Cognitive Science Foundations of Integer Understanding and Instruction

Varma, Sashank; Blair, Kristen Pilner; Schwartz, Daniel L.

doi:10.1007/978-3-030-00491-0_14

Cited by 2 publications

(3 citation statements)

References 45 publications

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“…Early numerical development involves understanding fundamental concepts and procedures for natural numbers {1, 2, 3 …}, including one-to-one correspondence and cardinality (Gelman & Gallistel, 1986;Sarnecka & Carey, 2008). Researchers in mathematical cognition and mathematics education have posited that natural number concepts serve as a foundation for later-acquired mathematical concepts, such as integer concepts (e.g., Bofferding, 2014;Varma et al, 2019). Integers extend the natural numbers with zero and negative whole numbers.…”

Section: Relations Between Concepts Of Zero and Integer Understandingmentioning

confidence: 99%

“…According to Varma and Schwartz (2011), the reflection point model should be manifested in an inverse distance effect on mixed comparisons (i.e., positivenegative comparisons)-that is, slower response times for pairs that are further apart. This pattern should arise due to the psychophysical scaling of magnitude representations of the mental number line for natural numbers and the corresponding reflection for negative numbers (Varma et al, 2019). Briefly, mixed pairs that are close together (e.g., -2, 1) correspond to magnitudes that are further apart in psychophysical terms (e.g., 2 vs. 1), and therefore fast and easy to distinguish, whereas mixed pairs that are farther apart (e.g., -5, 6) correspond to magnitudes that are closer together in psychophysical terms (e.g., 5 vs. 6), so they are slower and more difficult to distinguish, yielding an inverse distance effect (see Varma et al, 2019, for discussion).…”

Section: Relations Between Concepts Of Zero and Integer Understandingmentioning

confidence: 99%

“…Fourth, do children's initial conceptions of zero predict whether they activate the magnitudes of negative numbers when they compare positive and negative numbers? Based on past work, we expect that participants who represent the magnitudes of negative numbers in terms of their distance from zero (i.e., as symmetrical to the positive numbers) should demonstrate a distance effect for positive-negative pairs (specifically, an inverse distance effect; Varma et al, 2019). In contrast, participants who represent all negative numbers as "less than zero"--that is, who do not automatically activate the distinct magnitudes of negative numbers-should not demonstrate a distance effect.…”

Section: Research Questionsmentioning

confidence: 99%

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Is zero more than nothing? Relations between concepts of zero and integer understanding

Vest,

Alibali

2024

Preprint

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The concept of zero is challenging for many children. This study investigated conceptions of zero in fifth- through seventh-grade children (N = 72) in the United States and examined how children’s conceptions of zero related to other aspects of their integer knowledge. At the outset of the study, many participants held a null conception of zero (i.e., zero as “nothing”), and some participants held a conception of zero as the symmetry point between the positive and negative integers. Participants in higher grades were more likely to express a symmetry conception. We hypothesized that participants’ conceptions of zero would be related to other aspects of integer knowledge. Relative to participants who held a null conception of zero, participants who held a symmetry conception demonstrated greater knowledge of the additive inverse principle (for every number x, there exists an inverse, -x, such that the two numbers sum to zero) and greater integer arithmetic skill. Conceptions of zero were not related to integer magnitude understanding. We also examined whether a brief lesson focusing on zero as the symmetry point would lead to shifts in participants’ conceptions of zero and gains in understanding of the additive inverse principle. Participants were randomly assigned to receive a lesson about zero as null or zero as the symmetry point between positive and negative integers. The brief lesson did not lead to substantial changes in participants’ conceptions of zero or their additive inverse knowledge.

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Section: Relations Between Concepts Of Zero and Integer Understandingmentioning

confidence: 99%

Section: Relations Between Concepts Of Zero and Integer Understandingmentioning

confidence: 99%

Section: Research Questionsmentioning

confidence: 99%

See 1 more Smart Citation

Is zero more than nothing? Relations between concepts of zero and integer understanding

Vest,

Alibali

2024

Preprint

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Ninth-grade students’ conceptual understanding of the number line

Ünal,

Ala,

Kartal

et al. 2024

J. Numer. Cogn.

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Sixty (35 girls and 25 boys) 9th-grade students’ conceptual understanding of the number line was qualitatively assessed through verbal explanations and visual representations. The assessment included an open-ended question focused on students’ number line descriptions and the explanations coalesced around six features: sequential ordering (i.e., numbers are sequentially represented), positivity-negativity of numbers (i.e., the number line contains positive and negative numbers), non-centrality (i.e., zero does not have to be in the center), infinity, increment flexibility (i.e., number line increments can vary), and continuity (i.e., numbers can be placed anywhere between minus infinity and plus infinity without breaks). The students’ explanations show that these six features emerge in five successive stages in the conceptual understanding of the number line. These stages are (1) no knowledge, (2) sequential ordering and positivity-negativity, (3) infinity and non-centrality, (4) incremental flexibility, and (5) continuity. The last two stages were not found in most descriptions. The results suggest that students’ understanding of the number line is incomplete and may be overestimated by commonly used quantitative assessments of number line knowledge.

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Cognitive Science Foundations of Integer Understanding and Instruction

Cited by 2 publications

References 45 publications

Is zero more than nothing? Relations between concepts of zero and integer understanding

Is zero more than nothing? Relations between concepts of zero and integer understanding

Ninth-grade students’ conceptual understanding of the number line

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