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 1 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 b a times axb b b b where there are a bs MTE 494 Arizona State University 2 Re read the two problems does the idea of multiplication as repeated addition fit both problems Explain MTE 494 Arizona State University 3 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 4 When does multiplication NOT make bigger BUT what about say 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 a bs 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 5 Under this meaning of an amount is a part of an amount more specifically it is 3 times as large as one fourth of the amount Thus 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 6 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 7
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