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This paper describes an emerging approach to the design of task sequences and the theory that undergirds it. The approach aims at promoting particular mathematical concepts, understood as the result of reflective abstraction. Central to this approach is the identification of available student activities from which students can abstract the intended ideas. The approach differs from approaches in which learning to solve the problem posed is the intended learning. The paper illustrates the approach through data from a teaching experiment on division of fractions.

The goal of our research program is to explicate the learning of mathematical concepts in ways that are useful for instructional design and to develop design principles based on those explications. I review one type of concept and our elaboration of reflective abstraction, coordination of actions (COA) that accounts for its construction. I then postulate a second type of concept and a second type of reflective abstraction that accounts for its construction, the abstraction of commonality (AOC).

Mathematical understanding continues to be one of the major goals of mathematics education. However, what is meant by "mathematical understanding" is underspecified. How can we operationalize the idea of mathematical understanding in research? In this article, I propose particular specifications of the terms "mathematical concept" and "mathematical conception" so that they may serve as useful constructs for mathematics education research. I discuss the theoretical basis of the constructs, and I examine the usefulness of these constructs in research and instruction, challenges involved in their use, and ideas derived from our experience using them in research projects. Finally, I provide several examples of articulated mathematical concepts.

Reversibility of concepts, a key aspect of mathematical development, is often problematic for learners. In this theoretical paper, we present a typology we have developed for categorizing the different reverse concepts that can be related to a particular initial concept and explicate the relationship among these different reverse concepts. We discuss uses of the typology and how reversibility can be understood as the result of reflective abstraction. Finally, we describe two strategies for promoting reversibility that have distinct uses in terms of the types of reverse concepts they engender. We share empirical results which led to our postulation of these strategies and discuss their theoretical basis.

Tzur and Simon (2004) postulated 2 stages of development in learning a mathematical concept: participatory and anticipatory. The authors discuss the affordances for research of this stage distinction related to data analysis, task design, and assessment as demonstrated in a 2-year teaching experiment.

One of the challenges of learning ratio concepts is that it involves intensive quantities, a type of quantity that is more conceptually demanding than those that are evaluated by counting or measuring (extensive quantities). In this paper, we engage in an exploration of the possibility of developing reasoning about intensive quantities during the teaching of whole-number multiplication. The investigation is grounded in a measurement-based instructional design attempt that built on the Elkonyn-Davydov curricular approach. We report on this investigation and resulting implications for learning to reason about intensive quantities and understanding ratio concepts.

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). The mechanism is an elaboration of Piaget's (2001) reflective abstraction and is potentially useful for addressing some of the more intractable problems in teaching mathematics. Implications of the mechanism for lesson design are discussed and exemplified. (Contains 2 figures and 9 footnotes.)

Simon's (1995) development of the construct of hypothetical learning trajectory (HLT) offered a description of key aspects of planning mathematics lessons. An HLT consists of the goal for the students' learning, the mathematical tasks that will be used to promote student learning, and hypotheses about the process of the students' learning. However, the construct of HLT provided no framework for thinking about the learning process, the selection of mathematical task, or the role of the mathematical tasks in the learning process. Such a framework could contribute significantly to the generation of useful HLTs. In this article we demonstrate how an elaboration of reflective abstraction (i.e., reflection on activity-effect relationships), postulated by Simon, Tzur, Heinz, and Kinzel (in press), can provide such a framework and thus a theoretical elaboration of the HLT construct.

Outlines two mathematics teaching strategies: the exploration/discovery approach to mathematical concepts, and the use of pair group problem-solving. These require teachers to shift from the traditional role of lecture/demonstrator to a role demanding new skills in planning and facilitating student work. Includes 10 references. (MD)

Presented are three cases for intuitive understanding in secondary and college level geometry. Four ways to develop the intuition (physical experience, mutable manipulatives, visualization, and looking back) step are discussed. (YP)

For teachers to be effective in promoting understanding of mathematical concepts, they need to understand the nature of mathematical concepts. However, to date, the mathematics education literature has not focused on this important area of teacher understanding. One reason that attention to this issue has been lacking is that the term "mathematical concept" has generally been used with little specification. In recent theoretical work, I have explicated two types of mathematical (arithmetical) concepts. The first type involves knowledge of the logical necessity of a mathematical relationship. The second type specifies what it means to understand an arithmetic operation. In this commentary, I examine the implications of this work for primary teacher understanding about the nature of arithmetical concepts and challenges inherent in promoting those understandings. [ABSTRACT FROM AUTHOR]

Abstract This article was prompted by the thoughtful commentaries of Norton, Tzur, and Dreyfus. Their commentaries pointed out important ideas that were left implicit or only partially explained. The clarifications made here include how the Learning Through Activity (LTA) research program differs from related research programs, the relation of the LTA theoretical framework to scheme theory, our choice to not employ the construct of perturbation in explaining learning, and the structure of hypothetical learning trajectories. In addition, I discuss a type of mathematical concept that we have not discussed previously. [ABSTRACT FROM AUTHOR]