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July 2001, Volume 32, Issue 4


Predications of the Limit Concept: An Application of Repertory Grids- FREE PREVIEW!
Stephen 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.

The Impact of Two Standards-Based Mathematics Curricula on Student Achievement in Massachusetts- FREE PREVIEW!
Julie E. Riordan, Pendred E. Noyce
Since the passage of the Education Reform Act in 1993, Massachusetts has developed curriculum frameworks and a new statewide testing system. As school districts align curriculum and teaching practices with the frameworks, standards-based mathematics programs are beginning to replace more traditional curricula. This paper presents a quasi-experimental study using matched comparison groups to investigate the impact of one elementary and one middle school standards-based mathematics program in Massachusetts on student achievement. The study compares statewide standardized test scores of fourth-grade students using Everyday Mathematics and eighth-grade students using Connected Mathematics to test scores of demographically similar students using a mix of traditional curricula. Results indicate that students in schools using either of these standards-based programs as their primary mathematics curriculum performed significantly better on the 1999 statewide mathematics test than did students in traditional programs attending matched comparison schools. With minor exceptions, differences in favor of the standards-based programs remained consistent across mathematical strands, question types, and student sub-populations.

Using Everyday Knowledge of Decimals to Enhance Understanding- FREE PREVIEW!
Kathryn C. Irwin
The study investigated the role of students' everyday knowledge of decimals in supporting the development of their knowledge of decimals. Sixteen students, ages 11 and 12, from a lower economic area, were asked to work in pairs (one member of each pair a more able student and one a less able student) to solve problems that tapped common misconceptions about decimal fractions. Half the pairs worked on problems presented in familiar contexts and half worked on problems presented without context. A comparison of pretest and posttest results revealed that students who worked on contextual problems made significantly more progress in their knowledge of decimals than did those who worked on noncontextual problems. Dialogues between pairs of students during problem solving were analyzed with respect to the arguments used. Results from this analysis suggested that greater reciprocity existed in the pairs working on the contextualized problems, partly because, for those problems, the less able students more commonly took advantage of their everyday knowledge of decimals. It is postulated that the students who solved contextualized problems were able to build scientific understanding of decimals by reflecting on their everyday knowledge as it pertained to the meaning of decimal numbers and the results of decimal calculations.

Where's the Evidence?- FREE PREVIEW!
Jeremy Kilpatrick
During the last half century, school mathematics in North America has undergone two major waves of attempted reform: the new math movement of the 1950s through the early 1970s and the standards-based movement of the past two decades or so. Although differing sharply in their approach to curriculum content, these reform efforts have shared the aim of making mathematics learning more substantial and engaging for students. The rhetoric surrounding the more recent movement, however, has been much more shrill, the policy differences more sharply drawn, the participants more diverse. The so-called math wars of the 1960s (DeMott, 1962, ch. 9) were largely civil wars. They pitted advocates of rigor and axiomatics against those promoting applied, genetic approaches and were conducted primarily in journal articles and at professional meetings. Today's warfare ranges outside the profession and has a more strident tone; it is much less civil in both senses of the word.