Neuropsychological Evidence for the Role of Graphical and Algebraic Representations in Understanding Function

There are difficulties accessing students ’ thinking about mathematical concepts, although methods such as task observation and interviews provide some useful information. In recent years it has become possible to use functional magnetic resonance imaging (fMRI) techniques to access brain activity while students are thinking about mathematics. In this study we have used this technique to examine brain activity while students were processing graphical and algebraic representations of function. Results show some evidence for increased difficulty of translation between these formats for linear compared with quadratic functions. We also describe regions of the brain that are involved in the translations. Function is one of the fundamental concepts of school and tertiary mathematics, and yet it is often misunderstood by students and school teachers. This may be due to the number and apparent complexity of its representational manifestations and the concept images these may evoke. In a comparison of the function concept maps of eight professors having PhDs in mathematics with those of twenty-eight university mathematics students, Williams (1998) found the latter had an emphasis on minor detail, such as the variable used, algorithms, and the idea that functions are equations in the student maps. In contrast she found that “none of the experts demonstrated the students ’ propensity to think of a function as an equation. Instead, they defined it as a correspondence, a mapping, a pairing, or a rule. ” (ibid, p. 420). Some, including teachers, have a tendenc