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Quantifying exponential growth: three conceptual shifts in coordinating multiplicative and additive growth. (English)
J. Math. Behav. 39, 135-155 (2015).
Summary: This article presents the results of a teaching experiment with middle school students who explored exponential growth by reasoning with the quantities height ($y$) and time ($x$) as they explored the growth of a plant. Three major conceptual shifts occurred during the course of the teaching experiment: (1) from repeated multiplication to initial coordination of multiplicative growth in $y$ with additive growth in $x$; (2) from coordinating growth in $y$ with growth in $x$ to coordinated constant ratios (determining the ratio of $f(x_{2})$ to $f(x_{1})$ for corresponding intervals of time for $(x_{2}-x_{1})\geq 1$), and (3) from coordinated constant ratios to within-units coordination for corresponding intervals of time for $(x_{2}-x_{1})<1$. Each of the three shifts is explored along with a discussion of the ways in which students’ mathematical activity supported movement from one stage of understanding to the next. These findings suggest that emphasizing a coordination of multiplicative and additive growth for exponentiation may support students’ abilities to flexibly move between the covariation and correspondence views of function.
Classification: I23 M63
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