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March 2000, Volume 31, Issue 2

FEATURES

The Evidential Basis for Knowledge Claims in Mathematics Education Research
Frank K. Lester, Dylan William
During the past few years, diverse individuals and groups have begun to promote a variety of old and new instructional approaches, programs, and policies for mathematics education (Dixon, Carnine, Lee, Wallin, & Chard, 1998; Jacob, 1997; Kilpatrick, 1997; Wu, 1997). Researchers are being exhorted to gather and analyze data for evaluating the efficacy of various instructional approaches and curricula. Moreover, individuals both within and outside of the mathematics education research community have offered "evidence" to support specific agendas. In view of this turbulent state of affairs, it seems especially timely for researchers and those interested in research to engage in discussions of the notions that are at the heart of all educational research activity. This essay addresses one of these fundamental notions--evidence.

The Nature and Roles of Research in Improving Achievement in Mathematics
Douglas Carnine, Russell Gersten
In describing the role research should play in the new National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics: Discussion Draft (1998), the authors noted, "With an emphasis on understanding mathematics that is fundamental, educators can be supported as they move beyond some of the superficial interpretations that have been made of ideas from the previous Standards documents. Currently, there are instructional programs that have emphasized solely some of the pedagogical intentions of standards--such as discourse, worthwhile mathematical tasks, or learning from problems--without sufficient attention to students' learning of mathematics content. (p. 17)" However, a major issue in the field of mathematics education is what type or types of research should play this critical role (Kelly & Lesh, 1999). In particular, a major focus in the current debates is whether controlled experimental or quasi-experimental research--conducted in real classroom settings--should be the predominant determinant of policy and practice

Young Children's Intuitive Understanding of Rectangular Area Measurement
Lynne N. Outhred, Michael C. Mitchelmoore
The focus of this article is the strategies young children use to solve rectangular covering tasks before they have been taught area measurement. One hundred fifteen children from Grades 1 to 4 were observed while they solved various array-based tasks, and their drawings were collected and analyzed. Children's solution strategies were classified into 5 developmental levels; we suggest that children sequentially learn 4 principles underlying rectangular covering. In the analysis we emphasize the importance of understanding the relation between the size of the unit and the dimensions of the rectangle in learning about rectangular covering, clarify the role of multiplication, and identify the significance of a relational understanding of length measurement. Implications for the learning of area measurement are addressed.

Teachers' and Researchers' Beliefs About the Development of Algebraic Reasoning
Mitchell J. Nathan, Kenneth R. Koedinger
Mathematics teachers and educational researchers ordered arithmetic and algebra problems according to their predicted problem-solving difficulty for students.   Predictions deviated systematically from algebra students' performances but closely matched a view implicit in textbooks. Analysis of students' problem-solving strategies indicates specific ways that students' algebraic reasoning differs from that predicted by most teachers and researchers in the sample and portrayed in common textbooks. The Symbol Precedence Model of development of algebraic reasoning, in which symbolic problem solving precedes verbal problem solving and arithmetic skills strictly precede algebraic skills, was contrasted with the Verbal Precedence Model of development, which provided a better quantitative fit of students' performance data. Implications of the findings for student and teacher cognition and for algebra instruction are discussed.

Interaction or Intersubjectivity? A Reply to Lerman
Leslie P. Steffe, Patrick W. Thompson
Lerman, in his challenge to radical constructivism, presented Vygotsky as an irreconcilable opponent to Piaget's genetic epistemology and thus to von Glasersfeld's radical constructivism. We argue that Lerman's stance does not reflect von Glasersfeld's opinion of Vygotsky's work, nor does it reflect Vygotsky's opinion of Piaget's work. We question Lerman's interpretation of radical constructivism and explain how the ideas of interaction, intersubjectivity,   and social goals make sense in it. We then establish compatibility between the analytic units in Vygotsky's and von Glasersfeld's models and contrast them with Lerman's analytic unit. Consequently, we question Lerman's interpretation of Vygotsky. Finally, we question Lerman's use of Vygotsky's work in mathematics education, and we contrast that use with how we use von Glasersfeld's radical constructivism.

A Case of Interpretations of Social: A Response to Steffe and Thompson
Stephen Lerman
In their response to my (1996) article, Steffe and Thompson argued that I have taken an early position of Vygotsky's and that his later work is subsumed in and developed by von Glasersfeld. I argue that the two theories, Vygotsky's and radical constructivism, are, on the contrary, quite distinct and that this distinction, when seen as a dichotomy, is productive. I suggest that radical constructivists draw on a weak image of the role of social life. I argue 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 of the last 2 decades.

Dichotomies or Binoculars: Reflections on the Papers by Steffe and Thompson and by Lerman
Thomas E. Kieren
Mathematics education in schools can be viewed either as primarily a sociocultural phenomenon or as a nurturing of the individual's mathematical development.   However, instead of taking the dichotomous view, contrasting the Vygotskian and the Pigetian perspectives, one may see the two as separate "truths," providing different lenses through which to attain a more complete reciprocal embodied view of mathematics education.

A Comparison of Problems That Follow Selected Content Presentations in American and Chinese Mathematics Textbooks
Yepling Li
To illuminate the cross-national similarities and differences in expectations related to students' mathematics experiences between the United States and China, I compared all relevant problems that followed the content presentation of addition and subtraction of integers in several American and Chinese mathematics textbooks. A 3-dimensional framework (for mathematical features, contextual features, and performance requirements) was developed in this study to analyze these textbook problems. The results show that the percentage differences in problems' dimensions, mathematical and contextual features, were smaller than the difference in problems' performance requirements. Specifically, the differences found in problems' performance requirements indicate that the American textbooks included more variety in problem requirements than the Chinese textbooks. The results are relevant to documented cross-national differences in American and Chinese students' mathematical performances.

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