Ways of Thinking About Mul4plica4on Interes4ng 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 explana4on repeated addi on is oPen the only meaning for mul4plica4on that students learn and retain First meaning aSached 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 4mes axb b b b where there are a bs MTE 494 Arizona State University 2 Re read the two problems does the idea of mul4plica4on as repeated addi4on t both problems Explain MTE 494 Arizona State University 3 So what s the problem Overemphasizing mul4plica4on as repeated addi4on to the exclusion of other interpreta4ons leads many students to unwi ngly develop the following misconcep4on Mul plica on always makes bigger When does mul plica on NOT make bigger When rst factor in the product axb is a whole number repeated addi4on meaning makes sense Further mul4plica4on does make bigger when the rst factor is a whole number 1 MTE 494 Arizona State University 4 When does mul plica on NOT make bigger BUT what about say x200 Does it makes sense to think about adding 200 three fourths 4mes A more unifying meaning Mul4plica4on as imagining mul4plici4es of things amounts and making some number including a frac4onal 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 4mes as much as b MTE 494 Arizona State University 5 Under this meaning of an amount is a part of an amount more speci cally it is 3 4mes as large as one fourth of the amount Thus x200 is an amount that is 3 4mes as large as one fourth of 200 the result of making 3 copies of one fourth of 200 In this concep4on mul4plica4on de nitely 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 interpreta4on of mul4plica4on Their thinking 0 73 is less than 1 lb so it should cost less than 2 19 So I must do some calcula4on that gives less than 2 19 If mul4plica4on makes bigger was their guide then the opera4on of mul4plica4on is not an op4on MTE 494 Arizona State University 6 Summary Some concep4ons of mathema4cal ideas are more powerful and genera4ve than others We can approach mathema4cs instruc4on from an engineering perspec4ve to 1 Design mathema4cal concep4ons i e specify meanings and ways of understanding a mathema4cal idea to target as learning objec4ves 2 Create and engage students in ac4vi4es designed to foster those targeted concep4ons The making copies concep4on of mul4plica4on is an example of such an approach and e ort MTE 494 Arizona State University 7
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