Activity-based instructional strategies promote physical interaction with manipulatives to represent abstract concepts. As a means to improve student achievement in mathematics, educational researchers and practitioners often recommend instructional strategies that capitalize on the assumed benefits of manipulatives. Recent research has indicated…

In this proposal, I introduce a method for modeling the dynamics of a sixth-grade student's accommodation of his fractions scheme to include a disembedding operation (Steffe & Olive, 2010). I will describe a three-part approach consisting of a constructivist teaching experiment, retrospective analysis, and stochastic modeling of the student's…

Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory--the domains…

This study was conducted to investigate the immediate impacts of preschool attendance on Turkish children's mathematics achievement. The participants were 200 children who attended or did not attend preschool. The number and operation task and the geometric shapes sorting task were used as the data collection tools. The children who attended…

In light of conceptual metaphor theory, historical mathematicians' and students' difficulty with negative numbers reveals that the collecting objects metaphor may be a cognitive obstacle to those first learning about negative numbers. Moreover, consistency of physical motions with targeted ideas is a factor of cognition. Thus, this…

This study tested the hypothesis that subitizing ability may cause achievement differences in mathematics especially for students with mathematics learning disabilities. Students from 1st through 4th grade were applied to curriculum based math achievement tests (MAT). Based on MAT scores, they were divided into four groups as Mathematics Learning…

The maturation of multiplicative thinking is key to student progress in middle school as rational number, ratio, and proportion concepts are encountered. But many students arrive from the intermediate grades and falter in developing this essential disposition. Elementary students have historically learned multiplication and division as operation…

The purpose of the present study was to explore the 3rd-grade cognitive predictors of 5th-grade computational skill with rational numbers and how those are similar to and different from the cognitive predictors of whole-number computational skill. Students (n=688) were assessed on incoming whole-number calculation skill, language, nonverbal…

Like Pascal's triangle, Faulhaber's triangle is easy to draw: all you need is a little recursion. The rows are the coefficients of polynomials representing sums of integer powers. Such polynomials are often called Faulhaber formulae, after Johann Faulhaber (1580-1635); hence we dub the triangle Faulhaber's triangle.

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated…

Whether and in what way enumeration processes differ for small and large sets of objects is still a matter of debate. In order to shed light on this issue, EEG data were obtained from 60 normally developing elementary school children. Adopting a standard non-symbolic numerical comparison paradigm allowed us to manipulate numerical distance between…

The Trail-Making Test (TMT) is a widely used neuropsychological measure that assesses visuomotor abilities and cognitive flexibility. For the TMT-A condition participants are required to locate and connect numbers (i.e. 1-2-3...) while in the TMT-B condition participants perform the set-shifting task of locating and connecting numbers and letters…

For number-form synaesthetes, digits occupy idiosyncratic spatial locations. Atypical to the mental number line that extends horizontally, the synaesthete (L) experiences the numbers 1-10 vertically. We used a spatial cueing task to demonstrate that L's attention could be automatically directed to locations within her number-space--being faster to…

This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds…

Time perception has not been comprehensively examined in mathematics difficulties (MD). Therefore, verbal time estimation, production, and reproduction were tested in 13 individuals with MD and 16 healthy controls, matched for age, sex, and intellectual skills. Individuals with MD performed comparably to controls in time reproduction, but showed a…