The representation of fraction magnitudes and the whole number bias reconsidered

DeWolf, Melissa; Vosniadou, Stella

doi:10.1016/j.learninstruc.2014.07.002

Cited by 136 publications

(155 citation statements)

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“…However, even when the overall magnitude of the whole fractions is the most important predictor of response times and error rates, the components nevertheless exert an influence on the comparison process (e.g., DeWolf & Vosniadou, 2015;Meert et al, 2010;Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013). For instance, Meert et al found that the distance between numerators was also a significant predictor of response times, in addition to the overall distance of the whole fractions.…”

Section: Constructing Matched Fraction Pairsmentioning

confidence: 99%

“…Studies have employed either only single-digit fractions (Bonato et al, 2007;Huber et al, 2014;Meert et al, 2009Meert et al, , 2010 or a combination of singledigit and multidigit fractions (DeWolf & Vosniadou, 2015;Obersteiner et al, 2013;Schneider & Siegler, 2010). Ischebeck et al (2016) investigated explicitly whether single-digit and two-digit fractions are processed differently in a magnitude comparison task, and they found evidence that the results were similar for single-digit and two-digit fractions.…”

Section: Item Populationmentioning

confidence: 99%

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Using propensity score matching to construct experimental stimuli

et al. 2016

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Propensity score matching is widely used in various fields of research, including psychology, medicine, education, and sociology. It is usually applied to find a matched control group for a treatment group. In the present article, we suggest that propensity score matching might also be used to construct item sets matched for different parameters. We constructed stimuli to illustrate the use of propensity score matching in item construction for the exemplary cases of numerical cognition research and reading research. In particular, we provide a step-by-step approach, using the statistics software R, for how to apply propensity score matching for constructing matched stimuli. This approach involves deciding on a population of stimuli, determining and calculating the covariates, and finally applying the propensity-matching method to find a set of items matched to another predefined set. Thereby, we were able to construct well-matched item sets for both examples. Hence, we conclude that the propensityscore-matching method is useful for constructing matched stimuli. Further cases of application are discussed.

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Section: Constructing Matched Fraction Pairsmentioning

confidence: 99%

Section: Item Populationmentioning

confidence: 99%

Using propensity score matching to construct experimental stimuli

et al. 2016

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“…Research shows that school students (Meert, Grégoire, & Noël, 2010b;Van Hoof et al, 2013) Comparison problems are congruent when the larger fraction is composed of the larger components, as in "20/27 vs. 11/19". Researchers have referred to the performance differences between congruent and incongruent comparison problems as the "whole number bias" or "natural number bias" (Alibali & Sidney, 2015;DeWolf & Vosniadou, 2015;Ni & Zhou, 2005;Obersteiner et al, 2013). The existence of bias corroborates Obersteiner & Hofreiter 173 earlier findings suggesting that people activate natural number magnitudes automatically and unintentionally even if doing so is not necessary for solving the problem at hand (Hubbard, Piazza, Pinel, & Dehaene, 2005).…”

Section: Natural Number Bias In Number Comparisonmentioning

confidence: 54%

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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“…), and different populations (i.e., spanning from infants to adults, and from novices to expert Resnick et al, 1989); in the comparison of fractions (e.g., "the numerical value of a fraction increases when the value of the terms increases", Stafylidou & Vosniadou, 2004); and in arithmetical operations (e.g., "multiplication always makes bigger", Fischbein, Deri, Nello, & Marino, 1985). The interrelations among students' overreliance on natural number knowledge, their overconfidence in their erroneous answers, and the degree of resilience of their misconceptions have also been investigated within this research strand (Durkin & Rittle-Johnson, 2015; Fischbein, 1987;Merenluoto & Lehtinen, 2004).Another strand of research looks into the reasoning processes that underlie the manifestation of the bias.Reaction time studies have been used to investigate the observation that natural numbers "come to mind first",showing that-besides making more errors in incongruent tasks-people need more time to answer correctly in incongruent than congruent tasks (DeWolf & Vosniadou, 2015;Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013;Vamvakoussi, Van Dooren, & Verschaffel, 2012;Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). This line of research also investigates the possible connections between the manifestation of the bias and domain-general executive function skills such as inhibitory control (Gómez, Jiménez, Bobadilla, Reyes, & Dartnell, 2015;Vosniadou, 2014).…”

mentioning

confidence: 99%

“…Reaction time studies have been used to investigate the observation that natural numbers "come to mind first", showing that-besides making more errors in incongruent tasks-people need more time to answer correctly in incongruent than congruent tasks (DeWolf & Vosniadou, 2015;Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013;Vamvakoussi, Van Dooren, & Verschaffel, 2012;Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). This line of research also investigates the possible connections between the manifestation of the bias and domain-general executive function skills such as inhibitory control (Gómez, Jiménez, Bobadilla, Reyes, & Dartnell, 2015;Vosniadou, 2014).…”

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confidence: 99%

Bridging psychological and educational research on rational number knowledge

2018

Self Cite

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In this paper we focus on the development of rational number knowledge and present three research programs that illustrate the possibility of bridging research between the fields of cognitive developmental psychology and mathematics education. The first is a research program theoretically grounded in the framework theory approach to conceptual change. This program focuses on the interference of prior natural number knowledge in the development of rational number learning. The other two are the research program by Moss and colleagues that uses Case's theory of cognitive development to develop and test a curriculum for learning fractions, and the research program by Siegler and colleagues, who attempt to formulate an integrated theory of numerical development. We will discuss the similarities and differences between these approaches as a means of identifying potential meeting points between psychological and educational research on numerical cognition and in an effort to bridge research between the two fields for the benefit of rational number instruction.Keywords: number cognition, natural number bias, conceptual change, rational number development, educational implications Numerical cognition is a research area that appeals to mathematics education researchers, to cognitivedevelopmental psychologists and to neuroscientists. However, the researchers coming from these different fields approach numerical cognition in different ways in terms of theoretical perspectives, questions asked, methodologies used, and most importantly, of end goals (Berch, 2016). Thus there is great need for dialogue between psychological and educational research, particularly when it comes to implications for instruction. In With this article we contribute to this effort, arguing that some of the research that lies in the intersection of cognitive-developmental psychology and mathematics education can be fruitful for both fields and very relevant for instruction. We focus on the development of rational number knowledge and we present our research jnc.psychopen.eu | 2363-8761 program that is theoretically grounded in the framework theory approach to conceptual change (Vosniadou, Vamvakoussi, & Skopeliti, 2008;Vosniadou & Verschaffel, 2004). This research program focuses on the adverse effects of prior number knowledge in the transition from natural to rational numbers (aka the natural number bias) and the challenges it presents for students' numerical and algebraic reasoning. This phenomenon has been long noticed and studied by mathematics education researchers and has more recently attracted much attention by psychologists and neuroscientists. Apart from being a phenomenon of interest to both fields, research on the natural number bias has many important implications for instruction. We will compare our research program to two other programs that also illustrate the possibility of bridging research between psychology and mathematics education. The first is a program by Moss and colleagues who developed and tested a curriculum fo...

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The representation of fraction magnitudes and the whole number bias reconsidered

Cited by 136 publications

References 29 publications

Using propensity score matching to construct experimental stimuli

Using propensity score matching to construct experimental stimuli

Do we have a sense for irrational numbers?

Bridging psychological and educational research on rational number knowledge

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