Unexpected classroom episodes can teach mathematics

Unexpected classroom episodes can teach mathematicsAuthor(s): ROCHELLE WILSON MEYERSource: The Arithmetic Teacher, Vol. 24, No. 3 (MARCH 1977), pp. 230-232Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41189253 .

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Unexpected classroom episodes can teach mathematics

ROCHELLE WILSON MEYER

An assistant professor of mathematics at Suffolk County (New York)

Community College, Rochelle Meyer is involved primarily with

the development and teaching of remedial courses. She first became concerned with remediation during her association

with the City University of New York.

^^ ourses in mathematics content are standard fare for prospective and prac- ticing teachers. Most of these courses are necessarily taught without an integrated component of classroom experience. Al- though the primary focus of a content course is mathematics not methodology, the content course is a proper forum to demonstrate an interplay between the mathematics understood by the teacher and the teaching methods used by that teacher. For example, a teacher who is insecure in his knowledge of an area is often very rigid in his presentation of that area and does not recognize or follow interesting student ideas. Having students play the role of classroom teacher is a technique that has helped students in my classes to see the interplay between understanding content and teaching it and to examine their depth of understanding in a particular area. I call it the "episode technique."

I present the students with a short classroom episode in which an elementary school pupil reacts in a mathematically unexpected way. Then each of the students records in writing how he would respond to the situation if he were the classroom teacher. After ten minutes or so, when all students are done writing, the responses are discussed, either in small groups or by the entire class.

One example of an episode that I have used with success is based on an example given by Robert B. Davis reporting the actions of a third-grade student called Kye (Davis 1969). I frame the actions this way:

A teacher, trying to motivate the need for regrouping in subtraction, presents a class with the problem

93 -47

and asks the class to try to solve the problem. After a few minutes, one child brings his paper to the teacher and says, "You can't take the 7 from the 3, so you take the 3 from the 7, getting 4. Then .you take the 40 from the 90, getting 50, which is 4 too big, so the answer is 46." What do you (as teacher) say or do in response to this student?

93 -47

4

50 46

(This episode can be nicely acted out rather than presented in written form.)

My students seem very interested in par- ticipating in a discussion of content when they have first "been a teacher." Most of them have reacted to this episode by doing

230 The Arithmetic Teacher



the indicated subtraction mentally and finding the answer 46. Beyond this, reactions differ on such points as telling the child to explain his method, trying the method on other examples (including examples not needing regrouping or examples involving more than two digits). The vari- ety of student responses reflects differences in student competence in the content involved - some students do not know how to evaluate the child's method and the discussion often helps some students improve their understanding of subtraction, place value, and signed numbers.

Some of my students respond to this subtraction episode with surprise - many do not believe a child could have come up with this idea. I have found that documentation of episodes is often useful in convincing students that events like these do occur. Many valuable documented episodes are reported in journals. One excellent example is the first part of the article "Nathan's Conjecture," by Arlene Fromewick, in the April 1973 issue of the Arithmetic Teacher.

A. Take a fraction that is not in its sim- plest form, and subtract the numerator from the denominator.

B. Simplify the fraction, and then subtract the numerator from the denominator.

C. Divide the result obtained in the first step by the result obtained in the sec- ond step.

D. The quotient obtained in the third step seems to be the greatest common factor (GCF) of the numerator and denominator of the given fraction.

For example, given therž, steps A 45

through D would yield the following results: A. 45 - 35 = 10

B %

= T and9-7 = 2

С 10 ̂ 2 = 5 D. The GCF of 35 and 45 is 5.

Nathan's conjecture is true and the proof takes but a few lines to write down. The

heart of the proof lies in understanding two concepts: fraction simplified or reduced to lowest terms and greatest common factor. The depth of the discussion of the content in this and other episodes can vary. When formal proofs of the issues that arise are not suitable; intuitive proofs and numerical examples would certainly be in order. An example of such a case is in an article "Catherine's Discovery," by Michael С Mitchelmore, which appeared in the Febru- ary 1974 issue of the Arithmetic Teacher. Catherine, at the age of 5, announced, "The 4-times table is in the 2-times table" and then proceeded to demonstrate:

2 © 6 © 10 © 14 © 18...

An investigation of this relationship can lead to the concepts of lowest common multiple (LCM) and greatest common divi- sor (GCD); the investigation can proceed informally or with the greater sophis- tication of modern algebra.

Occasionally I have to resort to inventing an episode when I need one to bring atten- tion to a particular area:

A little child says, "When I stand next to my Daddy he is taller than I am but when we lie down I am just as tall as he is." What do you respond?

I had intended this episode to focus atten- tion on orientation of objects in space and the relationship of tallness, thickness, and the vertical direction. Several of my students noted that when lying down, the child and father may have had their heads at the head of a bed, and it was to that that the child referred.

The exploration of the mathematics content of an episode sometimes leads to re- lated questions: What has the teacher taught before this episode? Why did the teacher pose a problem for which a method of solution had not already been presented? How do you cope with a question to which you do not know the answer without either cutting off the student or appearing igno- rant? This last issue has been crucial in persuading my students that the elementary

March 1977 231



school teacher needs to know more mathematics than just what is taught to the elementary school student. Discussions of var- ious episodes help them to see how an understanding of the underlying structure and spirit of mathematics provides the teacher with the knowledge and confidence to help the student explore areas not in the textbook.

The episode technique seems to help the students to see how their studies apply in actual classroom situations and it is a be- ginning of the kind of self-evaluation proc- ess all good teachers must engage in. The crucial element in this technique is that each student first reacts, and then he eval- uates his own reaction, not someone else's.

The technique also can be used by an individual as he reads. When an unexpected action has been described, the reader stops reading, imagines himself as a teacher who has just witnessed the actions, and reacts. The reader can then compare his reactions with the rest of the article. This procedure

is analogous to the way mathematicians often read mathematics - stopping and trying to anticipate the next result, trying to prove a theorem before reading the proof.

Two further notes: (1) With a change in the focus of the discussions, this technique can be successfully used in methods courses, and (2) I have used the episode technique as part of a final examination by having the response written (in duplicate, one for the student and one for me) during the last class meeting and a critical evaluation of the response written as part of the final examination.

References

Davis, Robert B. "Discovery in the Teaching of Mathematics." In Learning by Discovery: A Critical Appraisal, edited by Lee S. Shulman and Evan R. Keislar, pp. 120-21. Chicago: Rand McNally & Co., 1969.

Fromewick, Arlene. "Nathan's Conjecture." Arith- metic Teacher 20 (April 1973):289.

Mitchelmore, Michael C. "Catherine's Discovery." Arithmetic Teacher 21 (February 1974):90-91.

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232 The Arithmetic Teacher



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Unexpected classroom episodes can teach mathematics