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Steven R. Williams

Scholarly work in mathematics education has recently begun to look deeply at what mathematics is. This increased interest in what was once the purview of philosophers of mathematics grows from a recognition that both teaching and learning mathematics are intimately connected with doing mathematics.

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Steven R. Williams

This study documents 10 college students' understanding of the limit concept and the factors affecting changes in that understanding. Common informal models of limit were identified among the 10 students, who were then presented with alternative models of limit and with anomalous limit problems. The problems were designed to encourage students to make changes in their own models to reflect a more formal conception. Individual models of limit varied widely even among students who initially described limits in similar ways. The dynamic aspect of these models was extremely resistant to change. This resistance was influenced by students' belief in the a priori existence of graphs, their prior experiences with graphs of simple functions, the value they put on conceptually simple and practically useful models, and their tendency to view anomalous problems as minor exceptions to rules. These factors combined to inhibit students' motivation to adopt a formal view of limit.

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Steven R. Williams

This study uses repertory grid methodology together with a predicational view of human thinking to describe the informal models of the limit concept held by two college calculus students. It describes how their models, based on iteratively choosing points that get closer to the limiting value, are affected by experimental sessions designed to alter them. The informal models are based on the notion of actual infinity, which poses a severe cognitive obstacle to the learning of the formal definition of limit. The study also suggests that the predicational view of cognition, together with analysis of repertory grid data based on fuzzy set theory, can be useful in studying students' concept images of advanced mathematical concepts.

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Steven R. Williams

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Steven R. Williams and Keith R. Leatham

We present the results of 2 studies, a citation-based study and an opinion-based study, that ranked the relative quality of 20 English-language journals that exclusively or extensively publish mathematics education research. We further disaggregate the opinion-based data to provide insights into variations in judgment of journal quality based on geographic location, journal affiliations and publishing records, and experience in the field. We also report factors that survey respondents indicated were important indicators of journal quality. Finally, we compare our results to previous related rankings and conclude by discussing how our results might inform authors, editors, and evaluators in their efforts to publish and recognize quality research in mathematics education.

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Blake E. Peterson, Steven R. Williams and Penelope H. Dunham

The most crucial stage in the process of becoming a teacher occurs at the very outset, during the transition from student to student teacher to novice teacher. Many people can provide vital support to the new teacher: cooperating teachers, university supervisors, instructors of methods classes, and more experienced teachers in the school can all act as mentors. What is known about the mentoring process? What is unique to mentoring mathematics teachers? In this article we hope to outline what is known and offer some guidance for those wishing to be effective mentors.