The paper details a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts to force limit discussion into the language of individual real numbers and equality. The system of near-numbers…

Calls for reform in mathematics education around the world state that proofs should be part of school mathematics at all levels. Turning these calls into a reality falls on teachers' shoulders. This paper focuses on one secondary school teacher's reactions to students' suggested proofs and justifications in elementary number theory. To determine…

The author investigated the effects of the concrete-representational-abstract (CRA) instructional sequence on the computation performance of students with specific learning disabilities and students identified as at risk for failure in mathematics. Researchers have showed the CRA sequence to be effective for teaching basic mathematics facts,…

Student achievement in mathematics has been a focal concern in the United States for many years. The National Research Council's 2001 report and the recent report of the National Mathematics Advisory Panel (2008) both called attention to student achievement in mathematics, and both called for all students to learn algebra by the end of eighth…

This article introduces a framework for the remediation of number combination (NC) deficits. Research on the remediation of NC deficits is summarized, and research program studies are used to illustrate the 3 approaches to remediation. The Framework comprises a 2-stage system of remediation. The less intensive stage implementing 1 of 3…

Previous research has reported that children's numerical equivalence judgments are affected by surface similarity and counting ability (e.g. Mix, Huttenlocher, & Levine, 1996; Siegel, 1973), a pattern that suggests categorization processes play a role in numerical development. However, because these studies involved memory for sets, large set…

This study investigated "individual differences" in different aspects of early number concepts in preschoolers. Eighty 4-year-olds from Oxford nursery classes took part. They were tested on accuracy of counting sets of objects; the cardinal word principle; the order irrelevance principle; and predicting the results of repeated addition…

Hannula and Lehtinen (2001, 2005) defined spontaneous focusing on numerosity (SFON) as the tendency to notice the relatively abstract attribute of number despite the presence of other attributes. According to nativists, an innate concept of one to three directs young children's attention to these "intuitive numbers" in everyday situations--even…

Studies have reported high correlations in accuracy across estimation contexts, robust transfer of estimation training to novel numerical contexts, and adults drawing mistaken analogies between numerical and fractional values. We hypothesized that these disparate findings may reflect the benefits and costs of learning linear representations of…

Grapheme-color synesthesia, when achromatic digits evoke an experience of a specific color (photisms), has been shown to be consistent, involuntary, and linked with number concept in adults, yet there have been no comparable investigations with children. We present a systematic study of grapheme-color synesthesia in children aged between 7 and 15…

This is unit three of a fifteen-unit secondary mathematics textbook. This unit contains two chapters. The first chapter discusses integers and the second chapter discusses rational numbers. Operations with both types of numbers as well as the structure of the systems are discussed. (MK)

This is one in a series of SMSG supplementary and enrichment pamphlets for high school students. This series is designed to make material for the study of topics of special interest to students readily accessible in classroom quantity. Topics covered include natural numbers, positive integers, sets, well ordering, lower bound, upper bound,…

Several excerpts from Euclid's "Elements" are cited, and their applications to the natural, positive rational, and positive real number systems are discussed. (SD)