Who can escape the natural number bias in rational number tasks? A study involving students and experts

Obersteiner, Andreas; Hoof, Jo Van; Verschaffel, Lieven; Dooren, Wim Van

doi:10.1111/bjop.12161

Cited by 29 publications

(19 citation statements)

References 44 publications

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“…It is particularly in line with research on academic mathematicians who solved almost all fraction comparison problems correctly but showed a natural number bias in terms of response times in problems with common components (Obersteiner et al, 2013). It seems that the natural number bias is particularly likely to occur in problems in which participants focus strongly on the natural number components (Alibali & Sidney, 2015;Obersteiner, Van Hoof, Verschaffel, & Van Dooren, 2016). …”

Section: Discussionsupporting

confidence: 84%

Do we have a sense for irrational numbers?

Obersteiner

Hofreiter

2017

J. Numer. Cogn.

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Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers' whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers. Number sense refers to the adaptive and flexible use of numbers. A precondition of number sense is the ability to quickly assess magnitude information represented by number symbols (e.g., 12, -5, or 3/4). Curricula and educational standards for school mathematics (e.g., CCSSI, 2010) include an understanding of number magnitudes as an important goal of school mathematical learning. Furthermore, theories from cognitive psychology consider understanding of number magnitudes as a fundamental step of numerical development (Siegler, Thompson, & Schneider, 2011; see Siegler, Fazio, Bailey, & Zhou, 2013). The reason why understanding number magnitudes is considered so important is that possessing magnitudes is a shared property of all types of numbers which students learn about at school, whether natural, whole, rational, or real numbers. i Thus, magnitudes are a key feature of all real numbers.In spite of the widely accepted importance of number magnitude understanding, research into magnitude understanding has almost exclusively focused on natural numbers (i.e., the counting numbers) (e.g., De Smedt, Verschaffel, & Ghesquière, 2009), integers (including negative numbers, e.g., Varma & Schwartz, 2011;Young & Booth, 2015), and rational numbers in decimal notation (e.g., 0.2) (Desmet, Grégoire, & Mussolin, 2010) or fraction notation (e.g., 1/5) (Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). Studies have amply shown Journal of Numerical Cognition jnc.psychopen.eu | 2363-8761 that most people are able to assess magnitudes of these numbers, albeit with more difficulty and less automatically for larger and less common numbers (e.g., fractions with two-digit components such as 31/73) than for smaller numbers and numbers t...

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Section: Discussionsupporting

confidence: 84%

Do we have a sense for irrational numbers?

Obersteiner

Hofreiter

2017

J. Numer. Cogn.

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“…These response patterns align with dual process theories that rely on the distinction and interaction between intuitive and analytical reasoning processes (Gillard, Van Dooren, Schaeken, & Verschaffel, 2009;Vamvakoussi et al, 2013). Several reaction time studies with student and adult populations support the theories of co-existence and intuitive interference of natural number knowledge by showing statistically significant differences in accuracy rates and response times between tasks that were in-line with intuitions about the results of arithmetic operations and tasks that falsified these intuitions (Obersteiner, Van Hoof, Verschaffel, & Van Dooren, 2016;Vamvakoussi et al, 2012;Van Hoof et al, 2015).…”

supporting

confidence: 68%

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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“…These questions are particularly relevant in the domain of fraction learning. A large body of work demonstrates that children's prior knowledge of whole numbers negatively biases their understanding of fraction symbols (e.g., Hartnett & Gelman, 1998;Mack, 1995;Meert, Grégoire, & Noël, 2010;Smith, Solomon, & Carey, 2005;Stafylidou & Vosniadou, 2004; and operations (Obersteiner, Van Hoof, Verschaffel, & Van Dooren, 2016;Siegler & Pyke, 2013;Van Hoof, Vandewalle, Verschaffel, & Van Dooren, 2015). This well-documented phenomenon is known as the whole number bias (e.g., Ni & Zhou, 2005).…”

Section: Fractions and The Whole Number Biasmentioning

confidence: 99%

Creating a context for learning: Activating children’s whole number knowledge prepares them to understand fraction division

Sidney

Alibali

2017

J. Numer. Cogn.

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When children learn about fractions, their prior knowledge of whole numbers often interferes, resulting in a whole number bias. However, many fraction concepts are generalizations of analogous whole number concepts; for example, fraction division and whole number division share a similar conceptual structure. Drawing on past studies of analogical transfer, we hypothesize that children’s whole number division knowledge will support their understanding of fraction division when their relevant prior knowledge is activated immediately before engaging with fraction division. Children in 5th and 6th grade modeled fraction division with physical objects after modeling a series of addition, subtraction, multiplication, and division problems with whole number operands and fraction operands. In one condition, problems were blocked by operation, such that children modeled fraction problems immediately after analogous whole number problems (e.g., fraction division problems followed whole number division problems). In another condition, problems were blocked by number type, such that children modeled all four arithmetic operations with whole numbers in the first block, and then operations with fractions in the second block. Children who solved whole number division problems immediately before fraction division problems were significantly better at modeling the conceptual structure of fraction division than those who solved all of the fraction problems together. Thus, implicit analogies across shared concepts can affect children’s mathematical thinking. Moreover, specific analogies between whole number and fraction concepts can yield a positive, rather than a negative, whole number bias.

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Who can escape the natural number bias in rational number tasks? A study involving students and experts

Cited by 29 publications

References 44 publications

Do we have a sense for irrational numbers?

Do we have a sense for irrational numbers?

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

Creating a context for learning: Activating children’s whole number knowledge prepares them to understand fraction division

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