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May 2000, Volume 31, Issue 3

FEATURES

Mathematical Beliefs and Conceptual Understanding of the Limit of a Function
Jennifer Earles Szydlik
In this study, I investigated 27 university calculus students' mathematical beliefs and connections between those beliefs and their understandings of limit. Participants were selected on the basis of questionnaire and interview responses to real-number, infinity, function, and sources-of-conviction items. Data obtained in a subsequent limit interview suggest a relationship between sources of conviction and understanding of limit; students with external sources of conviction gave more incoherent or inappropriate definitions of limit, held more misconceptions of limit as bound or unreachable, and were less able to justify limit calculations than those with internal sources of conviction. The influence of content beliefs on understanding of limit is less evident.

Achievement Results for Second and Third Graders Using the Standards-Based Curriculum Everyday Mathematics
Karen C. Fuson, William M. Caroll, Jane V. Drueck
Students using Everyday Mathematics (EM), developed to incorporate ideas from the NCTM Standards, were at normative U.S. levels on multidigit addition and subtraction symbolic computation on traditional, reform-based, and EM-specific test items. Heterogeneous EM 2nd graders scored higher than middle- to upper-middle-class U.S. traditional students on 2 number sense items, matched them on others, and were equivalent to a middle-class Japanese group. On a computation test, the EM 2nd graders outperformed the U.S. traditional students on 3 items involving 3-digit numbers and were outperformed on the 6 most difficult test items by the Japanese children. EM 3rd graders outscored traditional U.S. students on place value and numeration, reasoning, geometry, data, and number-story items.

Steering (Dis)Course Between Metaphors and Rigor: Using Focal Analysis to Investigate an Emergence of Mathematical Objects
Anna Sfard
This study deals with students' construction of mathematical objects. The basic claim is that the need for communication--any attempt to evoke certain actions by others--is the primary driving force behind all human cognitive processes. Effectiveness of verbal communication is seen as a function of the quality of its focus. Material objects may serve as a basis for creation of such a focus, but in some discourses, focus-engendering objects must be created. Such discursive construction is observed in analysis of one classroom episode. Special attention is given to metaphor, which is the point of departure for the construction process, and to the subsequent dialectical process of closing the gap between the metaphor-induced expectations and the need for a well-defined construction procedure to ensure effective communication.

Effects of Standards-Based Mathematics Education: A Study of the Core-Plus Mathematics Project Algebra and Functions Strand
Mary Ann Huntley, Chris L. Rasmussen, Roberto S. Villarubi, Jaruwan Sangtong, James T. Fey
To test the vision of Standards–based mathematics education, we conducted a comparative study of the effects of the Core-Plus Mathematics Project (CPMP)¬† ¬†curriculum and more conventional curricula on growth of student understanding, skill, and problem-solving ability in algebra. Results indicate that the CPMP curriculum is more effective than conventional curricula in developing student ability to solve algebraic problems when those problems are presented in realistic contexts and when students are allowed to use graphing calculators. Conventional curricula are more effective than the CPMP curriculum in developing student skills in manipulation of symbolic expressions in algebra when those expressions are presented free of application context and when students are not allowed to use graphing calculators.

The Affective and Cognitive Dimensions of Math Anxiety: A Cross-National Study
Hsiu-Zu Ho, Deniz Senturk, Amy G. Lam, Judes M. Zimmer, Sehee Hong, Yukari Okamoto, Sou-Yung Chiu, Yasuo Nakazawa, Chang-Pei Wang
In this study we focus on math anxiety, comparing its dimensions, levels, and relationship with mathematics achievement across samples of 6th-grade students from China, Taiwan, and the United States. The results of confirmatory factor analyses supported the theoretical distinction between affective and cognitive dimensions of math anxiety in all 3 national samples. The analyses of structural equation models provided evidence for the differential predictive validity of the 2 dimensions of math anxiety. Specifically, across the 3 national samples, the affective factor of math anxiety was significantly related to mathematics achievement in the negative direction. Gender by nation interactions were also found to be significant for both affective and cognitive math anxiety.