Knowing_and_Teaching_Elementary_Mathemat (1)

Recommendations

Info

Subtraction with Regrouping: Approaches to Teaching a TopicI would give students bundles of popsicle sticks that are wrapped in rubber bands,with 10 in each bundle. And then I’d write a problem on the board, and I would havebundles of sticks, as well, and I would first show them how I would break it apart(italics added), to go through the problem, and then see if they could manage doingthe same thing, and then, maybe, after a lot of practice, maybe giving each pair ofstudents a different subtraction problem, and then they could, you know, demonstrate,or give us their answer. Or, have them make up a problem using sticks, breaking themapart and go through it. (Ms. Fiona)What Ms. Fiona reported was a typical method used by many teachers. Obviously, it isrelated more to subtraction with regrouping than the methods described by Ms. Florenceand Tr. Barry. However, it still appears procedurally focused. Following the teacher’s demonstration,students would practice how to break a bundle of 10 sticks apart and see how itwould work in the subtraction problems. Although Ms. Fiona described the computationalprocedure clearly, she did not describe the underlying mathematical concept at all.Scholars have noted that in order to promote mathematical understanding, it is necessarythat teachers help to make connections between manipulatives and mathematical ideasexplicit (Ball, 1992; Driscoll, 1981; Hiebert, 1984; Resnick, 1982; Schram, Nemser, &Ball, 1989). In fact, not every teacher is able to make such a connection. It seems that onlythe teachers who have a clear understanding of the mathematical ideas included in the topicmight be able to play this role. Ms. Faith, the beginning teacher with a conceptual understandingof the topic, said that by “relying heavily upon manipulatives” she would helpstudents to understand “how each one of these bundles is 10, it is 1 ten or 10 ones,” to knowthat “5 tens and 3 ones is the same as 4 tens and 13 ones,” to learn “the idea of equivalentexchange,” and to talk about “the relationship with the numbers”:What I would do, from that point, is show how each one of these bundles is 10, itis 1 ten or 10 ones. I would make sure that was clear. And what would happen if weundid this little rubber band and put 10 over here, how many ones would we have?And to get to the next step, I would show that now you have 1, 2, 3, 4 tens and 13ones and then subtract in that fashion …I would say to the child so you are tellingme that we have not added anything or subtracted anything to the 53, right? Right…Five tens and 3 ones is the same as 4 tens and 13 ones, and what happens when youtake 25 from that?Unlike the other teachers who used manipulatives to illustrate the computational procedure,Ms. Faith used them to represent the mathematical concept underlying the procedure.The only reason that Ms. Faith’s use of manipulatives could take her students “further” thanthat of other teachers was that she understood the mathematical topic in a deeper way thanothers. Using a similar method, teachers with different views of the subject matter wouldlead students to different understandings of mathematics.5
6Knowing and Teaching Elementary MathematicsTHE CHINESE TEACHERS’ APPROACH:“DECOMPOSING A HIGHER VALUE UNIT”Some of the Chinese teachers’ understandings of the topic overlapped these of the U.S.teachers. The group of Chinese teachers who held a “borrowing” conception had a focusvery similar to that of their U.S. counterparts:I will tell students that when you compute problems like 53–25, you first line up thenumbers and start the subtraction from the ones column. Since 3 is not big enoughfor you to subtract 5 from it, you should borrow a ten from the tens column and turnit into 10 ones. Adding the 10 ones to 3 you get 13. Subtract 5 from 13 you get 8. Put8 down in the ones column. Then you move to the tens column. Since 5 on the tenscolumn has lent a 10 to the ones column, there are only 4 tens left. You take 20 from40 and get 20. Put it down in the tens column. (Ms. Y.)Ms. Y. was in the middle of her second year of teaching. Her explanation was a variant ofMs. Fawn’s. She focused on the specific steps of the algorithm and did not show any interestin its rationale. The proportion of Chinese teachers who held such procedurally directedideas, however, was substantially smaller than that of the U.S. teachers (14% vs. 83%).Figure 1.1 shows the teachers’ different understandings of the topic.Most of the Chinese teachers focused on regrouping. However, in contrast to the U.S.teachers, about 35% of the Chinese teachers described more than one way of regrouping.These teachers not only addressed the rationale for the standard algorithm, but also discussedother ways to solve the problem that were not mentioned by the U.S. teachers. Let’s first takea look at the catchphrase of the Chinese teachers: decomposing a unit of higher value.“Decomposing a unit of higher value [tui yi *]” is a term in Chinese traditional arithmeticreckoning by the abacus. Each wire on an abacus represents a certain place value. Thevalue of each bead on the abacus depends on the position of the wire that carries the bead.The farther to the left a wire is located on the abacus, the larger the place value it represents.Therefore, the values of beads on left wires are always greater than those on right wires.FIG. 1.1. Teachers’ understanding of subtraction with regrouping.When subtracting with regrouping on the abacus, one needs to “take” a bead on a left wireand change it into 10 or powers of 10 beads on a wire to the right. This is called “decomposinga unit of higher value.”* The Chinese characters for this and other Chinese words appear in Appendix Fig. A.2.
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 32 and 33: Subtraction with Regrouping: Approa
Page 54 and 55: Multidigit Number Multiplication: D
Page 76 and 77: Chapter 3Generating Representations
Page 78 and 79: For some reason it is in the back o
Page 80 and 81: Generating Representations: Divisio
Page 82 and 83: We also encourage students to solve
Page 84 and 85:
Generating Representations: Divisio
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
of math.” She said she used to ha
Page 92 and 93:
Page 94 and 95:
been changed. Now what is known is
Page 96 and 97:
Page 98 and 99:
While the U.S. teachers tended to u
Page 100 and 101:
Page 102 and 103:
Exploring New Knowledge: The Relati
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Teachers’ Subject Matter Knowledg
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
105These four properties are interr
Page 136 and 137:
Chapter 6Profound Understanding Of
Page 138 and 139:
Profound Understanding of Fundament
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Chapter 7ConclusionAs I said at the
Page 154 and 155:
Conclusion 125as necessarily preced
Page 156 and 157:
Conclusion 127Time is an issue here
Page 158 and 159:
Conclusion 129evidence, students wo
Page 160 and 161:
AppendixFIG. A.1. The U.S. teachers
Page 162 and 163:
ReferencesArmstrong, B., & Bezuk, N
Page 164 and 165:
References 135Kaput, J., & Nemirovs
Page 166 and 167:
References 137Stein, M., Baxter, J.
Page 168 and 169:
Fang and Paine’s “Bridging pola
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Liping Ma’s “Knowing and Teachi
Page 190 and 191:
Author Index161Ferrucci, B., 150, 1
Page 192 and 193:
Author Index163Simon, M., xxviii, x
Page 194 and 195:
Subject Index165commutative law, 39
Page 196 and 197:
Subject Index167knowledge,see also
Page 198:
Subject Index169teacher preparation
show all

Knowing_and_Teaching_Elementary_Mathemat (1)

Create successful ePaper yourself

Delete template?

Save as template?