The public has renewed interest and concern about education and those of usin the field must be prepared to answer question. One of the question is usually related to student performance. The National Assessment of Educational Progress (NAEP) is a primary source to which we can turn for support of our answer.

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- Author or Editor: Edward A. Silver x

### Mary Montgomery Lindquist, Thomas P. Carpenter, Edward A. Silver and Westina Matthews

### Edward A. Silver, Joanna Mamona-Downs, Shukkwan S. Leung and Patricia Ann Kenney

In this study, 53 middle school teachers and 28 prospective secondary school teachers worked either individually or in pairs to pose mathematical problems associated with a reasonably complex task setting, before and during or after attempting to solve a problem within that task setting. Written responses were examined to determine the kinds of problems posed in this task setting, to make inferences about cognitive processes used to generate the problems, and to examine differences between problems posed prior to solving the problem and those posed during or after solving. Although some responses were ill-posed or poorly stated problems, subjects generated a large number of reasonable problems during both problem-posing phases, thereby suggesting that these teachers and prospective teachers had some personal capacity for mathematical problem posing. Subjects posed problems using both affirming and negating processes; that is, not only by generating goal statements while keeping problem constraints fixed but also by manipulating the task's implicit assumptions and initial conditions. A sizable portion of the posed problems were produced in clusters of related problems, thereby suggesting systematic problem generation. Subjects posed more problems before problem solving than during or after problem solving, and they tended to shift the focus of their posing between posing phases based at least in part on the intervening problem-solving experience. Moreover, the posed problems were not always ones that subjects could solve, nor were they always problems with “nice” mathematical solutions.

### Patricia Ann Kenney, Judith S. Zawojewski and Edward A. Silver

Suppose that you gave the problem in **figure 1** to your students to solve and asked them to explain how Marcy could determine the number of dots in the twentieth figure without drawing all twenty pictures. What kinds of strategies do you think that your students would suggest to Marcy? Will some strategies be more efficient than others? Will some students disregard the directive and draw all or some of the twenty pictures to solve the problem? Also, will their explanations be complete and clear?

### Jerry P. Becker, Edward A. Silver, Mary Grace Kantowski, Kenneth J. Travers and James W. Wilson

A U.S.–Japan Seminar on Mathematical Problem Solving was held at the East-West Center in Honolulu 14– 18 July 1986 (Becker and Miwa 1987). Among the seminar's proposals was that cross-cultural research on American and Japanese students' problem-solving behaviors be organized and carried out. The author were in Japan in the fall of 1988 to meet with their Japanese counterparts, plan research, and make visits to mathematics classrooms preliminary to conducting a two-year program of research. In addition to planning the research, we were on a fact-finding visit to classrooms to better acquaint ourselves with mathematics teaching and learning in Japan.

### Catherine A. Brown, Thomas P. Carpenter, Vicky L. Kouba, Mary M. Lindquist, Edward A. Silver and Jane O. Swafford

This article is the first of two articles reporting on the seventh-grade and eleventh- grade results of the fourth mathematics assessment of the National Assessment of Educational Progress (NAEP) administered in 1986. The elementary school results appear in companion articles in the *Arithmetic Teacher* (Kouba et al. 1988a, 1988b). Secondary school data from previous national assessments have been reported in the *Mathematics Teacher* (see, e. g., Carpenter et al. [1980, 1983))

### Catherine A. Brown, Thomas P. Carpenter, Vicky L. Kouba, Mary M. Lindquist, Edward A. Silver and Jane O. Swafford

This article is the second of two articles reporting on the seventh-grade and eleventh-grade results of the fourth mathematics assessment of the National Assessment of Educational Progress (NAEP) administered in 1986. The first article (Brown et al. 1988) presented the background, methodology, and the results of students' performance on discrete mathematics, data organization and interpretation, number and operations, and measurement. This article reports students' performance on variables and relations, geometry, fundamental methods of mathematics, and attitudes. An analysis of eleventh-grade students' performance by mathematics course background was possible, and these data will be reported here where appropriate.

### Edward A. Silver, Mary M. Lindquist, Thomas P. Carpenter, Catherine A. Brown, Vicky L. Kouba and Jane O. Swafford

This article is the third one to appear in the *Mathematics Teacher* reporting the results and analysis compiled by the NCTM Interpretive Team for the Fourth Mathematics Assessment of the National Assessment of Educational Progress (NAEP). The first two articles reported on the performance of seventh- and eleventh-grade students in specific content areas (Brown et al. 1988a. 1988b). This article discusses trends in performance across the last three NAEP mathematics assessments and reports on indicators of instructional activity. A companion article on the performance of third- and seventh-grade students appears in this month's *Arithmetic Teacher* (Carpenter et al. 1988).

### Vicky L. Kouba, Catherine A. Brown, Thomas P. Carpenter, Mary M. Lindquist, Edward A. Silver and Jane O. Swafford

This article is the first of two articles to appear in the *Arithmetic Teacher* reporting the third-grade and seventh-grade results of the fourth mathematics assessment of the National Assessment of Educational Progress (NAEP). The secondary school data appear in companion articles in the *Mathematics Teacher* (Brown et al. 1988a, 1988b). This article summarizes the major results of the performance on number, operations, and word-problem items. Some examples of the data are given to support conclusions about what students in general are and are not learning. The specific items reported are altered items that are parallel to the actual items and are illustrative of the results. The actual items are not included because they may be used in future assessments. Interpretations were made from analysis of the actual items. Not all items in the assessment were given to both third-grade and seventh-grade students. Some were administered only when appropriate for the given grade level. Results for eleventh-grade students are included when such information aids in understanding the performance of the third-grade or seventh-grade students.

### Vicky L. Kouba, Catherine A. Brown, Thomas P. Carpenter, Mary M. Lindquist, Edward A. Silver and Jane O. Swafford

### Thomas P. Carpenter, Mary M. Lindquist, Catherine A. Brown, Vicky L. Kouba, Edward A. Silver and Jane O. Swafford

This article is the third one to appear in the *Arithmetic Teacher* reporting the results of the fourth mathematics as essment of the National Asse sment of Educational Progress (NAEP). The first two article reported achievement of third-and seventh-grade students in pecific content area (Kouba et al. 1988a. 1988b). This article discusse general achievement trend and change in performance over time. A parallel article on trend in performance of older students appears in this month's *Mathematics Teacher* (Silver et al. 1988).