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### Ways of Thinking About Multiplication

Interesting empirical result: Success on Problem 2 is usually 35-40% less than on Problem 1

Many solvers think they should divide/subtract on Problem 2

MTE 494, Arizona State University

What might explain this result?

A plausible explanation: repeated addition is often the only meaning for multiplication that students learn and retain.

First meaning attached to “4x12” is: 4x12 = 12+12+12+12. Similarly, “3x8.71” means 8.71+8.71+8.71.

For students “axb” usually means “do something”, namely add ba times: axb = b+b+…+b, where there are abs.

MTE 494, Arizona State University

Re-read the two problems; does the idea of multiplication as repeated addition fit both problems? Explain.

MTE 494, Arizona State University

So what’s the problem?

- Overemphasizing multiplication as repeated addition to the exclusion of other interpretations leads many students to unwittingly develop the following misconception:

Multiplication always makes bigger

?? When does multiplication NOT make bigger?

When first factor in the product axb is a whole number, repeated addition meaning makes sense. Further, multiplication does “make bigger” when the first factor is a whole number >1

MTE 494, Arizona State University

?? When does multiplication NOT make bigger?

BUT what about, say, “(¾)x200”? Does it makes sense to think about adding 200 three-fourths times?

A more unifying meaning: Multiplication as imagining multiplicities of things/amounts and making some number (including a fractional part) of copies of things/amounts.

“axb” means imagine abs or a copies of amount b. The amount you get by making a copies of b is a times as much as b.

MTE 494, Arizona State University

Under this meaning “(¾) of an amount” is a part-of-an amount—more specifically, it is 3 times as large as one-fourth of the amount.

Thus, (¾)x200 is an amount that is 3 times as large as one-fourth of 200 [the result of making 3 copies of one-fourth of 200]. In this conception, multiplication definitely does not make bigger.

Students who thought they should subtract or divide in Problem 2 are thought to be missing this “part-of-an amount” interpretation of multiplication.

Their thinking: ”0.73 is less than 1 lb, so it should cost less than $2.19. So I must do some calculation that gives less than $2.19.” If multiplication-makes-bigger was their guide, then the operation of multiplication is not an option.

MTE 494, Arizona State University

Summary

- Some conceptions of mathematical ideas are more powerful and generative than others

- We can approach mathematics instruction from an “engineering” perspective to:

1. Design mathematical conceptions (e.g., specifying meanings and ways of understanding a mathematical idea) to target as learning objectives;

2. Create and engage students in activities designed to foster those targeted conceptions.

The “making copies” conception of multiplication is an example of such an approach and effort.

MTE 494, Arizona State University

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