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Amsterdam, 8 December 1993

Professor David L. Haury

JSTE

1929 Kenny Road

Columbus, OH 43210 - 1080U. S. A.

Dear Professor Haury:

Subject: Manuscript #920508 144

Thank you very much for your letter of November 18, 1993. Enclosed you will find the revised

version of my paper in hard as well as in soft copy (file BERG.WP5 in WordPerfect 5.1).

Hopefully, I have addressed the concerns of the reviewers to your satisfaction. The originalpaper was written in Indonesia. Since returning to the Netherlands, I have read a number of

papers about the use of alternative conceptions in teacher education, including some examples

from developing countries. I have incorporated these in the revised paper and supplied a better

rational and some more background information.

I dropped the example of the cognitive conflict demonstration and refer instead to the description

in White and Gunstone's Probing Understanding. I also dropped the abstract following your

suggestion.

Sincerely yours,

Ed van den Berg

Centre for Development Cooperation and Services (CDCS)

Free University

De Boelelaan 1115

1081 HV Amsterdam

Netherlands

E-mail: EDBERG@NAT.VU.NLfax: 31 (country code) 20-646 2320

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AN EXAMPLE OF USING ALTERNATIVE CONCEPTIONS IN PHYSICS

TEACHER EDUCATION IN A DEVELOPING COUNTRY

Euwe (Ed) van den Berg1

Centre for Development Cooperation and Services

Free University, De Boelelaan 1115

1081 HV, Amsterdam

Netherlands

1The article is based on the author's experience (1981-1991) in physics teacher education at Satya

Wacana Christian University, Salatiga, Indonesia.

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AN EXAMPLE OF USING ALTERNATIVE CONCEPTIONS IN PHYSICS

TEACHER EDUCATION IN A DEVELOPING COUNTRY

I. Pre-service teacher education

Introduction and context

In the past many science teaching methods courses were quite general. Students

studied various theories of learning, educational goals, instructional objectives,

teaching methods and evaluation. Topics specific to science education might have

included inquiry methods, however, approaches very specific to teaching science

content were few. The study of alternative conceptions (see Table 1 for definitions)

provides ways of making science teaching methods courses more content focussed

and thus more obviously relevant to pre-service students and in-service teachers. At

the same time, such courses provide an opportunity to discuss and perhaps remediate

problems of prospective teachers with high school level science. This paperdescribes a course on alternative conceptions in physics which has been used with

pre-service students as well as with in-service teachers.

In colleges for teacher education in Indonesia, the number of physics pre-service

students is usually sufficient to design special courses. In fact, teacher education

students declare their major upon first enrollment and collectively follow a program

designed for their major field of specialization. The program is designed at a national

level and implemented nation-wide. Unfortunately, almost all the nationally

prescribed science courses concern college level science and there is little opportunity

to remediate (secondary) school science and gain a deeper understanding of it.

McDermott (1990) clearly demonstrated the inadequacy of traditional college physics

courses by themselves to reach this deeper understanding of school science.

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However, many teacher education programs in Indonesia and elsewhere do not yet

offer the opportunity of special science courses for teachers of the type suggested by

McDermott. A solution might then be to create a special methods course on

alternative conceptions in school science and their remediation. In countries or

institutions where numbers of pre-service students in any discipline are too small for a

special course, the course could be broadened to include concepts from different

disciplines. What should be the purpose of such a course?

Many alternative conception researchers plead for methods of teaching which are

based on constructivist theories of learning such as the generative learning model

(Osborne & Wittrock, 1985). Typical constructivist strategies include concept

mapping, cognitive conflict, use of analogies, individual or small group laboratory

work, much discussion among peers in small groups, and much interaction with the

teacher. Some of these strategies have been well described in the bookProbing

Understandingby White and Gunstone (1992). More important yet is thought to be

meta-learning, e.g. that students become more aware of their own learning and learn

to consciously use learning strategies. Obviously meta-learning is very important for

pre-service students and it could be argued that the main goal of pre-service programs

is to change the pre-service students' conceptions of learning and teaching (Gunstone,

1993; Fensham & Northfield, 1993). Fensham and Gunstone (1993) emphasized

meta-learning in a 10 month residential program for prospective Filipino in-service

instructors and reported successes after much struggle to "convert" students to

meta-learning during the first three months of the program. Whether the Filipino

instructors were able to productively apply meta-learning in their own in-service

programs is not yet known.

The role envisaged for the teacher in constructivist classrooms differs greatly from

the conventional role of the teacher, and certainly from teacher behavior in classrooms

in Indonesia and other developing countries. Large classes (45 students/class), small

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classrooms, low salaries frequently necessitating second jobs, limited laboratory

facilities and equipment, limited availability of textbooks, and a crowded exam

syllabus make more individualized teaching next to impossible for the "average"

teacher under "average" conditions. Therefore our teacher education program at

Satya Wacana Christian University in Indonesia opted for a rather pragmatic

approach. On the one hand the program emphasized competency in conventional

whole class (rather than individualized) teaching methods with much emphasis on

activating students through interaction (questioning), seatwork, short demonstrations

using commonly available objects and materials (Liem, 1987, 1991), and linking

science to everyday-life experience. In a poor teaching environment, pre-service

students should at least become competent conventional teachers. On the other hand

pre-service students were also exposed to other methods such as small group work

and individual research projects in science and science education courses.

Pre-service students also tried out some non-conventional methods in teaching

assignments with small groups of students in nearby schools (in addition to student

teaching whole classes). In a better teaching environment pre-service students

should become teachers who are able to use a variety of creative methods.

Course purpose

Considering the pragmatic approach and its rational explained above, it was decided

to limit the course to the study of alternative conceptions and remediation strategies

rather than ambitiously try to go to the meta-cognitive level. In short: the purpose of

the course (17 weeks x 4 hours/week) is to acquaint pre-service physics teachers with

the existence of alternative conceptions, examples, origin of alternative conceptions,

cognitive psychology, and diagnosis and potential remediation techniques. At the

same time the course should offer the opportunity to make (third year) pre-service

students aware of their own conceptions with high school physics subject matter and

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should attempt to remediate these alternative conceptions.

Program

Initially (in 1986), the course was sequenced according to education aspects of

alternative conceptions: it would start with diagnosis of alternative conceptions

through written tests, interviews, and inspection of student's home work. Then, we

would discuss some topics in cognitive psychology but not in-depth. The second half

of the course would be spent on various remediation schemes/techniques, such as

cognitive conflict, anchor-bridges analogies, and the four phase approach of Osborne

and Freyberg (1985). Physics examples would be extensively used and discussed,

however, the physics topic used, might change every meeting: now mechanics, then

electricity, and then back again to mechanics.

Since 1989 the course is sequenced according to physics topics and the instructor

sticks to a certain physics topic for a number of meetings. Within each topic (say

forces) there is discussion of diagnosis and remediation of frequently encountered

alternative conceptions.

The current instructional objectives and order of topics as developed over four

versions of the course are described in tables 2 and 3.

---------------------------------------------------------------------------------------------

INSERT TABLES 1, 2 AND 3 ABOUT HERE

---------------------------------------------------------------------------------------------

Course assignments

Assignments consist of readings (in the Indonesian language) on physics and

educational aspects of alternative conceptions (Berg, 1991), diagnostic interviews

with junior and senior high school students, a research project (usually focussing on

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diagnosis), solving conceptually oriented physics problems selected from diagnostic

tests and textbooks such as Hewitt (1985) and Halliday & Resnick (1978), preparing

and executing (micro-teaching) physics demonstrations for achieving cognitive

conflict, analyzing and revising poor laboratory worksheets intended for teaching

concepts, and correcting and analyzing essay answers of high school students to

conceptual questions.

Cognitive conflict demonstrations

Many methods for remediation of alternative conceptions contain a conflict phase

(Nussbaum & Novick, 1982; Osborne & Freyberg, 1985). Research has shown that

cognitive conflict is not sufficient for conceptual change but has to be part of a wider

strategy which might combine various teaching methods. However, cognitive

conflict remains an important method for triggering student awareness of their

conceptions and subsequent classroom discussions. Two of the main methods in

creating cognitive conflict are student laboratory work and teacher-led

demonstrations. With both methods, students make predictions (based on their

conceptions) which are then tested in a laboratory or in demonstration experiments.

Demonstrations have some advantages over student laboratory work in achieving

cognitive conflict (Dekkers & Thijs, 1993). In demonstrations, it is easier to get the

students focussed on the relevant variables and keep them from being side-tracked by

the irrelevant ones. Furthermore, teacher control over the student activities is greater

and use of time is more efficient, and last but not least, teacher preparation time for

demonstrations is less than for practicals. However, one has to make sure students

are involved and that is done by inserting individual activity into the class

demonstration, for example, by having students write down their own predictions and

explanations before the actual demonstration takes place. The results of such

individual tasks are then used in discussion, with the advantage that far more students

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participate in the discussion than without the individual component. In the latter

case, the most vocal "volunteers" would dominate. The large class sizes (45 students

per class), curriculum/exam pressure and limited facilities in developing countries,

constitute another reason to (often) prefer demonstration over student laboratory.

In the two methods courses prior to this alternative conception course, much

emphasis is put on the use of demonstrations and teaching aids to show and explain

physics phenomena and link them to students' everyday lives. These courses include

some classroom practise in schools. In the alternative conception course a specific

kind of demonstration is exercised: the use of a demonstration to contrast student

predictions (based on their conceptions) with experimental results and to provide

students excercises in thinking back-and-forth between observations and implicit

beliefs or theories (Berg, 1988). This kind of demonstration is now often called

"Predict-Observe-Explain (POE)" demonstration. An excellent description can be

found in a recent book by White and Gunstone (1992).

Many of the pre-service students in our program do quite well in these

demonstrations as they have had demonstration training in their previous two methods

courses. However, in-service teachers clearly need training. In spite of warnings not

to announce the experimental results beforehand, many in-service teachers start with:

Today we will prove that..... That way, no cognitive conflict will occur and students

will be bored.

Demonstrations fit for this approach are those where students are quite confident

that they can predict the outcomes (even though their predictions will be wrong).

Examples can be found in many items of common alternative conception tests. Good

sources are also Liem (1987, 1991) and de Vries (1958, 1960). Obviously, a teacher

should not exclusively use POE's but also use other, non-counterintuitive, types of

demonstrations.

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Evaluation

Pre-service students are evaluated on 1) mastery of physics concepts discussed in

the course, 2) knowledge about frequently encountered alternative conceptions, 3)

the report of the individual research project, and 4) other assignments such as

performance in micro-teaching demonstrations. Mastery of concepts and knowledge

about common alternative conceptions constitute about 70% of the grade.

Results

In the course, participants discover many alternative conceptions in themselves. It

is disappointing to see that even pre-service students with good results in basic

physics courses sometimes still regress and exhibit some major misconceptions. On

the other hand, after encountering their own misconceptions (and feeling

embarrassed), many course participants are highly motivated to correct their

conceptions. Rather exciting discussions occur from time to time, between students

as much as between students and the instructor.

Most pre-service students make quite a bit of progress in mastery of high school

level concepts as evidenced by the final course exam (compared to pretests for various

physics topics). However, we do not know yet how long these effects last. The least

one hopes to achieve is that pre-service students will be more aware of their own

alternative conceptions so that warning lights start flashing in their heads when they

start to teach particularly sensitive topics.

Our best students (top 20%) conduct undergraduate thesis research and so far, 5

students have done their research on diagnosis and description of alternative

conceptions, resulting in 4 national level paper presentations and 7 publications.

These thesis students made great conceptual gains as evidenced from the diagnostic

tests and interview protocols they constructed and interpretation of data.

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The design and execution of the course on alternative conceptions made us take a

second look at some first year science courses. For example, a course on electricity

was completely overhauled, leading to welcome improvements of student

achievement (Berg et al., 1992).

II. In-service teacher education

Poor science subject matter mastery of teachers and lecturers is a problem in many

developing countries including Indonesia. In most cases, this poor subject matter

mastery has been caused by the poor quality of secondary and higher education and

NOT necessarily by lack of intellectual ability. One major cause is the lack of

monitoring of individual performance and proper feedback in the educational system

K - 12 plus college.

In-service programs where school subject matter is simply being repeated, are not

popular. Teachers feel they have already studied the subject matter and they are also

teaching it. It is felt as humiliating when they have to restudy school science.

Furthermore, if subject matter is simply taught again in the same way in-service

participants have studied the material before, it is unlikely that teachers will discover

their weaknesses and misconceptions.

On the other hand, the alternative conception research shows where student (and

teacher) problems are. So knowledgeable instructors of in-service training can go

straight to the potential weaknesses rather than waste time on repeating subject matter

already mastered by participants. Confrontation with their own alternative

conceptions in rather simple problems, can (if tactfully handled) lead to high

motivation of teachers in restudying high school level physics during an in-service

training. At the same time, teachers experience that physics is more than filling in

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formulas and juggling some mathematics. We have applied this approach in a

number of short in-service courses in Indonesia and managed to generate high

motivation. We have not had opportunities for evaluation of long-term conceptual

progress. The main benefit of a short in-service course might be increased sensitivity

of participants to their own alternative conceptions and that in itself might be a source

for a gradual long term improvement of concept mastery.

Short 3-4 day workshops we have run, are as follows: Typically we take one or two

physics subjects such as Heat or Electric Circuits. The choice of topics is usually

determined by the participants' teaching schedule right after the in-service.

Fortunately, that schedule is the same all over Indonesia. At the start of the

workshop teachers take an anonymous diagnostic test concerning these topics and

then they work through remedial laboratory exercises based on the items of the

diagnostic test and participate in subsequent classroom discussion of physics

concepts. Then there is a discussion of conceptual change and remediation strategies

but always within the context of the physics topics chosen. The remainder of the

workshop is spent on remedial physics exercises to improve subject matter mastery

and on micro-teaching in which teachers exercise remediation strategies and

techniques such as conceptual conflict demonstrations. Although the workshops use

individual or small group laboratory work, emphasis in remediation strategies is on

whole class teaching methods as class size (45 students/class), facilities, and teacher

background make the use of more individualized methods unlikely. Remedial

materials and readings are taken from the pre-service course. In-service teachers

usually are quite worried about the microteaching. Senior high school teachers are

more nervous about it than elementary school teachers. However, afterwards many

of them consider microteaching a useful experience.

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III. In-service education of college and university faculty

At the university level the alternative conception approach to improvement of basic

physics knowledge of lecturers could work well also. We once tried this in a national

workshop for lecturers of Physics and Physics Teacher Education Departments in

universities and colleges. Participants kept good motivation in exercises of high

school level concepts. Some have started research on alternative conceptions and

have made conference presentations already.

Lucas et al. (1992, 1993a, 1993b) have provided in-service for teacher educators in

Papua New Guinea (4 week course) and for teachers and in-service instructors in Fiji.

They report similar problems with subject matter mastery of participants. Lucas and

his colleagues also used alternative conceptions as a focus for the in-service training,

amongst others, working with concept maps and they report a high interest on the part

of participants and a self-sustained follow-up among part of them (such as continued

use of concept maps).

A major component of the work of expatriate science lecturers in developing

countries concerns course development and training of junior lecturers, not through

workshops, but through team teaching and other forms of collaboration. Knowledge

of alternative conceptions could help to identify the bottle necks in science courses

(often wrongly assumed to be only due to mathematics) and in junior lecturer's subject

matter mastery. Such bottle necks could be restructured with a different teaching

approach and home work assignments. Work on such matters with junior lecturers

will be slow, but potentially very rewarding. The expatriate scientist should realize

that science alternative conceptions among junior lecturers are also encountered in

industrial countries.

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References

Beeby, C.E. (1979). Assessment of Indonesian Education: A Guide to Planning.Wellington: New Zealand Council for Educational Research with Oxford University

Press.

Berg, E. van den (1988). The demonstration: A neglected teaching method.Journal of Science and Mathematics Education in S.E. Asia, XI(1), 19-25.

Berg, E. van den (1991). Miskonsepsi Fisika dan Remediasi (Misconceptions andremediation). Salatiga (Indonesia): Satya Wacana Christian University.

Berg, E. van den, Darjito, A., van den Berg, R. (1992). Misconceptions on electriccurrent and potential: Assessment and remediation. Journal of Science and

Mathematics Education in S.E. Asia, XV(1), 68-80.

Berg, E. van den, Lunetta, V.N. (1984). Science teacher diploma programs inIndonesia. Science Education, 68(2), 195-203.

Dekkers, P., Thijs, G.D. (1993). Effectiveness of practical work in the remediation ofalternative conceptions in mechanics with students in Botswana. Paper presented atthe Third International Seminar on Misconceptions and Educational Strategies inScience and Mathematics, August 1-4, 1993 (to be published in electronics proceedingsof the seminar).

Dupin, J.J., Johsua, S. (1989). Analogies and "modeling analogies" in teaching: Some

examples in basic electricity. Science Education, 73(2), 207-224.

Fensham, P.J., Gunstone, R.F. (1993). Cross-country cooperation in curriculum

change and professional development. Paper presented at the International Conferenceon Science Education in Developing Countries: From Theory to Practice. Jerusalem,

3-7 January 1993.

Fensham, P.J., Northfield, J.R. (1993). Pre-service science teacher education: Anobvious but difficult area for research. Studies in Science Education, 22, 67-84.

Gunstone, R.F. (1993). Can we prepare teachers to teach in the way that studentslearn? Key-note lecture, International Conference on Science Education inDeveloping Countries: From Theory to Practice. Jerusalem, 3-7 January 1993.

Halliday, D., Resnick, R. (1978). Physics (third edition). New York: John Wiley &Sons.

Hewitt, P. (1985). Conceptual Physics (fifth edition). Boston: Little, Brown andCompany.

Katu, N., Lunetta, V.N., van den Berg, E. (1992). The development of conceptions inbasic electricity: An application of "teaching experiment" methodology. Paperpresented at the Annual Meeting of the National Association for Research in Science

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Teaching, Cambridge, Massachusetts, March 21 - 25, 1992.

Liem, T.L. (1987). Invitations to Science Inquiry (2nd Edition). ISBN1-878106-21-X. Published by Science Inquiry Enterprises, 14358 Village View Lane,Chino Hills, California 91709, U.S.A. (Also included in the NSTA catalog).

Liem, T.L. (1991). Invitations to Science Inquiry, Supplement to First & SecondEdition. ISBN 1-878106-01-5. Publisher: see above.

Lucas, K.B., Swinson, K.V., Tulip, D. (1992). Effective professional development forteacher educators in Papua New Guinea. Paper presented at the International SeminarState-of-the-Art of Research in Science and Mathematics Education in Southeast Asiaand the Pacific, RECSAM, Penang, Malaysia, 17-19 February 1992.

Lucas, K.B., Tulip, D.F., Swinson, K.V. (1993a). Professional enrichment in thedisciplines of science and mathematics for teacher educators in Papua New Guinea.

Paper presented at the International Conference on Science Education in DevelopingCountries: From Theory to Practice. Jerusalem, 3-7 January 1993.

Lucas, K.B., Cook, A. (1993b). Transforming secondary science teaching practice inFiji: A collaborative staff development project. Paper presented at the InternationalConference on Science Education in Developing Countries: From Theory to Practice.Jerusalem, 3-7 January 1993.

McDermott, L. C. (1990). A perspective on teacher preparation in physics and othersciences: The need for special science courses for teachers. American Journal ofPhysics, 58(8), 734-742.

Nussbaum, J., Novick, S. (1982). Alternative frameworks, conceptual conflict andaccommodation: Toward a principled teaching strategy. Instructional Science, 11,

183-200.

Osborne, R., Freyberg, P. (1985). Learning in Science: The implications of children'sscience. Auckland: Heinemann.

Osborne, R., Wittrock, M. (1985). The generative learning model and itsimplications for science education. Studies in Science Education, 12, 59-87.

de Vries, L. (1958, 1960). The Book of Experiments, The Second Book of Experiments,The Third Book of Experiments. London: Murray.

Walberg, H. J. (1991). Improving school science in advanced and developing

countries. Review of Educational Research, 61(1), 25-69.

White, R., Gunstone, R. (1992). Probing Understanding. London: The Falmer Press.

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Table 1: Definitions

A conceptis the concept as currently intended by the community of scientists (the currently

dominant view of the concept) and described in good textbooks. Please note the importance ofrelationships between concepts. It is through its network of relationships with other concepts that aconcept acquires its meaning. Such relationships can be pictured in a concept map.

Conception refers to an individual's interpretation of the meaning of a concept. Such aninterpretation would usually have some idiosyncratic features, even if the individual is a scientist. Forexample, conceptions of scientists with regard to major concepts such as atom orforce might still be

slightly different. Conceptions could be pictured in conception maps, analogous to concept maps.A misconception oralternative conception is a conception which in some aspects is contradictory

to or inconsistent with the concept as currently intended . Such inconsistency often shows in one ormore relations of the conception with other conceptions, thus involves usually more than one concept.

Apreconception is a conception which has been formed without exposure to formal instruction(in school) about the concept. However, often the term preconceptions is used for the conceptions atthe beginning of a course, then preconceptions are free of influence of that course, but may have beeninfluenced by previous courses.

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Table 2: Instructional objectives

Students are able to:1. define and distinguish terms like concept, conception, misconception, alternative conception,

preconception, alternative framework, assimilation, accommodation.

2. make concept maps of the major concepts in the junior and senior high school physics syllabi, theseconcept maps concern relationships between concepts and between concepts and real worldexamples.

3. explain potential causes for alternative conceptions (constructivism).4. give examples of alternative conceptions in mechanics, electricity, heat, optics, modern physics, and

selected mathematics concepts and contrast the alternative conceptions with current scientificconcepts and their interrelations. Concepts included are amongst others Mechanics:position-velocity-acceleration, force-momentum-impulse, independence of horizontal andvertical motion of frictionless projectiles, Newtons laws and student conceptions of them, forceson objects at rest, work-energy; Electricity: current, voltage, energy, power, brightness of lamps,series and parallel circuits, various arrangements of voltage sources, examples of alternativeconceptions such as local/sequential reasoning, attenuation model, voltage source as constantcurrent source, etc.; Heat: heat-energy-temperature, intensive and extensive variables, flow ofheat energy, heat capacity versus specific heat; Optics: seeing ("radar" idea, eidola), propagationand speed of light, colors, reflection; Modern Physics: continuous versus atomic view of matter,student conceptions of atoms and bonding; Mathematics: length-area-volume, proportionalityand its application, differentiation and integration.

5. write items for a diagnostic test.6. explain the generative learning model of Wittrock and Osborne.7. explain briefly the basics of constructivism.

8. analyze and improve instructions for demonstration and laboratory exercises for concept learning.9. write demonstration and laboratory instructions for teaching concepts and remediating alternativeconceptions.

10.identify alternative conceptions through interviewing and diagnostic testing.

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Table 3: Contents of course on alternative conceptions in Physics

1. Several examples of alternative conceptions in physics.2. Assimilation and Accommodation in conceptual development.3. Concepts, Conceptions, Alternative conceptions, Concepts and Conception maps.

4. Alternative conceptions in Mechanics starting with forces on objects at rest. Activities includeworking through conceptual problems taken from the alternative conception literature and fromsources such as Hewitt (1985) and Halliday & Resnick (1978). Answers to part of these

problems are then tested in laboratory experiments or sometimes demonstrations, often leadingto conceptual conflict and extensive small group or class discussions.

5. Assessment of conceptions through interviews and diagnostic tests, students watch a video ofclinical interviews and then prepare and carry out an interview protocol to assess mechanics orelectricity conceptions.

6. Alternative conceptions in Electricity (mainly electric circuits). Activities start with a diagnostictest, followed by laboratory experiments and conceptual exercises.

7. Origin of alternative conceptions: constructivism, generative learning model as an example of aconstructivist theory of learning, consequences for classroom learning.

8. Remediation techniques: changing the traditional teaching sequence of concepts, conceptualconflict, use of analogies, meta-learning.

9. Alternative conceptions regarding Heat, Temperature, and related concepts. Demonstrationexperiments that could be conducted when teaching about heat and temperature.

10. Student project: Construction of diagnostic tests and/or interview protocols by individual class

participants. In the case of tests, items are screened, revised and pooled by topic. Items of theinstructor's test bank are added. Students then conduct an individual research project using oneof the tests and analyze their data using qualitative and quantitative item analysis for multiple

choice and essay items. In the case of interviews, the students submit a sample tape in additionto a report.11. Microteaching to exercise cognitive conflict demonstrations (three sessions).12. Remediation models such as Osborne and Freyberg's (1985) and others.13. Alternative conceptions in Optics.14. Potential of laboratory and demonstrations in remediation of alternative conceptions.15. Criticizing and revising worksheets for teaching concepts in the laboratory.16. Writing conceptual exercises and worksheets for conceptually oriented laboratory sessions.17. Alternative conceptions in Modern Physics.18. Alternative conceptions in other branches of Physics (such as fluids etc.).19. Examples of alternative conceptions in Mathematics which are relevant for Physics.