Natural number bias in operations with missing numbers

Christou, Konstantinos P.

doi:10.1007/s11858-015-0675-6

Cited by 25 publications

(33 citation statements)

References 26 publications

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“…There are quite extensive studies on pupils' and teachers' knowledge on the density of rational numbers (Christou, 2015;Depaepe et al, 2015;Durkin & Rittle-Johnson, 2015;McMullen et al, 2015;Stacey, Helme, Archer, & Condon, 2001;Widjaja et al, 2008;Vamvakoussi et al, 2011;Vamvakoussi & Vosniadou, 2004Van Hoof, Verschaffel, & Van Dooren, 2015). Most studies show that pupils and teachers often over-generalise the discrete nature of natural numbers to rational numbers.…”

Section: Background and Research Questionsmentioning

confidence: 99%

“…Many previous studies (McMullen, Laakkonen, Hannula-Sormunen & Lehtinen;2015;Prediger, 2008;Vamvakoussi, Christou, Mertens & Van Dooren, 2011;Vamvakoussi & Vosniadou, 2004 have argued that there is conceptual change involved in the process of passing from natural to rational numbers. This means that learning rational numbers requires one to change one's prior conceptions of something, like numbers, in order to be compatible with a new mathematical situation -they cannot simply be adapted but need more fundamental revision.…”

Section: Introductionmentioning

confidence: 99%

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Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia

Putra

2019

EURASIA J MATH SCI T

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The present study aims to introduce the notion of praxeological change, developed based on the Anthropological Theory of the Didactic, to describe a necessity of changing mathematical praxeologies when passing from natural to rational numbers. It is applied to study and compare Danish and Indonesian pre-service teachers' (PSTs) knowledge of the density of rational numbers. They work in pairs to solve and discuss a hypothetical teacher task, which involves both mathematical and didactical tasks, related to the density of rational numbers. The findings highlight significant differences of the mathematical and didactical knowledge which are shared by the Danish and Indonesian PSTs. In particular, the Danish PSTs are more successful than the Indonesian PSTs in proposing didactical praxeologies to support pupils' praxeological change. They use the mathematical idea of converting fractions into decimals or vice versa and representing fractions and decimals on the same number line, while the Indonesian pairs tend to suggest pupils to order fractions and decimals based on the ordering properties of natural numbers.

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Section: Background and Research Questionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia

Putra

2019

EURASIA J MATH SCI T

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“…The role of inhibition in mathematics tasks was the topic of a Special Issue of ZDM Mathematics Education (Van Dooren & Inglis, 2015). The studies in this Special Issue contrasted the participants' performance in tasks consisting of a consistent and an inconsistent condition and in which the inconsistent trials involved salient but irrelevant perceptual stimuli (e.g., comparing perimeters of geometric shapes as in Babai, Shalev, & Stavy, 2015), simple intuitions (e.g., longer length implies larger number, as in Lubin, Simon, Houde, & De Neys, 2015), or conceptual change learning, where the whole number bias had to be overcome (e.g., Christou, 2015;McMullen, Hannula-Sormunen, & Lehtinen, 2015;Van Hoof, Janssen, Verschaffel, & Van Dooren, 2015). The results confirmed the hypothesis that the employment of mathematics concepts in the consistent condition was more accurate and faster than in the inconsistent condition, suggesting that the extra time is used to inhibit the interfering stimuli.…”

Section: The Recruitment Of Ef Skills In Conceptual Change Processesmentioning

confidence: 99%

The Recruitment of Shifting and Inhibition in On‐line Science and Mathematics Tasks

et al. 2018

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Prior research has investigated the recruitment of inhibition in the use of science/mathematics concepts in tasks that require the rejection of a conflicting, nonscientific initial concept. The present research examines if inhibition is the only EF skill recruited in such tasks and investigates whether shifting is also involved. It also investigates whether inhibition and/or shifting are recruited in tasks in which the use of science/mathematics concepts does not require the rejection of an initial concept, or which require only the use of initial concepts. One hundred and thirty-three third- and fifth-grade children participated in two inhibition and shifting tasks and two science and mathematics conceptual understanding and conceptual change (CU&C) tasks. All the tasks were on-line, and performance was measured in accuracy and RTs. The CU&C tasks involved the use of initial concepts and of science/mathematics concepts which required conceptual changes for their initial formation. Only in one of the tasks the use of the science/mathematics concepts required the concurrent rejection of an initial concept. The results confirmed that in this task inhibition was recruited and also showed that the speed of shifting was a significant predictor of performance. Shifting was a significant predictor of performance in all the tasks, regardless of whether they involved science/mathematics or initial concepts. It is argued that shifting is likely to be recruited in complex tasks that require multiple comparisons of stimuli and the entertainment of different perspectives. Inhibition seems to be a more selective cognitive skill likely to be recruited when the use of science/mathematics concepts requires the rejection of a conflicting initial concept.

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“…The three categories included: a) congruent tasks, in which the results of the operations were in-line with students' intuitions (i.e., bigger results for multiplication and smaller for division), and the missing numbers were natural numbers (e.g., 3 × _ = 12); b) tasks which were again in-line with students' intuitions about the results of Natural Number Bias in Arithmetic Operations 24 the operations, but the missing numbers were rational numbers larger than 1 instead of natural numbers (e.g., 3 × _ = 11); and c) tasks in which the magnitude of the results of the operations falsified students' intuitions that multiplication makes bigger and division makes smaller (e.g., 3 × _ = 2), and in which the missing number was a rational number smaller than 1. In all those studies (Christou, 2015a(Christou, , 2015b(Christou, , 2017, students had statistically significantly higher accuracy rates for the items aligned with their intuitions about the size of the results of each operation than items that falsified these intuitions. Additional studies have shown similar performance differences (Obersteiner et al, 2016;Vamvakoussi et al, 2013;Van Hoof et al, 2015).…”

Section: The Dual Effect Of Nnb In Arithmetic Operations With Missingmentioning

confidence: 85%

“…Students' initial conception of numbers, coupled with their intuitions for the results of arithmetic operations, suggests the NNB may affect how students solve arithmetic problems with missing operands. To test this, Christou administered paper and pencil tasks to primary (Christou, 2015a(Christou, , 2015b and secondary school students (Christou, 2017). The tasks included arithmetic equations with operations between a given number and a missing number (e.g., 9 ÷ _ = 4).…”

Section: The Dual Effect Of Nnb In Arithmetic Operations With Missingmentioning

confidence: 99%

Natural number bias in arithmetic operations with missing numbers – A reaction time study

et al. 2020

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When reasoning about numbers, students are susceptible to a natural number bias (NNB): When reasoning about non-natural numbers they use properties of natural numbers that do not apply. The present study examined the NNB when students are asked to evaluate the validity of algebraic equations involving multiplication and division, with an unknown, a given operand, and a given result; numbers were either small or large natural numbers, or decimal numbers (e.g., 3 × _ = 12, 6 × _ = 498, 6.1 × _ = 17.2). Equations varied on number congruency (unknown operands were either natural or rational numbers), and operation congruency (operations were either consistent – e.g., a product is larger than its operand – or inconsistent with natural number arithmetic). In a response-time paradigm, 77 adults viewed equations and determined whether a number could be found that would make the equation true. The results showed that the NNB affects evaluations in two main ways: a) the tendency to think that missing numbers are natural numbers; and b) the tendency to associate each operation with specific size of result, i.e., that multiplication makes bigger and division makes smaller. The effect was larger for items with small numbers, which is likely because these number combinations appear in the multiplication table, which is automatized through primary education. This suggests that students may count on the strategy of direct fact retrieval from memory when possible. Overall the findings suggest that the NNB led to decreased student performance on problems requiring rational number reasoning.

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Natural number bias in operations with missing numbers

Cited by 25 publications

References 26 publications

Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia

Praxeological Change and the Density of Rational Numbers: The Case of Pre-service Teachers in Denmark and Indonesia

The Recruitment of Shifting and Inhibition in On‐line Science and Mathematics Tasks

Natural number bias in arithmetic operations with missing numbers – A reaction time study

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