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

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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.

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Notes

  1. In mathematics gradient is usually seen as a calculated, exact value rather than a perceived quantity, even for graphs. However cognitive neuroscience research on visual processing shows that line orientation is processed very early in the visual stream in primary visual cortex (e.g. Tootell et al., 1998), and thus estimating it should require no conscious calculation.

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Acknowledgments

We thank Texas Instruments and the University of Auckland Research Committee for funding support. We also thank the staff at the Center for Advanced MRI, as well as Ed Hubbard, Donna Addis, Philippe Pinel, and Nick McNair for advice on fMRI analyses. Partial results from this study were previously presented at two conferences: as a poster at Human Brain Mapping (Wilson, Thomas, Lim & Corballis, 2008) and as a paper at a Mathematics Education Research Group of Australasia Conference (Thomas, Wilson, Corballis & Lim, 2008).

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Authors and Affiliations

  1. Department of Mathematics, The University of Auckland, Auckland Mail Centre, Private Bag 92019, Auckland, 1142, New Zealand

    Michael O. J. Thomas

  2. School of Educational Studies and Human Development, University of Canterbury, Christchurch, New Zealand

    Anna J. Wilson

  3. Department of Psychology, The University of Auckland, Auckland, New Zealand

    Michael C. Corballis & Vanessa K. Lim

  4. School of Science, Mathematics and Technology Education, The University of Auckland, Auckland, New Zealand

    Caroline Yoon

  5. Department of Psychology, The University of Auckland, Auckland, New Zealand

    Anna J. Wilson

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  1. Michael O. J. Thomas
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Correspondence to Michael O. J. Thomas.

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M. O. J. Thomas and A. J. Wilson contributed equally to the work.

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Thomas, M.O.J., Wilson, A.J., Corballis, M.C. et al. Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function. ZDM Mathematics Education 42, 607–619 (2010). https://doi.org/10.1007/s11858-010-0272-7

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