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Scaling-continuous variation: supporting students’ algebraic reasoning


This paper introduces a new mode of variational and covariational reasoning, which we call scaling-continuous reasoning. Scaling-continuous reasoning entails (a) imagining a variable taking on all values on the continuum at any scale, (b) understanding that there is no scale at which the continuum becomes discrete, and (c) re-scaling to any arbitrarily small increment for x and coordinating that scaling with associated values for y. Based on the analysis of a 15-h teaching experiment with two 12-year-old pre-algebra students, we present evidence of scaling-continuous reasoning and identify two implications for students’ understanding of rates of change: seeing constant rate as an equivalence class of ratios, and viewing instantaneous rate of change as a potential rate. We argue that scaling-continuous reasoning can support a robust understanding of function and rates of change.

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The research reported in this paper was supported by the National Science Foundation (grant no. DRL-1419973).

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Correspondence to Amy Ellis.

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Ellis, A., Ely, R., Singleton, B. et al. Scaling-continuous variation: supporting students’ algebraic reasoning. Educ Stud Math 104, 87–103 (2020).

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  • Student reasoning
  • Algebra
  • Middle school
  • Teaching experiments