Some of the difficulties which students have in developing mathematical problem solving ability are (1) realizing that prob lem solving is more than answer getting, (2) failing to consider exceptional cases, and (3) failing to review the solution in order to (*a*) validate it, *(b)* improve it, (*c*) generalize it, (*d*) consolidate what has been Iearned, and (*e*) suggest what else might be learned. Additional difficulties may come front the misleading presentation of a problem.

# Search Results

### James W. Wilson and Jerry P. Becker

### Thomas A. Romberg and James W. Wilson

This study investigated the effect of an advance organizer (AO), cognitive set (CS), and post organizer (PO) in learning and retention of written mathematics materials. Self-instructional booklets were designed so that the pages containing the AO, CS, and PO varied according to experimental treatment in a 2 × 2 × 2 factorial design. Booklets were randomly assigned within 9 Algebra 2 classes.

A significant (p<.05p<.05) interaction between the AO and the PO and a significant main effect (p<.05p<.05) for CS was found on the retention test. Examination of the cell means indicated (a) a facilitating effect for CS, and (b) either an AO or a PO was facilitating, but used together they were not facilitating.

### Wayne H. Martin and James W. Wilson

In this issue, which focuses on evaluation in mathematics education, it is both timely and appropriate to consider the status of the first assessment of mathematics conducted by the National Assessment of Educational Progress (NAEP). During the 1972–73 academic year, a national sample of 9-year-olds, 13-yearolds, 17-year-olds (including high school dropouts and early graduates), and young adults between the ages of 26 and 35 were assessed to determine their levels of attainment in certain mathematical knowledges and skills. National Assessment is now in the process of analyzing the mathematics data and has begun work on a cooperative project with the National Council of Teachers of Mathematics (NCTM) to consider the interpretations and curricular implications of the mathematics results. The purpose of this article is to explain how NAEP's mathematics assessment differs from the usual normreferenced testing programs and to preview what types of information will result from the assessment data.

### Maria L. Fernandez, Nelda Hadaway, James W. Wilson and Anna O. Graeber

Teachers and researchers have found that students presented with mathematics problems often begin performing calculations without giving much thought to the problem (e.g., Artzt and Armour-Thomas [1990]; Lester [1985]; Schoenfeld [1983, 1985a, 1985b, 1987]). Students often attend primarily to such surface features of the problem as context or type and the size of the given numbers. Such students often pursue unfruitful directions without monitoring or assessing their knowledge or actions, only to become frustrated and withdraw from the problem-solving activity. Schoenfeld (1983, 1985a, 1985b, 1987) presents examples of students' Jack of monitoring or managing their actions or knowledge during problem-solving episodes, which result in their pursuit of “wild mathematical geese.” Kilpatrick (1985) indicates that school instruction often reinforces students' impulsivity. Typically, students are not given much time to think about and make sense of the mathematics concepts they are learning. They are not encouraged to invoke managelial processes to monitor, regulate, or assess their understanding and actions. This article includes a blief review of research dealing with the individual's managelial processes during problem solving and discusses the implications of this research for problem-solving instruction.

### Thomas P. Carpenter, Terrence G. Coburn, Robert E. Reys and James W. Wilson

During the 1972-73 academic year. the National Assessment of Educational Progress (NAEP) conducted its first assessment in mathematics. Representative national samples of 9-year-olds, 13-year-olds, 17-year-olds (including high school dropouts and early graduates), and adults between the ages of 26 and 35 were assessed to determine their levels of attainment in mathematical concepts and skills.

### James A. Siders, Jane Z. Siders and Rebecca M. Wilson

The study was designed to examine the potential of using repeated rate-of-response measures to identify students exhibiting difficulties in arithmetic. Thirty subjects were followed over 2 years. Pearson product-moment correlations were computed to determine the ability of rate-of-response measures to predict achievement in arithmetic. A follow-up evaluation indicated that the predictive accuracy of the rate of response was 97%.

### Thomas P. Carpenter, Terrence G. Coburn, Robert E. Reys and James W. Wilson

Learning to solve word problems is a significant hurdle in learning mathematics. Failure to learn the skills needed to solve such problems can significantly inhibit a child's progress in school mathematics. Therefore, it is important that the solving of word problems occurs as a meaningful and successful part of each child's mathematical education.

### Thomas P. Carpenter, Terrence G. Coburn, Robert E. Reys and James W. Wilson

The National Assessment of Educational Progress (NAEP) has now reported its first mathematics assessment. This article will examine the NAEP mathematics assessment for the two youngest age groups: 9-year-olds and 13-year-olds. The mathematics assessment was done in 1972-73 and will be repeated each five years, with about half of the exercises repeated from one assessment to the next. Previous articles in the *Arithmetic Teacher* (Foreman and Mehrens, 1968; Martin and Wilson, 1974) have provided information on the nature of the exercises, the procedures for assessment, the purposes of assessment, and general information on NAEP.

### Terrence G. Coburn, Robert E. Reys and James W. Wilson

### Edited by Thomas P. Carpenter

A direct result of the increased attention being focused on the metric system is the reexamination of the entire area of measurement in the curriculum. For many this reex amination has led to the conclusion that the real problem in teaching metric measurement will not be with the metric units but with basic concepts of measurement. This contention is supported by the results of selected measurement exercises from the mathematics assessment of the National Assessment of Educational Progress (NAEP). This article focuses on two of these exercises that tested fun damental area and volume concepts.

### Thomas P. Carpenter, Terrence G. Coburn, Robert E. Reys and James W. Wilson

This is the second article based on the results of the National Assessment of Educational Progress (NAEP) to consider the process of measurement. In the October issue two NAEP exercises dealing with the primary concept of partitioning regions and sol ids into units of measure were examined. The resu lts of these exercises and the other research reviewed indicated that many younger students do not have any understanding of this fundamental concept that underlies area and volume measurement, and some misconceptions persist with many older students.