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Martin A. Simon

Prospective teachers' knowledge of division was investigated through an open-response written instrument and through individual interviews. Problems were designed to focus on two aspects of understanding division: connectedness within and between procedural and conceptual knowledge and knowledge of units. Results indicated that the prospective teachers' conceptual knowledge was weak in a number of areas including the conceptual underpinnings of familiar algorithms, the relationship between partitive and quotitive division, the relationship between symbolic division and real-world problems, and identification of the units of quantities encountered in division computations. The research also characterized aspects of individual conceptual differences. The research results suggest conceptual areas of emphasis for the mathematical preparation of elementary teachers.

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Martin A. Simon

All fields of research need to reconsider, on an ongoing basis, their standards for research quality. At the present time in mathematics education, such discussion is particularly critical to the ongoing growth and strength of the field. Mathematics education research has evolved significantly in the last 20 years. The most sweeping change has been the acceptance and subsequent predominance of qualitative research methodologies. With this change has come a plethora of new and adapted methodologies. On one hand, this is a sign of the field's vitality. On the other hand, the rapid changes and diversity in how research is conducted have posed major challenges with respect to standards for research quality (Lesh, Lovitts, & Kelly, 2000).

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Martin A. Simon

Constructivist theory has been prominent in recent research on mathematics learning and has provided a basis for recent mathematics education reform efforts. Although constructivism has the potential to inform changes in mathematics teaching, it offers no particular vision of how mathematics should be taught; models of teaching based on constructivism are needed. Data are presented from a whole-class, constructivist teaching experiment in which problems of teaching practice required the teacher/researcher to explore the pedagogical implications of his theoretical (constructivist) perspectives. The analysis of the data led to the development of a model of teacher decision making with respect to mathematical tasks. Central to this model is the creative tension between the teacher's goals with regard to student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students.

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Martin A. Simon

In my article, I framed a developing model of mathematics teaching. As such, the model presented was neither complete nor elaborated in detail The article was designed to generate discussion that can contribute to the further development of this and other models. Steffe and D'Ambrosio's response contributes in important ways to this discussion. They indicate an acceptance of many of the components of the model and elaborate several of them.

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Martin A. Simon

If the current mathematics education reform is to have significant impact on the mathematical development of students, the mathematical understandings encompassed by the multiplicative conceptual field are paramount. These mathematical understandings range from initial conceptions of multiplication and division through concepts of ratio and proportion. fractions, and linear and nonlinear functions. Not only is an understanding of multiplicative relationships important in the modeling of real world situations, but such relationships form the basis of the mathematics of probability, of similarity of geometric figures, and of rate of change.

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Martin A. Simon

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 about, and work with, the multiple theories available. I present alternatives to views of the competition or supersession of theories and indicate the kinds of discussions that will support effective theory use in mathematics education research. I describe the potential for mathematics education researchers to make informed, justified choices of a theory or theories to address particular research agendas.

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Martin A. Simon and Ron Tzur

In this article we articulate a methodology for studying mathematics teacher development in the context of reform. The generation of accounts of teachers' practice, an adaptation of the case study, provides an approach to understanding teachers' current practice and to viewing their current practice in the context of development toward envisioned reforms. The methodology is an alternative both to studies that focus on teachers' deficits and to teachers' own accounts of their practice. Conceptual frameworks developed within the mathematics education research community are applied to the task of investigating the nature of practice developed by teachers in transition. We characterize this methodology as explicating the teacher's perspective from the researchers' perspectives.

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Martin A. Simon and Glendon W. Blume

This study focused on understanding the evaluation of the area of a rectangular region as a multiplicative relationship between the lengths of the sides. The analysis of data from a whole-class teaching experiment involving prospective elementary teachers resulted in the development of a description of the quantitative reasoning involved. Important aspects of this reasoning include the anticipation of a rectangular array of units as the structure of the area quantity and the subsequent constituting of the units based on the linear measures. The analysis of the prospective teachers' understandings proved useful in generating additional problems. The teachers' engagement with these problems resulted in their developing more complete understandings of the constitution of area units.

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Martin A. Simon, Nicora Placa and Arnon Avitzur

Tzur and Simon (2004) postulated 2 stages of development in learning a mathematical concept: participatory and anticipatory. In this article, we 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. We describe our modifications to and further explicate and exemplify the theoretical underpinnings of these stage constructs. We introduce a representation scheme and use it to trace the development of a concept from initial activity, through the participatory stage, and to the anticipatory stage.

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Martin A. Simon, Ron Tzur, Karen Heinz, Margaret Kinzel and Margaret Schwan Smith

We postulate a construct, perception-based perspective, that we consider to be fundamental to the practices of many teachers currently participating in mathematics education reform in the United States. The postulation of the construct resulted from analyses of data from teaching experiments in teacher education classes with a combined group of prospective and practicing teachers and from case studies with individuals from that group. A perception-based perspective is grounded in a view of mathematics as a connected, logical, and universally accessible part of an ontological reality. From this perspective, learning mathematics with understanding requires learners' direct (firsthand) perception of relevant mathematical relationships. Analyses of data are presented and implications of the construct for mathematics teaching and mathematics teacher education are discussed.