Edward A. Silver

The Role of Task Format, Mathematics Knowledge, and Creative Thinking on the Arithmetic Problem Posing of Prospective Elementary School Teachers

A Test of Arithmetic Problem Posing was developed by the authors to examine the arithmetic problem-posing behaviours of sixty-three prospective elementary school teachers. Results of analysis were then used to examine task format (i.e., the presence or absence of specific numerical information) on subjects’ problem posing and the relationship between subjects’ problem posing and their mathematics knowledge and verbal creativity. The major findings were that the test effectively evaluated arithmetic problem posing, and that most subjects were able to pose solvable and complex problems. In addition, problem-posing performance was better when the task contained specific numerical information than when it did not, and that problem-posing performance was significantly related to mathematical knowledge but not to verbal creativity.

Generating Multiple Solutions for a Problem: A Comparison of the Responses of U.S. and Japanese Students.

A task involving simple mathematics, yet complex in its call for the generation of multiple solution methods, was administered to about 150 U.S. students, most of whom were in fourth grade. Written responses were examined for correctness, evidence of strategy use and mode of explanation. Results for the U.S. sample were also compared to those obtained from about 200 Japanese fourth-grade students. Students in both countries (a) produced multiple solutions and explanations of their solutions, (b) exhibited almost identical patterns and frequency of strategy use across response occasions, and (c) used the same kinds of explanations, with a majority of the responses involving solution explanations that combined both visual and verbal/symbolic features. Nevertheless, Japanese students tended to produce explanations involving more sophisticated mathematical ideas (multiplication rather than addition) and formalisms (mathematical expressions rather than verbal explanations) than did U.S. students.

Assessment in the Context of Mathematics Instruction Reform: The Design of Assessment in the Quasar Project

Recent high-level political interest in the improvement of mathematics education in the United States has led to the increased prominence of reports by the National Academy of Sciences (National Research Council, 1989), the American Association for the Advancement of Science (1989) and National Council of Teachers of Mathematics (1989). These reform-oriented reports have focused the attention of educational practitioners and policy makers on new goals for mathematics education and new descriptions of mathematical proficiency, in which terms like reasoning, communication, problem solving, conceptual understanding, and mathematical power are used frequently to describe an expanded view of mathematical proficiency that goes beyond memorization and mere competence in the basic skills of rational number computation. The reform discussion has thus led naturally to considerations of how to assess students’ attainments with respect to this new version of mathematical proficiency and how to assess improvements that may result from curricula and instructional reforms that might be undertaken3. This paper focuses on the efforts of one project to deal with the interface between assessment and instructional reform.

Equity and Validity Considerations in the Design and Implementation of a Mathematics Performance Assessment: The Experience of the QUASAR Project

The attention of educational practitioners and policy makers has recently focused on mathematics education reform in the United States. Reports by the National Academy of Sciences (National Research Council, 1989) and the National Council of Teachers of Mathematics (1989) have specified a new vision of mathematical proficiency—sometimes referred to as “world class standards” for mathematics education—and they have been issued at a time when concerns have been expressed publicly about evidence that American students are unable to perform at acceptable levels in mathematics and about the implications of this low level of performance for the nation’s long-term economic competitiveness.

Fairness and Equity in Measuring Student Learning Using a Mathematics Performance Assessment: Results from the Quasar Project

Performance assessments that have the capability of measuring a broad range of reasoning and thinking skills are considered to be fundamental tools in the educational reform movement (Linn, 1993). They are being used to document the need for educational reform as well as to monitor the impact of reform at the national, state, and local levels. Many advocates of performance assessments are optimistic that these assessments will be sensitive to measuring the impact of educational reform and that the differences in performance on these assessments among ethnic, linguistic, and gender subgroups will be narrower than those observed on multiple-choice tests. Thus, performance assessments are considered by many to be fairer and more equitable assessments of achievement for various subgroups than multiple-choice tests. It is important, however, to ensure that empirical evidence is obtained to support the fairness of the assessment for various subgroups of students.

Chapter 6 Assessment and mathematics education reform in the united states

Abstract Amid calls to reform U.S. school mathematics instruction, current assessment practices are also being examined, challenged and modified. Standardized, multiple-choice tests are widely used in the U.S. and have been reported to influence negatively the content and form of both classroom instruction and classroom assessment. Evidence that teachers may modify their instruction based on the content and form of externally-mandated tests has led some reformers to devise an assessment-driven strategy to change mathematics instruction. Many efforts are underway to develop alternatives to multiple-choice items for the assessment of mathematical proficiency. The promise of this strategy is illustrated by examining the California Assessment Program. Limitations of this strategy include the dilemma of linking educational reform to externally-mandated testing; the technical difficulty of constructing reliable, valid alternatives to multiple-choice items; and the likelihood that changes in assessment will not alone lead to the educational reform being sought.