ability grouping and sex differences in mathematics achievement

Ability Grouping and Sex Differences in Mathematics AchievementAuthor(s): Maureen T. Hallinan and Aage B. SørensenSource: Sociology of Education, Vol. 60, No. 2 (Apr., 1987), pp. 63-72Published by: American Sociological AssociationStable URL: http://www.jstor.org/stable/2112582 .

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ABILITY GROUPING AND SEX DIFFERENCES IN MATHEMATICS ACHIEVEMENT

MAUREEN T. HALLINAN AAGE B. S0RENSEN

University of Notre Dame Harvard University

Sociology of Education 1987, Vol. 60 (April):63-72

In this paper we argue that previous research has overlooked structural and organizational factors as possible explanations of sex differences in the mathematics achievement of schoolchildren. In particular, we focus on ability grouping as a possible mechanism through which differential opportunities to learn mathematics are presented to males and females and as an instructional practice generating social-psychological processes that differentially affect students by sex. We test the argument on longitudinal data from a larger sample of students in fourth- through sixth-grade classes in both desegregated and segregated schools. The results show that sex is a factor in the assignment of students to ability groups: Males are more likely than females to be assigned to the high-ability group. Nevertheless, the analyses do not reveal an effect of ability-group level on growth in mathematics achievement.

For some time now, researchers and educational practitioners have been concemed about observed differences in the mathematics achievement of male and female students. These differences are well documented. Several studies report that in the elementary grades, girls have higher mathematics achievement than boys but that in junior and senior high school, boys are more likely than girls to elect mathematics courses and are more likely to succeed in those courses. (For a review of this literature, see Fennema and Peterson [1985].)

A number of social-psychological explanations for sex differences in mathematics achievement have been offered. Researchers have suggested that females develop negative attitudes toward mathematics because they view it as a stereotypically male subject, that they have fewer role models of successful mathematicians than males, and that they receive pressure from male peers not to excel in mathematics. Structural explanations point to the typical avoidance of advanced mathematics courses by females as a reason for their lack of success in mathematics and the low probability that they will study mathematics in college or pursue it as a career option.

While these explanations are reasonable, they ignore other important dimensions of schooling that may affect sex differences in mathematics achievement. In particular, virtually no attention has been given to how organizational characteristics of a classroom may produce sex differ-

ences in student outcomes. This is surprising, since it is in the classroom that opportunities for learning are provided, that attitudes toward particular subjects are formed, and that peer and teacher interactions about mathematics and other subjects occur. Consequently, it seems impera- tive to examine whether classroom characteristics and pedagogical practices differentially influence the mathematics achievement of male and female students.

ABILITY GROUPING AND ACHIEVEMENT

A major organizational characteristic of a classroom is the organization of students for instruction. Ordinarily, teachers either instruct the class as a whole or divide the students into smaller groups. When teachers form small instructional groups, they generally rely on measures of aptitude or prior achievement in assigning students to groups. Ability grouping is common for reading instruction at the elementary level and occurs somewhat frequently for mathematics instruction.

The rationale for ability grouping is straight- forward. When students are grouped by ability, teachers are able to gear the level and pace of instruction to the aptitudes of the students. This lessens the likelihood that students become discouraged by material that is too difficult for them or bored by material that is too easy. Also, during the actual period of instruction, students in small groups receive more individual attention from the teacher than students in larger groups. These advantages of ability grouping are believed to offset the disadvantages of reduced overall instructional time and unsuper- vised seatwork. In short, while ability grouping provides fewer opportunities for learning than

This research was funded by the Wisconsin Center for Education Research, which is supported in part by National Institute of Education grant NIE-G-81-0009. Address all correspondence to Professor Maureen T. Hallinan, Department of Sociology, University of Notre Dame, 400 Decio Hall, Notre Dame, IN 46556.

63



64 HALLINAN AND S0RENSEN

whole-class instruction, it also provides greater utilization of those opportunities.

A large and growing literature on tracking and ability grouping examines the effects of these pedagogical practices on student achievement (e.g., Barr and Dreeben 1983; Gamoran 1986; Oakes 1985; Rosenbaum 1976; S0rensen and Hallinan 1986). After conducting a comprehen- sive meta-analysis of studies on grouping effects in elementary and middle schools, Slavin (1986) concluded that tracking has no effect on achievement but that within-class ability grouping has a positive effect on mathematics achievement. These conclusions are supported by a large number of fairly rigorous studies.

S0rensen and Hallinan (1986) examined the effects of ability grouping on reading achievement in a large longitudinal data set. The results show no advantage of ability grouping over whole-class instruction in reading. However, in ability-grouped classes, students assigned to the high group had greater gains in reading achievement than those assigned to the low group. In addition, small homogeneous ability groups produced greater gains in achievement than larger, more heterogeneous groups.

A number of other studies also show that ability grouping favors students assigned to the high group and disadvantages students assigned to the low group (see Good and Marshall [1984] and Bossert, Barnett and Filby [1984] for reviews of this work). A differential impact of ability grouping on students varying in aptitude or achievement may explain the frequently reported absence of a main effect of ability grouping. High-ability groups are often charac- terized as having a positive learning context created by students who are highly motivated to learn and who provide strong learning models. In contrast, the climate of low-ability groups appears to be less conducive to learning; students are more easily distracted, have more behavioral problems, and provide weaker instructional models. Moreover, if teachers need to devote more instructional time to organization and discipline in low-ability groups, the amount of 'instruction presented to these students is further reduced.

ABILITY GROUPING AND SEX DIFFERENCES IN MATHEMATICS ACHIEVEMENT

One aspect of ability grouping that has been almost completely ignored by social scientists is the possible differential effect of this pedagogical practice on students who differ in ascribed rather than achieved characteristics. That is, controlling for aptitude, does ability grouping favor students of one sex or race over another?

A differential impact of ability grouping by sex or race could occur in one of two ways.

First, net of prior achievement, sex or race could be a factor in the assignment of students to ability groups. Teachers may consciously or unconsciously take ascribed characteristics of students into account in their choice of ability- group levels for students. For example, teachers may judge that girls are less interested in mathematics than boys and hence favor boys in assignments to the high-ability groups. To the extent that the level of an ability group affects growth in achievement, sex or race biases in the assignment process could be one of the mechanisms that create differences in achievement between males and females and between blacks and whites.

S0rensen and Hallinan (1984) examined how race affects the assignment of students to ability groups in several desegregated elementary schools and found no discrimination toward black or white students in assignments to the high reading group. However, the high-ability groups tended to be larger in majority black classrooms than in majority white classrooms, giving black students an increased probability of being assigned to the high group.

A second way ability grouping could have a differential effect on student achievement by sex or race is through the social-psychological processes that occur during ability-group instruction. One such process is related to student status. The bases for status may include ascribed characteristics, and females or blacks may have lower status than males or whites. Status differences interfere with student participation in classroom interactions and activities by limiting the frequency, content, and duration of the interactions of lower-status students. If ability grouping places heavier demands on students for group participation than whole-class instruction, which seems likely, then it disadvantages lower-status students.

Another process that could link ability grouping to sex or race differences in achievement is modelling. One way students learn is by imitation. Students in the high-ability group are apt to be the strongest learning models in the classroom. If students are disproportionately assigned to the high-ability group by sex or race, then high-ability students of one sex or race will interact more with these learning models than other students. Moreover, ability grouping constrains the visibility of strong learning models to the remaining students in the class. Again, if females or blacks are disproportionately excluded from the high-ability group, they will have fewer opportunities than males or whites to observe these models. These con- straints that ability grouping places on the visibility of model students may produce differential achievement by sex or race for students in ability-grouped classes.



ABILITY GROUPING AND SEX DIFFERENCES 65

S0rensen and Hallinan (1986) examined the differential impact of ability grouping on reading achievement by race. They showed that the difference between white and black gains in reading achievement over a school year was greater in grouped classes than in ungrouped classes. Since the mean achievement scores were greater for whites than for blacks, this result suggests that ability grouping disadvantages black students. No advantage of ability grouping for reading for males or females was found, however.

In this paper, we investigate the effects of ability grouping on the mathematics achievement of elementary school students. We pay particular attention to sex differences in the effects of ability grouping on gains in mathematics achievement. The analyses are based on a subset of the longitudinal data set that was used in the analysis of grouping effects on reading achievement reported in S0rensen and Hallinan (1986). Comparisons between the reading and mathematics results will be made.

METHODOLOGY

Sample

The sample for this study is a subset of a large, longitudinal data set obtained from 1,477 students in 48 classes in 10 public and private schools in northern California. The schools were selected to maximize variation in racial composition and in such pedagogical practices as instructional grouping. The sample included 10 fourth grades, 12 fifth grades, 10 sixth grades, 5 seventh grades, and 11 combined grades. The mean class size was 30.7 with a standard deviation of 5.8. The racial distribution was 658 blacks (44.5 percent), 679 whites (47.2 percent), 75 Asians (5.1 percent), and 47 Chicanos (3.2 percent). Since the number of Asians and Chicanos was too small to permit separate analyses, they were coded as white for this study.1 The black students were from lower- to middle-class backgrounds, and the white students were from middle- to upper-class backgrounds.

Standardized achievement tests were administered to all the classes in the fall of the school year and to about half of the classes again in the spring. The schools used three different batteries of tests, including the California Achievement Test, the Iowa Test of Basic Skills, and the Metropolitan Achievement Test. The reading

test scores were transformed into a single metric, using the Anchor test procedures (see S0rensen and Hallinan 1986), but no transformations were available for the mathematics tests. Consequently, caution must be exercised in comparing mathematics scores across schools. This problem is reduced in our analysis because we use grade-equivalent scores rather than raw achievement scores. If the populations on which the tests were normed are similar, these scores should be comparable.2 The comparability of the mathematics scores is not a concern in the analyses that rely on class rank rather than achievement test scores.

The teachers in the sampled classes indicated whether or not the class was grouped for instruction, the basis of assignment to instructional groups, and the level and membership of the ability groups. Nineteen of the 48 classes had ability groups for both mathematics and reading instruction. The composition of the mathematics and reading groups varied consid- erably in each class, indicating that few students were equally proficient in both subjects. An additional 13 classes had ability grouping only for reading. In this paper, we contrast the 19 classes that had mathematics ability groups with the 29 classes that did not.

Of the 1,477 students in the sample, 242 (16 percent) were absent the day the standardized achievement test was administered at the beginning of the school year. Further reduction in data occurred because not all the students in the ability-grouped classes were assigned to within-class ability groups. A few students in each class were sent out of the classroom for advanced work in mathematics or for remedial work. These students were not included in the sample. Finally, some of the analyses required mathematics scores at two points in time. The

1 The Asians and Chicanos in the sample resembled whites more than blacks in socioeconomic status. Moreover, classroom observation revealed that Asian and Chicano students were more socially distant from blacks than from whites. These factors motivated the decision to code all nonblack students as whites.

2 Comparisons of grade-equivalent scores across grades or schools must be interpreted with caution. The likelihood of misinterpretation of change in grade- equivalent scores appears to be greatest when comparisons are made among groups of students at different grade levels. The problem is that variance in grade- equivalent scores usually increases with grade, so that low- and high-scoring students appear to be more distant from the median than their raw scores indicate. In our analyses, we are less concerned with comparing growth in achievement across grades than with identifying the determinants of this growth. Use of grade-equivalent scores for this purpose is less controversial. A greater problem for our analysis is the unavailability of transformations from one mathematics test battery to another. Although most of the students in the sample took the CTBS battery, the presence of some test scores from other batteries makes the analysis less rigorous than we would like. Therefore, we recommend that the hypotheses, presented in this paper be further tested on a more adequate data set.




spring test scores were available for 192 students in ability-grouped classes and for 376 students in ungrouped classes. Because of the large amount of missing data, we used recently developed statistical techniques (Heckman 1979) to determine whether the missing data were associated with a nonrandom subset of students, which could bias parameter estimates. The results revealed no systematic differences between students who were in the sample and those excluded because of missing data.3

In addition to the mathematics achievement test scores and the level and membership of the mathematics ability groups, the data set contains information obtained from school records, including the sex and race of the students and the size, grade, and racial composition of the classes. Class rankings and such class-level variables as the mean and standard deviation of the test scores were constructed from the data.

Procedures

It is possible that unmeasured characteristics relevant to learning might differentiate between students in grouped classes and those in ungrouped classes. If this type of sample selection bias were present in the data, it could lead to incorrect conclusions about the effects of grouping. To avoid this possibility, we again used Heckman's statistical techniques to derive sample selection correction terms. However, because the magnitude of these terms was too small to affect the results of a multivariate analysis, they are not included.

Four sets of multivariate analyses are presented. In the first set, the probability of assignment to the high-ability mathematics group is regressed on classroom and individual characteristics, with special attention to sex. Since the dependent variable is dichotomous, a logistic regression model is estimated. The second set of analyses examines the effects of ability grouping on gains in mathematics achievement. The third set estimates the effects of membership in the high- or low-ability group on gains in mathematics achievement. The final

set of analyses investigates whether sex differences exist in the effects of ability grouping on mathematics achievement. These last three sets of analyses are based on ordinary least squares regression.

RESULTS

Assignment to the High-Ability Group

Table 1 presents five logistic regression models in which assignment to the high-ability mathematics group is regressed on selected individual and classroom characteristics. The independent variables are added sequentially to the model. In model 1, we find no effect of sex or race on assignment to the high group. Both coefficients are positive, indicating a tendency for females and blacks to be preferred in the assignment process, but the results are not statistically significant. In model 2, a student's grade-equivalent mathematics score at the beginning of the school year is added. This score has a strong positive effect on group assignment: The higher a student's standardized test score in mathematics, the higher the likelihood that the student will be assigned to the high-ability group. The effects of sex and race are not altered by controlling for mathematics achievement at the beginning of the school year.

In model 3, the percentage of blacks in the class and the relative size of the high-ability mathematics group are added to the model. The latter variable is calculated as the proportion of students in the class who are in the high-ability group. Both of these classroom characteristics have a positive effect on assignment and do not change the effects of the other variables in the model. The higher the percentage of black students in the class, the higher the likelihood that a student will be assigned to the high-ability group, undoubtedly because the high-ability groups tended to be larger in the majority black classrooms than in the other classrooms in the sample. The relative size of the high-ability group also has a strong positive effect on assignment, as would be expected. Net of these classroom characteristics, sex and race continue to have no effect on assignment.

Model 4 replaces grade-equivalent mathematics scores with mathematics scores standardized within the classroom (z scores). We have argued elsewhere (S0rensen and Hallinan 1984) that a student's mathematics achievement relative to other class members is more important in the assignment process than a student's actual score. The only variables affecting assignment to the high group in model 4 are the relative size of the ability group and the z score. The higher a student's mathematics test score relative to his or her classmates, the higher the likelihood that the student will be assigned to the high-abilitv

3 A detailed explication of the application of Heckman's technique to the reading achievement data in our sample is found in S0rensen and Hallinan (1986, p. 531). A similar procedure was followed for the mathematics data analyzed in this paper. All the individual- and classroom-level variables in our analyses were used to estimate three models to obtain a correction for sample selection. The results indicated that no correction was necessary. Though a measure for SES was not available, SES was obviously highly correlated with race in our sample, and the inclusion of race in the models indicated that the students for whom data were missing did not differ from other students in race or other ascribed or achieved characteristics.




Table 1. Logistic Regressions of Assignment to the High-Ability Mathematics Group on Individual and Classroom Characteristics (Standard Errors in Parentheses)

Independent Model Variables 1 2 3 4 5

Constant - .725 - 2.369 4.487 - 2.063 -2.329 (.252) (.581) (.932) (.702) (.732)

SEX .457 .486 .564 .567 .573 (.287) (.295) (.308) (.313) (.312)

RACE .391 .051 .642 .421 .456 (.288) (.313) (.429) (.444) (.453)

MATH .036* .034* (.011) (.012)

PCTBL 1.723* .904 1.193 (.655) (.677) (.707)

RELH 3.723* 3.371* 3.564* (1.210) (1.230) (1.250)

Z .736* 1.122* (.183) (.285)

Z*SEX -745* (.374)

L 273.67 262.49 251.07 244.97 240.86

df 201 200 198 198 197

NOTE: MATH is the grade-equivalent mathematics score at the beginning of the year, PCTBL is the percentage of the class that is black, RELH is the relative size of the ability group, Z is the math scores standardized within classroom, Z*SEX is an interaction term.

* Coefficient is at least twice its standard error.

group. The percentage of blacks no longer has an effect on assignment, and the effects of sex and race remain insignificant.

Finally, in model 5, an interaction between z score and sex is added. This term should reveal whether high-ranking females are more likely to be assigned to the high-ability group than high-ranking males-that is, whether high achievement benefits one sex in the assignment process more than the other. The model shows a statistically significant negative effect of Z*SEX on assignment. This indicates that females who rank high in the mathematics achievement distribution in the class are less likely to be assigned to the high-ability group than are males who rank high. In other words, high achievement benefits males more than females in the assignment process. Since models 1-4 show that sex, per se, is not a determinant of assignihent, these results suggest that teachers rely more heavily on other characteristics in assigning females to the high group than they do for males. It could be, for example, that teachers favor males in the assignment process because they believe them to be more mature or harder workers than females. Whatever the rationale, the results show that teachers do consider sex in assigning students to the high-ability group and that high-achieving males are given an advantage over high-achieving females.

In this final model in Table 1, the effects of relative mathematics achievement and relative size of the high-ability group remain statistically significant, but sex, race, and percentage black are not significant. The major determinants of assignment to the high-ability group, then, are rank in the mathematics achievement distribution of the class, relative size of the high-ability group, and an interaction between sex and rank in class. This analysis was repeated for assignment to the low-ability group, and the results were virtually the same. That is, rank in class and size of the low-ability group had direct effects on assignment, in the direction expected. An observed negative interaction between sex and rank indicates that low-ranking males are more likely to be assigned to the low-ability group than low-ranking females. Thus, while their sex disadvantages females in assignment to the high-ability group, it protects them from placement in the low-ability group.

Effects of Ability Grouping on Mathematics Gains

Table 2 presents a multivariate regression of growth in mathematics achievement on individual and classroom characteristics for the total sample and for grouped and ungrouped classes. The purpose of this analysis is to determine




Table 2. Effects of Ability Grouping on Gains in Mathematics Achievement (Standard Errors in Parentheses)

Independent All Grouped Ungrouped Variables Classes Classes Classes

Constant 27.120 37.100 28.420 (4.450) (12.100) (4.870)

GROUP -3.793 (2.610)

MATH*GROUP .065 (.057)

MATH .798* .827* .811* (.038) (.051) (.041)

RACE 3.798* 5.151* 3.913* (1.370) (2.360) (1.660)

SEX .333 1.224 .107 (.787) (1.220) (.988)

PCTBL -13.600* 4.018 -13.490* (3.150) (7.220) (3.570)

GRADE5 1.334 1.559 2.916 (1.640) (3.560) (1.910)

GRADE6 4.695* 4.960 5.240* (1.910) (4.060) (2.220)

GRADE7 .300 -5.608 2.560 (2.070) (4.240) (2.540)

CLSIZE .180 -1.617* .193 (.100) (.402) (.112)

CLMEAN -.147 .122 -.156 (.090) (.189) (.106)

CLSD .054 1.658* -.199 (.127) (.332) (.168)

R 2 .740 .740 .750

N 568 192 376

NOTE: GROUP is a dichotomous variable coded one if the class is grouped for mathematics instruction, zero otherwise; MATH*GROUP is an interaction term; GRADE5, GRADE6, and GRADE7 are dichotomous variables indicating membership in grades 5, 6, and 7; CLSIZE is class size; CLMEAN and CLSD are the mean and standard deviation of the class distribution of mathematics scores at the beginning of the school year.


whether, ability grouping affects gains in mathematics achievement.

In the first equation, individual-level mathematics achievement at the end of the year is regressed on two grouping terms and on other individual and classroom variables. The first grouping term (GROUP) is a dichotomous variable coded one if the class is grouped for mathematics instruction and zero otherwise. The second grouping variable (MATH*GROUP) measures an interaction between mathematics achievement at the beginning of the school year and the existence of mathematics ability groups in the class. The coefficient of this variable will determine whether the possible advantages or

disadvantages of grouping depend on a student's achievement level in mathematics. The other variables in the model include mathematics achievement at the beginning of the year, race, sex, percentage of blacks in the class, grade (treated as a dummy variable with grade 4 the omitted category), class size, and the mean and standard deviation of the class distribution of mathematics scores at the beginning of the school year.

The results show that ability grouping for mathematics instruction has no direct effect on gains in mathematics. The parameter estimate of the grouping variable is negative, implying an advantage of whole-class instruction over ability grouping, but the coefficient is not statistically significant. Nor is there a significant interaction between mathematics achievement at the beginning of the year and the existence of mathematics ability groups in the class. This result is consistent with much prior research showing no direct effect of grouping on achievement. It is also consistent with the results of our study of ability-group effects on reading achievement.

The classroom and individual factors that influence growth in mathematics achievement are mathematics score at the beginning of the year, race, percentage of blacks in the class, and grade. Net of prior mathematics achievement, whites show greater achievement gains over the year than blacks. The positive effect of being in the sixth grade on mathematics gains may be due to the influence of one or two outstanding teachers in the sample rather than to a grade effect per se. The negative effect of percentage black appears to be a contextual effect possibly due to a smaller number of high-ability learning models in majority black classes.4

When this analysis is repeated for the ability-grouped classes, prior mathematics achievement, race, class size, and the standard deviation of the distribution of mathematics scores emerge as statistically significant predic- tors of mathematics gains. Net of prior mathematics achievement, blacks show less gain than whites over the school year in the ability- grouped classes. Class size has a negative effect on growth in achievement, possibly because more students are assigned to each ability group in larger classes. One of the justifications for ability grouping is that the reduction in instructional time is offset by the improved instruction that can be offered when students are taught in small homogeneous groups. Students

4 It should be noted that the difference in mean mathematics achievement scores between blacks and whites is statistically significant; whites scored signifi- cantly higher than blacks. This difference was found in both grouped and ungrouped classes.




assigned to large, somewhat heterogeneous ability groups may lose this advantage. This was demonstrated for reading groups in Hallinan and S0rensen (1983).

Finally, the standard deviation of the class distribution of mathematics scores has a strong, statistically significant, positive effect on mathematics gains in ability-grouped classes. This result is an anomaly, since greater class diversity implies more heterogeneous ability groups (assuming fairly equal-sized groups), which should have a negative effect on learning. We believe that this result, which does not appear in the other two regressions in Table 2, is idiosyncratic. It will be examined again in the next analysis.

Another important finding in Table 2 is that the percentage of blacks in the class has no effect on mathematics achievement in ability- grouped classes. This is probably due to the resegregation of students into ability groups, which is due to the correlation between race and achievement. The racial composition of the ability groups in our sample did not mirror the racial composition of the classes, and it is the former that is more likely to affect student achievement in ability-grouped classes.

The determinants of growth in mathematics achievement for students in ungrouped classes are prior mathematics achievement, race, percentage of blacks, and grade. Net of fall mathematics scores, black students learn less mathematics in ungrouped classes than white students, as was the case in grouped classes. The percentage of blacks has a negative effect on mathematics achievement in ungrouped classes, probably, as stated above, because there are fewer high-ability students and thus fewer learning models in majority black classes than in majority white or all-white classes. This finding contrasts with the absence of an effect of racial composition on achievement in ability-grouped classes, where the racial composition of the ability group is likely more relevant than the racial composition of the class. Finally, being in the sixth grade is associated with greater gains in mathematics, presumably because of teacher effects.

These three analyses of the effects of ability grouping on growth in mathematics achievement show that ability grouping has no main effect on mathematics achievement. Net of relevant individual and classroom characteristics, students in ability-grouped classes make as much progress in mathematics as students in ungrouped classes. Whites experience greater gains in mathematics achievement than blacks, but sex has no statistically significant effect on mathematics achievement. This is true in the grouped as well as the ungrouped classes. Hence, while ability grouping disadvantages

black students, it does not benefit one sex more than the other.

Effects of Membership in High- or Low-Ability Group on Growth in Mathematics Achievement

The results, so far, indicate that teachers take sex into account in assigning students to ability groups for mathematics instruction but that enrollment in a grouped or ungrouped class does not affect growth in achievement. Two ques- tions remain: (1) Does assignment to the high (or low) group have greater benefits for males than for females? (2) Is being in an ability- grouped class more beneficial for males than for females?

Despite the absence of a main effect of ability grouping on achievement, students assigned to the high-ability group may make greater gains in mathematics than those assigned to lower groups. Moreover, the level of an ability group may have a differential impact on the learning of males and females. It is very important that we determine whether or not this is true, since our prior results show that sex is a factor in the assignment process.

In Table 3, we regress gains in mathematics achievement on individual and classroom characteristics, including dichotomous variables for membership in the high-ability mathematics group (HIGH) and membership in the low- ability group (LOW). Three interaction terms

Table 3. Effects of Membership in High- or Low-Abil- ity Group on Gains in Mathematics Achieve- ment

Independent Variables b SE

Constant 9.273 12.97 MATH .799* .073 RACE 4.800* 2.313 SEX .773 1.308 GRADE5 -3.720 3.844 GRADE6 -.888 3.557 GRADE7 -9.685* 3.478 HIGH 1.190 5.501 LOW 1.530 6.224 SEX*HIGH .004 .107 HIGH*MATH .004 .108 LOW*MATH -.099 .156 CLSIZE .239 .764 CLSIZE*GRSIZE -.390* .061 CLMEAN .316 .625 CLSD .606 .625 R2= .71 N= 192

NOTE: HIGH and LOW are dichotomous variables indicating membership in the high- and low-ability groups; GRSIZE is the size of the ability group; SEX*HIGH, HIGH*MATH, LOW*MATH, and CLSIZE*GRSIZE are interaction terms.





are included in the model: SEX*HIGH to determine whether females or males benefit more by assignment to the high-ability mathematics group, HIGH*MATH to ascertain whether assignment to the high-ability group benefits high-achieving students more than lower- achieving students, and LOW*MATH to determine whether membership in the low-ability group has a differential impact on students by their prior mathematics achievement. Ability- group size is also included in the analysis, since it is expected to have a negative effect on learning (Hallinan and S0rensen 1985).

The results show no effect of group level on growth in mathematics achievement. Contrary to a number of previous studies, we find no advantage of being in the high-ability group and no disadvantage of being in the low-ability group. Moreover, we find no interaction between mathematics achievement and assignment to the high or low group. Thus, the benefits of ability grouping for mathematics instruction are experienced by all students regardless of their prior achievement. These results are similar to our previous findings, which showed that ability-group level did not affect growth in reading and did not interact with other variables to affect learning.

Our main goal in this analysis is to determine whether assignment to the high-ability group for mathematics instruction differentially affects males and females. Despite the absence of a direct effect of ability-group level, an interaction between sex and assignment to the high- or low-ability group could influence learning. However, the model presented in Table 3 shows that this is not the case. The parameter estimate of the effect of SEX*HIGH is virtually zero. (The effects of SEX*LOW were also found to be insignificant and are not included in this model.) Thus, in our sample at least, females are not benefitted or disadvantaged more than males by being assigned to the high- or low-ability group for mathematics instruction.

Four independent variables-prior mathematics achievement, race, grade, and ability-group size-have statistically significant effects on gains in mathematics achievement for the grouped classes presented in Table 3. Blacks learn less mathematics than whites in ability- grouped classes (as in ungrouped classes), and prior mathematics achievement is the strongest predictor of future mathematics achievement. Being in seventh grade has a positive effect on mathematics learning, but since we find no consistent effects of grade across the different analyses, we do not attach much importance to this finding. Ability-group size has a negative effect on growth in mathematics achievement. This is as expected, since students receive less individual attention in larger groups than in

smaller groups and since heterogeneity is likely to be greater in larger groups, making it more difficult for teachers to accommodate the learning needs of all the students. It should be noted that the curious effect of class standard deviation observed in Table 2 disappears in this model, as does the negative effect of class size reported in Table 2 for the grouped classes.

Differential Effect of Ability Grouping by Sex

The final analysis, presented in Table 4, examines whether classes grouped for mathematics instruction provide more learning opportunities for males than for females. In this analysis, the independent variables include an interaction between enrollment in an ability-grouped class and sex (GROUP*SEX) and a two-way interaction between enrollment in a grouped class, sex, and prior mathematics achievement (GROUP*SEX*MATH). The interaction term MATH*GROUP, which was included in Table 2 but found to be insignificant, is also included in this analysis.

The effects of the individual- and class-level variables in the model are very similar to those reported in Table 2 for the model without the sex interactions. An examination of the new variables shows that the effect of the GROUP*SEX interaction is virtually zero. That is, the way a class is organized for instruction (ability grouped or ungrouped) does not have a differential effect on students by sex. This is true even when prior mathematics achievement is taken into account. The failure of the two-way

Table 4. Effects of Individual and Classroom Charac- teristics on Gains in Mathematics Achievement in Ability-Grouped Classes

Independent Variables b SE

Constant 27.250 4.468 MATH .799* .038 GROUP -3.751 3.381 MATH*GROUP .052 .076 RACE 3.817* 1.380 SEX .008 .970 PCTBL -13.570* 3.155 GRADE5 1.273 1.655 GRADE6 4.625* 1.921 GRADE7 .277 2.075 CLMEAN -.145 .090 CLSD .055 .127 CLSIZE .177 .099 GROUP*SEX .099 4.447 GROUP*SEX*MATH .020 .096 R2= .74 N= 568

NOTE: GROUP*SEX and GROUP*SEX*MATH are interaction terms.





GROUP*SEX*MATH interaction to attain statistical significance reveals that grouping practices do not affect males more than females, regardless of their mathematics aptitude. Thus, we find no support for the argument that the organization of students for instruction differentially affects the mathematics gains of males and females.

CONCLUSIONS

The first set of analyses presented in this paper show that the assignment of students to ability groups for mathematics instruction is indeed influenced by sex. Teachers are more likely to assign high-ranking boys to the high-ability group than high-ranking girls. This suggests that teachers use idiosyncratic considerations as well as objective test scores in assigning students to ability groups. Decisions regarding the placement of girls may reveal discrimination: Girls with high aptitude in mathematics are less likely to be assigned to the high-ability group than boys, and girls, in general, are more likely to be misassigned than boys.

The remaining analyses show that ability grouping in general does not affect gains in mathematics. Students derive the same benefits from whole-class instruction and ability-grouped instruction. This result has been reported in previous research. Apparently, the advantages of ability grouping (i.e., a more appropriate level and pace of instruction) are offset by the disadvantages (i.e., reduced instructional time). What is surprising is that ability-group level did not affect achievement for the students in our sample. This result is inconsistent with a number of studies showing that assignment to the high-ability group accelerates learning and that assignment to the low-ability group retards learning. The relationship between ability-group level and mathematics achievement deserves further attention in future research.

In addition, our results show that females are not disadvantaged by the way the class is organized for instruction and do not experience a different effect of membership in the high (or low) group than boys. If these results are generalizable, then sex considerations in the assignment process do not affect mathematics achievement and, consequently, fail to explain observed differences in the mathematics achievement of males and females.

Five considerations must be made in interpret- ing our findings, however. First, sex had no effect on growth in mathematics achievement for the students in our sample. Since the students were primarily in grades 4 through 6, the sex difference typically observed in junior and senior high school students may not yet

have occurred. Second, our sample was small, particularly for analysis of the grouped classes. This made it difficult for the parameter estimates to attain statistical significance and constrained the number of controls that could be included in the analysis. Third, ability-group level did not affect the achievement of the students in our sample. Sex effects may have emerged had group-level effects been present. Fourth, the achievement data are based on different standardized tests. If the procedures for norming these tests were not similar or if the populations differed, then comparisons across tests may be inappropriate. Finally, a majority of the students in our sample came from majority black classes. These classes tended to be larger than the majority white or all-white classes and were more likely to have whole-class instruction. Further, the black students showed slower gains in mathematics than the white students. Failure to find both group-level effects and sex differences in responses to grouping practices in these data may be due to a confounding of the effects of race, class size, racial composition, and grouping practices.

Because of the limitations of our sample, the present work should not be considered conclu- sive. A larger data set is needed to further test the hypotheses outlined here. The primary importance of this paper lies in its examination of previously ignored ways in which sex differences in mathematics achievement may occur. Nevertheless, the empirical analysis demonstrates that a differential impact of the organization of instruction on males and females and a sex effect on the ability of students to respond to opportunities for learning created by pedagogical practices should not be assumed.

REFERENCES

Barr, Rebecca, and Robert Dreeben. 1983. How Schools Work. Chicago: University of Chicago Press.

Bossert, Steven T., Bruce G. Barnett, and Nikola N. Filby. 1984. "Grouping and Instructional Organiza- tion." Pp. 39-51 in The Social Context of Instruction, edited by Penelope Peterson, Louise Cherry Wilkin- son, and Maureen T. Hallinan. Orlando: Academic Press.

Fennema, Elizabeth, and Penelope Peterson. 1985. "Autonomous Learning Behavior: A Possible Explana- tion of Gender-Related Differences in Mathematics." Pp. 17-35 in Gender Influences in Classroom Interaction, edited by Louise Cherry Wilkinson and Cora B. Marrett. Orlando: Academic Press.

Gamoran, Adam. 1986. "The Stratification of High School Learning Opportunities." Paper presented at the annual meetings of the American Educational Research Association, San Francisco.

Good, Thomas L., and Susan Marshall. 1984. "Do Students Learn More in Heterogeneous or Homoge- neous Groups?" Pp. 15-38 in The Social Context of




Instruction, edited by Penelope Peterson, Louise Cherry Wilkinson and Maureen T. Hallinan. Orlando: Academic Press.

Hallinan, Maureen T., and Aage B. S0rensen. 1983. "The Formation and Stability of Instructional Groups." American Sociological Review 48:838-51.

. 1985. "Class Size, Ability Group Size, and Student Achievement." American Journal of Educa- tion 94:71-89.

Heckman, James J. 1979. "Sample Selection Bias as a Specification Error." Econometrica 47:153-62.

Oakes, Jeannie. 1985. Keeping Track: How Schools Structure Inequality. New Haven: Yale University Press.

Rosenbaum, James E. 1976. Making Inequality. New York: Wiley.

Slavin, Robert E. 1986. "Ability Grouping and Student Achievement in Elementary Schools: A Best Evidence Synthesis." Technical Report No. 1. Baltimore: Johns Hopkins University, Center for Research on Elemen- tary and Middle Schools.

S0rensen, Aage B., and Maureen T. Hallinan. 1984. "Effects of Race on Assignment to Ability Groups." Pp. 85-103 in The Social Context of Instruction, edited by Penelope Peterson, Louise Cherry Wilkinson, and Maureen T. Hallinan. Orlando: Academic Press.

. 1986. "The Effects of Ability Grouping on Gaowth in Academic Achievement." American Educa- tional Research Journal 23:519-42.

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ability grouping and sex differences in mathematics achievement

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