Students’ understanding of basic relational concepts in mathematics (function, equivalence and order relation) 

Project hopefully supported by Academy of Finland/DAAD 2009-2010

University of JoensuuDr. 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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