Many researchers agree that children's difficulty with fraction and rational numbers is associated with their whole number knowledge, but they disagree on the origin of the whole number bias. This article reviews three explanations of the nature of the bias. These accounts diverge on the questions of whether or not early quantitative…

In this article we describe the mathematics curriculum and teaching practices in a purposive sample of high-poverty elementary schools working with 3 of the most widely disseminated comprehensive school reform programs in the United States. Data from 19,999 instructional logs completed by 509 first-, third-, and fourth-grade teachers in 53 schools…

Patients with corticobasal degeneration (CBD) appear to have impaired number knowledge. We examined the nature of their number deficit while we tested the hypothesis that comprehension of larger numbers depends in part on verbal mediation. We evaluated magnitude judgments and performance on number conservation measures rooted in Piagetian theory…

This paper is written on a level accessible to college/university students of mathematics who are taking second-year, algebra based, mathematics courses beyond calculus I. This article combines material from geometry, trigonometry, and number theory. This integration of various techniques is an excellent experience for the serious student. The…

Two experiments investigated developmental changes in large number discrimination with visual-spatial arrays. Previous studies found that 6-month-old infants were able to discriminate arrays that differ by a ratio of 1:2 but not 2:3. We found that by 10 months, infants were able to reliably discriminate 8 from 12 elements (2:3) but not 8 from 10…

We investigate the possibility of approximating the value of a definite integral by approximating the integrand rather than using numerical methods to approximate the value of the definite integral. Particular cases considered include examples where the integral is improper, such as an elliptic integral. (Contains 4 tables and 2 figures.)

The meaning of International Standard Book Numbers (ISBN) and the mathematics involved in the use of a check digit to catch errors in number transcription are discussed. (JM)

Bases such as 5 and 12 provide the same structural place value benefits as base 10. However, when numbers less than one are concerned, base 10 provides friendly decimals for the most common fractions of half, quarter, three-quarters. Base 5 is not user friendly at all in this regard. Base 12 would provide nice dozenimals(?) for the same…

There are two different conceptions of the innate basis for numerical abilities. On the one hand, it is claimed that infants possess a "number module" that enables them to construct concepts of the exact numerosities of sets upon which arithmetic develops (e.g. Butterworth, 1999 ; Gelman & Gallistel, 1978). On the other hand, it has been proposed…

This study investigated the effect of preschool experience (two types of preschool: Madrasa and non-Madrasa) on the cognitive development of children in East Africa. In the three countries studied (Kenya, Uganda, and Tanzania/Zanzibar) preschool education is burgeoning and government standards are being set. This quasi experimental evaluation used…

The number line estimation task captures central aspects of children's developing number sense, that is, their intuitions for numbers and their interrelations. Previous research used children's answer patterns and verbal reports as evidence of how they solve this task. In the present study we investigated to what extent eye movements recorded…

In this note, we will focus on several applications on the Dirichlet's box principle in Discrete Mathematics lesson and number theory lesson. In addition, the main result is an innovative game on a triangular board developed by the authors. The game has been used in teaching and learning mathematics in Discrete Mathematics and some high schools in…

Elapsed-time problems are notoriously difficult for children. Instruction on techniques for teaching and learning elapsed time is not emphasized in current mathematics education literature. Nor is it addressed in "Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence" (NCTM 2006). This absence of…

Do children draw upon abstract representations of number when they perform approximate arithmetic operations? In this study, kindergarten children viewed animations suggesting addition of a sequence of sounds to an array of dots, and they compared the sum to a second dot array that differed from the sum by 1 of 3 ratios. Children performed this…