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…

This article considers the question of what specific actions a teacher might take to create a culture of inquiry in a secondary school mathematics classroom. Sociocultural theories of learning provide the framework for examining teaching and learning practices in a single classroom over a two-year period. The notion of the zone of proximal…

We articulate and explicate a mechanism for mathematics conceptual learning that can serve as a basis for the design of mathematics lessons. The mechanism, reflection on activity-effect relationships, addresses the learning paradox (Pascual-Leone, 1976), a paradox that derives from careful attention to the construct of assimilation (Piaget, 1970).…

Explores some of the possible contributions of complexity science to discussions of the teaching of mathematics. Focuses on the use of the vocabulary of complexity in the re-description of mathematical communities and the application of principles of complexity to the teaching of mathematics. Highlights compatible and complementary discussions…

Presents a case study in which two 8th grade students developed knowledge for modeling a physical device called a winch. Demonstrates that students have and can use criteria for evaluating algebraic representations. Explains how students can develop modeling knowledge by coordinating criteria with knowledge for generating and using algebraic…

Describes one pattern of reasoning observed in a mathematical activity with students in a grade 5 class and discusses ways in which the pattern is or is not mathematical. Describes patterns of reasoning in terms of ways of reasoning and degree of formulation of the reasoning. Distinguishes between mathematical reasoning and scientific reasoning in…

Introduces crossdisciplinarity as a strategy for highlighting the discrete notions of learning that psychology thus far has succeeded in coherently articulating. This strategy positions teachers to consult their own values, interests, and strengths in defining their own teaching priorities while at the same time marshaling accessible, theory-based…

Suggests that radical constructivists draw on a weak image of the role of social life. Argues that a thick notion of "social" leads to a complexity of sociocultural theories concerning the teaching and learning of mathematics, a perspective that is firmly located in the debates surrounding cultural theory over the last two decades. (Contains 63…

Mathematics education in schools can be viewed either as primarily a sociocultural phenomenon or as a nurturing of the individual's mathematical development. Argues that instead of taking the dichotomous view, contrasting the Vygotskian and the Piagetian perspectives, the two may be seen as separate "truths" providing different lenses through…

Utilizes the psychological and sociocultural components of a constructivist paradigm to provide a detailed analysis of how the cognitive constructions students make as they enumerate 3D arrays of cubes develop and change in an inquiry-based, problem-centered mathematics classroom. Describes the classroom work of three pairs of 5th-grade students…

Investigates, by conducting individual clinical interviews of 97 children ages 3 to 6, the criteria preschool children use to distinguish members of a class of shapes from other figures, emphasizing identification and descriptions of shapes and reasons for these identifications. Concludes that young children initially form schemas on the basis of…

Analyzes the use of the commutative property for solving multiplication problems by children who learn about multiplication in schools and by street vendors who solve multiplication problem through repeated addition. Concludes that the use of commutativity to solve multiplication problems is closely related to the use of multiplication. Contains…