Christian R Hirsch

Student preferences for representations of functions

Incorporating multiple representations enhanced through technology is a theme in reformed mathematics curricula. While the need for students to use and reason on multiple representations is widely accepted, the research‐based knowledge on how to best accomplish this goal is just beginning to emerge. Current cognitive theories indicate that students’ knowledge of and preference for various representations is crucial to the decision process. This project develops and validates an instrument for determining student preferences for multiple representations of functions and establishes base‐line data for future research. The results indicate that students do have preferences for representations of functions and that these preferences vary between contextualized and non‐contextualized settings. Furthermore, students’ preferences between these two settings are more similar for students in a graphics calculator‐based calculus course than for students in a traditional calculus course. If a student has a cognitive ...

Responding to Calls for Change in High School Mathematics: Implications for Collegiate Mathematics

This article briefly discusses the recent history of mathematics reform in high school, and then reports on research evidence for one of the new NSF-funded curriculum projects, the Core-Plus Mathematics Project (CPMP). Implications for collegiate mathematics are also discussed. A summary of the results of several studies using a range of achievement measures comparing CPMP students to comparable students in more conventional high school mathematics curricula is given on page 114: “Thus, research to date indicates that CPMP students perform particularly well [and better than the comparison students] on measures of conceptual understanding, interpretation of mathematical representations and calculations, and problem solving in applied contexts. Their performance is also relatively strong in content areas like statistics and probability that are emphasized in the curriculum. On measures of algebraic manipulative skills, CPMP students usually, but not always, score as well as students in more conventional curricula.” A study of student performance on a mathematics placement test used at a major university is summarized on page 116: “On the algebra subtest, the means of the [conventional] precalculus students and the CPMP Course 4 students were virtually identical. On the intermediate algebra subtest, the mean of the precalculus group was greater than that of the Course 4 group. The only statistically significant difference in means was on the calculus readiness subtest (t = 4.93, p < 0.01). That difference favored the CPMP students.” Responding to Calls for Change in High School Mathematics: Implications for Collegiate Mathematics Harold L. Schoen and Christian R. Hirsch

Alternative models for mathematics assignments

This paper discusses three alternative models for structuring homework assignments in college level mathematics. A distributive model which incorporates both early and late review of previously learned material into the daily assignment schedule is explicated in detail. The relative efficacy of the conventional method for assigning homework and the distributive method in promoting achievement and favourable attitudes toward mathematics in a first semester course in calculus was investigated in a controlled experiment. Significant (p < 0.03) interactions between mathematical ability and assignment model were found on three of four unit tests. In each case, pupils with the weaker pre‐calculus background profited more from the distributive assignment schedule. There were no significant interactions or differences on a comprehensive post‐test or on attitudes toward mathematics. Implications for research and practice in college level mathematics, physics, and engineering courses are suggested.