A look at some ways that teachers can improve the mathematics education of girls, as well as that of boys.

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### Elizabeth Fennema

On Monday, March 23, I was in a somewhat unusual and comfortable situation for a university professor. (Perhaps one should really substitute the word *lazy* for *comfortable*.) I had written a chapter for the 1982 NCTM Yearbook on the middle school, and it could serve as my NCTM speech at St. Louis. True, I had not said anything very new, innovative, or controversial; but after all, this was the third year in a row that I had talked about sex-related differences at NCTM, so no one could expect me to have much new information or insight. Then, as I was reading the newspaper and having a leisurely cup of tea before going to school, I read the following: “ Due to evolutionary factors, males are also better at doing math problems … ” (Wisconsin State Journal, March 23, 198 1).

### Elizabeth Fennema

It has long been accepted as true that boys learn mathematics better than girls do. To determine the validity of this belief, 36 studies concerned basically or tangentially with sex differences in mathematics achievement were reviewed and two others were analyzed in depth. The data from one study (Parsley, et al., 1964), which often has been quoted as supportive of boys' mathematics superiority, was reevaluated with the conclusion that the data from this study do not support the idea that boys are superior to girls in mathematics achievement. Data concerned with sex differences in achievement from the National Longitudinal Study of Mathematics Achievement were also presented.

No significant differences between boys' and girls' mathematics achievement were found before boys and girls entered elementary school or during early elementary years. In upper elementary and early high school years significant differences were not always apparent. However, when significant differences did appear they were more apt to be in the boys' favor when higher-level cognitive tasks were being measured and in the girls' favor when lower-level cognitive tasks were being measured. No conclusion can be reached concerning high school learners.

### Elizabeth H. Fennema

Although there is reasonable agreement about which mathematical ideas should be taught in the elementary school (Begle 1966) and that these ideas should be taught meaningfully (Dawson and Rud-dell [a] 1955), there is little agreement on how learning environments should be structured to facilitate this learning. One reason for this lack of agreement is an inadequate recognition of the role that concrete and symbolic models can and should play in facilitating the learning of mathematical ideas.

### Elizabeth Fennema

Statements concerning the efficacy of manipulative materials in facilitating the learning of mathematical ideas abound in current mathematics education literature. Partly in response to such statements, mathematics laboratories, or activity-based curricula that include a heavy reliance on the use of such materials, are becoming more and more prevalent. This belief that manipulative materials do indeed enhance the learning of mathematics has gained much validity from learning theories such as those suggested by Bruner, Dienes, and Piaget. These theories strongly support the idea that children need physical involvement, such as might be provided by hands-on experiences with manipulative materials, in order to add new ideas to their cognitive structure. However, attempts to translate such theories into classroom practice and to empirically measure the results have not provided evidence that the use of such materials by teachers does indeed result in better learning than the use of symbols alone (Suydam and Weaver 1970; Kieren 1971; Fennema 1972). This difference between theory and research findings indicates that manipulative materials are no panacea; the use of materials does not automatically ensure that mathematics learning will follow. However, even without strong empirical support, a strong case can be built for the inclusion of manipulative materials in an elementary school mathematics program.

### Elizabeth H. Fennema

Mathematical ideas taught in the elementary school can be represented by at least two types of models: concrete (manipulative or en active) and symbolic. Although there has been much theoretical discussion concerning the value of concrete models in enhancing learning (Kieren, 1969), empirical data are just beginning to accumulate that give some information concerning the efficacy of concrete models (Kieren, 1971), and such data are far from conclusive.

### Elizabeth Fennema and Thomas P. Carpenter

The second mathematics assessment of the National Assessment of Educational Progress provides new insight into the problems of sex-related differences in mathematics. Information about course taking and achievement in specific content areas and at different cognitive levels is available from a representative national sample of over 70 000 9-, 13-, and 17-year-olds. The purpose of this article is to report the sex-related differences that were found in this assessment and to explore the significance of these differences.

### Elizabeth Fennema and Lindsay A. Tartre

This longitudinal study investigated how girls and boys who were discrepant in their spatial and verbal performance used spatial visualization skills in solving word problems and fraction problems. The subjects, 36 girls and 33 boys, were interviewed annually in Grades 6, 7, and 8. Each student was asked to read a problem, draw a picture to help solve it, solve it, and then explain how the picture was used in the solution. Students who differed in spatial visualization skill did not differ in their ability to find correct problem solutions, but students with a higher level of spatial visualization skill tended to use spatial skills in problem solving more often than students with a lower level of skill. Girls tended to use pictures more during problem solving than boys did, but this did not enable them to get as many correct solutions. Low spatial visualization skill may be more debilitating to girls' mathematical problem solving than to boys'.