ONE POINT OF VIEW: A Different Perspective

ONE POINT OF VIEW: A Different PerspectiveAuthor(s): Doug JonesSource: The Arithmetic Teacher, Vol. 41, No. 2 (OCTOBER 1993), pp. 73-74, 120Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195918 .

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ONE POINT OF VIEW

A Different Perspective Doug Jones

the quintes- sential first grader,

is delightfully bratty and philosophical at the same time. He has much to say to mathematics teachers about valuing different per-

spectives and points of view (see fig· 1 ). As teachers, we are asked to help students

develop their mathematical power - their abilities to explore, conjecture, and reason logically, as well as to use a variety of mathematical methods effectively to solve nonroutine problems (NCTM 1989, 1991). Our role is to be more of a facilitator of

learning and less a dispenser of facts and skills - to get our students involved in doing mathematics. But by itself, getting students to "do" mathematics is not enough. We and our students need to make sense of our mathematical activity, and sense- making in mathematics is fostered by the collaboration of perspectives (Davis and Hersh 1981; Tymoczko 1986). This collaboration takes myriad forms: some are public, like getting feedback from others about one' s own mathematical ideas (Lakatos 1976); some are private, like trying to come up with examples, counterexamples, and nonexamples for a certain concept (Wertheimer 1 959). In the end, the interplay between perspectives adds appreciation and depth of understanding of concepts.

Helping students appreciate different perspectives and different ways of doing mathematics may help them to deepen their understanding. Indeed, one theme of this article is that we need to emphasize the

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OCTOBER 1993 73



search for, and purposeful use of, different perspectives by students. The second theme is that we as teachers need to expand our own perspectives to include those of our students.

We often do not understand what students are doing or thinking (Thompson 1 982). For instance, consider John, who, when asked to put -5, 0, 4, -2, -8 in order from smallest to largest, wrote 0, -2, 4, -5, -8. He was apparently considering the magnitude of the numbers, not the relative "positiveness" of the numbers. We may try to clear up John's difficulties by reminding him that ordering refers to arrangement on a number line, hoping that he will somehow correct him- self. But often we do not explore what kind of sense John's work makes to him, and in that omission we are wrong. McCleary and McKinney (1986) point out, "Mathemati- cians do not reject the results of former communities of mathematicians; indeed, they try to bring as much of the past as is reasonable into their present picture of the discipline" (p. 5 1 ). Likewise, we as teachers need not reject our students' mathematics; we need to try to understand it so that it can enrich our own view of mathematics. We need to see the mathematical world through their eyes.

One way to do so is to try to make the familiar strange (Prince 1970; Elbow 1986). We must ask, "What can this student be thinking so that her actions make sense from her perspecti ve?" The materials, tools, skills, and even blueprints for a student' s construc- tion are apt to differ from our own. We can begin by trying to understand a student's framework, but we must then extend it - we must try to construct similar meaning for ourselves and learn to value it. "Seeing" that John is thinking of magnitude instead of number-line position just tells us that his interpretation of the task is different from ours. By exploring his view as it relates to our own, we enrich our understanding of magnitude and order, of students, and of meaningful ways to teach.

But a catch is involved - trying to see John's way of making sense of order and magnitude, let alone trying to balance it with our own, is a very hard task. As illustrated by Calvin (fig· 1), it can easily lead to too much information and a retreat to a view that is comfortable. The danger is that if too much of mathematics as we know it has to be given up, then the importance of understanding students' mathematics is also

likely to be given up. The other extreme, accepting all perspectives as valid, is equally dangerous.

We need to develop our own mathematics-for-teaching power more fully. To apply the student goals of the NCTM's Curricu- lum and Evaluation Standards (1989) to teaching, we need to develop our ability to explore students' understanding, conjecture about their difficulties, reason logically about the kinds of sense that students make of

mathematical ideas, solve nonroutine problems - both mathematical and peda- gogical, communicate about and through mathematics, and make connections between students' mathematics and that of the wider mathematical community. We also need to develop our disposition to seek, evaluate, and use alternative perspectives in all phases of our teaching.

(Continued on роде 1 20)

74 ARITHMETIC TEACHER



^■^ЩМ ONE POINT OF VIEW

■■■^И (Continued from page 74)

We need to think of our students as mathematicians struggling with the creation of new mathematical ideas. Sometimes they will find connections that we did not see or that do not make sense to us. But whatever their involvement, we need to think of them as striving to make sense of what they are doing. We must work hard at temporarily putting aside our teacher perspective so that we can explore a student's perspective in much the same way that we want our students to explore their own mathematical ideas with patterning blocks, numbers, or functions. However, we must also work hard at avoiding the cognitive overload of seeing things in too many ways. We cannot afford the single perspective that states, "You're still wrong" or the unexamined acceptance of multiple perspectives that states, in effect, "Anything goes." By striving to make the familiar strange and toggling back and forth between students' mathemati-

cal thoughts and our own, we deepen our understanding of mathematics, students, and what it means to teach.

References

Davis, Phillip, and Reuben Hersh. The Mathematical Experience. New York: Houghton Mifflin Co., 1981.

Elbow, Peter. Embracing Contraries: Explorations in Learning and Teaching. New York: Oxford Uni- versity Press, 1986.

Lakatos, Imre. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge University Press, 1976.

McCleary , John, and Audrey McKinney . "What Math- ematics Isn't." Mathematical Intelligencer 8 (1986):51-53,77.

National Council of Teachers of Mathematics. Cur- riculum and Evaluation Standards for School Math- ematics. Reston, Va.: The Council, 1989.

. Professional Standards for Teaching Math- ematics. Reston, Va.: The Council, 1991.

Prince, George. The Practice of Creativity. New York: Collier Books, 1970.

Thompson, Patrick W. "Were Lions to Speak, We Wouldn't Understand." Journal of Mathematical Behavior 3 (Summer 1982):147-65.

Tymoczko, Thomas. "Making Room for Mathemati- cians in the Philosophy of Mathematics." Math- ematical Intelligencer 8 (1986):44-50.

Wertheimer, Max. Productive Thinking. New York: Harper & Brothers, Publishers, 1959.

The author thanks William S. Bush and Jonathan Prasse for their helpful comments in the preparation of this article.

Doug Jones teaches at the University of Kentucky, Lexington, KY 40506-00 1 7. His interests include teachers ' beliefs, context, and understanding factors that affect teacher change.

The views expressed in "One Point of View" do not necessarily reflect the views of the Editorial Panel of the Arithmetic Teacher or the National Council of Teachers of Mathematics. Readers are encouraged to respond to this editorial by sending double-spaced letters to the Arithmetic Teacher/or possible publica- tion in "Readers ' Dialogue.

" Manuscripts of approxi-

mately six hundred words are welcomed for review for "One Point of View. " Щ

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120 ARITHMETIC TEACHER

We can expand our perspective to

include that of students.



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ONE POINT OF VIEW: A Different Perspective