Introducing Additional Concrete Models of Operations: A Discovery Approach

Introducing Additional Concrete Models of Operations: A Discovery ApproachAuthor(s): Floyd VestSource: The Arithmetic Teacher, Vol. 25, No. 7 (April 1978), pp. 44-46Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191497 .

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Introducing Additional Concrete Models of Operations: A Discovery Approach By Floyd Vest

Most mathematics educators agree that children should have a broad view of the operations on whole numbers (Ashlock 1971, Reys 1972). Children, in other words, should not be limited to a single interpretation or model of an operation. There are several advan- tages to giving children experience with more than one interpretation of an operation, including the belief that children will become better problem solvers when they have a broader view of the operations.

After children have become familiar with one model of an operation, the study of additional interpretations in a discovery mode is an interesting experience for both teacher and pupil. The discovery approach allows students to explore some of the important relation- ships that might otherwise be obscured by familiar notation and rapid instruction (Vest 1974). The following student/teacher dialogue, recorded from actual lessons with children, illustrates such a discovery mode. In this particular dialogue, the comparison model for addition and subtraction is being in- troduced to a student whose dominant view of subtraction is the take-away model (Vest 1969).

Teacher. I am going to draw some pictures, and it's your job to try to figure out the pattern that these pictures have, and to describe them. You are going to do a lot of thinking and I am going to try not to help you.

Student. Okay. Teacher. The first thing I am going to

do is to draw two pictures. (The teacher makes drawings like those shown in figures I and 2.) I would like for you to make up a third picture like these.

An associate professor in the mathematics depart- ment of North Texas State University in Dentón, Texas, Floyd Vest teaches mathematics and mathematics education courses for elementary and sec- ondary school teachers.

Fig. 1

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Fig. 2

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Student. Just draw anything - like? Teacher. As long as it's like those

two (pointing to the figures that have been drawn) in some way.

( The student constructed a drawing similar to the one in figure 3.)

Teacher. How many lines do you mean there to be?

Student. Three. Teacher. Okay. What about your

picture is like the two I drew? Name everything you can.

Student. Well, there are two oval shapes and dots and lines.

Teacher. All right, good. Is that all there is? Each of them has two oval shapes, each of them has dots, and each of them has lines?

Student. Well, one of the sets has more dots than the other one, and all of yours do, too.

Teacher. All right. If you were going to draw one of these pictures, the first thing you do is what?

Student. You make two oval shapes. Teacher. All right. What is the sec-

ond thing you do? Student. Put dots in them.

Teacher. Okay. Student. And then you draw lines to

the ones that are even with each other. Teacher. Okay. When you are draw-

ing the lines, does it matter how many lines you draw? Could you have drawn one line instead of three in your picture (referring to figure 3) and still have a picture that fits with my two pictures?

Fig.3

o--- - 4o/ о v^

Student. I don't know. Teacher. Okay. What about my two

pictures, when I went to match the dots? How did I know when to stop matching the dots?

Student. You have to run out of dots on one side.

Teacher. All right. So we are going to draw pictures where you have to run out of dots on one side.

( The teacher has the student draw two or three examples to be sure the student has the right idea.)

Teacher. Let's look at this picture (pointing to figure I) and talk about the numbers. What numbers would you say are involved in this picture?

Student. What numbers? Teacher. I'll give you a hint. There is

a five involved. Student. (Immediately.) And there is

a two. Teacher. Okay. How did you get

two? Where did I get the five? Student. There is a set of five. Teacher. And you could have gotten

the two from two places. There are. . . . Student. Yeah. There are two lines

and two dots.

44 A rithmetic Teacher



Teacher. All right. There are two numbers, five and two. Do you see any other numbers that are involved in the picture?

( The student and the teacher look at the other figures and discuss the two kinds of numbers in the other pictures.)

Teacher. I claim that each picture has one more number involved, one that you have overlooked. Study the pictures and see if you can find the kind of number you haven't named yet. (Pause) We have named two numbers, the number in the larger set and the number matched. But besides the dots that are matched, there is something else.

Student. The dots that aren't matched.

Teacher. All right. Student. Three (pointing at figure I).

Two (pointing at figure 2). And two (pointing at figure 3).

Teacher. If I give you a picture, how would you know whether it is like these or not? What would you look for in the picture.

(After the student had summarized the characteristics of the illustra- tions of the comparison model, naming the three kinds of numbers that were found in each, the teacher introduces a nonstandard notation.)

Teacher. For this picture (pointing to figure 1) I am going to write a 5 and a 2.

( The teacher then wrote the following symbols under the drawing in figure I.)

~^' <-» (5,2)

Teacher. All right, in this picture (pointing to figure 2), what numbers are involved?

Student. Four and six. ( The teacher then wrote the following symbols under the drawing in figure 2.)

_] <- ► (6,4) Teacher. In the third picture (point-

ing to figure 3), what numbers do you see that are involved?

Student. Five and three. ( The teacher then wrote the following symbols under the drawing in figure 3.)

HI <- ► (5,3)

Teacher. Now I claim that each picture has one more number.

Student. The dots that aren't matched.

Teacher. All right. ( The student volunteered the answers three, two, and two for the boxes in the nonstandard notation, and wrote them in. The teacher then wrote the following:)

^] <-► (7,2) Teacher. Can you make a picture like

the others to find out what goes in the box?

( The student constructed a drawing similar to the one in figure 4. )

Fig. 4

о о о о о/ v;

Teacher. Can you fill in the box then? The student then wrote 5 in the box.

Teacher. Tell me, what part of the picture goes with what part of the ex- pression that I have written?

( The student discusses the meaning of the numerals in the nonstandard form. The teacher encourages gen- eralization of meanings of the numerals and encourages mental im- agery. The teacher can also expect discovery of the operation represented (Vest 1973).)

Teacher. Okay. If you had a friend and you wanted to tell him about these pictures, what kinds of rules would you give him to explain how to write down the numbers if he had a picture to go by?

Student. Well- Teacher. Don't use any specific num-

bers. Just tell me in general how your friend would know what to write.

Student. Well, if you had the, uh, numbers you would make how many dots in the first number and draw a circle around it, and for the second number, you would make how many dots and put a circle around it. And then, you would match the number of

dots with the first one with the number of dots in the second, until you got down to the last dot. And that's all.

Teacher. How about the third number?

Student. It would already be there after you got through.

Teacher. Okay, but how would your friend know what the third number would be?

Student. It would be the ones left over from matching.

Teacher. Okay, good. ( The teacher then wrote the following:)

^] <- ► (8,5)

Teacher. I want you to try to figure out a way to fill in that box without drawing a picture.

( The student wrote 3 in the box.) Teacher. Can you explain to me how

you knew to put a 3 there? Student. Subtract. Teacher. All right, did that work on

all the other pictures? Student. I don't know; it worked on

that one (referring to figure I). Yeah, it works ( after a pause to look at the other figures).

Teacher. What made you guess subtraction would work?

Student. Well, because the number was - I don't know. I guess it was obvi- ous.

Teacher. Why didn't you guess that it would be multiplication or addition? What made you guess subtraction?

Student. Well, because it was subtraction in these pictures, I guess.

Teacher. So far it worked four times. How could we tell if it would work every time?

Student. I don't know. (After the student's discovery that the operation of subtraction works, the teacher lets the student use the standard equation form for subtraction.)

Teacher. Well, how about trying an- other one, and see if it works. We might as well write subtraction equations now.

Student. Like what? Teacher. Well, just take a sub-

traction equation that we haven't done and see if it works.

( The student wrote the following: 10-4 =6)

April 1978 45



Teacher. Okay, try it. ( The student made a drawing similar to the one in figure 5.)

Fig. 5

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о о о о/ W

Student. It works. Teacher. Well, is there something

about the picture itself that looks like subtraction? What is subtraction?

Student. Taking away numbers. Tak- ing away a smaller number from a big- ger number.

Teacher. If we had just looked at the pictures and never written down any numbers, do you think you ever would have guessed that this might work like subtraction?

Student. Well, if you match up every number, the number left over will be the answer. It just works that way.

Teacher. Okay. Here is an equation, a subtraction equation: 100 - 64 = 36. I could ask you to draw a picture for that but it might take a little bit of time because the numbers are so big. So instead, let me just ask you to explain to me how you would draw the picture.

Student. We would put- uh- 100 things, dots, and put a circle around them. And then put 64 dots and a circle around them. And we would put lines between the 64 dots and the 100 dots until we ran out of dots in the lowest side. There would be 36 dots left unmatched.

Teacher. Are you sure there would be 36 dots left unmatched? What is it that tells you that?

Student. Well, those 64 dots are gone on the 100 side since they are already matched up, and so there has to be 36 left.

Teacher. Why do you think it's al- ways subtraction?

Student. It just is. Teacher. Can you tell me more than

"it just is"? Student. Well, you take away how

many are all on this side by making lines across.

Teacher. Suppose I were to say it's addition?

Student. It is (pause) cause 5 + 2 = 7 (referring to figure 4).

Teacher. Okay. So maybe it could be addition.

Student. Yeah, you can add the ones that aren't. Let me try it down here. It works right there (referring to figure 3). It works there (referring to figure 1 ). It works there (referring to figure 2). It's addition and subtraction.

Teacher. All right, and what is the reason for that?

Student. I don't know (laughter and pause). Cause here you are adding the ones that aren't matched plus the - the ones that - I don't know. It's just both of them.

Teacher. Okay, I think that is a pretty good explanation all around.

In the foregoing episode we have il- lustrated an inductive, discovery approach to introducing a subsequent interpretation (model) of subtraction and addition. This approach is appropriate for students who have previously learned a model for the particular operation. The procedure involved presenting instances of the selected model, presenting nonstandard notation, and allowing the student to inductively gen- eralize the model and discover the rela- tionship between the model and operation that it represents (discover the isomorphism). We observe that the basis for the student's discovery of the operation represented was by induction on number patterns (the model pro- duced the same answers as subtraction) and by relating concrete elements of two models. The student viewed the

comparison model through the take- away model by imagining the subset of matched elements in the larger set being "taken away" (Härtung et al. 1960, p. 35).

The customary, more direct method of introducing subsequent models usu- ally involves presenting together an il- lustration and standard equation, and declaring the rules for connecting the two. It appears to the author that this customary approach has the dis- advantage of artificially mediating the isomorphism by familiar equations and masking the characteristics of the subsequent model. In the discovery approach, there is less resistance on the part of students to careful consid- eration of the new model.

References

Ashlock, Robert B. "Teaching the Basic Facts: Three Classes of Activities." Arithmetic Teacher 18 (October 1 971 ):359-64.

Härtung, Maurice L., Henry Van Engen, Lois Knowles, and E. Glenadine Gibb. Charting the Course for Arithmetic. Chicago: Scott, Foresman and Co., 1960.

Reys, Robert E. "Mathematics, Multiple Em- bodiment, and Elementary Teachers." Arith- metic Teacher 19 (October 1972):489-93.

Vest, Floyd. "A Catalog of Models for the Oper- ations of Addition and Subtraction of Whole Numbers." Educational Studies in Mathemat- ics 2 (July 1969):59-68.

. "Mapping Models of Operations and Equations." School Science and Mathematics 72 (May 1972):449-57.

. "Using Models of Operations and Equa- tions." Educational Studies in Mathematics 5 (June 1973):147-55.

. "Behavioral Correlates of a Theory of Abstraction." Journal of Structural Learning 4 (1974): 175-86.

. "Problem Solving As Viewed through a Theory of Models." Educational Studies in Mathematics 6 ( 1976):395-408.D

COUISVICCE M'E'EJINQ Make plans noví to attend the National Council oj Oeachers of Mathematics Name-

of-Site Meeting on 19-21 October 1978, at the Executive Inn, Houisule, Ken-

tucky. Come ani hear, along With many others, these outstanding speakers: Shirley Hill, Charles Jordan, Ъш Maletsky, Ceroy Dalton, Betty Beaumont, and Ernest Юипсап. An interesting session on uBack to Basics" that Will be different and in-

formatile is planned. Ohe program speakers Will attempt to deal with some pertinent issues faced by today's teachers of mathematics at all levels, of instruction.

See the September 1978 issue of the NCOM Newsletter for additional information and housing reservation form.

19-21 October 1978

46 Arithmetic Teacher



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Introducing Additional Concrete Models of Operations: A Discovery Approach