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Ideas foundational to calculus learning and their links to students’ difficulties

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Abstract

Existing literature reviews of calculus learning made an important contribution to our understanding of the development of mathematics education research in this area, particularly their documentation of how research transitioned from studying students’ misconceptions to investigating students’ understanding and ways of thinking per se. This paper has three main goals relative to this contribution. The first goal is to offer a conceptual analysis of how students’ difficulties surveyed in three major literature review publications originate in the mathematical meanings and ways of thinking students develop in elementary, middle and early high school. The second goal is to highlight a contribution to an important aspect that the articles in this issue make but was overshadowed by other aspects addressed in existing literature reviews: the nature of the mathematics students experience under the name “calculus” in various nations or regions around the world, and the relation of this mathematics to ways ideas foundational to it are developed over the grades. The third goal is to outline research questions entailed from these articles for future research regarding each of them.

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Notes

  1. We readily admit the power of using symbols to represent “unknowns”. For example, Schoenfeld (1988) posed this problem: Reverse the digits in 39 and 62. Then 39 × 62 = 93 × 26. Are there other pairs of numbers having this property? By representing digits’ place value you get (10a + b)(10c + d) = (10b + a) (10d + c), which reduces to ac = bd.

  2. Here we finesse the matter of function notation, which is notoriously confusing for students and teachers (Carlson, 1998; Even, 1990; Sajka, 2003).

  3. Thompson and Carlson (2017) were careful to define “variable” as a symbol which represents the value of a quantity (concrete or abstract) whose value varies. They distinguished among using a symbol as a variable, as a parameter, and as a constant, where the distinction resides in what the person intends to represent.

  4. Two quantities conceived as having fixed values cannot be thought to have a rate of change. It would be like asking, “What is the rate of change of 5 with respect to 3?”.

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  1. University of California at San Diego, San Diego, USA

    Patrick W. Thompson

  2. Arizona State University, Tempe, USA

    Guershon Harel

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Thompson, P.W., Harel, G. Ideas foundational to calculus learning and their links to students’ difficulties. ZDM Mathematics Education 53, 507–519 (2021). https://doi.org/10.1007/s11858-021-01270-1

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