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July 2000, Volume 31, Issue 4

FEATURES

A Study of Proof Conceptions in Algebra
Lulu Healy, Celia Hoyles
After surveying high-attaining 14- and 15-year-old students about proof in algebra, we found that students simultaneously held 2 different conceptions of proof: those about arguments they considered would receive the best mark and those about arguments they would adopt for themselves. In the former category, algebraic arguments were popular. In the latter, students preferred arguments that they could evaluate and that they found convincing and explanatory, preferences that excluded algebra. Empirical argument predominated in students' own proof constructions, although most students were aware of its limitations. The most successful students presented proofs in everyday language, not using algebra. Students' responses were influenced mainly by their mathematical competence but also by curricular factors, their views of proof, and their genders.

Mathematicians Writing
Leone Burton, Candia Morgan
In this article we report on part of a study of the epistemological perspectives of practicing research mathematicians. We explore the identities that mathematicians present to the world in their writing and the ways in which they represent the nature of mathematical activity. Analysis of 53 published research papers reveals substantial variations in these aspects of mathematicians' writing. The interpretation of these variations is supported by extracts from interviews with the mathematicians. We discuss the implications for students and for novice researchers beginning to write about their mathematical activity.

Problem Solving as a Means Toward Mathematics for All: An Exploratory Look Through a Class Lens
Sarah Theule Lubienski
As a researcher-teacher, I examined 7th-graders' experiences with a problem-centered curriculum and pedagogy, focusing on SES differences in students' reactions to learning mathematics through problem solving. Although higher SES students tended to display confidence and solve problems with an eye toward the intended mathematical ideas, the lower SES students preferred more external direction and sometimes approached problems in a way that caused them to miss their intended mathematical points. An examination of sociological literature revealed ways in which these patterns in the data could be related to more than individual differences in temperament or achievement among the children. I suggest that class cultural differences could relate to students' approaches to learning mathematics through solving open, contextualized problems.

Read how to use this article as part of a Professional Development Experience.

Computational Estimation Skill of College Students
Sheri A. Hanson, Thomas P. Hogan
We examined computational estimation skill of 77 college students who estimated answers to problems presented in brief intervals. We categorized 23 "think-aloud" estimation strategies used by 45 participants in individual follow-up sessions. Some categories were based on strategies found in previous studies; others were based on responses in this study. Although students correctly estimated answers to most problems on addition and subtraction of whole numbers, they performed poorly on multiplication and division of decimals and subtraction of fractions. Students were more successful in solving computational problems than in estimating answers. Scores on the estimation tests showed substantial correlation with SAT Mathematics scores and with a direct measure of computational skill, but they did not significantly correlate with SAT Verbal scores.

Student Understanding of the Cartesian Connection: An Exploratory Study
Eric J. Knuth
The topic of multiple representations of functions is important in secondary school mathematics curricula, yet many students leave high school lacking an understanding of the connections among these representations. Research results are presented from a study in which students' understandings of the connections between algebraic and graphical representations of functions were examined. Responses from 178 students, enrolled in 1st-year algebra through calculus, revealed an overwhelming reliance on algebraic representations, even on tasks for which a graphical representation seemed more appropriate. The findings indicate that for familiar routine problems many students have mastered the connections between the algebraic and graphical representations; however, such mastery appeared to be superficial at best.