Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function

Abstract

Using traditional educational research methods, it is difficult to assess students' understanding of mathematical concepts, even though qualitative methods such as task observation and interviews provide some useful information. It has now become possible to use functional magnetic resonance imaging (fMRI) to observe brain activity whilst students think about mathematics, although much of this work has concentrated on number. In this study, we used fMRI to examine brain activity whilst ten university students translated between graphical and algebraic formats of both linear and quadratic mathematical functions. Consistent with previous studies on the representation of number, this task elicited activity in the intra-parietal sulcus, as well as in the inferior frontal gyrus. We also analysed qualitative data on participants' introspection of strategies employed when reasoning about function. Expert participants focused more on key properties of functions when translating between formats than did novices. Implications for the teaching and learning of functions are discussed, including the relationship of function properties to difficulties in conversion from algebraic to graphical representation systems and vice versa, the desirability of teachers focusing attention on function properties, and the importance of integrating graphical and algebraic function instruction.

Cognitive Processing of Cryptography Concepts: An fMRI Study

“…Though cryptography is an applied discipline of mathematics, brain activation patterns noted in our results are not consistent with the areas of activation noted in prior fMRI studies of cognitive processing of basic mathematical operations (Delazer, et al, 2006) or simple linear and quadratic functions (Thomas, et al, 2010), even when questions were represented mathematically. The results of this analysis show that most of the activation in response to the stimuli of different representations is in the visual cortex.…”

Handheld technology for mathematics education: flashback into the future

“…In these mental schemes, technical and conceptual aspects are intertwined and codevelop. We cannot look into the heads of our students [even if neuroscientists are advancing (Thomas et al, 2008)! ], so schemes cannot be observed directly but have to be inferred from what can be observed, the instrumented techniques.…”

Traveling down the road: from cognitive neuroscience to mathematics education … and back

“…For example, Butterworth and Laurillard (2010) highlighted that children with dyscalculia had difficulties with representing numerosities, which might hinder their development of mathematical skill. Other contributions indicate that some solution methods are cognitively more demanding than others (Lee et al, 2010;Thomas et al, 2010). These data suggest that it might not be appropriate to teach these methods at young ages, when functions of working memory and attentional control have not fully developed yet (Luna, Garver, Urban, Lazar, & Sweeney, 2004), or that teaching of these solution methods should involve appropriate scaffolds to reduce cognitive load, for example, by increasing awareness to appropriate features of the task and ignoring the inappropriate ones (see also Stavy & Babai, 2010).…”

“…It is the first special issue containing a collection of studies that use neuroscientific methods to examine mathematics learning that is published in a leading mathematics education journal. Moreover, this issue is not restricted to elementary number processing, but addresses explicitly arithmetical and mathematical skills that are formally taught in school, such as arithmetic (Menon, 2010), word problem solving (Lee et al, 2010;Obersteiner et al, 2010), geometry (Stavy & Babai, 2010;Bornemann et al, 2010), algebra (Thomas et al, 2010), and connecting, at least in terms of the mathematics subjects that are investigated, clearly to school mathematics.…”