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Page 1
YESS 4 Trabzon, August 22, 2008
Theory versus PracticeRevisiting 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
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YESS 4 Trabzon, August 22, 2008
0. Contents
1. Introduction
2. Searching for the relevant literature
3. What are the main characteristics of the theory-practice-dichotomy?
4. Excursion: The theory-practice-dichotomy in mathematics
5. A paradigm change: The practice-theory-challenge
6. Professional development
7. The project: Mathematics done differently
8. Looking back
9. Conclusion: What I would like to remark...
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YESS 4 Trabzon, August 22, 2008
1. IntroductionSubjective view and profile
Research mathematician
Discrete mathematics, coding theory, 2D codes etc.
Noncommutative algebra (MSC 2000: 16W60, 16W99)
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YESS 4 Trabzon, August 22, 2008
1. IntroductionMathematics Subject Classification (MSC 2000)
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YESS 4 Trabzon, August 22, 2008
1. IntroductionMathematics Subject Classification (MSC 2000)
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YESS 4 Trabzon, August 22, 2008
1. IntroductionMathematics Subject Classification (MSC 2000)
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YESS 4 Trabzon, August 22, 2008
1. IntroductionSubjective view and profile
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.
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YESS 4 Trabzon, August 22, 2008
1. IntroductionSubjective view and profile
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
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YESS 4 Trabzon, August 22, 2008
1. IntroductionThe 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.
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YESS 4 Trabzon, August 22, 2008
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.
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YESS 4 Trabzon, August 22, 2008
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
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YESS 4 Trabzon, August 22, 2008
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
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YESS 4 Trabzon, August 22, 2008
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
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YESS 4 Trabzon, August 22, 2008
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.
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YESS 4 Trabzon, August 22, 2008
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?
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YESS 4 Trabzon, August 22, 2008
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).
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YESS 4 Trabzon, August 22, 2008
4. ‘Excursion’: The TPD in mathematicsContrasts 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.
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YESS 4 Trabzon, August 22, 2008
5. Changing the challenge: The practice-theory-challengeThe ‘practice side’ of our problem
Mathematics Done Differently
a German Initiative for Teachers‘ Professional
Development
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YESS 4 Trabzon, August 22, 2008
6. Continuous Professional Development (CPD)Provocative Statements of Tenorth (2006)
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YESS 4 Trabzon, August 22, 2008
6. 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)
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YESS 4 Trabzon, August 22, 2008
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)
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YESS 4 Trabzon, August 22, 2008
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 ...
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YESS 4 Trabzon, August 22, 2008
6. Continuous Professional Development (CPD) Conflicting Contexts
Professional development takes place in a field of tension
Teachers‘ needs
Educational demands
„Administration should ...“
„Teachers should ...“
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YESS 4 Trabzon, August 22, 2008
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
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YESS 4 Trabzon, August 22, 2008
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)
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done DifferentlyIn-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
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done DifferentlyIdentifying 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
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done DifferentlyConstitutive 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
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done DifferentlyConstitutive 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)
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done Differently Course Statistic
Courses
Performed courses 60
Scheduled courses 26
In preparation 30
Waiting list 20
Courses “on demand”
Performed courses 7
Scheduled courses 5
In preparation 11
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done DifferentlyCourses 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 Schüler“ anbieten könnten.
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done DifferentlyCourses 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
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done DifferentlyConclusions (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
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YESS 4 Trabzon, August 22, 2008
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
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done Differently Geometry Unplugged (I)
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done Differently Geometry Unplugged (II)
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YESS 4 Trabzon, August 22, 2008
7. Mathematics Done Differently Geometry Unplugged (III)
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YESS 4 Trabzon, August 22, 2008
8. Looking backTheoretical 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
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YESS 4 Trabzon, August 22, 2008
8. Looking backRequirements 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
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YESS 4 Trabzon, August 22, 2008
8. Looking backTo 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
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YESS 4 Trabzon, August 22, 2008
8. Looking backWhat 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
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YESS 4 Trabzon, August 22, 2008
8. Looking backWhat 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.
Hersh (1997) used the formulation:
Philosophies (e.g. Platonism, Formalism) may have devasting effects on teaching.
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YESS 4 Trabzon, August 22, 2008
Thanks for your patience