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József Kontra: Flexibility in Thinking and Mathematical Achievement

The main purpose of this study is to report the attempts made to identify components involved in mathematical grades. With its focus on solutions rather than on strategies or processes, the current assessment procedure (grading) provides inadequate information about performance and if used reflects a fragmented view of mathematics rather than a view of knowledge as an integrated whole. What makes students differ in their performance in mathematics problems? In solving mathematical problems there are situations in which students do not already know what to do. They decide to abandon one approach and start another. To be efficient, the problem solvers need skills in how to manage the resources at their disposal. Specifically, the use of the term flexibility is addressed here. In our puzzle-problem approach to flexibility subjects were tested with challenging problems that do not lend themselves to obvious solutions. The introduction of this paper considers what a focus on insight offers us with respect to selecting new problems typified by the „six mathsticks problem” alleged to measure flexible thinking. Thus, for the tests several problems were collected from the literature on insight using a taxonomy (proposed by Weisberg, 1995). The other problems were constructed by a three-step procedure (see Gick & Lockhart, 1995). The research was carried out in 16 schools. Altogether 826 students were tested. The 5th grade sample consisted of 346 students and the 7th grade sample contained 480 students. It was found that students who had better grades were more likely to solve the testproblems correctly. Of course, any mathematical performance is built on a foundatian of basic mathematical knowledge. Within this context, we develop implications for measuring comprehension, learning strategies, and metacognitive processes. Taken together, these recommendations move us toward a broader concept of assessment.

MAGYAR PEDAGÓGIA 99. Number 2. 141-155. (1999)

Levelezési cím / Address for correspondence: Kontra József, H–7400 Kaposvár, Ezredév u. 10. E-mail:


Magyar Tudományos Akadémia