Interference of Instrumental Instruction in Subsequent Relational Learning
Summary (2 min read)
Design
- The authors objective in this study was to compare the effects of students' receiving instrumental instruction prior to relational instruction (the I-R treatment) with the effects of relational instruction only (the R-O treatment).
- This decision was based on two considerations.
- The design included a written pretest for all students, 5 days of instrumental instruction given to half the students (I-R group), an intermediate test covering the instrumental instruction for the I-R group, a 3-day relational treatment given to all students (I-R and R-O groups combined), a posttest following the relational instruction, and a retention test 2 weeks later.
- The interview data supplemented the quantitative data to provide us with a better understanding of any interference effects that might be revealed in the test scores.
- While controlling for the effects of the two covariables, pretest and CAT score, the authors conducted an analysis of covariance to test whether or not the two treatments differed in effectiveness.
Participants
- Six intact regularly scheduled fifth-grade mathematics classes were used for this study; three classes were taught by each of 2 fifth-grade mathematics teachers in a middle-class, semirural school.
- All these classes were grouped heterogeneously for mathematics instruction.
- To control for the class variable, the authors separated each class into two treatment groups by using random stratification by gender and Page 529 achievement level.
- In selecting six students from each treatment group to be interviewed, the authors sought stratified random samples that included three students of high ability (as measured by the CAT score) and three students of low ability, as well as three girls and three boys.
Instructional Content?
- The mathematical content chosen for this study was area and perimeter of squares, rectangles, triangles, and parallelograms.
- In the relational instruction, the authors tried to use students' intuitions about size and distance to develop measurement strategies for area and perimeter in the context of these geometric shapes.
- The students worked three problems with the instructor and then five additional problems in cooperative groups.
- Gradually most students used increasingly sophisticated methods, though a few did not progress beyond using counting strategies.
- No pencil-and-paper calculations were taught during this relational unit.
Instruments
- Intermediate test, posttest, and retention test.the authors.
- The pretest, posttest, and retention test were nearly identical.
- In 8 of these items, shapes were presented in a diagram with only necessary measures provided (e.g., "Find the area of the following square" with the length of one side given).
- The posttest mean score for students receiving only relational instruction was 16.42 in comparison with a mean score of 14.36 for students who were given instrumental instruction prior to relational instruction.
- The Minitab program was used to assess the statistical power of the analysis.
Qualitative Analysis
- To gain further insight into effects of sequencing instructional modes, the first author conducted three interviews with 12 stratified, randomly selected students (6 from each group including 3 boys and 3 girls of varying mathematical achievement).
- The authors used the following coding system for the final interview.
- After data for all topics had been summarized in this fashion, the transcripts were separated according to the treatment group of the student.
Qualitative Results
- In the final-interview data the authors found further evidence of an interference of instrumental instruction on subsequent relational understanding, and these data helped us identify cognitive, attitudinal, and metacognitive characteristics of the interference.
- The concepts of area and perimeter appear to have been less clearly differentiated for the I-R students than for the R-O students.
Authors
- Dolores D. Pesek, Associate Professor, Mathematics Education, Southeastern Louisiana University, Department of Teacher Education, SLU-749, Hammond, LA 70402.
- David Kirshner, Associate Professor, Mathematics Education, Department of Curriculum and Instruction, Louisiana State University, 223 Peabody Hall, Baton Rouge, LA 70803-4728; dkirsh@lsu.edu.
- 1The terms relational and instrumental were coined by the late Richard R. Skemp, to whom the authors dedicate this report for his contributions to the topic and his encouragement of this study.
- The study reported here was conducted by the first author as part of her dissertation (Simoneaux, 1992) at Louisiana State University under the supervision of the second author.
- A brief report (Simoneaux & Kirshner, 1994) of the dissertation was presented at PME-NA XVI.
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Citations
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Cites background from "Interference of Instrumental Instru..."
...Advocates for a conceptual-to-procedural sequence cite a study that compared procedural-then-conceptual instruction to only conceptual instruction (Pesek and Kirshner 2000)....
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137 citations
Cites background from "Interference of Instrumental Instru..."
...Some take the perspective that learning procedures before concepts interferes with attaining conceptual understanding (Byrnes & Wasik, 1991; Mack, 1990; Pesek & Kirshner, 2000; Skemp, 1987; Wearne & Hiebert, 1988)....
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References
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9,000 citations
"Interference of Instrumental Instru..." refers result in this paper
...This analysis is consistent with psychological models of limited cognitive resources (e.g., Kahneman, 1973) as well as with studies that document the degradation of performance when attentional resources are divided between two tasks that both…...
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1,194 citations
"Interference of Instrumental Instru..." refers background in this paper
...The hard truth is that real reform in mathematics education calls for a reorientation of classroom norms and practices (Cobb, Boufi, McClain, & Whitenack, 1997; Lampert, 1990)....
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1,158 citations
"Interference of Instrumental Instru..." refers background or methods in this paper
...Because relational instruction usually is assumed to take more time than instrumental instruction to implement (Hiebert & Carpenter, 1992; Skemp, 1987), time constraints are often cited as a principal reason for teachers' relying mostly on instrumental instruction....
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...In contrast, according to some theories of learning, "the degree of understanding is determined by the number and strength of the connections [among representations]" (Hiebert & Carpenter, 1992, p. 67)....
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...Second, using a Page 528 shorter relational treatment provided the possibility of highlighting the effectiveness of relational teaching, challenging the usual assumption that meaningful instruction requires more time for addressing the same content (Hiebert & Carpenter, 1992; Skemp, 1987)....
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1,088 citations
"Interference of Instrumental Instru..." refers background or methods in this paper
...shorter relational treatment provided the possibility of highlighting the effectiveness of relational teaching, challenging the usual assumption that meaningful instruction requires more time for addressing the same content (Hiebert & Carpenter, 1992; Skemp, 1987)....
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...Because relational instruction usually is assumed to take more time than instrumental instruction to implement (Hiebert & Carpenter, 1992; Skemp, 1987), time constraints are often cited as a principal reason for teachers' relying mostly on instrumental instruction....
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...Students should develop relational understanding--"understand[ing] both what to do and why" (Skemp, 1987, p. 9)....
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...Second, using a Page 528 shorter relational treatment provided the possibility of highlighting the effectiveness of relational teaching, challenging the usual assumption that meaningful instruction requires more time for addressing the same content (Hiebert & Carpenter, 1992; Skemp, 1987)....
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Frequently Asked Questions (12)
Q2. What are the possible mechanisms of interference?
Three possible mechanisms of interference were outlined in the introduction: cognitive interference, in which deeply rooted prior understandings of the content domain obtrude into attempts to construct new understandings; attitudinal interference, in which students' attitudes and beliefs about themselves or about the domain in question serve to prevent their full engagement in learning activities; and metacognitive interference, in which maintenance of prior instrumental understanding requires rehearsal or other mental effort and new learning is rejected (perhaps unconsciously) as disruptive to the existing competencies.
Q3. How many items could be solved using the formulas presented in the instrumental instructional unit?
Twenty-four of the items on the tests could be solved using one of the formulas presented in the instrumental instructional unit.
Q4. How many classes were used for this study?
Six intact regularly scheduled fifth-grade mathematics classes were used for this study; three classes were taught by each of 2 fifth-grade mathematics teachers in a middle-class, semirural school.
Q5. What was the effect of the intermediate test and CAT score on the posttest score?
Because the students in the I-R group took an intermediate test following the instrumental instruction (in addition to the pretest and the posttest), the effects of this intermediate test score and CAT score on the posttest score also were examined using multiple regression analyses.
Q6. What did the students say they did not know?
When asked to explain why the perimeter of a rectangle and the area of a triangle are given by the formulas P = 2(l + w) and A = 1/2 bh, respectively, all I-R students either said that they did not know or gave incorrect explanations.
Q7. What were the reliability coefficients for the pretest, intermediate, and retention tests?
The reliability coefficients using Cronbach's alpha on the pretest, intermediate test, posttest, and retention test were .699, .754, .873, and .840, respectively.
Q8. What were the strategies students used to solve the problems?
In all cases, the strategies students used evolved out of their intuitive understandings of area and perimeter and were sensible to them.
Q9. Why was the p value slightly above the indicated alpha level?
Because the p value only slightly exceeded the indicated alpha level of .05, the strong likelihood is that the results would have been adjudged to be significant.
Q10. How many lessons did the first author teach?
The first author taught each intact class (consisting of both the I-R and R-O students) in three 1-hour lessons over a 3-day period.
Q11. What did the students explain about the triangle area formula?
In explaining the triangle area formula, one of the I-R students simply said that he did not understand, and four students gave confusing explanations (e.g., "I think one measure is one half the other"; "you take the number and you add them together"; "you take one half away"; and "one half [is there] because the sides are not always going to be the same.").
Q12. What did the students use to describe the difference between area and perimeter?
To describe the difference between area and perimeter, all the I-R students used the term inside in defining area, and three of the six used the word outside for perimeter.