The arithmetical performance of typically achieving 5- to 7-year-olds (N=29) was measured at four 6-month intervals. The same seven tasks were used at each time point: exact calculation, story problems, approximate arithmetic, place value, calculation principles, forced retrieval, and written problems. Although group analysis showed mostly linear…

In two experiments, 330 3- to 4-year-olds competed for stickers in a game in which the optimal response strategy was to point to an empty box that their opponent would receive in order to obtain a baited box for themselves. When the baited box contained stickers, children showed a strong tendency to point at the baited box and therefore lose the…

A number derivative is a numerical mapping that satisfies the product rule. In this paper, we determine all number derivatives on the set of integers modulo n. We also give a list of undergraduate research projects to pursue using these maps as a starting point.

Developing proficiency in algebra is the focus of instruction in high school mathematics courses and is a minimum expectation for high school completion for all students including those with learning difficulties. However, the foundation for success is laid in grades 4-8 (aged 9-14). In this paper, we assert that students' development of algebraic…

Researchers (Ball, 2003; Ma, 1999; Schulman, 1986) have long investigated how a teacher's subject matter and pedagogical content knowledge impact on students' learning of mathematics. In an attempt to account for the relationship between teacher knowledge and student learning, this study examined four Romanian first grade teachers' knowledge about…

The present study examined predictive power of behavioral self-regulation, family and child characteristics on children's number sense. The participants consisted of 101 kindergarten children. A subsample of 30 children was randomly chosen for the reliability procedures of Assessing Number Sense and Head, Toes, Knees and Shoulders instruments.…

This study investigated the effects of computer-based practice on the acquisition and maintenance of basic academic skills for two children with moderate to intensive disabilities. The special education teacher created individualized computer games that enabled the participants to independently practice academic skills that corresponded with their…

This article describes the affordances of using clinical interviews with struggling mathematics learners to inform intervention and instruction. A case study of a fourth grade student, Danny, is presented to give an illustrative example of the complexity of the student's mathematical understandings in terms of place value. From the clinical…

Small numbers are spontaneously associated with left space and larger numbers with right space (the SNARC effect), for example when classifying numbers by parity. This effect is often attributed to reading habits but a causal link has so far never been documented. We report that bilingual Russian-Hebrew readers show a SNARC effect after reading…

This article discusses students' perceptions as a source for understanding education. It addresses what didactically experienced children, called "didactikids", taught us about the empty number line as a didactical model for teaching whole number calculations. The article mainly reports on a student consultancy study carried out in the…

Given a sequence g[subscript k] greater than 0, the "g-factorial" product [big product][superscript k] [subscript i=1] g[subscript i] is extended from integer k to real x by generalizing properties of the gamma function [Gamma](x). The Euler-Mascheroni constant [gamma] and the beta and zeta functions are also generalized. Specific examples include…

Cohen Kadosh, Tzelgov, and Henik [Cohen Kadosh, R., Tzelgov, J., & Henik, A. (2008). A synesthetic walk on the number line: The size effect. "Cognition", 106, 548-557] present a new paradigm to probe properties of the mental number line. They describe two experiments which they argue to be inconsistent with the exact small number model proposed by…

Are small and large numbers represented similarly or differently on the mental number line? The size effect was used to argue that numbers are represented differently. However, recently it has been argued that the size effect is due to the comparison task and is not derived from the mental number line per se. Namely, it is due to the way that the…

In this article, we will show that the Pythagorean approximations of [the square root of] 2 coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers…

This book focuses on essential knowledge for teachers about rational numbers. It is organized around four big ideas, supported by multiple smaller, interconnected ideas--essential understandings. Taking teachers beyond a simple introduction to rational numbers, the book will broaden and deepen their mathematical understanding of one of the most…