In a recent paper, Alcock and Inglis (in press) noted a distinction between the way that Weber (in press) and they defined syntactic and semantic proof productions. Weber argued that "a syntactic proof production occurs when one works predominantly within the representation system of proof [...] Alternatively, a semantic proof production occurs…

Weber (2009) suggested that counterexamples can be generated by a syntactic proof production, apparently contradicting our earlier assertion (Alcock & Inglis, 2008). Here we point out that this ostensible difference is the result of Weber working with theoretical definitions that differ slightly from ours. We defend our approach by arguing that…

Presents a perspective on the nature of the use of proofs in high school geometry. Compares three currently used approaches to the geometry curriculum: (1) traditional geometry with no explanation of the axiomatic system; (2) hands-on geometry with no proofs until the end of the course; and (3) experimental geometry with no proofs. (DDR)

Argues that the organization of cognitive structures for technical domains can be visualized as a network of connected thinkable models. Describes a taxonomy of models that has been developed and discusses the issue of how representations relate to human modes of perception and action. Contains 25 references. (DDR)

Profiles a middle school mathematics teacher and examines her use of two problems from a pilot version of a 6th grade unit developed by a mathematics curriculum project. Reports that problem-solving-oriented curricula provide opportunities for students to make mathematics connections and leads to student confusion and uncertainty. Contains 35…

The status of mathematics instruction, especially in the elementary school, is discussed. A meaningful, holistic approach is advocated, rather than an emphasis on rules and procedures. (MNS)

Examples of relatively novel computer software environments from the representation perspective are described. Even more novel approaches to curriculum reform to cultivate higher-order thinking skills are then discussed. (MNS)

How multiplication is usually taught in school and how it could be taught are discussed. Development of understanding is illustrated through children's words and work. (MNS)

How the notion of limit can be developed through a meaningful approach is discussed. Selected portions of the high school calculus course are described, and errors on a test are analyzed. (MNS)

How physiological disciplines can contribute to the study of how people learn mathematics is considered. Manipulative and experiential learning, sequential versus hierarchical organization, declarative versus procedural knowledge, and short-term versus long-term memory are among the points discussed. (MNS)

Presents a constructivist view of mathematics, mathematics teaching, and mathematics achievement, and differentiates this view as an alternative to the typical traditional perspective of these terms. Discusses the roots of this approach in contemporary information-processing psychology. (TW)

The first in a series of three articles, successful instructional materials from a 15-year software development effort are analyzed and characterized with special attention given to educational experiences intended to shape students' perceptions of the fundamental nature, interconnectedness, and usefulness of mathematics. The software programs…

The second in a series of three articles, some themes of an undercurrent of educational objectives that move beyond the evident content goals are outlined. The themes relate the computer software materials analyzed in Part I to the cognitive experiences that students should have in learning about subject matter. (MDH)

Provides a brief summary of current research in mathematics education at the college level. Explores the current needs of college-level faculty. Suggests ways of coping with the apparent perception that some of the best contemporary research is useless or irrelevant from the practitioner's point of view. (22 references) (JJK)

Presents a look at the history of algebra with a historical-psychological framework. This article is one of the keynote addresses of the Algebra Working Group of the Seventh International Conference on Mathematical Education held in Quebec City, Canada in August 1992. (33 references) (MKR)