What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density

Vamvakoussi, Xenia; Christou, Konstantinos P.; Mertens, L.; Dooren, Wim Van

doi:10.1016/j.learninstruc.2011.03.005

Cited by 42 publications

(34 citation statements)

References 12 publications

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“…However, it will lead to an incorrect answer on items that are incongruent (i.e., relying on natural number knowledge leads to a different response than relying on natural number knowledge), unless type 2 processing inhibits the intuitive response tendency. Using paper–pencil tests, previous studies found that people were in fact systematically more accurate on congruent than incongruent items (Vamvakoussi et al ., ; Van Hoof et al ., ). This performance pattern is an indicator of the natural number bias.…”

Section: Introductionmentioning

confidence: 99%

“…In the matter of density, students have been found to believe that, as with natural numbers, rational numbers have predecessors and successors (e.g., that 4/5 comes after 3/5), although this is not the case. Similarly, students seem to believe that there are no, or only a finite number of numbers, between two pseudo‐consecutive rational numbers such as 3/5 and 4/5 (Vamvakoussi, Christou, Mertens, & Van Dooren, ; Vamvakoussi & Vosniadou, ). In fact, however, there are always infinitely many numbers between any two rational numbers.…”

Section: Introductionmentioning

confidence: 99%

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

Obersteiner

Hoof

Verschaffel

et al. 2015

British J of Psychology

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Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth-grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions. We conclude that it is possible for experts to be unaffected by the natural number bias on rational number tasks when they use strategies that do not involve natural numbers.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Obersteiner

Hoof

Verschaffel

et al. 2015

British J of Psychology

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“…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%

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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“…They were also more likely to say that these infinite intermediate numbers have the same representational form as the interval end points (i.e., infinitely many decimals between decimals, and infinitely many fractions between fractions). These results were replicated in a cross-cultural comparison with Flemish secondary students (Vamvakoussi, Christou, Mertens, & Van Dooren, 2011).…”

Section: Psychological Experimentsmentioning

confidence: 57%

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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What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density

Cited by 42 publications

References 12 publications

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

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

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

Bridging psychological and educational research on rational number knowledge

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