The processing of number scales beyond whole numbers in development: Dissociations in arithmetic in Turner’s syndrome

The arithmetical skills in two children with Turner’s syndrome (TS), each the focus of a case study, were analysed in whole numbers and other number scales that have not been systematically explored previously, fractions, decimals, percentages, and negative numbers. The intention was to identify the fractionation of arithmetical skills. The two girls with TS showed dissociations of arithmetical skill in the calculation system of whole numbers that support its modular organization. Fractionation of skills was observed in some components of the other number scales, suggesting an analogous organization within these scales. The operational specificity of impairment within number scales but not others argued against a unitary arithmetical system but rather for autonomous operational scales within distinct number scales. A general model of arithmetic is proposed

Using resource graphs to represent conceptual change

We introduce resource graphs, a representation of linked ideas used when reasoning about specific contexts in physics. Our model is consistent with previous descriptions of resources and coordination classes. It can represent mesoscopic scales that are neither knowledge-in-pieces or large-scale concepts. We use resource graphs to describe several forms of conceptual change: incremental, cascade, wholesale, and dual construction. For each, we give evidence from the physics education research literature to show examples of each form of conceptual change. Where possible, we compare our representation to models used by other researchers. Building on our representation, we introduce a new form of conceptual change, differentiation, and suggest several experimental studies that would help understand the differences between reform-based curricula.Comment: 27 pages, 14 figures, no tables. Submitted for publication to the Physical Review Special Topics Physics Education Research on March 8, 200

Increasing cognitive inhibition with a difficult prior task:implications for mathematical thinking

Dual-process theories posit two distinct types of cognitive processing: Type 1, which does not use working memory making it fast and automatic, and Type 2, which does use working memory making it slow and effortful. Mathematics often relies on the inhibition of pervasive Type 1 processing to apply new skills or knowledge that require Type 2 processing. In two studies, we demonstrate that giving participants a difficult task (Raven’s Matrices) before a task that requires the inhibition of intuitive responses (the Cognitive Reflection Test) significantly improves performance. Our findings suggest that encountering a difficult task that requires Type 2 processing before completing a task that requires inhibition of Type 1 processing may encourage an enduring ‘Type 2’ mindset, whereby participants are more likely to spontaneously use Type 2 processing for a period of time. Implications for mathematics education are discussed

How many decimals are there between two fractions? aspects of secondary school students' understanding of rational numbers and their notation

We present an empirical study that investigated seventh-, ninth-, and eleventh-grade students' under-standing of the infinity of numbers in an interval. The participants (n = 549) were asked how many (i.e., a finite or infinite number of numbers) and what type of numbers (i.e., decimals, fractions, or any type) lie between two rational numbers. The results showed that the idea of discreteness (i.e., that fractions and decimals had "successors" like natural numbers) was robust in all age groups; that students tended to believe that the intermediate numbers must be of the same type as the interval endpoints (i.e., only decimals between decimals and fractions between fractions); and that the type of interval endpoints (natural numbers, decimals, or fractions) influenced students' judgments of the number of intermediate numbers in those intervals. We interpret these findings within the framework theory approach to conceptual change. © Taylor & Francis Group, LLC

Understanding the structure of the set of rational numbers: A conceptual change approach

In the present article, we argue that the conceptual change approach to learning can apply in the case of mathematics, taking into consideration the particular nature of mathematical knowledge and the neurobiological bases of mathematical cognition. In the empirical study that is reported in this article, we investigated ninth graders' understanding of algebraic and structural properties of rational numbers, from a conceptual change perspective. We make the point that understanding rational numbers is not indiscriminately difficult. We show that prior knowledge about natural numbers supports students dealing with algebraic properties of rational numbers, while the idea of discreteness is a fundamental presupposition, which constrains students' understanding of density. © 2004 Elsevier Ltd. All rights reserved

Bridging the Gap Between the Dense and the Discrete: The Number Line and the "Rubber Line" Bridging Analogy

In two experiments we explored the instructional value of a cross-domain mapping between "number" and "line" in secondary school students' understanding of density. The first experiment investigated the hypothesis that density would be more accessible to students in a geometrical context (infinitely many points on a straight line segment) compared to a numerical context (infinitely many numbers in an interval). The participants were 229 seventh to eleventh graders. The results supported this hypothesis but also showed that students' conceptions of the line segment were far from that of a dense array of points. We then designed a text-based intervention that attempted to build the notion of density in a geometrical context, making explicit reference to the number-to-points correspondence and using the "rubber line" bridging analogy (the line as an imaginary unbreakable rubber band) to convey the no-successor principle. The participants were 149 eighth and tenth graders. The text intervention improved student performance in tasks regarding the infinity of numbers in an interval; the "rubber line" bridging analogy further improved performance successfully conveying the idea that these numbers can never be found one immediately next to the other. © 2012 Copyright Taylor and Francis Group, LLC

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

It is widely documented that the density property of rational numbers is challenging for students. The framework theory approach to conceptual change places this observation in the more general frame of problems faced by learners in the transition from natural to rational numbers. As students enrich, but do not restructure, their natural number based prior knowledge, certain intermediate states of understanding emerge. This paper presents a study of Greek and Flemish 9th grade students who solved a test about the infinity of numbers in an interval. The Flemish students outperformed the Greek ones. More importantly, the intermediate levels of understanding-where the type of the interval endpoints (i.e., natural numbers, decimals, or fractions) affects students' judgments-were very similar in both groups. These results point to specific conceptual difficulties involved in the shift from natural to rational numbers and raise some questions regarding instruction in both countries. © 2011 Elsevier Ltd

Teachers' attitudes to and beliefs about web-based Collaborative Learning Environments in the context of an international implementation

Fifty-six teachers, from four European countries, were interviewed to ascertain their attitudes to and beliefs about the Collaborative Learning Environments (CLEs) which were designed under the Innovative Technologies for Collaborative Learning Project. Their responses were analysed using categories based on a model from cultural-historical activity theory [Engeström, Y. (1987). Learning by expanding: An activity-theoretical approach to developmental research. Helsinki: Orienta-Konsultit; Engeström, Y., Engeström, R., & Suntio, A. (2002). Can a school community learn to master its own future? An activity-theoretical study of expansive learning among middle school teachers. In G. Wells & G. Claxton (Eds.), Learning for life in the 21st century. Oxford: Blackwell Publishers]. The teachers were positive about CLEs and their possible role in initiating pedagogical innovation and enhancing personal professional development. This positive perception held across cultures and national boundaries. Teachers were aware of the fact that demanding planning was needed for successful implementations of CLEs. However, the specific strategies through which the teachers can guide students' inquiries in CLEs and the assessment of new competencies that may characterize student performance in the CLEs were poorly represented in the teachers' reflections on CLEs. The attitudes and beliefs of the teachers from separate countries had many similarities, but there were also some clear differences, which are discussed in the article. © 2005 Elsevier Ltd. All rights reserved