With democracy in mind, promoting students' cognitive, personal, and social development can inform and shape the mathematics curriculum and classroom practice with the goal of their becoming more capable, self-reflective, and socially aware human beings. Toward that realization, their mathematics experience could include: heuristics, as it…

How people see the world, even how they research it, is influenced by beliefs. Some beliefs are conscious and the result of research, or at least amenable to research. Others are largely invisible. They may feel like "common knowledge" (though myth, not knowledge), unrecognized premises that are part of the surrounding culture. As we…

In this review, we attempt to integrate two crucial aspects of numerical development: learning the magnitudes of individual numbers and learning arithmetic. Numerical magnitude development involves gaining increasingly precise knowledge of increasing ranges and types of numbers: from non-symbolic to small symbolic numbers, from smaller to larger…

This article overviews the theoretical and applied works of the psychologist and pedagogue Leonid Zankov. Zankov's model of teaching is based on Vygotsky's theory that appropriate teaching methods stimulate cognitive development, whose core notion is the Zone of Proximal Development. This educational psychology research was verified by large scale…

The dictionary definition of constraint is one-sided, solely restrictive. The problem-solving definition is two-sided. Constraints come in pairs. One retains its restrictive function, precluding something specific; the other directs search for its substitute. The paired constraint model is applied to both domain and classroom. I discuss the…

Bayesian orthodoxy posits a tight relationship between conditional probability and updating. Namely, the probability of an event "A" after learning "B" should equal the conditional probability of "A" given "B" prior to learning "B". We examine whether ordinary judgment conforms to the orthodox view. In three experiments we found substantial…

The purpose of this article is to elaborate Cottrill et al.'s (1996) conceptual framework of limit, an explanatory model of how students might come to understand the limit concept. Drawing on a retrospective analysis of 2 teaching experiments, we propose 2 theoretical constructs to account for the students' success in formulating and understanding…

A multiple-probe-across-problem-sets (tasks) design was used to evaluate the effects of the Detect, Practice, and Repair (DPR) on multiplication-fact fluency development in seven third-grade students nominated by their teacher as needing remediation. DPR is a multicomponent intervention and begins with a group-administered, metronome-paced…

A content analysis of over 28,000 pages from 141 elementary school mathematics textbooks published between 1900 and 2000 shows that widely used mathematics textbooks have changed substantially. Textbooks from the early part of the century were typically narrow in content but presented substantial amounts of advanced arithmetic and also asked…

This study investigated the comprehension and effectiveness of teaching formal, probabilistic decision-making skills to middle school students. Two specific objectives were to determine (1) if middle school students can comprehend a probabilistic decision-making approach, and (2) if exposure to the modeling approaches improves middle school…

The CAME project was inaugurated in 1993 as an intervention delivered in the context of mathematics with the intention of accelerating the cognitive development of students in the first two years of secondary education. This paper reports substantial post-test and long-term National examination effects of the intervention, yet, by discussing the…

This paper presents the results of a study of the structural development of young students' drawings of arrays, and in particular, the significance of using lines instead of drawing individual squares. Students' array drawings were classified on basis of numerical properties, and perceived structural similarities that reflected the spatial…

An evolution-based framework for understanding biological and cultural influences on children's cognitive and academic development is presented. The framework, which defines biologically primary and secondary classes of cognitive ability, is a foundation for examining current approaches to reform and mathematics instruction in the United States.…

Research advocating classroom use of the computer programming language LOGO is based on relatively long periods of instruction, plentiful provision of computers, and knowledgeable teachers. Recent research shows that LOGO may confuse children's understanding of angles. Considers implications for Scottish primary schools using LOGO as recommended…

This study investigated the effects of teaching basic math skills to 16 children (11 typically developing, 5 with developmental delays) in an integrated university preschool. "Connecting Math Concepts Level K (CMC--K"; Engelmann & Becker, 1995) was delivered by the classroom teacher to small groups of 4 to 6 children over 6.5 weeks.…