Universiteit Leiden

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PhD project

Graphing algebraic formulas

Many students have difficulties to read and give meaning to algebraic formulas. In this study it is explored whether graphing formulas with pen and paper will improve students’ability to reason with algebraic formulas.

Peter Kop


  • drs. Peter Kop - PhD candidate
  • prof.dr. J.H. van Driel (Melbourne Graduate School of Education, The University of Melbourne) - supervisor
  • prof.dr. P.H.M. Drijvers (Universiteit Utrecht)
  • dr. F.J.J.M. Janssen

Project description

Graphing formulas with paper and pen helps learners to connect the algebraic and graphical representation of a function, which is important in learning to read algebraic formulas. To be able to graph formulas, a repertoire of functions that can be instantly visualized by a graph (IGFs) are necessary, as well as heuristics to tackle more complex functions.

A two-dimensional framework was identified to describe the variety of strategies for graphing formulas. Five experts and three mathematics school teachers performed three tasks: a graphing a complex formula, a matching formula-graph task and a card sorting task. The framework was validated by these participants. We found some differences between experts‘ performances but big differences between teachers‘ performances. Not all teachers participating in this study turned out to have sufficient expertise in graphing formulas to complete this task.

The framework can be used to compare teachers’ and learners’ strategies to those of experts and to design learning trajectories for teachers and learners. The experts’ repertoires of IGFs resembled the basic functions taught in secondary school. In recognizing IGFs, the experts efficiently used function-family, prototype, and attribute reasoning. Not all participating teachers showed expert behaviour in graphing formulas.

In a next study it is investigated whether teaching general thinking methods in recognition will help students to reason with formules in relation to their graphs.

Foto: STOCK4B / Blend / Learning Pictures / Universal Images Group

Kop, P.M.G.M., Janssen, F.J.J.M., Drijvers, P.H.M., Veenman, M.,V.J. & Van Driel, J.H. (2015). Identifying a framework for graphing formulas from expert strategies. The Journal of Mathematical Behavior 39(3), 121–134.

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