan er waarden van β waarvoor het stelsel Ax = 0 strijdig is?

(b) Bepaal alle waarden van β waarvoor het stelsel Ax = 0 precies 1 oplossing heeft,en bepaal die oplossing.

(c) Bepaal alle waarden van β waarvoor het stelsel Ax = 0 oneindig veel oplossingenheeft, en bepaal de oplossingsverzameling.

Stel b is de vector gegeven door

b =

001

.

(d) Bepaal alle waarden van β waarvoor het stelsel Ax = b strijdig is.

(e) Bepaal alle waarden van β waarvoor het stelsel Ax = b consistent is, en bepaalde oplossingsverzameling.

1. (English) [4+4+4+4+4 Points.]

Consider the matrix

A =

1 2 02 3 11 1 β

with β ∈ R.

(a) Are there values for β for which the system Ax = 0 is inconsistent?

(b) Determine all values of β for which the system Ax = 0 has exactly 1 solution,and determine this solution.

(c) Determine the values of β for which the system Ax = 0 has infinitely manysolutions, and determine the solution set.

1

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Let b be the vector given by

b =

001

.

(d) Determine all values of β for which the system Ax = b is inconsistent.

(e) Determine all values of β for which the system Ax = b is consistent, anddetermine the solution set.

2. (Nederlands) [4+4+4+4 Punten.]

Stel A is een m × n matrix en b ∈ Rm. Beschouw het stelsel lineaire vergelijkin-gen Ax = b, met onbekende x ∈ Rn. Laat K de oplossingsverzameling zijn vandit stelsel. De nulruimte N(A) is de oplossingsverzameling van het bijbehorendehomogene stelsel Ax = 0. R(A) is de kolomruimte van A.

(a) Toon aan dat N(A) een lineaire deelruimte is van Rn.

(b) Toon aan: K 6= ∅ dan en slechts dan als b ∈ R(A).

(c) Laat v ∈ K. Toon aan dat geldt

K = {v + w | w ∈ N(A)}.

(d) Is K een lineaire deelruimte van Rn? Leg uit.

2. (English) [4+4+4+4 Points.]

Let A be a m × n matrix and b ∈ Rm. Consider the linear system Ax = b, withunknowns x ∈ Rn. Let K be the solution set of this system. The nullspace N(A) isthe solution set of the solution set of the corresponding homogeneous system Ax = 0.R(A) is the column space of A.

(a) Show that N(A) is a linear subspace of Rn.

(b) Show that: K 6= ∅ if and only if b ∈ R(A).

(c) Let v ∈ K. Show that

K = {v + w | w ∈ N(A)}.

(d) Is K a linear subspace of Rn? Explain.

3. (Nederlands) [4+4+4 Punten.]

Stel dat A en B n× n matrices zijn, en definieer een 2n× 2n matrix door

M =

(A I0 B

).

(a) Toon aan: als A en B niet-singulier zijn dan is M niet-singulier.

(b) Stel dat A en B niet-singulier zijn. Bepaal de inverse van M .

(c) Bekijk nu de matrixvergelijking(A I0 B

)(XY

)=

(AB

),

met als onbekenden de n × n matrices X en Y . Stel dat A en B niet-singulierzijn. Bepaal X en Y .

2

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3. (English) [4+4+4 Points.]

Let A and B be n× n matrices, and define a 2n× 2n matrix as

M =

(A I0 B

).

(a) Show that: if A and B are nonsingular then M is nonsingular.

(b) Suppose A and B are nonsingular. Determine the inverse of M .

(c) Consider the matrix equation(A I0 B

)(XY

)=

(AB

),

for the unknown n×n matrices X and Y . Suppose that A and B are nonsingular.Determine X and Y .

4. (Nederlands) [4+4+4+4+4 Punten.]

P3 is de vectorruimte van alle polynomen van graad kleiner dan 3, met reele coeffi-cienten. Definieer de afbeelding T : P3 → R3 door

T (p(x)) :=

p(1)p(2)p(3)

.

(a) Toon aan dat T een lineaire afbeelding is.

(b) Bepaal de matrix A van T ten opzichte van de geordende basis E := {1, x, x2}in P3 en de standaard basis F = {e1, e2, e3} in R3.

(c) Bepaal de rang van A.

(d) Wat volgt hieruit over de dimensie van de kern ker(T ) van T?

(e) Bepaal ker(T ).

4. (English) [4+4+4+4+4 Points.]

P3 is the vector space of all polynomials of degree less than 3 with real coefficients.Define the map T : P3 → R3 as

T (p(x)) :=

p(1)p(2)p(3)

.

(a) Show that T is a linear map.

(b) Determine the matrix A of T with respect to the ordered basis E := {1, x, x2}in P3 and the standard basis F = {e1, e2, e3} in R3.

(c) Determine the rank of A.

(d) What does this imply for the dimension of the kernel ker(T ) of T?

(e) Determine ker(T ).

3

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5. (Nederlands) [4+4+4 Punten.]

Laat x en y lineair onafhankelijke vectoren in Rn zijn, en definieer een deelruimte Svan Rn door S := span(x,y). Definieer een n× n matrix A door A := xyT + y xT .

(a) Toon aan dat A symmetrisch is.

(b) Bewijs dat N(A) = S⊥.

(c) Toon aan dat de rang van A gelijk is aan 2.

5. (English) [4+4+4 Points.]

Let x en y be linearly independent vectors in Rn, and define a subspace S of Rn asS := span(x,y). Define an n× n matrix A as A := xyT + y xT .

(a) Show that A is symmetric.

(b) Show that N(A) = S⊥.

(c) Show that the rank of A is equal to 2.

6. (Nederlands) [4+4+4+4 Punten.]

Gegeven is de matrix

A =

4 −5 11 0 −10 1 −1

.

(a) Bepaal het karakteristieke polynoom van A.

(b) Bepaal de eigenwaarden van A.

(c) Bepaal de eigenvectoren van A.

(d) Diagonaliseer A (d.w.z. geef een niet-singuliere matrix X en een diagonaalmatrixD zodat D = X−1AX).

6. (English) [4+4+4+4 Points.]

Consider the matrix

A =

4 −5 11 0 −10 1 −1

.

(a) Determine the characteristic polynomial of A.

(b) Determine the eigenvalues of A.

(c) Determine the eigenvectors of A.

(d) Diagonalize A (i.e. determine a nonsingular matrix X and a diagonal matrix Dsuch that D = X−1AX).

4

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Large group teaching observation #2 February 21st, 2006

Quantum Mechanics, about 22 students

Summary of lecturing techniques: Holger started off with a summary of the lecture from the previous Thursday. He gave this summary using the overhead, as he did in the previous lecture I observed. This worked very well as he briefly recapitulated the previous lecture, ending up at exactly the formula that he needed to start today’s lecture. After about 10 minutes of summary, he began to lecture on the board about the new material. He was discussing the transmission and reflection probabilities of a particle encountering a potential step. The lecture was clear and Holger explained what the mathematics meant in physical terms with diagrams of the particle approaching and being reflected from, or transmitted over the barrier. In the middle of the lecture, after having gone through the case when the particle’s energy is greater than the height of the potential step, Holger set out the problem for the case when the energy is less than the height of the potential step. He then asked the students to work out the form of the wavefunction themselves. This involved two steps, the first being to write the general form of the wavefunction, and the second being to find out what conditions were imposed on this wavefunction by matching up both it and its derivative at x=0, where the potential step lies. Holger gave the students about five minutes to work on this problem individually or in groups. He visited the groups of students and gave encouragement to those who were managing to get started on the problem. When a group said that they had no idea where to start, he didn’t have time to help them, but moved on to check on the next group. After a few minutes, he proceeded to describe the answer on the board. About 40 minutes into the lecture Holger switched to the overhead projector again to carry out a calculation that was more involved and would have taken a lot of time on the board. He handed out notes for the students so that they would not have to copy down everything on the transparencies. Unfortunately at the end of this he had run out of time to show the students a maple worksheet on transmission probabilities. The worksheet covered one of the problems set on the homework. However, he had emailed the worksheet to the students so they can try it out on their own. This was the last lecture that Holger was teaching (subsequent lectures would be taught by someone else) and so it was understandable that there was a certain amount of material that Holger needed to cover.

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Analysis of effectiveness of techniques used: Once again I think that the use of the overhead to run over what was done in the previous lecture was very effective. This is particularly true because this was the first lecture of the week and the students would probably have forgotten a lot over the weekend. Also, today’s lecture was a continuation of the same topic as the previous lecture, and so it was crucial that the students remember the methods used in the earlier lecture. I think this worked well and the students appeared to be paying attention. However, there wasn’t any chance for them to give any input or feedback allowing Holger to gage their understanding of the topic at this point in the lecture. He didn’t ask them any questions, just laid out the summary himself. This is not necessarily a bad thing, since there was a chance for the students to take part later on, and time pressures would not have allowed for any extensive discussion. Once Holger arrived at new material, he switched to lecturing on the board. The lecturing went at a good, steady pace and I think he was clear about what he wanted to get across. He worked in full detail through the case that the particle’s energy is greater than the height of the potential step. He then gave the students a few minutes to work out for themselves the case when the energy is less than the height of the potential step. This was a good choice of problem because in structure it followed exactly the case just presented in class. Everything the students needed to know had been covered in the summary or in the part of the lecture they had just listened to. Holger announced that they had three minutes to write down the general form of the wavefunction and the matching conditions at x=0. If there had been more time available, it might have been better to have given them longer, or not to have stated a time at all, as the 3 minutes might have seemed alarmingly short to the students! I watched the response of the students. I was sitting at the back, and several of the students around me clearly had no idea at all where to begin. Groups of students near the front seemed to be able to start working on the problem. Holger waited a minute or two before starting to walk round amongst the students. I think this was good as it allowed them to think about how to start the problem on their own and without him standing over them. However, it was good that after a few moments he started round the room because I could see that the students at the back had been prepared just to sit there and not think about the problem, but once they saw that Holger was going to approach them, they got out their notes and started looking back over them. They were still unable to start the problem, but were at least reviewing the relevant material. When Holger encountered groups of students who were able to make a start on the question he was encouraging and enthusiastic. Coming across a group who had no idea how to start, however, he didn’t try to help them get started. I think the best course of action would have been to help the students at least make the first step. Since they had

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admitted that they had no idea at all how to start, it’s unlikely that just leaving them for longer would lead to anything productive. I think that time pressure was a problem here as well, since in order to stop and help a student you have to know that you have a few minutes available to give a decent explanation, and if there are several other groups also similarly stuck, it must seem like an impossible task. Still, I think it would have been preferable if he had at least tried to start explaining the question. Otherwise it must look to the student as though they are not worth the effort, or as if he has already given up on them. If time had permitted, it would also have been better for Holger to have called on the students who did manage to make some progress to help him put the answer on the board. Unless the students had had time to work through the problem thoroughly it might be rather intimidating to ask one of them to go to the board, but I think it could have worked well if he had prompted them to give him the answer bit by bit as he wrote it on the board. This would have the advantage that it would keep the students focused on the problem, rather than turning off their own reasoning and just copying from the board, and it might have kept the progress of solving the problem nice and slow so that those students who hadn’t been able to do the question would have a better chance to understand. Since the question came in two parts, perhaps the perfect way to have approached it, if time had permitted, would have been to let the students have some time to think about the problem, then get them to tell him the form of the wavefunction, which he could write on the board for everyone to use, and then to give them more time to find the matching conditions. This second part wouldn’t be too hard once they all had the form of the wavefunction. That way even students who hadn’t been able to start the problem might have managed to do the second part and so would go away feeling as if they had understood and accomplished something. Using the overhead projector for the final example was appropriate, as it was a complicated calculation and would have taken an unwarranted amount of time for Holger to write it all on the board. It was very good that he handed out copies of the notes so that the students could concentrate on what he was saying and not worry about taking notes. This part of the lecture was a little rushed, but there were only 10 minutes left and this was important material that needed to be covered. Perhaps better time planning during earlier lectures would have left this last lecture a bit more relaxed, but given the point Holger was at in the material at the beginning of the lecture, he did a commendable job of finishing his part of the course. The explanations were still clear and it was certainly not so rushed that there was any chance that the students were unable to copy all the notes.

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Summary: Holger made a good attempt at getting the students more involved in the lecture. This was not completely successful, but this was in a large part due to lack of time. In more relaxed circumstances I would recommend that Holger give himself some time to help at least a few groups to get started on the problem he has set them to work on in class, and I would suggest that getting the students to help him to write up the answer to the exercise would also end up with them learning more. A question with two parts, such as the one he set them, would also lend itself well to him giving a solution of the first part and then letting the students work further to see if they could find the solution to the second part themselves once they’ve seen the solution to the first part. Extra thoughts: I feel very sympathetic to Holger’s struggle to get the students more involved in the lecture, and the opposing pressure of time. I know from past experience with my own teaching that there is a certain amount of material to get through and there simply isn’t time to exchange 10 or 15 minutes of lecturing during each lecture for something that is more interactive. The only solution I can think of to this problem (apart from teaching a slimmer syllabus) is to give the students some of the material on handouts or in some other form that they must work through on their own, outside of lecture time. I actually think that this could work well if the students were used to this approach. It hands a bit more of the learning over to the students, and that most likely means that they will learn the topic better than just blindly copying notes. However, in the current atmosphere of teaching and learning in the department, I am very much afraid that many students simply would not read the extra material and would therefore never learn it. The students are too used to having the material fed to them on the board in standard lecture format. It would certainly be good to change this expectation, but it may not be up to Holger to do it single-handed! He certainly made an attempt in the right direction and I think with the support of the whole department we could all learn to teach mathematics in a manner less passive for the students.

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# length of the intervala := 1;

# massm := 1;

# hbarhbar := 1;

a := 1m := 1hbar := 1

E := n -> n^2 * Pi^2 * hbar^2 / ( 2 * m * a ) ;

E := n/ 12 n2 p2 hbar2

m a

# nth eigenfunction of time independent SEpsi := (x,n) -> sqrt(2/a)*sin(n*Pi*x/a);

# nth eigenfunction of SEPSI := (x,t,n) -> exp(-I*E(n)*t/hbar)*sqrt(2/a)*sin(n*Pi*x/a);

y := x, n /2a sin

n p xa

PSI := x, t, n /eK

I E n thbar 2

a sinn p xa

psi0 := x -> 1/sqrt(a);

y0 := x/ 1a

c := n -> int( psi0(x)*psi(n,x,0) , x = 0..a );

c := n/0

ay0 x y n, x, 0 dx

psi_Fourier := ( x, k ) -> sum( c(n) * psi(x,n) , n = 1..k);

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PSI_Fourier := ( x, t, k ) -> sum( c(n) * PSI(x,t,n) , n =1..k );

psi_Fourier := x, k />n = 1

k

c n y x, n

PSI_Fourier := x, t, k />n = 1

k

c n PSI x, t, n

plot({psi0(x),psi_Fourier(x,100)},x=0..1);

x0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

# period is 4.0/Pi;

plot({psi0(x),Re(PSI_Fourier(x,0.1,1000))},x=0..1);

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x0.2 0.4 0.6 0.8 1

K0.2

0

0.2

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0.6

0.8

1.0

1.2

1.4

1.6

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with(plots):

# length of the intervala := 2;

# height of the barrierV := 3;

# mass of the particlesm := 1;

# Planck's constant/(2 pi)hbar := 1;

a := 2V := 3m := 1hbar := 1

V_is := x -> piecewise( x < 0, 0, x > 0 and x < a, V, x > a, 0);

V_is := x/piecewise x! 0, 0, 0 ! x and x! a, V, a! x, 0

plot( V_is(x) , x = -2 .. a + 2 , thickness = 3 );

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xK2 K1 0 1 2 3 4

1

2

3

# for 0 < x or x > a, and arbitrary energies E > 0

k0 := E -> sqrt(2*m*E)/hbar;

# for 0 < x < a, and energies E > V

k1 := E -> sqrt(2*m*(E-V))/hbar;

# for 0 < x < a, and energies 0 < E < V

k1_tilde := E -> sqrt(2*m*(V-E))/hbar;

# note that k1_tilde = i k1

k0 := E/2 m Ehbar

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k1 := E/2 m EKVhbar

k1_tilde := E/2 m VKEhbar

T := E -> piecewise( E<=V, # lecture (1 + ( k0(E)^2 + k1_tilde(E)^2 )^2 * sinh( k1_tilde(E) * a )^2 /( 4* k0(E)^2 * k1_tilde(E)^2))^(-1),

E>V, # exercise 8, problem sheet 3(1 + ( k0(E)^2 - k1(E)^2 )^2 * sin( k1(E) * a )^2 /( 4* k0(E)^2 * k1(E)^2 ))^(-1)

);

# note that the distinction between the cases E < V and E > V is not necessary because # sin(ix)^2 = -sinh(x)^2 and # k1_tilde^2 = - k1^2

T := E/piecewise E% V,1

1C14

k0 E 2 C k1_tilde E 2 2

sinh k1_tilde E a 2

k0 E 2 k1_tilde E 2

, V

! E,1

1C14

k0 E 2 K k1 E 2 2

sin k1 E a 2

k0 E 2 k1 E 2

plot({ T(E), 1 # reference line } , E = 0 .. 6*V , thickness = 3);

# save the plot in a variable plot1

plot1 := plot({ T(E), 1 # reference line} , E = 0 .. 6*V , thickness = 3 ):

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E0 2 4 6 8 10 12 14 16 18

0

0.2

0.4

0.6

0.8

1

E_res := n -> n^2 * Pi^2 * hbar^2 / ( 2 * m * a^2 ) + V ; # n = 1,2,3,...

E_res := n/12

n2 p

2 hbar2

m a2 CV

plot({ [E_res(1),t,t=0..1], [E_res(2),t,t=0..1], [E_res(3),t,t=0..1]});

# save the plot in a variable plot2

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plot2 := plot({ [E_res(1),t,t=0..1], [E_res(2),t,t=0..1], [E_res(3),t,t=0..1]}):

5 6 7 8 9 10 11 12 13 140

0.2

0.4

0.6

0.8

1

display(plot1,plot2);

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E0 2 4 6 8 10 12 14 16 18

0

0.2

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1

# normalise incoming flux of particles to 1A0 := 1;

# energy just below barrier V E_is := 0.95*V;

# or energy just above barrier V# E_is := 1.2*V;

# or energy equal to a resonance# E_is := E_res(1);

# solve matching conditions at x = a and x = 0sols := solve({ # continuity of psi at x = 0

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A0+B0 = A1+B1, # continuity of psi' at x = 0I*k0(E_is)*(A0-B0) = -k1_tilde(E_is)*(A1-B1), # continuity of psi at x = a A1*exp(-k1_tilde(E_is)*a)+B1*exp(k1_tilde(E_is)*a)=A2*exp(I*k0(E_is)*a), # continuity of psi' at x = a -k1_tilde(E_is)*(A1*exp(-k1_tilde(E_is)*a)-B1*exp(k1_tilde(E_is)*a))=I*k0(E_is)*A2*exp(I*k0(E_is)*a) },{ B0 , A1 , B1 , A2 });

A0 := 1E_is := 2.85

sols := A1 = 1.979788654K 0.6549887792 I, A2 = K0.2560402168C 0.1780236907 I, B0= 0.8124758756K 0.4925788510 I, B1 = K0.1673127787C 0.1624099283 I

psi := x -> subs(sols,piecewise( x<0, A0 * exp( I * k0(E_is) * x ) + B0 * exp( -I*k0 (E_is) * x ), x>0 and x<a, A1 * exp( -k1_tilde(E_is) * x ) + B1 * exp( k1_tilde(E_is) * x ), x>a, A2 * exp( I * k0(E_is) * x )));

y := x/subs sols, piecewise x! 0, A0 eI k0 E_is xCB0 eKI k0 E_is x, 0 ! x and x! a,

A1 eKk1_tilde E_is xCB1 ek1_tilde E_is x, a! x, A2 eI k0 E_is x

plot({ Re(psi(x)), V_is(x)}, x = -5 .. a+5 , thickness = 3);

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xK4 K2 0 2 4 6

K1

1

2

3

plot({ abs(psi(x))^2, V_is(x) } , x = -5 .. a+5 , thickness = 3);

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Evaluatie Integrerend Project Dynamische Systemen, 4e periode 2008-2009 docent: H. Waalkens Verplicht vak 2e jaar bachelor Wiskunde en Technische Wiskunde Dit blad geeft een korte samenvatting van het volledige evaluatierapport. Bij de verwerking van de scores zijn hiervoor telkens de twee uiterste antwoordpercentages bij elkaar opgeteld; dus bv % mee eens en % helemaal mee eens worden samengenomen, evenals % oneens en % helemaal mee oneens. Wanneer meer dan 30% van de studenten een negatief oordeel heeft over een onderwerp, dan wordt dit als een overschrijding beschouwd. Aantal respondenten: 20 Aantal overschrijdingen: 4 Studierichting: Wiskunde 95% Overig 5% Hoorcollege: 1 overschrijding. - De uitleg van de docent was duidelijk (94%). - Het tempo tijdens de hoorcolleges was goed (90%). - De opbouw van de colleges was logisch (95%). - De docent gaf duidelijk aan hoe de collegestof in de projecten moest worden

toegepast (71%). - Het aantal colleges was niet voldoende om de projecten te kunnen uitvoeren (40%). - Mijn voorkennis om dit vak te kunnen volgen, was voldoende (95%). Leermiddelen: 1 overschrijding. - Het boek was geschikt voor zelfstudie (83%). - De kwaliteit van het boek was goed (95%). - De practicumhandleiding was niet duidelijk (33%). Projecten: 0 overschrijdingen. - De formulering van de opdracht(en) was duidelijk 79%). - De begeleiders waren deskundig (100%). - De omvang van de opdracht(en) was goed (89%). - Mijn kennis van en vaardigheden met de computer voor het maken van de

opdracht(en) was voldoende (94%). - Het niveau van de opdracht(en) was goed (79%). Communicatieve vaardigheden: 0 overschrijdingen. - De onderlinge taakverdeling bij de groepsopdrachten verliep goed (85%). - Ieder groepslid leverde een evenredige bijdrage (85%). Schriftelijke uitdrukkingsvaardigheid - Het commentaar van de docent op mijn schrijfvaardigheid was nuttig (100%).

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Mondelinge uitdrukkingsvaardigheid - Het commentaar van de docent op mijn mondelinge uitdrukkingsvaardigheid was

nuttig (89%). Toetsen/ Beoordeling: 0 overschrijdingen. - De beoordelingscriteria voor de onderdelen die meetellen voor het eindcijfer, waren

duidelijk (89%). - Het niveau van de eindopdracht was goed (93%). Tentamen: 1 overschrijding. - Het niveau van het tentamen was goed (84%). - Het was niet voldoende duidelijk wat je voor het tentamen moest kennen (45%). - Het tentamen was een goede afspiegeling van de leerstof (84%). - 89% van de studenten vond het tetamen redelijk te doen. Algemeen: 1 overschrijding. - De informatievoorziening over dit vak was voldoende (90%). - De andere studieverplichtingen vormden een belemmering om voldoende aandacht

aan dit vak te kunnen besteden (31%). - Over het algemeen heb ik dit vak met plezier gevolgd (85%). Tijdsbesteding: - 68% van de studenten heeft 67%-100% van de hoorcolleges bijgewoond. - 65% van de studenten heeft 30-60 uur aan het project besteed. - 55% van de studenten heeft 1-3 uur per week aan zelfstudie besteed voor dit vak.

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