Knowing_and_Teaching_Elementary_Mathemat (1)

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Chapter 1Subtraction With Regrouping:Approaches To Teaching A TopicScenarioLet’s spend some time thinking about one particular topic that you may work withwhen you teach, subtraction with regrouping. Look at these questions ( etc.).How would you approach these problems if you were teaching second grade? Whatwould you say pupils would need to understand or be able to do before they couldstart learning subtraction with regrouping?When students first learn about subtraction, they learn to subtract each digit of the subtrahendfrom its counterpart in the minuend:To compute this, they simply subtract 2 from 5 and 1 from 7. However, this straightforwardstrategy does not work all the time. When a digit at a lower place value of the subtrahendis larger than its counterpart in the minuend (e.g., 22−14, 162−79), students cannot conductthe computation directly. To subtract 49 from 62, they need to learn subtraction withregrouping:Subtraction, with or without regrouping, is a very early topic anyway. Is a deep understandingof mathematics necessary in order to teach it? Does such a simple topic even involvea deep understanding of mathematics? Would a teacher’s subject matter knowledge makeany difference in his or her teaching, and eventually contribute to students’ learning? Thereis only one answer for all these questions: Yes. Even with such an elementary mathematicaltopic, the teachers displayed a wide range of subject matter knowledge, which suggeststheir students had a corresponding range of learning opportunities.
2Knowing and Teaching Elementary MathematicsTHE U.S. TEACHERS’ APPROACH:BORROWING VERSUS REGROUPINGConstruing the TopicWhen discussing their approach to teaching this topic, the U.S. teachers tended to beginwith what they expected their students to learn. Nineteen of the 23 U.S. teachers (83%)focused on the procedure of computing. Ms. Fawn, a young teacher who had just finishedher first year of teaching, gave a clear explanation of this procedure:Whereas there is a number like 21−9, they would need to know that you cannot subtract9 from 1, then in turn you have to borrow a 10 from the tens space, and whenyou borrow that 1, it equals 10, you cross out the 2 that you had, you turn it into a10, you now have 11−9, you do that subtraction problem then you have the 1 left andyou bring it down.These teachers expected their students to learn how to carry out two particular steps: taking1 ten from the tens place, and changing it into 10 ones. They described the “taking”step as borrowing. By noting the fact that “1 ten equals 10 ones,” they explained the step of“changing.” Here we can see the pedagogic insight of these teachers: Once their studentscan conduct these two key steps correctly, they will very likely be able to conduct the wholecomputation correctly.The remaining four teachers, Tr. Bernadette, Tr. Bridget, Ms. Faith, and Ms. Fleur,however, expected their students to learn more than the computational procedure. Theyalso expected their students to learn the mathematical rationale underlying the algorithm.Their approach emphasized two points: the regrouping underlying the “taking” step and theexchange underlying the “change” step. Tr. Bernadette, an experienced teacher, said:They have to understand what the number 64 means…I would show that the number64, and the number 5 tens and 14 ones, equal the 64. I would try to draw the comparisonbetween that because when you are doing the regrouping it is not so muchknowing the facts, it is the regrouping part that has to be understood. The regroupingright from the beginning.Ms. Faith, another teacher at the end of her first year of teaching, indicated that studentsshould understand that what happens in regrouping is the exchange within place values:They have to understand how exchanges are done…he base 10 blocks when youreach a certain number—10, in base 10, in the ones column that is the same as, say,10 ones or 1 ten…they have to get used to the idea that exchanges are made withinplace values and that it does not alter the value of the number….Nothing happens tothe actual value, but exchanges can be made.What teachers expected students to know, however, was related to their own knowledge.The teachers who expected students merely to learn the procedure tended to have a proce-
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 32 and 33: Subtraction with Regrouping: Approa
Page 54 and 55: Multidigit Number Multiplication: D
Page 76 and 77: Chapter 3Generating Representations
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Generating Representations: Divisio
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We also encourage students to solve
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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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