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THE EFFECTS OF COOPERATIVE HOMEWORK ON MATHEMATICS ACHIEVEMENT OF CHINESE HIGH SCHOOL

STUDENTS

ABSTRACT. This study examined the effects of cooperative homework on mathematics achievement, taking into account team characteristics. Results of analysis of covariance (ANCOVA) showed that three-member teams high in ability (three high achievers or two high achievers plus one middle/low achiever) seem preferable in organizing cooperative learning for mathematics homework. Middle and low achievers all benefited from coopera- tive mathematics homework, whereas high achievers did not although they still maintained their top position in mathematics achievement.

Cooperative learning refers to the learning that takes place in cer- tain environment where students work cooperatively in small groups to share ideas and to complete certain academic tasks (Davidson and Kroll, 1991). In their meta-analysis on cooperative, competitive, and individ- ualistic learning, Johnson et al. (1981) found that cooperative learning strategies are the most effective in enhancing productivity and academic achievement. Other reviews of literature have similar conclusions (Sha- ran, 1980; Slavin, 1980, 1983a, 1983b, 1985). As far as mathematics is concerned, Davidson and Kroll (1991) summarized that 'less than half of studies comparing small-group and traditional methods of mathematics instruction have shown a significant difference in student achievement; but when significant differences have been found, they have almost always favored the small-group procedure' (p. 362).

However, there are concerns about research on the effectiveness of cooperative learning. The common procedure is to have students, in small groups of four to six members, work cooperatively on some task structures in either a carefully controlled laboratory (experiment) environment or a carefully designed field (classroom) environment. Teacher-led instructions dominate cooperative learning in most cases (Stevens et al., 1991). There is also the exclusive control of researchers on team formation. In most coop- erative learning strategies for classroom use, students of heterogeneous abilities are assigned into one group with the purpose to make groups roughly equal in their average academic ability (Chambers and Abrami, 1991). The other purpose is to promote interactions between students so that high achievers are able to help low achievers. It is not clear, however,

Educational Studies in Mathematics 31: 379-387, 1996. (~) 1996 Kluwer Academic Publishers. Printed in Belgium.

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whether this is an absolute assumption to follow in the use of cooperative learning methods.

Sharan (1990) noted that cooperative learning has such a general def- inition that it may actually include various models that are different in their assumptions about the nature of teaching and learning and about the role of teachers and students in cooperative learning. This study was designed to determine the effects of cooperative learning on mathematics achievement under conditions of no control on team formation, no control on learning environment, and the absence of teachers. Doing mathematics homework collaboratively in teams was used as a special case to resemble this direction in modeling cooperative learning. The research questions were:

1. Does cooperative homework (in self-built teams, in self-selected loca- tions, and with the absence of teachers) affect mathematics achieve- ment?

2. Does students' prior level of mathematics achievement affect the coop- erative learning method characterized in Question 1 ?

3. What are the effects of the number of students in a team, team formation (homogeneous or heterogeneous) and team ability (high, middle, or low) on the cooperative learning method characterized in Question 1 ?

4. What kinds of teams are potentially effective in the cooperative learn- ing method characterized in Question 1, considering the number of students in a team, team formation, and team ability?

METHOD

Subjects in this study were 182 high school senior students (92 males and 90 females) from an urban high school with four classes in grade 12 in Hebai, China. The sample was homogeneous in students' family background, but heterogeneous in their academic background. Overall, this high school was slightly above average in academic achievement. There were four treatment levels based on the four classes: in class 1 with 47 students, 22 teams of two members each were built; in class 2 with 46 students, 15 teams of three members each (1 team from class 1) were built; in class 3,with 44 students, 12 teams of four members each (1 team from class 2) were formed; in class 4 with 45 students, 9 teams of five members each were formed. Students built their own teams in each class, based on friendship, common interest, or motivation for help. Teachers negotiated around a few left ungrouped to build teams.

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Students had completed their high school mathematics and were receiv- ing a systematic review. One teacher reviewed algebra oriented content; the other reviewed geometry oriented content. Each teacher gave three lessons to each class in each week (45 minutes for each lesson). Lessons and homework were kept the same for all students. Teachers marked stu- dents' homework with 'pass' or 'fail', but kept pointing out problems in solutions and giving suggestions for improvement. All of the 182 students were to take the annual nation-wide College Entrance Examinations. The school has a policy that demands college-bound senior students to attend a two-hour homework session after school in which students usually do various homework individually. Altering this tradition slightly, students were asked to first finish their mathematics homework cooperatively in teams. Students decided where they would like to do their mathematics homework.

Students were to take their first city-wide simulation examinations in eight weeks. Therefore, an eight-week intervention of cooperative learn- ing was carried out on those students. Mathematics achievement as a result of cooperative learning was measured through the simulation math- ematics examination (coefficient alpha was 0.83) in which 80% of the marks were allocated to open-ended items on problem solving I and 20% to multiple-choice items on conceptual and procedural understanding 2. Prior mathematics achievement was obtained by averaging scores from previous middle and final examinations on high school mathematics which had a similar structure to the simulation examination. The coding of each team on formation and ability was carried out separately by two mathe- matics teachers (interrater agreement was 89%). Team formation (either homogeneous or heterogeneous) was based on teachers' teaching expe- rience with individual students and their prior mathematics achievement, and team ability (high, middle, or low) was based on the team average in prior mathematics achievement. Team formation and ability were finally effect coded. Effect coding was also used to represent the four treatment levels.

The present study examined the effects of cooperative homework on mathematics achievement taking into account team characteristics. Analy- sis of covariance (ANCOVA) was deemed as the appropriated statisti- cal method, and the multiple regression/correlation (MRC) approach to ANCOVA (see Cohen and Cohen, 1983; Pedhazur, 1982) was used. The analysis included one dependent variable of mathematics achievement; one independent variable of treatment (two-member vs. three-member vs. four-member vs. five-member); one covariate of prior mathematics achievement; two moderator variables of team formation (homogeneous

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TABLE I R square, change in R square. F ratio for the change, and significance of the change for

the covariate, moderator, treatment, and interaction (N = 182).

Hierarchical predictors R z Change in R 2 F ratio Significance of F

Covariate (prior mathematics 0.8995 0 .8995 1610.768 0.000

achievement)

Moderator (team formation and team 0.9091 0.0096 6.243 0.000

ability) Treatment (two vs. three vs. four vs. 0.9201 0.0110 7.948 0.000

five team members) Interaction (treatment by covariate) 0.9212 0.0012 0.829 0.480

Note. p<0.05. Categorical variable of team formation is dummy coded: (0) for homogeneous team; (1) for heterogeneous team. Categorical variables of team ability and treatment are effect coded. For team ability: high (10); middle (01); low (-1-1). For treatment, two-member teams (100); three-member teams (010); fourmember teams (001); five,member teams (-1-1-1). Moderator variables are entered as a set into the regression.

vs. heterogeneous) and team ability (high vs. middle vs. low). The inter- action term between treatment and covariate was tested first for statistical

significance. If it was statistically significant and explained a meaningful proportion of variance in the dependent variable, the analysis would follow the procedure of aptitude treatment interaction (ATI) (see Pedhazur, 1982). If it was not statistically significant, treatment effects would be examined with covariate and moderator variables being statistically controlled. This procedure would not only remove the effect of the covariate on the depen-

dent variable but also uncover the effects of moderator variables on various

treatments.

RESULTS AND DISCUSSION

The results of hierarchical MRC are presented in Table I. The interaction term was not statistically significant. Therefore, the analysis followed the procedure of ANCOVA rather than ATI. The hierarchical MRC was rerun without the interaction term (see Table II).

When covafiate and moderator variables were controlled, treatment showed statistically significant effects on mathematics achievement. Table II indicates that the mean mathematics achievement for the treatment of three member teams was significantly above the grand mean of mathemat-

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TABLE II

Beta weights, F ratios for beta weights, and significance of beta weights for the treatment, covariate, and moderator variables

383

Predictor variables Beta weight F ratio Significance of F

Prior mathematics 0.9113 1469.232 0.000

achievement

Team formation (vector one) 0.0279 1.327 0.250

Team ability (vector one) 0.1093 17.132 0.000

Team ability (vector two) -0.0041 0.035 0.852

Treatment (vector one) -0.2168 0.672 0.414

Treatment (vector two) 0.1010 14.306 0.000

Treatment (vector three) 0.0256 0.949 0.331

Note. p<0.05. Covariate variable = prior mathematics achievement. Dummy coding for the moderator variable of team formation: homogeneous (0); heterogeneous (1). Effect coding for the moderator variable of team ability. Vector one: high (1); middle (0); low (-1). Vector two: high (0); middle (1); low (-1). Effect coding for the treatment variable. Vector one: two-member (1); three-member (0); four-member (0); five-member (-1). Vector two: two-member (0); three-member (1); four-member (0); five-member (-1). Vector three: two-member (0); three-member (0); four-member (1); five-member (-1).

TABLE III

Original and adjusted means by treatment, team formation, and team ability (N = 182)

Homogeneous team Heterogeneous team

High Middle Low High Middle Low

OM AM OM AM OM AM OM AM OM AM OM AM

Treatmentl 84 76 65 71 85 79 70 72 63 67

Treatment2 86 78 71 74 82 78 71 76 64 69

Treatment3 80 73 70 73 78 75 72 74 68 76

Treatment4 81 73 68 72 73 70 73 72 65 70

Note. OM = Original mean. AM = Adjusted mean. Treatment = {two-member team, three-member team, four-member team, five-member team}. Team formation = {homogeneous team, heterogeneous team}. Team ability = {high, middle, low}.

ics a c h i e v e m e n t , w h e r e a s m e a n s for o the r t r ea tmen t s were no t s ign i f i can t ly

d i f f e ren t f r o m the g r a n d mean . The re fo re , th ree m e m b e r s t eams s e e m m o s t

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preferable for cooperative mathematics homework. Teams of four or five members may well be too large to be effective, while teams of two mem- bers may well be too small to promote meaningful interactions between members.

When the covariate variable was controlled, the moderator effects were statistically significant. Table II, however, demonstrates no statistically significant differences between homogeneous and heterogeneous teams which had similar moderator effects on treatments, and via treatments, on mathematics achievement. Table II also shows that the mean for teams high in ability was significantly above the grand mean, whereas means of teams middle and low in ability were not significantly different from the grand mean. Therefore, students in teams high in ability benefited from cooperative mathematics homework. Middle and low achieving students benefited significantly from cooperative mathematics homework and made considerable progress in mathematics achievement when they were in teams of high ability. The coding in this study suggests that such teams usually had several high achievers but one middle or low achiever.

The covariate variable of prior mathematics achievement was statisti- cally significant in Table I, and accounted for the most amount of variance in mathematics achievement. Table II further shows its importance in the explanation of mathematics achievement. Therefore, prior mathematics achievement was the most important predictor of mathematics achieve- ment. However, the treatment effect and the moderators effects were also statistically significant. To examine the treatment effect as it was moderated by the moderator variables, separate regression equations were built from the hierarchical MRC presented in Table II. Because there were no homoge- neous teams low in ability, obtained were 20 separate regression equations of mathematics achievement on prior mathematics achievement under the four treatments that were moderated by team formation and team ability. Based upon these equations, adjusted means for mathematics achievement under the moderated treatments were calculated (see Table III).

Table III shows that for the two-member treatment, heterogeneous teams high in ability benefited most from cooperative mathematics homework; for the three-member treatment, homogeneous teams high in ability did; for the four-member treatment, homogeneous teams high in ability did; for the five-member treatment, heterogeneous teams high in ability did. Among homogeneous teams, three members teams high in ability bene- fited most from cooperative homework; among heterogeneous teams, two members teams high in ability and four members teams high in ability did. For high ability teams, three-member homogeneous teams were the most efficient in cooperative homework; for middle ability teams, three-member

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heterogeneous teams were; for low ability teams, four-member heteroge- neous teams were. These findings provide generic clues for organizing cooperative homework teams.

Specifically, when the effect of covariate was removed and the effects of moderators were taken into account, the three-member homogeneous teams high in ability showed significantly higher mathematics achievement in 14 out of ! 9 comparisons than other teams 3. Students in the two-member heterogeneous teams high in ability and the three-member heterogeneous teams high in ability also demonstrated significantly higher mathematics achievement in 13 out of 19 comparisons than students in other teams. There were no statistically significant differences among these three top teams. In contrast, students in the five-member heterogeneous teams low in ability scored significantly lower in mathematics achievement in 13 out of 19 comparisons than students in other teams. Therefore, three-member teams high in ability (three high achievers; two high achievers and one middle or low achiever) seem to be preferable in organizing cooperative learning for mathematics homework.

Table III also illustrates the most revealing finding in this study. A careful comparison between original and adjusted means shows that middle and low achievers all benefited from cooperative mathematics homework, whereas high achievers did not although they still maintained their top position in mathematics achievement. High achievers in heterogeneous teams clearly have to spend much time helping other students. As a result, cooperative homework is unable to engage high achievers in the kind of in depth thinking frequently observed in individualised learning. But, it is worthy of further investigation why high achievers in homogeneous teams also benefit less from cooperative homework than expected. One potential reason is that normal mathematics homework is not challenging enough for high achievers because most of them do seek additional, more difficult mathematics exercises in individualised homework. If it is the case, students in homogeneous teams high in ability ought to be given more challenging mathematics homework in order to promote meaningful discussions among members.

Overall, the findings do seem to support the notion that heterogeneous teams are promising in arranging cooperative learning if the educational goal is to improve performance of a class as a group. Since the school in this study was only slightly above average in academic achievement and was relatively representative in terms of students' family background, the findings might be considered typical of a large number of Chinese urban high schools. Finally, the cooperative learning strategy outlined in this study seems feasible and at least as effective as the commonly referred

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coopera t ive learning strategies. D o i n g ma themat i c s h o m e w o r k coopera -

t ively in t eams can m a k e a d i f ference in ma themat i c s achievement . Fur ther

research should assess the effects o f coopera t ive ma themat i c s h o m e w o r k

on di f ferent ma themat ica l areas (algebra, geomet ry , analysis , etc.) and on

different ma thema t i ca l tasks (appl icat ion o f s tandard procedures , conc e p -

tual unders tanding , p rob l em solving, etc.).

N O T E S

1Two of the open ended items on problem solving are provide here as examples: (I) Let F(x)= x 2, solve the equation F(x 2 ) = x 2 + 6. (II) On the sea, ship A travels at a speed of vl (nautical miles/hour) west-bound. At

certain time, it is k nautical miles away from island P which is located Wc~~ from ship A. Ship B leaves the island at that time to intercept ship A. Suppose the speed of ship B is v2 (nautical miles/hour), demonstrate the direction in angle that ship B has to take as well as the time ship B needs in order to meet ship A. Also discuss the condition(s) that will cause no solution for this problem. 2Two of the multiple choice items on conceptual and procedural understanding are provided here as examples:

(I) Three different lines A, B, and C in certain space. A and B are perpendicular; A and C are perpendicular. What is the relationship between B and C?

(a) Parallel (b) Coincident

(c) Perpendicular (d) Need more information to decide (II) Point A (3, 4) is in a plane defined by axes X and Y. If a translation of axes is made so that the central point O moves from (0, 0) to (2,1), the new coordinates of point A will be:

(a) (0, 0) (b) (1, 3)

(c) (5, 5) (d) (6, 4)

3Adjusted means were tested against one another for significance by the modified Scheffe

method (see Pedhazur, 1982). Statistical results are available from the author.

R E F E R E N C E S

Chambers, B. and Abrami, E C.: 1991, 'The relationship between student team learn- ing outcomes and achievement, causal attributions, and affect', Journal of Educational Psychology 83, 140-146.

Cohen, J. and Cohen, E: 1983, Applied Multiple Regression~Correlation Analysis in Behav- ioral Sciences. Lawrence Edbaum Associates, Hillsdale, NJ.

Davidson, N. and Kroll, D. L.: 1991, 'An overview of research on cooperative learning related to mathematics', Journal for Research in Mathematics Education 22, 362-365.

Johnson, D. W., Maruyama, G., Johnson, R. T., Nelson, D. and Skon, L.: 1981, 'Effects of cooperative, competitive, and individualistic goal structures on achievement: A meta- analysis', Psychological Bulletin 89, 47-62.

Pedhazur, E. J.: 1982, Multiple Regression in Behavioral Research: Explanation and Pre- diction. Holt, Rinehart and Winston, New York.

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Sharan, S.: 1980, 'Cooperative learning in small groups: Recent methods and effects on achievement, attitudes, and ethnic relations', Review of Educational Research 10, 241- 271.

Sharan, S. (Ed.): 1990, Cooperative Learning: Theory and Research. Praeger, New York. Slavin, R. E.: 1980, 'Cooperative learning', Review of Educational Research 50, 315-342. Slavin, R. E.: 1983a, Cooperative Learning. Longman, New York. Slavin, R. E.: 1983b, 'When does cooperative learning increase student achievement?'

Psychological Bulletin 94, 429-445. Slavin, R. E.: 1985, 'An introduction to cooperative learning research'. In R. Slavin, S. Sha-

ran, S. Kagan, R. Hertz-Lazarowitz, C. Webb and R. Schmuch (Eds.), Learning to coop- erate, cooperate to learn (pp. 5-15). Plenum, New York.

Stevens, R. J., Slavin, R. E. and Famish, A. M.: 1991, 'The effects of cooperative learning and direct instruction in reading comprehension strategies on main idea identification'. Journal of Educational Psychology 83, 8-16.

Department of Mathematics and Science Education The University of British Columbia Vancouver, BC, Canada and

The Atlantic Center for Policy Research in Education, The University of New Brunswick Fredericton, NB, Canada