We report on an innovative design of algorithmic analysis that supports automatic online assessment of students' exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic…

One of the challenges of understanding the complexity of so-called reform mathematics instruction lies in the observational tools used to capture it. This article introduces a unique tool, drawing from commognitive theory, for describing classroom discussions. The Realization Tree Assessment tool provides an image of a classroom discussion,…

Quantitative and qualitative analyses of 37 first-graders' solutions to addition problems were conducted to re-examine inconsistencies in children's progress from counting-all to counting-on. The authors applied the constructivist's participatory-anticipatory stage distinction as a tool for fine-grained assessment. Among solutions given by…

Currently, there are more theories of learning in use in mathematics education research than ever before (Lerman & Tsatsaroni, 2004). Although this is a positive sign for the field, it also has brought with it a set of challenges. In this article, I identify some of these challenges and consider how mathematics education researchers might think…

The goal of this article is to examine students' mathematical development that occurs as a teacher works within each of 2 zones of learning: students' zones of proximal development (ZPD) and students' zones of potential construction (ZPC). ZPD, proposed by Vygotsky, is grounded in a social constructivist perspective on learning, whereas ZPC,…

In this rejoinder to Baroody and colleagues (2007), I point out that there are several areas of agreement between my position and that of Baroody et al.--most notably that both procedural knowledge and conceptual knowledge are of critical importance in students' learning of mathematics. However, there are issues on which Baroody et al. and I do…

The paper discusses how complex mathematical behaviors might possibly be accounted for in terms of a finite set of rules. Behaviors involved in knowing mathematical systems are extended to logical interrelationships between properties of mathematical systems. (JG)

Discusses intersubjectivity and examines limitations of radical constructivism. Compares Piaget's positioning of the individual in relation to social life with that of Vygotsky in support of the claim that radical constructivism does not offer enough as an explanation of children's learning of mathematics. (Author/MKR)

Self-instructional booklets containing the advance organizer, cognitive set, and post organizer varying according to experimental treatment were randomly assigned within nine Algebra 2 classes. Results showed a significant interaction between advance and post organizers: either one used separately facilitated learning and retention, but when used…

The authors challenge some of the conclusions reached about Piaget by Brian Rotman in his book. (MK)

The ability of students in grades 1-12 to generate proofs of theorems in an unfamiliar, one axiom, abstract system was investigated. There were no significant differences in performance of two levels of secondary students; fourth- through sixth-grade students were also able to develop proofs, but needed more time. (SD)

Programed units of instruction designed with four different sequences of characterizations and exemplifications of disjunctive concepts were given to preservice elementary school teachers. Students were tested on three levels of concept attainment, with differences between strategies resulting, as one moves through the levels of cognitive…

An outline of a model that provides a new perspective on the conceptual foundations of mathematics and a new approach for the study of children's acquisition of numerical operations is proposed. Attention is viewed as a pulse-like succession of moments and is supported by recent data from neurophysiology. (MP)