Yess 4 Günter Törner

Theory versus Practice Revisiting an outstanding research problem in mathematics education The case of teacher professional development - from the view of a mathematician - Prof. Dr. Günter Törner, Universität Duisburg-Essen

0. Contents

Introduction

Searching for the relevant literature

What are the main characteristics of the theory-practice-dichotomy?

Excursion: The theory-practice-dichotomy in mathematics

A paradigm change: The practice-theory-challenge

Professional development

The project: Mathematics done differently

Looking back

Conclusion: What I would like to remark...

Introduction Subjective view and profile

Research mathematician

Discrete mathematics, coding theory, 2D codes etc.

Noncommutative algebra (MSC 2000: 16W60, 16W99)

Introduction Mathematics Subject Classification (MSC 2000)

Research mathematician

Doing applied research (small projects with firms - e.g. steel factory ThyssenKrupp, DST Cargo loading etc.)

Mathematics is communicating, convincing, marketing

Mathematics is modeling

See Burton, L. (2004). Mathematicians as enquirers: learning about learning mathematics. Dordrecht: Kluwer Academic

See Byers, W. (2007). How Mathematicians Think . Princeton: Princeton University Press .

Mathematics Educator

Doing research in didactics (various domains and subject matters):

didactics of calculus, didactics of linear algebra, didactics of stochastics,

problem solving,

beliefs, attitudes, emotions,

professional development of teachers

... And further: Secretary in the German math society (DMV); a highly demanding and interesting job; my personal objective: ‘reconciling’(?) mathematicians and educators...

Involved in the German Mathematical year 2008 (5 million € paid by government and Deutsche Telekom Foundation)

Cooperation with the German Bureau of Statistics (Wiesbaden) to reveal figures and data around mathematics at university, extremely high drop-out rate, interesting gender aspects

Introduction The topic of the talk and its structure

Explicitly or implicitly we all have worked under such a headline.

Theory versus practice, bridging theory and practice, a convincing dichotomy, really?

Will the coherence between theory and practice do our job?

Hersh, Reuben. (1991). Mathematics has a front and a back. New directions in the philosophy of mathematics. Synthese 88 (2), 127 - 133.

Invitation to all: Please study papers of Hersh!

See also (recently printed): Byers, W. (2007). How Mathematicians Think - The Role of Ambiguity in Mathematics . Princeton: Princeton University Press.

2. Searching for the relevant literature: Theory-practice-dichotomy (TPD) 1

What should be taken in account?

National Council of Teachers of Mathematics (NCTM) (1953). The learning of mathematics. Its theory and practice. 21st yearbook 1953. Natonal Council of Teachers of Mathematics: Reston, VA.

IDM (1976). Relating Theory to Practice in Educational Research. Materialien und Studien, No. 6. Bielefeld: Institut für Didaktik der Mathematik.

Davis, Robert B. (1987). Theory and Practice. Journal of Mathematical Behavior 6 (1), 97-126.

Brown, S.J. & Cooney, T.J. (1991). Stalking the dualism between theory and practice. International Reviews Mathematical Education (ZDM) 23 (4), 112 - 117.

Cooney, T. (1994). Teacher education as a crucible for systematic cooperation between theory and practice. In Bazzine, L. (Ed), Theory and practice in mathematics education. Proceedings of the 5. International conference on systematic cooperation between theory and practice in mathematics education, Grado (Italy) 23 -27 May 1994, (pp. 67 - 79). Pavia: ISDAF.

Lerman, S. (1994). Towards a unified space of theory-and-practice in mathematics teaching: a research perspective. In L. Bazzini (Ed.), Proceedings of the Fifth International Conference on Systematic Co-operation between Theory and Practice in Mathematics Education (pp. 133-142). Pavia: Universita degli Studi di Pavia.

Steiner, H.G. (1987). Philosophical and epistemological aspects of mathematics and their interaction with theory and practice in mathematics education. For the Learning of Mathematics 7 (1), 7 - 13.

Steinbring, H. (1994). Dialogue between theory and practice in mathematics education. In Biehler, R.; Scholz, R.W., Straesser, R. & Winkelmann, B. (Eds.), Didactics of mathematics as a scientific discipline (pp. 89 - 102). Dordrecht: Kluwer Academic Publishers.

Malara, N.; Zan, R. (2002). The problematic relationship between theory and practice. In English, L. et al. Handbook of international research in mathematics education (pp. 553-580). Lawrence Erlbaum Associates: Mahwah, NJ.

Zilliox, J. (2003). Voyaging from theory to practice in learning: Teacher professional development. In Pateman, N.A., Dougherty, B.J., & Zillliox, J.T. (Eds.), 2003. Proceedings of the 27th International Conference for the International Group for the Psychology of Mathematics Education (PME) held jointly with the 25th Conference of PME-NA 25at Honolulu (Hawaii) (July 13 - 18, 2003) (Vol. 1, pp. 25-31).

Bussi, M.G.B.; Bazzini, L. (2003). Research, practice and theory in didactics of mathematics: towards dialogue between different fields. Educational Studies in Mathematics 54 , 203-223.

And many, many further papers in journals and articles in handbooks

2. Searching for the relevant literature: Theory-practice-dichotomy (TDP) 2

Be cautious with such a prototyping!

The lack of binary explanations (black-white): Simplified reduction of variables

Janus-faced phenomenon: There is no space for additions...

How to handle priorities: Danger of hen-egg-misunderstandings...

The standard questions: What is theory (TME)? What is practice?

All these terms are philosophically high loaded and there are separate discussions in our community.

3. What are the main characteristics of the TPD? The structure of the problem

3. What are the main characteristics of the TPD? The search for an adequate model

What are the leading paradigms for the dichotomy? Question of modeling:

See Greeno (1978): ‘pipe-line model’

... Crude oil which gets pumped out of the ground in basic research [...] finally ... Who use the knowledge in making products for use in school and send the stuff around to school users. This is analogous to sending refined gasoline to filing stations, where customers can drive up and fill their tanks.

See Greeno (1978), Gelbach (1979): complementarity model

... Always almost impossible to discriminate between basic and applied research.

Kilpatrick (1981): integrated model, working in iterated cycles

... Teachers as participants in research

See the article of Kilpatrick (1981), it still contains important messages

3. What are the main characteristics of the TPD? In which metaphor do we believe?

See Berry, J., & Sahlberg, P. (1996). Investigating pupils' ideas of learning. Learning and Instruction 6 (1), 19 - 36.

3. What are the main characteristics of the TPD? Are there neighboring problem fields?

The problem of efficiency of research in mathematics education

Communalities and differences in various countries

The persistent influence of long-lasting traditions

The question of ‘identity’ of our community

The relation to mathematicians?

Lacking confidence in our work

Partial failure of teacher education

Changes followed by changes

The Non-communication with administration (in schools, ministries etc.)

See General Assembly of IMU

‘ Crisis’ of mathematics education?

Symptoms?

3. What are the main characteristics of the TPD? Search for relevant literature

Where do we stand today?

Kilpatrick, J. (1981). The reasonable ineffectiveness of research in mathematics education. For the learning of mathematics 2 (2), 22-29.

Oser, F.K., Dick, A. & Patry, J.-L. (1992). Responsibility, effectiveness and the domains of educational research. In F.K. Oser, A. Dick & J.-L. Patry (Eds.), Effective and responsible teaching. San Francisco: Jossey-Bass.

Garet, M.; Porter, A.C. et al. (2001). What makes professional development effective? Results form a National sample of teachers. American Educational Research, 38 , 2001, 4, pp. 915-945.

Houtveen, A.A.M. et al. (2004). Effective School Improvement in Mathematics. School Effectiviness and School Improvement 5 (3-4), 337-376.

Berliner, D.C. (2002). Educational Research: The Hardest Science of All. Educational Researcher 31 , 18 - 20.

Törner, G.; & Hoechsmann, K.,. (2004). Schisms, breaks, and islands - seeking bridges over troubled waters: a subjective view from Germany. In McDougall, D. (Eds.), 2004. Proceedings of the 26th Conference for the International Group for the Psychology of Mathematics Education (PME) - North America Chapter (PME-NA) at Toronto (October 2004), Vol. 3, (p. 993--1001).

4. ‘Excursion’: The TPD in mathematics Contrasts and similarities

Misleading terminology: Pure mathematics versus applied mathematics

F. Klein’s proposal: Precision mathematics versus approximation mathematics

Wide spread attitude: Applied mathematics is dirty, bad mathematics.

Facts: The estimations of the famous (pure) mathematician Halmos turned out to be wrong! Platonism may fail!

What can be learnt from the mathematicians‘ dichotomy?

Contrast between mathematics and mathematics education

Hamming, R. W. 1980. The unreasonable effectiveness of mathematics. American Mathematical Monthlv, 87 , 81-90.

Lack of a unifying solution of the theory-practice-dichotomy.

‘ Practice’ is setting the objectives, standards and demands, since ‘practice’ is ordering and has to pay and live by.

5. Changing the challenge: The practice-theory-challenge The ‘practice side’ of our problem Mathematics Done Differently a German Initiative for Teachers‘ Professional Development

Continuous Professional Development (CPD) Provocative Statements of Tenorth (2006)

Continuous Professional Development (CPD) Lifelong Learning

...

Instead of teaching, I told stories.

Anything to keep them quite and in their seats.

They thought I was teaching.

I thought I was teaching.

I was learning.

...

Teacher Man, Frank McCourt (2005)

6. Continuous Professional Development (CPD) Everyday Learning

In-service training is at first, just another name for the everyday life of the job (Tenorth, 2007).

Professional development takes place, when

talking to colleagues, participating in school conferences

reflecting about teaching practice

planning next teaching

... (e.g. Loucks-Horsley et. al., 2003)

In-service training initiatives are implemented in a context of learning (cf. Guskey, 2000; Tenorth, 2007)

6. Continuous Professional Development (CPD) Educational Demands

Educational reforms continuously constitute demands that teachers are supposed to meet (Day & Sachs, 2004)

Teachers are assigned a key role since only they can change the way mathematics will be taught (Sowder, 2007; Stigler & Hiebert, 1997).

Currently, many changes take place in the German educational system:

Implementation of educational standards (Blum et. al., 2007)

Central assessments in Grade 10

Central exams at the end of high school

Reduction of time at school ...

6. Continuous Professional Development (CPD) Conflicting Contexts Professional development takes place in a field of tension Teachers‘ needs Educational demands „ Administration should ...“ „ Teachers should ...“

6. Continuous Professional Development (CPD) Challenges

“ An adherence to them might tend to oversimplify or skew in-service provision towards meeting the needs of the system whilst ignoring, at their peril, the needs of the teacher within it” (Day, 1997, p. 41)

Top-down implementation is easy to get started but ...

Whose project is it? Who owns it? (Kohonen, 2007)

Statement of an interviewed teacher reflecting new movements in Germany:

“ Set theory came, set theory went” ...

Consequence: No changes in the classroom

6. Continuous Professional Development (CPD) Different Approaches

Traditional Approaches:

Focus on content

Well arranged frame

Narrow goals

Based on bringing outside knowledge to the teachers (Krainer, 1996)

Grounded in a knowledge-for-praxis conception (Cochran-Smith & Lytle, 1999)

Innovative Approaches:

Sensitive to teachers’ needs

Consider community aspects

Are not of the type ”either/or”, but ”both/and" (Lieberman, 2007)

Grounded in a knowledge-of-practice conception (Cochran-Smith & Lytle, 1999)

7. Mathematics Done Differently In-Service Training Initiative

Concerned with spreading and broadening existing local or regional PD programs in different thematic fields under one roof and to design new ones accordingly to teachers‘ needs.

Sponsored by Telekom’s Charitable Foundation

Project duration: 01/2007 - 08/2009

In charge of the project: Jürg Kramer, Humboldt University Berlin, Günter Törner, University of Duisburg-Essen

7. Mathematics Done Differently Identifying Teachers‘ Needs

Quantitative Data

Data collection by a questionnaire

Center for Educational Research (zepf, University of Koblenz-Landau)

Participants were teachers from overall Germany

Information about interesting topics and conditions

Factors structuring teachers‘ needs and expectations were identified

7. Mathematics Done Differently Constitutive Characteristics (I)

Showing appreciation for the teachers

Connecting research and practice : “trainers” operate as tandems of researchers and teachers

Involving expertise of our colleagues: courses “à la carte“

Mathematical content oriented models

Pedagogical methods oriented models

International models

7. Mathematics Done Differently Constitutive Characteristics (II)

Considering teachers needs: courses “on demand”

Addressing groups of teachers from one school or neighboring ones

Offering elaborated course material on the homepage

Evaluating the single course and all courses:

On the “trainers” level (T1, T2)

On the participants level (T0, T1, T2)

7. Mathematics Done Differently Course Statistic Courses 20 Waiting list 30 In preparation 26 Scheduled courses 60 Performed courses Courses “on demand” 11 In preparation 5 Scheduled courses 7 Performed courses

7. Mathematics Done Differently Courses on Demand (I)

Example for a course “on demand“, taken from a teacher’s email:

Ich arbeite an einer integrierten Haupt-/Realschule in Berlin; wir unterrichten weitestgehend ohne innere Differenzierung. Als pädagogische Schwerpunkte für das nächste Jahr haben wir uns für die Binnendifferenzierung (Individualisierung) und die Förderung des selbstständigen Arbeitens der Schüler entschieden. Meine Anfrage lautet jetzt, ob Sie uns im ersten Schulhalbjahr 2007/08 eine Fortbildung zum Thema „Förderung des selbstständigen und eigenverantwortlichen Arbeitens der Sch ül er“ anbieten könnten.

7. Mathematics Done Differently Courses on Demand (II)

Initiated procedure:

Supporting and encouraging teachers to concretize their needs

Classifying the request with regard to research

Literature review

Searching for experts that may serve as “trainers”

Designing the course

Offering the course on the project homepage

...

Long term goal: Generating a net of experts

7. Mathematics Done Differently Conclusions (I)

Observations

Great demand for courses

Courses well-accepted

Courses “on demand” are challenging to design and implement

More courses for Elementary schools and at the interfaces are needed

Unfamiliar roles for

... teachers when formulating their needs

... “trainers” when formulating their goals for the course

7. Mathematics Done Differently Conclusions (II)

Main conclusion: WE ARE A LEARNING SYSTEM

Success of in-service training depends on many factors that play together but always in different strength

Challenge to understand teachers‘ needs from a research perspective

Flexibility is a very important factor in order to be able to respond to teachers‘ needs

Interesting dynamic in specific courses like group effects

7. Mathematics Done Differently Geometry Unplugged (I)

7. Mathematics Done Differently Geometry Unplugged (II)

7. Mathematics Done Differently Geometry Unplugged (III)

8. Looking back Theoretical settings

Three ‘C’s (Lachance & Confrey, 2003; Krainer, 2006)

Content: Challenging mathematical activities, observations and reflections

Community: Cooperation and communications with other teachers

Context: Establishing a fostering framework (administration, resources, etc.)

have to be intertwined with

Professional development: individuum as well as of the group

Development of school: ‘School’ cannot stand aside

Development of the educational system: ‘Administration’ must be part of the development

8. Looking back Requirements and ambiguity in CPD

... Please accept, although we have conflicting categories...

Diversity: (Krainer, 2007)

Overestimation of CPD: It is expected that..., accept the challenges

Be modest: It is not easy to quantify change; some steps seem small, but there are a first step, followed by others

Centrality of mathematics: the role of subject matter

Relativize the dominance of the domain

8. Looking back To accept ‘dirty’ and ‘dusty’ practice

Implementing an in-service teaching initiative ‘Mathematics done differently’

The unknown recipients of our intervention initiative

The dependence on mediators

The decisive social variables

Many variables are discontinuous: Small changes imply large changes

Define ‘sustainable’?

The various frameworks: School administration, curriculum

8. Looking back What I would personally like to remark...

Beliefs ... A never ending story

Beliefs ... the scapegoats?

Beliefs ... we live by!

Beliefs ... as possession systems (Abelson)

Beliefs ... as inertia forces

Beliefs ... as keys

8. Looking back What I would personally like to remark...

Don‘t believe in classifications leading to finite categories!

Be skeptical if someone claims to be able to change teachers.

R. Thom’s (1973), a Field’s medalist and highly sensitive observer:

In fact, whether one wishes it or not, all mathematical pedagogy, even if scarcely coherent, rest on a philosophy of mathematics. (page 204)

Corollary: You cannot change the teaching of mathematician (teacher) unless he/she has changed his/her philosophy on mathematics.

The real problem which confronts mathematics teaching is not that of rigour, but the problem of the development of 'meaning', of the ''of mathematical obiects.