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Some examples for research in mathematics education from own experience

Explore research on problem-solving processes in mathematics education, analyzing outcomes and trends. Investigate beliefs of students and teachers, historical heuristics, and changing patterns in mathematical beliefs.

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Some examples for research in mathematics education from own experience

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  1. Some examples for research in mathematics education from own experience Bernd Zimmermann Friedrich-Schiller Yliopisto Jena Turun Yliopisto 27.11.2003

  2. Steps of research • Idea/Goal/question • Framework • Design • Outcome/Results • Discussion • Outlook Prof. Bernd Zimmermann Turun Yliopisto 2003

  3. Analyzing problem solving processes (BZ Diss. 1977) • Idea/goal: Lüer/Dörner; searching for a basic rhythm of thinking connected with problems from incidence geometry • Design: exploratory study; developing a complex category-system, describing activities with respect to 12 activities/aspects, sequential analysis • Outcome: also a basic rhythm, two types of problem solvers, advantage and disadvantage of perception Prof. Bernd Zimmermann Turun Yliopisto 2003

  4. Trends in German Mathematics Education • Idea/goal: Constituting sound basis for research (in m.e.) • Design: development of categories constituting a trend: general goals; view on math, learning and teaching of maths, on research. Analysis of literature. • Outcome: elementarization, operatoric approach; genetic; “Bourbakism”; application; problem-oriented; computer oriented: Framework for development of questionnaire for teachers Prof. Bernd Zimmermann Turun Yliopisto 2003

  5. Beliefs of students and teachers • Idea/goal: Getting information about initial and boundary conditions of (math) instruction • Design: development of questionnaires for pupils and teachers by using trend analysis and further literature. • Outcome: Teachers: different clusters (e. g. Bourbaki-c.; problem.-or.-c., conventional c.). Pupils: often correspondence to attitudes of their teachers; explanations of teachers are most important, also for gender-differences. Prof. Bernd Zimmermann Turun Yliopisto 2003

  6. History of mathematical heuristics • Idea/goal: What kind of thinking methods (heuristics) proved to be most successful within 5000 years? What kind of typical difficulties occurred across time? (Invariants) • Design: Analysis of relevant (original) literature with respect to thinking methods and their outcome. • Outcome: Heuristics: successive approximation, working backwards, analogy, atomistic methods. Use/application first implicitly; later on explicitly. Prof. Bernd Zimmermann Turun Yliopisto 2003

  7. Changing patterns in the history of mathematical beliefs • Idea/goal: (1) Searching for motivating and stimulating activities involved in creating new maths. (2) To what extent these activities were involved in local (mathematical) beliefs? • Design: Analysis of literature. Rating the amount of importance. • Outcome: Eight important activities. Different profiles of beliefs at different times and for different cultures, supported by a “web-representation”. Prof. Bernd Zimmermann Turun Yliopisto 2003

  8. Studies of primary school children, who are potentially mathematically gifted (Käpnick 1997) • Idea/goal: (1) Development of a conceptual system for mathematically gifted pupils. (2) To search for different profiles of mathematically giftedness • Design: Analysis of literature. Empirical study by using cluster analysis. • Outcome: (1) Conceptual system for giftedness (2) Test for identification (3) Different profiles of mathematically giftedness. Prof. Bernd Zimmermann Turun Yliopisto 2003

  9. Different profiles of giftedness in mathematics – attempts to improve differentiation with respect to styles of thinking (Rehlich 1995) • Idea/goal: To get more information about different profiles of mathematical giftedness by use of a computerprogram • Design: (1) conceptual framework (2) program (3) subjects explore problem modeled by computer • Outcome: Different profiles of mathematical giftedness. Prof. Bernd Zimmermann Turun Yliopisto 2003

  10. Evaluation of the quality of mathematical problem solving processes by computer support (Rehlich) • Idea/goal: Evaluate problem solving processes by support of a computer program • Design: (1) conceptual framework (2) program (ready) (3) select appropriate problems • Outcome: standardized process analysis. Prof. Bernd Zimmermann Turun Yliopisto 2003

  11. Analyzing turning points and changes in mathematical problem solving processes (Heinrich 2003) • Idea/goal: To characterize situations of change in problem solving situations • Design: (1) Development of a sound theoretical framework (2) Selection of a representative problems (3) Problem solving sessions in small groups (4) Categorize, classify and analyze different situations of change • Outcome: Classification of situations of change; possible use for instruction Prof. Bernd Zimmermann Turun Yliopisto 2003

  12. Teacher students sensitivity for the complexity (sfc) of problem oriented (mathematics) instruction(Fritzlar 2003) • Idea/goal: Constructing an instrument to measure students sfc. • Design:. (1) Development of a sound theoretical framework concerning sensitivity for complexity of p.o.m.i. (2) Selection of a representative problem and gaining experience/material by teaching 50 classes (3) Development of a computer program (4) Development of a category system for sfc. (5) Analyzing students ps behavior when using this computer scenario • Outcome: (1) Programm (2) Category system (3) Determination of the sfc of a sample of 20 teacher students. Prof. Bernd Zimmermann Turun Yliopisto 2003

  13. Ordering and multimodality in thinking processes of mathematically gifted pupils: sequential and topological properties of cognitive microstates (Seidel 2001) • Idea/goal: To get brain-physiological evidence for giftedness related to (1) effective entropy-reduction and (2) flexibility in changing the mode of representation • Design: (1) sample of gifted students (2) set of problems (3)analyzing problemsolving processes by using EEG and appropriate software for evaluation • Outcome: Support of the aforementioned hypothesis. Prof. Bernd Zimmermann Turun Yliopisto 2003

  14. Future research: e.g. analyzing problem solving processes in (mathematically) gifted pupils? • Idea/goal: Analyze problem solving behavior of gifted students • Design: ?? • Outcome: ?? • Your suggestions? Prof. Bernd Zimmermann Turun Yliopisto 2003

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