Knowing_and_Teaching_Elementary_Mathemat (1)

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Subtraction with Regrouping: Approaches to Teaching a Topicdural understanding. To explain why one needs to “borrow” 1 ten from the tens place, theseteachers said, “You can’t subtract a bigger number from a smaller number.” They interpretedthe “taking” procedure as a matter of one number getting more value from anothernumber, without mentioning that it is a within-number rearrangement:You can’t subtract a bigger number from a smaller number…You must borrow fromthe next column because the next column has more in it. (Ms. Fay)But if you do not have enough ones, you go over to your friend here who has plenty.(Tr. Brady)“We can’t subtract a bigger number from a smaller one” is a false mathematical statement.Although second graders are not learning how to subtract a bigger number from a smallernumber, it does not mean that in mathematical operations one cannot subtract a biggernumber from a smaller number. In fact, young students will learn how to subtract a biggernumber from a smaller number in the future. Although this advanced skill is not taught insecond grade, a student’s future learning should not be confused by emphasizing a misconception.To treat the two digits of the minuend as two friends, or two neighbors living next doorto one another, is mathematically misleading in another way. It suggests that the two digitsof the minuend are two independent numbers rather than two parts of one number.Another misconception suggested by the “borrowing” explanation is that the value ofa number does not have to remain constant in computation, but can be changed arbitrarily—ifa number is “too small” and needs to be larger for some reason, it can just “borrow”a certain value from another number.In contrast, the teachers who expected students to understand the rationale underlyingthe procedure showed that they themselves had a conceptual understanding of it. Forexample, Tr. Bernadette excluded any of the above misconceptions:What do you think, the number, the number 64, can we take a number away, 46?Think about that. Does that make sense? If you have a number in the sixties can youtake away a number in the forties? OK then, if that makes sense now, then 4 minus 6,are we able to do that? Here is 4, and I will visually show them 4. Take away 6, 1,2, 3,4. Not enough. OK, well what can we do? We can go to the other part of the numberand take away what we can use, pull it away from the other side, pull it over to ourside to help, to help the 4 become 14.For Tr. Bernadette, the problem 64–46 was not, as suggested in the borrowing explanation,two separate processes of 4–6 and then 60–40. Rather, it was an entire process of “takingaway a number in the forties from a number in the sixties.” Moreover, Tr. Bernadettethought that it was not that “you can’t subtract a bigger number from a smaller number,”rather, that the second graders “are not able to do that.” Finally, the solution was that “wego to the other part of the number” (italics added), and “pull it over to our side to help.”The difference between the phrases “other number” and “the other part of the number” issubtle, but the mathematical meanings conveyed are significantly different.3
4Knowing and Teaching Elementary MathematicsInstructional Techniques: ManipulativesTeachers’ knowledge of this topic was correlated not only with their expectations aboutstudent learning, but also with their teaching approaches. When discussing how they wouldteach the topic, all except one of the teachers referred to manipulatives. The most popularmaterial was bundles of sticks (popsicle sticks, straws, or other kinds of sticks). Otherswere beans, money, base 10 blocks, pictures of objects, and games. The teachers said thatby providing a “hands-on” experience, manipulatives would facilitate better learning thanjust “telling”—the way they had been taught.A good vehicle, however, does not guarantee the right destination. The direction thatstudents go with manipulatives depends largely on the steering of their teacher. The 23teachers had different ideas that they wanted to get across by using manipulatives. A fewteachers simply wanted students to have a “concrete” idea of subtraction. With the problem52–25 for example, Tr. Belinda proposed “to have 52 kids line up and take 25 away and seewhat happens.” Ms. Florence reported that she would use beans as “dinosaur eggs” whichmight be interesting for students:I would have them start some subtraction problems with maybe a picture of 23 thingsand tell them to cross out 17 and then count how many are left… I might have themdo some things with dinosaur eggs or something that would sort of have a little moremeaning to them. Maybe have them do some concrete subtraction with dinosaureggs, maybe using beans as the dinosaur eggs or something.Problems like 52−25 or 23−17 are problems of subtraction with regrouping. However, whatstudents would learn from activities involving manipulatives like taking 25 students awayfrom 52 or taking 17 dinosaur eggs away from 23 is not related to regrouping at all. On thecontrary, the use of manipulatives removes the need to regroup. Tr. Barry, another experiencedteacher in the procedurally directed group, mentioned using manipulatives to getacross the idea that “you need to borrow something.” He said he would bring in quartersand let students change a quarter into two dimes and one nickel:A good idea might be coins, using money because kids like money… . The idea oftaking a quarter even, and changing it to two dimes and a nickel so you can borrow adime, getting across that idea that you need to borrow something.There are two difficulties with this idea. First of all, the mathematical problem in Tr. Barry’srepresentation was 25–10, which is not a subtraction with regrouping. Second, Tr.Barry confused borrowing in everyday life—borrowing a dime from a person who has aquarter—with the “borrowing” process in subtraction with regrouping—to regroup theminuend by rearranging within place values. In fact, Tr. Barry’s manipulative would notconvey any conceptual understanding of the mathematical topic he was supposed to teach.Most of the U.S. teachers said they would use manipulatives to help students understandthe fact that 1 ten equals 10 ones. In their view, of the two key steps of the procedure, takingand changing, the latter is harder to carry out. Therefore, many teachers wanted to showthis part visually or let students have a hands-on experience of the fact that 1 ten is actually10 ones:
Page 2 and 3: Knowing and TeachingElementary Math
Page 4 and 5: Knowing and TeachingElementary Math
Page 6 and 7: To Jianfeng, Sushu, and John,and to
Page 8 and 9: viii Contents5. Teachers’ Subject
Page 10 and 11: xAuthor’s Preface to the Annivers
Page 12 and 13: xiiAuthor’s Preface to the Annive
Page 14 and 15: xivSeries Editor’s Introduction t
Page 16 and 17: ForewordLee S.ShulmanThe Carnegie F
Page 18 and 19: xviii Forewordbe tests that assess
Page 20 and 21: AcknowledgmentsAbout 30 years ago,
Page 22 and 23: xxii AcknowledgmentsI would like to
Page 24 and 25: xxiv Introductionthe measurement mo
Page 26 and 27: xxvi Introductionschool or college
Page 28 and 29: xxviii IntroductionChapters 1 throu
Page 30 and 31: Chapter 1Subtraction With Regroupin
Page 34 and 35: Subtraction with Regrouping: Approa
Page 54 and 55: Multidigit Number Multiplication: D
Page 76 and 77: Chapter 3Generating Representations
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We also encourage students to solve
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Generating Representations: Divisio
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of math.” She said she used to ha
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been changed. Now what is known is
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While the U.S. teachers tended to u
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Exploring New Knowledge: The Relati
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Teachers’ Subject Matter Knowledg
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105These four properties are interr
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Chapter 6Profound Understanding Of
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Profound Understanding of Fundament
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Chapter 7ConclusionAs I said at the
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Conclusion 125as necessarily preced
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Conclusion 127Time is an issue here
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Conclusion 129evidence, students wo
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AppendixFIG. A.1. The U.S. teachers
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ReferencesArmstrong, B., & Bezuk, N
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References 135Kaput, J., & Nemirovs
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References 137Stein, M., Baxter, J.
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Fang and Paine’s “Bridging pola
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Liping Ma’s “Knowing and Teachi
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Author Index161Ferrucci, B., 150, 1
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Author Index163Simon, M., xxviii, x
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Subject Index165commutative law, 39
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Subject Index167knowledge,see also
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Subject Index169teacher preparation
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