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November 2000, Volume 31, Issue 5


Interference of Instrumental Instruction in Subsequent Relational Learning
Dolores D. Pesek, David Kirshner
To balance their professional obligation to teach for understanding against administrators' push for higher standardized test scores, mathematics teachers sometimes adopt a 2-track strategy: teach part of the time for meaning (relational learning) and part of the time for recall and procedural-skill development   (instrumental learning). We explore a possible negative effect of this dual approach when relational learning is preceded by instrumental learning. A group of students who received only relational instruction outperformed a group of students who received instrumental instruction prior to relational instruction. Interview data show aspects of cognitive, metacognitive, and attitudinal interference that may have been caused by the juxtaposition of instructional modes.  

Language Development and Concept Flexibility in Dyscalculia: A Case Study
Kristine K. Montis
Dyscalculia is a psychological and medical term that refers to extreme difficulty in learning mathematics and to deficits in the production of accurate, efficient arithmetic calculations, in particular. In this article I report on a yearlong qualitative case study of a 12-year-old student who displayed many characteristics of dyscalculia. The results of the study are discussed as they relate to recent medical and learning-disability research. This student's learning experiences during her school mathematics and tutoring sessions demonstrate the vital role language processes play in the development of the concept flexibility necessary for success in mathematics. Outlined in the closing section are implications of this study for pedagogy in classrooms that include mainstreamed students with learning disabilities.  

A Calculus Graphing Schema
Bernadette Baker, Laurel Cooley, Marma Trigueros
In this study, we analyzed students' understanding of a complex calculus graphing problem. Students were asked to sketch the graph of a function, given its analytic properties (1st and 2nd derivatives, limits, and continuity) on specific intervals of the domain. The triad of schema development in the context of APOS theory was utilized to study students' responses. Two dimensions of understanding emerged, 1 involving properties and the other involving intervals. A student's coordination of the 2 dimensions is referred to as that student's overall calculus graphing schema. Additionally, a number of conceptual problems were consistently demonstrated by students throughout the study, and these difficulties are discussed in some detail.  

Characterizing a Perspective Underlying the Practice of Mathematics Teachers in Transition
Martin A. Simon, Ron Tzur, Karen Heinz, Margaret Kinzel, 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.

Making Sense of the Total of Two Dice
Dave Pratt
Many studies have shown that the strategies used in making judgments of chance are subject to systematic bias. Concerning chance and randomness, little is known about the relationship between the external structuring resources, made available for example in a pedagogic environment, and the construction of new internal resources. In this study I used a novel approach in which young children articulated their meanings for chance through their attempts to "mend" possibly broken computer-based stochastic gadgets. I describe the interplay between informal intuitions and computer-based resources as the children constructed new internal resources for making sense of the total of 2 spinners and 2 dice.  

Who's Counting? A Survey of Mathematics Education Research 1982-1998
Sarah Theule Lubienski, Andrew Bowen
This study provides a broad look at mathematics education research published between 1982 and 1998. The ERIC database was utilized to count and categorize more than 3,000 articles from 48 educational research journals. We identified the number of articles relating to gender, ethnicity, class, and disability that were published in journals from various categories. Attention was also given to grade levels, mathematical topics, and general educational topics in conjunction with each equity group. We conclude that, in comparison with research on ethnicity, class, and disability, research on gender was more prevalent and integrated into mainstream U.S. mathematics education research. Overall, the majority of research seemed to focus on student cognition and outcomes, with less attention to contextual or cultural issues.