Some examples for research
in mathematics education
from own experience
Bernd Zimmermann
Friedrich-Schiller Yliopisto Jena
Turun Yliopisto 27.11.2003
Steps of research
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Idea/Goal/question
Framework
Design
Outcome/Results
Discussion
Outlook
Prof. Bernd Zimmermann Turun
Yliopisto 2003
Analyzing problem solving
processes (BZ Diss. 1977)
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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
Trends in German
Mathematics Education
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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
Beliefs of students and
teachers
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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
History of mathematical
heuristics
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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
Changing patterns in the
history of mathematical beliefs
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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 “webrepresentation”.
Prof. Bernd Zimmermann Turun
Yliopisto 2003
Studies of primary school children, who
are potentially mathematically gifted
(Käpnick 1997)
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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
Different profiles of giftedness in mathematics –
attempts to improve differentiation with respect
to styles of thinking (Rehlich 1995)
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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
Evaluation of the quality of mathematical
problem solving processes by computer
support (Rehlich)
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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
Analyzing turning points and changes in
mathematical problem solving processes
(Heinrich 2003)
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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
Teacher students sensitivity for the
complexity (sfc) of problem oriented
(mathematics) instruction (Fritzlar 2003)
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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
Ordering and multimodality in thinking processes of
mathematically gifted pupils: sequential and topological
properties of cognitive microstates (Seidel 2001)
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Idea/goal: To get brain-physiological evidence
for giftedness related to (1) effective entropyreduction 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
Future research: e.g. analyzing
problem solving processes in
(mathematically) gifted pupils?
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Idea/goal: Analyze problem solving
behavior of gifted students
Design: ??
Outcome: ??
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Your suggestions?
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Prof. Bernd Zimmermann Turun
Yliopisto 2003