Episodes with Several Models of MultiplicationAuthor(s): Floyd VestSource: Mathematics in School, Vol. 14, No. 4 (Sep., 1985), pp. 24-26Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216527 .

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After children have become familiar with several models of an operation, the study of the relationships between the models and the operation is an interesting experience for both the teacher and the pupil. Important and useful underlying relationships can be discovered which might otherwise be obscured. Although textbooks for younger children include several models for multiplication, they rarely contain materials and explicit suggestions for teaching that models of an operation represent each other (are isomorphic), that they illustrate the same operation, and in turn that an operation is an abstraction derived from its several interpretationst. The following pupil/teacher dia- logue, recorded from actual lessons, illustrates these rela- tionships and their discovery by a ten-year-old pupil who had completed four years of schooling. The pupil had previously studied multiplication of whole numbers in terms of several models including Repeated Addition, Set Union, the Number Line, and the Cartesian Prod- uct Model found commonly in school mathematics. He had learned to write equations for models and to use models to solve equations.

The pupil was taught by a questioning, interview method as illustrated in the following dialogues. The teacher first conducted probes with models of a very similar structure. In the first episode the pupil gives a response which illustrates the correspondences between instances of two models. The correspondences are basic to the general isomorphism. The teacher supplied the pupil with an instance of the Set Union Model for multiplication and asked the pupil to construct an isomorphic instance of the Number Line Model. The pupil and teacher referred to the Set Union Model as "towers" or "chips".

24

(The teacher laid out 4 sets (towers) with 3 chips in each set.)

CC (Teacher): "Did you see what I did with the chips?" JO (Pupil): "Uh-huh." CC: "Good. Would you do something like it with

the number line?" JO: "Okay." (The pupil drew a number-line

illustration with 4 jumps of 3 each.)

0 1 2 3 4 5 6 7 8 9 10 11 12

CC:

JO:

CC:

"Could you explain to me how you knew what to do on the number line?' " There are 3 in each set so 3 in each jump. There are 4 sets so 4 jumps." "Fine."

In this episode the pupil has declared two of the three correspondences that indicate how the two instances are mathematically alike. The pupil has gone a step beyond working with the models separately. A form of closure exists in that the pupil has cited a direct mathematical relationship between instances of the Set Union Model and the

Number Line Model. After similar probes with models of a very similar

structure, the teacher conducted probes between the Repeated Addition Model and the Cartesian Product Model which are quite dissimilar in structure. In the

Mathematics in School, September 1985

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following episode the teacher and pupil referred to the Cartesian Product Model as "pairing".

CC:

JO:

CC: JO:

CC: JO: CC: JO:

(The teacher wrote an instance of the repeated ad- dition model, 4 + 4 + 4= 12.) "See if you can think of a pairing situation that would be like this." (Unexpectedly, the pupil drew an instance of the Cartesian Product Model for 4 x 8=32, but later corrected it as indicated by the figure below.) "Okay. How are those two situations alike?" "Like if this was a set and this was a set, there would be 8 in this set and 4 in this set. (Pause.) And?" " What's the matter?" "That's wrong. (Pause.) There aren't 12 lines." "Oh, I see. What would you do?" "Oh, I get it! (The pupil drew a correct instance of Cartesian Product.) Yeh. There are 4 here and there are 4 in each thing"(referring to 4 elements in one set in Cartesian Product instance and addends of 4 in the Repeated Addition instance).

CC: JO: CC:

JO:

CC: JO:

"And it gives you the same answer?" "Uh-huh." "You said there are 4 there (referring to Cartesian Product) and there are 4 in each one there (referring to Repeated Addition)?" "And there are 3 here and there are 3 of those (referring to the 2 instances)." "12 lines? (referring to Cartesian Product)." "And there are 12 in all (referring to the Repeated Addition instance)."

In this episode, not only has it been implied by usage that the models of dissimilar structure both represent multiplication but the pupil has cited the explicit connec- tions that provide for the isomorphism between instances of the models.

We now examine the pupil's ability to work with models of a general level above that of specific instances. The following probe, involves with the Cartesian Product Model (refer- red to as "pairing") and the Set Union Model (referred to as "using towers" or "using chips"):

CC:

JO:

CC: JO:

" You are going to have to think without being able to look at pictures. Suppose you have a friend who knows how to do pairing and who knows how to do towers and you are to tell him the rules for making a pairing situation from towers. What would the rules be?" " Well, (pause) you would put the number of towers in one column, you would put that many in one column in the pairing." (Translation: The number of sets in the Set Union Model would be equal to the number of elements in one set in the Cartesian Product Model.) "All right." "And you would take how many in each tower and put it (the number) in another column (pause) and then you would pair the pairing and count the lines and you would know the answer." (Translation: The number of elements in each set in the Set Union Model equals the number of elements in the other set in the Car- tesian Product Model.)

CC:

JO:

" When you count the lines, what does that go with in the towers?" "How many counters there are altogether." (Transla- tion: The number in all the sets combined in the Set Union Model equals the number of pairing lines in the Cartesian Product Model.)

The above general isomorphism probe was then followed immediately by a probe for abstraction:

CC: JO:

CC:

JO:

"All right; good. How is the pairing like the towers?" " Well, they are alike in (pause) that some pairings and some towers equal the same multiplication equation." "Or even if they don't both equal the same multiplication equation?" "They can both equal one." (Translation: They both are instances of a model of multiplication.)

In this episode, in response to leading questions, the pupil has supplied the rules for connecting the Cartesian Prod- uct Model and the Set Union Model at the level of the general isomorphism between the two models. He has shown the rules by which the two models represent each other.

In the probe for abstraction, when asked, "How is the pairing like the towers?", the pupil replied, "They are alike in that some pairing and some towers equal the same multiplication equation." Thus it is implied that the two distinct models represent the same operation.

We next illustrate that children can view multiplication in a more general and abstract manner by using more than two models.

For the following probe, the pupil was given preparatory instruction including experience with several models for multiplication including the Balance Model, the Rectan- gular Array Model, referred to as "blocks", and a form of the Cartesian Product Model referred to a "intersec- tion." The pupil was given a "pairing" instance of the Cartesian Product Model and asked to make several different isomorphic instances of other models. This was followed by a discussion aimed at encouraging abstraction of the operation.

CC: (The teacher drew a type of instance of the Cartesian Product Model referred to as "pairing")

CC: "Now. Here is a pairing situation. Can you make the balance situation that goes with it?" (With a balance beam and weights, the pupil con- structed a balance situation similar to the following.)

10 9 8 7 6 5 4 3 2 10 1 2 3 4 5 6 7 8 91,0

Mathematics in School, September 1985 25

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CC: "Good. Can you make the block situation that goes along with it?" (The student drew an instance of the Rectangular Array Model.)

For the Rectangular Array Model, the instance was always in the shape of a rectangular array and the factors in the equation indicated the "length" and "width" of the array.

CC: "Good. Can you make the counter or tower situation that goes along with it?" (The pupil constructed 2 sets (towers) of 3 chips each.)

CC: "Fine, and last of all can you make the intersection picture that goes along with it?" (The pupil drew a type of instance of the Cartesian Product Model referred to as "intersection.")

CC: "Good. These are all alike in what way?" JO: " They can all be multiplication, and they are all using

2, 3, and 6." CC: "All right. What multiplication equation? The same one

or different ones?" JO: "The same one." CC: " What would it be?" JO: "2 times, uh, 2 times 3 equals 6." CC: "Good." JO: "They are different because they are using different

ways to find the answer."

In this episode, the pupil has abstracted multiplication by specifying that it can be interpreted in terms of several models. It is reasonable to infer that in the pupil's view, multiplication need not be limited to a single model.

Another interesting teaching episode would involve en- couraging the pupil to abstract the meaning of whole numbers in the context of models of operations and beyond the notion of cardinal number (how many).

Multiple embodiment episodes such as those reported here are interesting, and as indicated in this report, are feasible for children of younger ages. Some teachers and authors feel that such experiences provide "better" under- standing of the mathematical concepts reflected2'3. Some feel that children's notion of multiplication should not be

limited to a single model (i.e. multiplication is an easy way to add) but instead should be abstract and general as is illustrated in the described episodes. '" Other feel that chil- dren should not only have experience with several models of an operation, but that they should learn that these models represent each other (are isomorphic). Perhaps learning to manipulate the relationship between models, as has been described, is important to children's developing problem solving skills6. It may even be that abstracting experiences will minimise the negative transfer from the concrete lessons of the earlier grades and develop pupils more capable of dealing with higher level abstractions in the intermediate grades.

In any case, if these activities seem to have some value, the reader may wish to try them with his or her pupils Those mathematics educators interested in research may desire to refine and investigage some of the questions related to resultant psychological savings and improved problem-solving skills.

References 1. Vest, F. (1971) A Catalog of Models for Multiplication and Division of

Whole Numbers, Educational Studies in Mathematics, 3, 2. 2. Dienes, Z. P. (1964) Building Up Mathematics, Hutchinson and Co.,

London. 3. Reys, R. E. (1972) Mathematics, Multiple Embodiments, and Elemen-

tary Teachers, Arithmetic Teacher, 19, 6. 4. Vest, F. (1978) Introducing Additional Concrete Models of Operations:

A Discovery Approach, Arithmetic Teacher, 25, 4. 5. Vest, F. (1982) A Concrete Structure of Arithmetic, Focus on Learning

Problems in Mathematics, 4, 2. 6. Vest, F. (1976) Teaching Problem Solving as Viewed Through a

Theory of Models, Educational Studies in Mathematics, 6, 4.

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