Experience, Problem Solving, and Discourse as Central Aspects of ConstructivismAuthor(s): Erna Yackel, Paul Cobb, Terry Wood, Graceann Merkel, Douglas H. Clements andMichael T. BattistaSource: The Arithmetic Teacher, Vol. 38, No. 4 (DECEMBER 1990), pp. 34-35Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195035 .

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RESEARCH INTO PRACTICE

Experience, Problem Solving, and Discourse as Central Aspects of Constructivism

the past five years, we have collaborated with teachers to

develop forms of instructional prac- tice in elementary school mathematics that are compatible with a construc- tivist view of teaching and learning. Two key aspects of our work form the basis for this discussion: first, the pro- cess of developing instructional activ- ities, and second, the importance of engaging students in mathematical discussion.

Serious attempts to develop in- structional practices compatible with a constructivist perspective require confronting taken-for-granted assump- tions that underlie current educational practices. Few, if any, would disagree with the view that learning occurs not as students take in mathematical knowledge in ready-made pieces but as they build up mathematical mean- ing on the basis of their experiences in the classroom. This aspect of the con- structivist perspective is easily em- braced by anyone interested in mean- ingful learning and is reflected in the NCTM's Curriculum and Evaluation

Prepared by Erna Yackel Purdue University/Calumet Hammond, IN 46323 Paul Cobb and Terry Wood Purdue University West Lafayette, IN 47907 Graceann Merkel Tippe canoe School Corporation West Lafayette, IN 47906

Edited by Douglas H. Clements State University of New York at Buffalo Buffalo, NY 14260 Michael T. Battista Kent State University Kent, OH 44242

34

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Standards for School Mathematics {Standards) (1989). Von Glasersfeld (1988) describes the instructional im- plications of this perspective as fol- lows:

If you believe that knowledge has to be con- structed by each individual knower, . . . teach- ing becomes a very different proposition from the traditional notion where the knowledge is in the head of the teacher and the teacher has to find ways of conveying it or transferring it to the student. . . . I'm primarily interested in devel- oping ways of thinking in the student. And if you want to do that, you are constantly working with conjectures . . . about what goes on in the student's head, and on those . . . you base your strategies. . . . What you present is never something that you expect the student to adopt as it is, but what you present is something that you think will make it possible for the student to find his or her own way of constructing.

In line with von Glasersfeld's rec- ommendations, we have attempted to use constructivism as a guiding frame- work within which to develop in- structional situations that facilitate students' progressive construction of increasingly abstract mathematical conceptions and procedures. In con- cert with the guided-discovery ap- proach, our goal is to have students develop mathematical concepts and relationships in ways that are person- ally meaningful. Important differences can be cited, however, between guided

discovery and our own approach in terms of the basic assumptions that guide the development of instructional activities and the corresponding in- structional strategies. The guided-dis- covery approach typically starts with a mathematical analysis of the relation- ships that students are supposed to dis- cover and attempts to embody them in manipulative materials or in activities in a readily apprehensible form (Treffers 1987). In contrast, a constructivist ap- proach acknowledges that individual students interpret instructional situa- tions in profoundly different ways. In this approach, our understanding of students' mathematical experiences as informed by cognitive models (Steffe, von Glasersfeld, and Cobb 1988) rather than a mathematical analysis consti- tutes the starting point. Consequently, the instructional activities were not de- signed to ensure that every student make the same preselected mathemati- cal constructions or apprehend the same relationships. Instead, we at- tempted to develop instructional activ- ities so that the interpretations of stu- dents at different conceptual levels would be productive for their individual learning. In particular, because learning often occurs as students attempt to re-

ARITHMETIC TEACHER

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solve cognitive conflicts, we used the cognitive models to anticipate which types of situations might potentially be problematic for students at different conceptual levels (Cobb, Wood, and Yackel, in press). As an example, con- sider the pair of problems shown in figure 1 that were used in a second- grade classroom in February. The fol- lowing solutions for the second task were suggested by three students working together.

Brenda: I'd say it's the same thing as this (the first task) because there's twelve - and six and six - and six and six make twelve.

Consuela: Wait a minute. Let me do it. Forty and forty makes eighty. Take these sixes away makes eighty, eighty-one, eighty- two, eighty-three, eighty-four, eighty-five, . . . , ninety- two (counting by ones).

Dai-soon: Forty-seven, forty-eight, forty-nine, .... (While listening to the girls, Dai-soon counts by ones starting with forty-seven, but he loses track of the number of times he has counted. Later he gets a hundred board to use on the subsequent tasks.) Brenda' s problem was relating the two tasks. Consuela and Dai-soon both solved the problem of adding 46 and 46, but their solutions indicate dif- ferences in their interpretations and in their personal mathematical concep- tions. Consuela's solution shows that she is capable of operating on two- digit numbers by thinking of them as made up of tens and ones. Dai-soon's solution of attempting to count on for- ty-six ones is more primitive.

Although students construct their own mathematical understandings, they do not do so in isolation. Inter- actions with both other students and the teacher give rise to crucial learn- ing opportunities. For this reason, the students typically work in pairs or small groups and are expected to col- laborate in their solution attempts. Collaboration involves much more than combining solution procedures to develop a joint solution. It involves developing explanations that are mean- ingful to someone else and trying to interpret and make sense of another's ideas and solution attempts as they evolve. As part of this process, stu-

dents attempt to verbalize and inter- pret only partially formed ideas, thereby engaging in what Barnes (1976) calls exploratory talk as op- posed to final-draft talk.

During subsequent whole-class dis- cussion, students are expected to give coherent explanations of their prob- lems, interpretations, and solutions and to respond to questions and chal- lenges posed by their peers. They are also expected to listen to, and try to make sense of, explanations given by others, to pose appropriate questions, and to ask for clarifications.

When students engage in this type of discourse, not only is the amount of time they spend participating in prob- lem-solving activities increased, but the nature of their problem-solving ac- tivity is itself extended to encompass learning opportunities that rarely arise in traditional instructional settings (Yackel, Cobb, and Wood 1989). These opportunities have their gene- sis in social interaction. As students participate in the discourse of what Richards (in press) calls inquiry math- ematics, they learn to develop math- ematical explanations and justifica- tions and engage in what is typically termed analytical reasoning. Con- sider, for example, the following ex- planation of Jerome's method for solving 56 - 10 = : 'Ten. Take away the six; that'd be fifty. And if six and four is ten, if you take away four that'd leave forty-six." On this occa- sion, unlike on previous ones, Jerome is describing and explaining his meth- od, not just using it to get an answer. He does not say, "Since six and four is ten ..." but "If six and four is ten. ..." Thus he is explaining his methodology to justify it; his result is secondary.

The teacher's role is to facilitate the development of this type of mathe- matical discourse (Wood and Yackel 1990) by helping students in their at- tempts to express their mathematical thinking while also encouraging them to conceptualize situations in alter- native ways. More generally, the teacher facilitates the students' mathe- matical development by subtly high- lighting selected aspects of their math- ematical contributions. In the process, he or she initiates and guides in the

classroom the evolution of taken-to- be-shared mathematical understand- ings that are compatible with those of the wider society.

In conclusion, we note that our em- phasis on mathematical discourse, with its resultant development of ex- planation and justification, is consis- tent with the National Council of Teachers of Mathematics' s standards on communication, reasoning, and connections. Our experience in the classroom suggests that these stan- dards are both realistic and attainable. Furthermore, both students and teach- ers find learning mathematics in this manner to be an exciting adventure.

References

Barnes, Douglas. From Communication to Cur- riculum. London: Penguin, 1976.

Cobb, Paul, Terry Wood, and Erna Yackel. "A Constructivist Approach to Second Grade Mathematics." In Constructivism in Mathe- matics Education, edited by Ernst von Gla- sersfeld. Dordrecht, Netherlands: Reidel, in press.

National Council of Teachers of Mathematics, Commission on Standards for School Mathe- matics. Curriculum and Evaluation Stan- dards for School Mathematics. Reston, Va.: The Council, 1989.

Richards, John. "Mathematical Discussions." In Constructivism in Mathematics Educa- tion, edited by Ernst von Glasersfeld. Dor- drecht, Netherlands: Reidel, in press.

Steife, Leslie P., Ernst von Glasersfeld, and Paul Cobb. Construction of Arithmetical Meanings and Strategies. New York: Spring- er- Verlag, 1988.

Treffers, A. Three Dimensions: A Model of Goal and Theory Description in Mathematics Instruction-the Wiskobas Project. Dor- drecht, Netherlands: Reidel, 1987.

von Glasersfeld, Ernst. Using Mathematical Thinking. Tape 1. BBC audiotape of an inter- view by Barbara Jaworski at the Open Uni- versity, produced by John Jaworski. Re- corded at the International Congress of Mathematics Education, Budapest, Hungary, July 1988.

Wood, Terry, and Erna Yackel. "The Develop- ment of Collaborative Dialogue within Small Group Interactions." In Transforming Chil- dren's Mathematics Education: An Interna- tional Perspective. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1990.

Yackel, Erna, Paul Cobb, and Terry Wood. "Small-Group Interactions as a Source of Learning Opportunities in Second-Grade Mathematics." 1989.

The research reported in this article was sup- ported by the National Science Foundation un- der grants nos. MDR 847-0400 and MDR 885- 0560. All opinions and recommendations expressed are, of course, solely those of the authors and do not reflect the position of the foundation, p

DECEMBER 1990 M

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