Abstract
This chapter considers psychological and neuroscience research on how people understand the integers, and how educators can foster this understanding. The core proposal is that new, abstract mathematical concepts are built upon known, concrete mathematical concepts. For the integers, the relevant foundation is the natural numbers, which are understood by reference to a mental number line (MNL). The integers go beyond the natural numbers in obeying the additive inverse law: for any integer x, there is an integer −x such that x + (−x) = 0. We propose that practicing applying this law, such as when students learn that the same quantity can be added or subtracted from both sides of an equation, transforms the MNL. In particular, perceptual mechanisms for processing visual symmetry are recruited to represent the numerical symmetry between the integers x and −x. This chapter reviews psychological and neuroscience evidence for the proposed learning progression. It also reviews instructional studies showing that the hypothesized transformation can be accelerated by novel activities that engage symmetry processing compared to conventional activities around number lines and cancellation. Ultimately, these instructional insights can guide future psychological and neuroscience studies of how people understand the integers in arithmetic and algebraic contexts.
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Notes
- 1.
Cultural differences may influence the left-right orientation of the number line, based on whether numbers are read from left-to-right or right-to-left in one’s native language. However, there is also evidence that a left-to-right orientation may be innate. (See Zohar-Shai, Tzelgov, Karni, and Rubinsten (2017) for a review of this literature.) This chapter does not address this culture difference. Instead, it assumes a left-right orientation, consistent with the data from English-speaking countries.
- 2.
In addition to mixed comparisons, zero comparisons can also differentiate the analog+ and symbol+ models. These are comparisons where one of the two numbers is zero (e.g., −2 vs. 0). See Varma and Schwartz (2011) for further discussion.
- 3.
SPL and IPS are also associated with visuospatial reasoning (e.g., Zacks, 2008). Thus, it is possible that they are recruited here not to process the magnitudes of positive integers and negative integers, but rather to process their symmetric relationship about zero. We consider the role of symmetry processing in integer understanding below, when describing the analog-x model.
- 4.
- 5.
This study is notable in testing children much younger than those in prior psychological and educational studies.
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Varma, S., Blair, K.P., Schwartz, D.L. (2019). Cognitive Science Foundations of Integer Understanding and Instruction. In: Norton, A., Alibali, M.W. (eds) Constructing Number. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-00491-0_14
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