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Students’ understanding of basic relational concepts in mathematics (function,
equivalence and order relation)
Project hopefully supported by Academy of Finland/DAAD
2009-2010
**Participants **
University of Joensuu: Dr.
Martti E. Pesonen and Docent Timo Tossavainen
University of Kiel (IPN):
Prof. Dr. Aiso
Heinze, Dr. Timo Ehmke
and Daniel Wagner
**Project Description** (from the application)
The Finnish pupils have shown, according to PISA surveys, a remarkably
better performance in everyday maths reasoning skills than for example
the German pupils. In this piece of research we try to study the university
mathematics freshmen’s conceptual understanding about functions and the
problems they have in adopting new mathematical concepts. The student samples
from the two countries are smallish, 50-100 individuals, so that comparison
of overall performance is only guiding, but supposed to be large enough
in identifying conceptual deficiencies and problems in learning abstract
concepts.
Function, binary operation, equivalence relation and partial order
are central mathematical concepts, especially in the academic mathematics
studies. The university freshmen are somewhat familiar with functions and
arithmetic binary operations, but in general not with equivalence nor order
relation. The first signs of the gap between school and university mathematics
content are seen in the theoretical point of view and emphasis; the concept
image achieved in school is not accurate enough to face the definitions
and using them. An earlier study showed that viable understanding of the
binary operation concept can very well be predicted from the students'
ability to recognize the cases which fulfil the axioms of the abstract
concept. The same is supposed to occur with the function concept, since
binary operation are special cases of two variable functions.
Equivalence and partial order relations are conceptually near functions,
they are special cases of relations between two sets. However, it is hypothesized
that the greatest differences - especially difficulties - in learning the
concept of function and the two other concepts comes from the fact that
functions are learned first already at school. This concept image is usually
very limited and vague but it is very permanent. It is not easy to replace
nor to correct a deficient and distorted concept image.
The aims of this project are to develop test instruments for assessing
the students' knowledge about the concepts function, equivalence relation
and order relation, and to investigate empirically the students' difficulties
in understanding these concepts. This is done using computer-based online
worksheets, which are designed to describe the strengths and weaknesses
of the students' concept knowledge in recognizing, identifying and producing
cases of the concepts, given in various representations (verbal, symbolic,
graphic). The same worksheets can be used as online learning material,
when they are used in adaptive mode with detailed feedback.
The test samples are university freshmen in Kiel, Joensuu and Savonlinna,
and this allows us to compare exploratively the students' understanding
of basic relational concepts in mathematics between the two countries.
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