Rational Number Project Home Page Cramer, K., Post, T., & Behr, M. (1989, January). Cognitive Restructuring Ability, Teacher Guidance and Perceptual Distracter Tasks: An Aptitude Treatment Interaction Study. Journal for Research in Mathematics Education, 20(1), 103-110.

Cognitive Restructuring Ability, Teacher Guidance, and Perceptual Distracter Tasks:

An Aptitude-Treatment Interaction Study

KATHLEEN A. CRAMER. University of Wisconsin-River Falls
THOMAS R. POST. University of Minnesota
MERLYN J. BEHR; Northern Illinois University

The aptitude-treatment interaction (ATI) study reported here explored the relationship between cognitive restructuring ability, as measured by the Group Embedded Figures Test (GEFT), and treatments varying in amounts of teacher guidance. It specifically investigated how these two variables affected performance on rational number tasks involving perceptual distracters.

Perceptual distracter problems evolved from the work of the Rational Number Project (Behr & Post, 1981). Such problems contain visual-stimulus information inconsistent with the demands of the task that must be identified and ignored prior to solving the problem. In addition to perceptual distracter tasks with inconsistent cues, Behr and Post defined three other types of rational number tasks in which the visual cues are more consistent with the tasks' demands. Table 1 illustrates these problem types.

 Table 1 Examples of Rational Number Tasks for the Problem:   Shade 2:3 Problem cue type Continuous model Discrete model Solution strategy Complete Shade 2 of 3 equal parts given. Incomplete Divide into 3 equal parts and proceed as in complete case. Extraneous Regroup the 6 parts (sections or circles) into 3 groups of 2 and proceed as in complete case. Perceptual distracter Ignore the divisions given, draw in new ones, and proceed as in complete case.

Behr and Post (1981) hypothesized that the ability to solve perceptual distracter problems is one index of the stability of the learner's rational number concepts. Strategies that children develop to deal with the cognitive conflict inherent in perceptual distracter problems will result in the development of more stable rational number concepts. Behr and Post further suggested that the ability to deal with distracters is related to at least two variables: (a) the learner's experience with problems requiring restructuring, and (b) the learner's cognitive style.

Textbook-dominated school programs are static. The children have little opportunity to manipulate materials and to vary other aspects of the problem situation. Problems generally are presented in a ready-to-solve form, and the children learn to take what is given, manipulate it according to predetermined rules, and proceed directly to a solution. The children may generalize incorrectly that problem conditions are always consistent with, and relevant to, the intended solution. As a result, they have difficulty with problems requiring restructuring.

The extent to which a child displays traits of the field-dependent/ independent cognitive style may also influence his or her success with perceptual distracter tasks. The specific aspect of field independence that is important here is cognitive restructuring ability. Cognitive restructuring is defined as actions that involve changing the perceptual or cognitive field rather than taking it "as is." Acting on a field entails (a) breaking up an organized field so parts are experienced as separate from the background, (b) providing an organization to a field that lacks one, or (c) imposing a different organization on a field than the one suggested by its inherent organization. Field-independent learners have a more highly developed cognitive restructuring ability than field-dependent learners do (Witkin & Goodenough, 1981). Field-independent people use their cognitive restructuring ability as a mediating device in processing perceptual and cognitive tasks. They "go beyond" the information given, restructuring or adding structure. Field-dependent people, because of their lack of cognitive restructuring ability, are more likely to "go along" with the perceptual or cognitive field, cuing in on the most dominant features.

Solutions to perceptual distracter tasks require cognitive restructuring. The learner needs to ignore salient cues that are inconsistent with the demands of the solution. Because cognitive restructuring is required to solve perceptual distracter problems, field-dependent students may have more difficulty solving these problems than field-independent students.

Cognitive style, and in particular cognitive restructuring ability, may also influence the type of instruction the student needs so as to learn to solve distracter problems. Research has suggested that field-dependent students have benefited from learning environments with high teacher guidance, whereas field-independent students have benefited from learning environments with low teacher guidance (McLeod & Adams, 1979; McLeod & Briggs, 1980; McLeod, Carpenter, McCormick, & Skvarcius, 1978; Witkin, Moore, Goodenough, & Cox, 1977). This research is the basis for our hypothesis that cognitive restructuring ability and levels of teacher guidance interact with performance on rational number tasks involving visual perceptual distracters.

1. Do students with low cognitive restructuring ability as measured by the GEFT learn to solve rational number perceptual distracter problems better with high teacher guidance than with low teacher guidance?
2. Do students with high cognitive restructuring ability as measured by the GEFT learn to solve rational number perceptual distracter problems better with low teacher guidance than with high teacher guidance?

METHOD

Subjects

Twenty-two fourth graders from one class in a suburban midwestern school district participated in the study. The school is in a predominantly white, middle-to-upper-income district. The students in the school were grouped for mathematics by ability measures. The students in the sample were ranked second out of four classes in ability. As participants in the Rational Number Project, the children, just prior to this study, were instructed in various fraction concepts for four class periods a week for 15 weeks. The instruction used a variety of manipulative materials. Care was taken not to include perceptual distracter items in any lesson prior to the experimental treatment.

Instruments

Witkin and Goodenough initially considered the Embedded Figures Test and its group form, the GEFT, to be measures of the cognitive style of field independence; they now consider the two tests to be measures of cognitive restructuring ability (Witkin & Goodenough, 1981). The GEFT was developed for use with subjects 10 years or older. The subjects in this study were 9 or 10 years old. A group form of the Children's Embedded Figures Test was initially used, but as this did not discriminate among the students along the cognitive restructuring continuum, the GEFT was given. The GEFT results were used to identify students with high and low cognitive restructuring abilities. A published reliability coefficient from a college-age sample was .84 (Witkin, Oltman, Raskin, & Karp, 1971). The KR-20 reliability coefficient calculated from the scores of the subjects in the study was .82.

We developed two versions of a rational number concepts test. One was used as a pretest and one as a posttest. Each 48-item test included 12 items (6 continuous, 6 discrete) in each of the problem cue types in Table 1. The test items used denominators of fourths through ninths; all fractions were less than 1. Two test formats were used: (a) given a fraction, circle or shade the amount, and (b) given a shaded or circled amount, name the fraction. The students wrote directly on the test booklet. Each question had the response options, "I don't know how to solve the problem" and "Nobody can solve this problem." KR-20 reliability coefficients calculated from the scores of the subjects in this study were .91 for the pretest and .79 for the posttest.

Procedure

The GEFT was administered to a class of 29 fourth graders. Their scores ranged from 2 to 16, with a score of 18 being the highest possible. Twenty-two students at the extreme ends of the score distribution were identified as having either high or low cognitive restructuring ability. The scores of the 11 students in the low cognitive restructuring group ranged from 0 to 6, with a mean of 4.91. The scores of the 11 students in the high cognitive restructuring group ranged from 10 to 16, with a mean of 12.55. The mean IQ of the low group (117.2) was not significantly different from that of the high group (119.0), t(20) = -0.508, p < .62. The mean score on a mathematics achievement test (Comprehensive Test of Basic Skills) for the low group (683.9) was not significantly different from that for the high group (690.7), t(20) = -1.406, p < .18.

The students were randomly assigned to two treatment conditions differing in levels of teacher guidance. Six students with high cognitive restructuring ability and five with low were assigned to the low guidance group. Five students with high cognitive restructuring ability and six with low were assigned to the high guidance group. An investigator was assigned to each treatment by a flip of a coin.

Treatments

Four parallel lessons were developed for the high guidance and low guidance treatments. The same examples were used in each treatment. Lesson 1 reintroduced the students to incomplete examples using discrete and continuous models. Lesson 2 reintroduced extraneous examples using discrete and continuous models. Lessons 3 and 4 presented for the first time problems with perceptual distracter cues. Lesson 3 used a continuous model, and Lesson 4 used a discrete model.

In Lessons 1 and 2, the instruction highlighted how complete, incomplete, and extraneous problems were alike and how they were different. The children were instructed to identify each problem by name and to identify a restructuring procedure for each one. Perceptual distracter problems were introduced as another instance where restructuring was needed. In contrast to the other problem types, the students were told that the information given in perceptual distracter problems was not helpful and could distract them from obtaining the correct solution. The lessons emphasized the idea that some information given in a problem may not be useful when solving it.

The treatments lasted for six class periods of 40 minutes each. Lessons 1 and 2 were completed in the first three periods; Lessons 3 and 4 in the last three periods.

Previous studies have differentiated high guidance from low guidance instruction by providing more structure in the student materials used by the high guidance group. In this study, the role of the teacher was used to distinguish high guidance from low guidance.

Each lesson had two parts: a lesson development phase and a practice phase. The treatments differed with respect to the lesson development phase. The students in both treatments completed the same two practice pages after the lesson development; these pages were corrected with teacher assistance in both groups. The lesson development dominated the class time; the practice phase of the lesson was only 10 minutes long.

The high guidance treatment was teacher centered, with little student choice. It used large-group instruction with the teacher pacing each example at the overhead projector. Through the teacher's questions, the students identified restructuring procedures for solving each problem type. The teacher provided immediate feedback on the answer to each example.

The teacher's role in the low guidance treatment was to provide the students with an opportunity to investigate problems on their own. After an initial introduction, the students worked through examples at their own rates. Leading questions were embedded in the materials to keep students involved and to lead them to "discover" characteristics of incomplete, extraneous, and perceptual distracter problems. Evidence of success was not immediate because the teacher did not monitor student work step by step. The teacher led a brief discussion at the end of the development phase of the lesson, at which time the students shared their findings.

RESULTS

Multiple regression techniques (Kerlinger & Pedhauzer, 1973) were used to determine the presence of Cognitive Restructuring Ability x Treatment interactions. The continuous perceptual distracter items were analyzed separately from the discrete items. Table 2 presents means and standard deviations for continuous and discrete perceptual distracter items for each Cognitive Restructuring Ability x Treatment group.

Table 3 gives information about the interactions. The squared multiple correlation for the full model was calculated using three independent variables: treatment group, GEFT score, and interaction. The change in R2 reflects the difference between the squared multiple correlation calculated using the full model and the squared multiple correlation when the interaction term was dropped. The interaction was significant (p < .025) for the continuous perceptual distracter items but not for the discrete items. Because the interaction was not significant, the intercepts for the discrete items were checked to determine if a treatment effect was present; the F ratio was less than 1. Figure I presents the interaction for the continuous perceptual distracter items. The regression equation using the GEFT score as the only predictor variable is reported for each treatment group.

 Table 2 Means and Standard deviations for Continuous and Discrete Perceptual Distracter Items on the Posttest Cognitive restructuring ability High guidance Low guidance M SD M SD Continuous High 2.83 1.47 4.20 1.09 Low 3.60 1.14 2.50 1.37 Discrete High 5.33 1.21 5.40 0.554 Low 5.00 0.71 4.83 1.47 Note. Maximum score = 6.

 Table 3 Regression Tests for Cognitive Restructuring Ability X Treatment Interactions Dependent variable R2 for full model Change in R2 F (1.18) Continuous perceptual distracter .4898 .2393 8.455* Discrete perceptual distracter .2506 .0002 0.014 *p<.025

Figure 1

Figure 1. Interactions of continuous perceptual distracter items with high or low teacher guidance.

DISCUSSION

An interaction between cognitive restructuring ability and levels of teacher guidance was significant for items having continuous perceptual distracters but not for items having discrete perceptual distracters. For students to perform well on the continuous tasks, the optimal amount of teacher guidance was inversely related to the students' level of cognitive restructuring ability. The students with high cognitive restructuring ability performed better with low levels of teacher guidance, whereas the students with low cognitive restructuring ability performed better with high levels of teacher guidance.

That the students found continuous perceptual distracters more difficult than discrete perceptual distracters may explain why an interaction was found for one kind of task and not the other. To help clarify why continuous tasks were more difficult, actual responses were examined and solution strategies identified. We found that the most frequent mode of responding to discrete perceptual distracter problems involved no overt restructuring of the diagram. The students generally circled the correct number of objects without changing the picture. This observation suggests that they may have used a solution strategy at the symbolic level: Probably a division algorithm was used because division was used to introduce the discrete model. The most common strategy for solving the continuous perceptual distracter tasks was to ignore the unneeded lines and draw in the correct ones. A solution to the continuous perceptual distracter problems required a physical restructuring of the diagram. The perceptual features of the task could not be avoided by solving the problem at a symbolic or algorithmic level.

The continuous perceptual distracter tasks may have been more difficult because they required cognitive restructuring; the discrete tasks could be solved using an algorithm. This difference between the tasks may help account for the difference in interactions. Future studies investigating higher order interactions (cognitive restructuring ability by levels of teacher guidance by task) might prove fruitful additions to the aptitude-treatment interaction literature if the tasks taught are differentiated by difficulty level and by need for cognitive restructuring.

REFERENCES

Behr, M. J., & Post, T. R. (1981). The effect of visual perceptual distracters on children's logical-mathematical thinking in rational-number situations. In T. Post & M. P. Roberts (Eds.), Proceedings of the Third Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 8-16). Minneapolis: University of Minnesota.

Kerlinger, F., & Pedhauzer, E. (1973). Multiple regression in behavioral research. New York: Holt, Rinehart & Winston.

McLeod, D., & Adams, V. (1979). The interaction of field independence with discovery learning in mathematics. Journal of Experimental Education, 48(1), 32-35.

McLeod, D., & Briggs, J. (1980). Interactions of field independence and general reasoning with inductive instruction in mathematics. Journal for Research in Mathematics Education, 11, 94-103.

McLeod, D., Carpenter, T., McCormick, R., & Skvarcius, R. (1978). Cognitive development and cognitive style as factors in mathematics achievement. Journal of Educational Psychology, 72, 326-330.

Witkin, H. A., & Goodenough, D. R. (1981). Cognitive styles: Essence and origins. New York: International Universities Press.

Witkin, H. A., Moore, C. A., Goodenough, D. R., & Cox, P. W. (1977). Field dependent and field independent cognitive styles and their educational implications. Review of Educational Research, 47, 1-64.

Witkin, H. A., Oltman, P. K., Raskin, E., & Karp, S. A. (1971). A manual for the embedded figures tests. Palo Alto, CA: Consulting Psychologists Press.

AUTHORS

KATHLEEN A. CRAMER. Assistant Professor, University of Wisconsin-River Falls. C107 Ames Teacher Education Center, River Falls, WI 54022

THOMAS R. POST, Professor of Mathematics Education, College of Education, 240 Peik Hall, University of Minnesota, Minneapolis, MN 55455

MERLYN J. BEHR, Professor, Department of Mathematical Sciences, Northern Illinois University. DeKalb, IL 60115

This paper is based in part on research supported by the National Science Foundation under Grant Nos. SED 81-12643 and DPE 84- 70077 (The Rational Number Project). Any opinions, findings, and conclusions expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.