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How Students Learn: Mathematics in the Classroom (2005)
Board on Behavioral, Cognitive, and Sensory Sciences and Education (BCSSE)

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217 5 Mathematical Understanding: An Introduction Karen C. Fuson, Mindy Kalchman, and John D. Bransford For many people, free association widb dbe word "madbematics" would produce strong, negative images. Gary Larson published a cartoon entitled "Hell's Library" that consisted of nodling but book after book of madh word problems. Many students and teachers resonate strongly with this cartoon's message. It is not just funny to Them; it is true. Why are associations with madhematics so negative for so many people? If we look through The lens of How People Learn, we see a subject that is rarely taught in a way dhat makes use of the dlree principles dhat are the focus of dais volume. Instead of connecting with, building on, and refining The madhematical understandings, intuitions, and resourcefulness dhat stu- dents bring to The classroom (Principle 1), madhematics instruction often overrides students' reasoning processes, replacing them with a set of rules and procedures that disconnects problem solving from meaning making, Instead of organizing The skills and competences required to do mathemat- ics fluently around a set of core mathematical concepts (Pnnciple 2), those skills and competencies are often themselves the center, and sometimes the whole, of instruction. And precisely because The acquisition of procedural knowledge is often divorced from meaning making, students do not use metacognitive strategies (Principle 3) when They engage in solving madh- ematics problems. Box 5-1 provides a vignette involving a student who gives an ar swer to a problem dhat is quite obviously impossible. When quizzed, he can see Blat his answer does not make sense, but he does not consider it wrong because he believes he followed the rule, Not only did he neglect to use metacognitive strategies to monitor whedher his answer made sense, but he believes that sense making is irrelevant.

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218 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM BOXS-] Computation Without Comprehension: An Observation by John Holt One boy, quite a good student, was working on the problem, "If you have 6 jugs, and you want to put 2/3 of a pint of lemonade into each jug, how much lemonade will you need7 " His answer was 18 pints. I said, " How much in each jug7" "Two- thirds of a pint." I said, "Is that more or less that a pint7" "Less." I said, "How many jugs are there7" "SIX." I said, "But that [the answer of 18 pints] doesn't make any sense." He shrugged his shoulders and said, "Well, that's the way the system worked out." Holt argues: "He has long since quit expecting school to make sense. They tell you these facts and rules, and your job is to put them down on paper the way they tell you. Never mind whether they mean anything or not. "' A recent report of The National Research Council p Adding It Up reviews a broad research base on The teaching and teaming of elementary school madhematics. The report argues for an instructional goal of "madhematical proficiency," a much broader outcome than mastery of procedures. The report argues that five intertwining strands constitute madhematical profi- ciency 1. Conceptual understanding comprehension of mathematical con- cepts, operations, and relations 2. Procedura/.fluenc~skill in carrying out procedures flexibly, accu- rately, efficiently. and appropriately 3. Strategic competence ability to fommulate, represent, and solve math- ematical problems 4. Adaptive reasoning capacity for logical thought, resection, expla- nation, and justification 5. Productive disposition habitual inclination to see Illathell sties as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy These strands map dilectly to the principles of How People Learn Pnn- ciple 2 argues for a foundation of factual knowledge (procedural fluency), tied to a conceptual framework (conceptual understanding), and organized in a way to facilitate retrieval and problem solving (strategic competence). Metacognition and adaptive reasoning bodh describe The phenomenon of ongoing sense making, reflection, and explanation to oneself and odhers. And, as we argue below, the preconceptions students bring to The study of madhematics affect more Than their understanding and problem solving; Those preconceptions also play a major role in whedher students have a productive

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MATHEMAT CAE UNDERSTAND FIG: AN INTFDDUCT ON 219 disposition toward mathematics, as do, of course, their experiences in learn- ing mathematics. The chapters that follow on whole number, rational number, and func- tions look at The principles of How People Learn as they apply to Those specific domains. In tills introduction, we explore how those principles ap- ply to the subject of madhematics more generally. We draw on examples from the Children's Madh World project, a decade-long research project in urban and suburban English-speaking and Spanish-speaking classrooms.3 PRINCIPLE #1: TEACHERS MUST ENGAGE STUDENTS' PRECONCEPTIONS At a very early age, children begin to demonstrate an awareness of number.4 As widh language, that awareness appears to be universal in nor- mally developing children, Though the rate of development vanes at least in part because of environmental influences s But it is not only the awareness of quantity dhat develops without formal training. Bodh children and adults engage in madhematical problem solving, developing untrained strategies to do so successfully when formal experi- ences are not provided. For example, it was found that Brazilian street chil- dren could perform madhematics when making sales in The street, but were unable to answer similar problems presented in a school context 6 Likewise, a study of housewives in California uncovered an ability to solve madhemati- cal problems when comparison shopping, even though the women could not solve problems presented abstractly in a classroom dhat required the same madhematics.7 A similar result was found in a study of a group of Weight Watchers, who used strategies for solving madhematical measure- ment problems related to dieting that They could not solve when The prob terns were presented more abstracdy.9 And men who successfully handi- capped horse races could not apply The same skill to securities in The stock market.9 These examples suggest that people possess resources in The form of informal strategy development and mathematical reasoning that can serve as a foundation for learning more abstract madhematics. But They also suggest that The link is not automatic. If There is no blidge between informal and formal madhematics, The two often remain disconnected. The fret principle of How People Learn emphasizes bodh the need to build on existing knowledge and The need to engage students preconcep- tions particularly when they interfere with learning. In mathematics, cer- tain preconceptions dhat are often fostered early on in school settings are in fact counterproductive. Students who believe them can easily conclude that The study of madhematics is "not for them ~ and should be avoided if at all possible. We discuss These preconceptions below.

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220 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM Some Common Preconceptions About Mathematics Preconception #1: Mathematics is about learning to compute. Many of us who attended school in The United States had madhematics instruction dhat focused primarily on computation, with little attention to learning with understanding. To illustrate, try to answer The following ques- tion: What, approximately, is the sum of 8/9 plus 12/13? Many people immediately try to find The lowest common denominator for the two sets of fractions and Then add Them because dhat is The procedure they learned in school. Finding The lowest common denominator is not easy in tills instance, and The problem seems difficult. A few people take a con- ceptual radher than a procedural (computational) approach and realize dhat 8/9 is almost 1, and so is 12/13, so The approximate answer is a little less than 2, The point of this example is not that computation should not be taught or is unimportant; indeed, it is very often critical to efficient problem solv- ing. But if one believes dhat madhematics is about problem solving and dhat computation is a tool for use to that end when it is helpful, Then the above problem is viewed not as a "request for a computation," but as a problem to be solved that may or may not require computation—and in this case, it does not. If one needs to find The exact answer to The above problem, computa- tion is The way to go. But even in This case, conceptual understanding of The nature of the problem remains central, providing a way to estimate The cor- rectness of a computation. If an answer is computed that is more than 2 or less Than 1, it is obvious dhat some aspect of problem solving has gone awry. If one believes that madhematics is about computation, however, Then sense making may never take place. Preconception #2: Mathematics is about "foDowing rules~to guarantee correct answers. Related to The conception of madhematics as computation is dhat of madh- ematics as a cut-and-dried discipline dhat specifies rules for finding The light answers Rule following is more general Than performing specific computa- tions. When students learn procedures for keeping track of and canceling units, for example, or learn algebraic procedures for solving equations, many

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MATHEMAT CAE UNDERSTAND FIG: AN INTFDDUCT ON 221 view use of These procedures only as following the rules. But The "rules" should not be cor fused widh The game itself The authors of The chapters in dais part of the book provide important suggestions about the much broader nature of madhematical proficiency and about ways to make The involving nature of mathematical inquiry visible to students. Groups such as The National Council of Teachers of Madhematics'° and The National Research Council" have provided important guidelines for The kinds of mathematics instruction That accord widh what is currendy known about The principles of How People Learn. The authors of The following chapters have paid careful attention to dais work and illustrate some of its important aspects. In reality, madhematics is a constantly evolving field that is far from cut and dried. It involves systematic pattern finding and continuing invention, As a simple example, consider the selection of units that are relevant to quantify an idea such as The fuel efficiency of a vehicle. If we choose miles per gallon, a two-seater sports car will be more efficient Than a large bus. If we choose passenger miles per gallon, the bus will be more fuel efficient (assuming it carries large numbers of passengers). Many disciplines make progress by inventing new units and metrics that provide insights into previ- ously invisible relationships, Attention to The history of madhematics illustrates That what is taught at one point in time as a set of procedures really was a set of clever inventions designed to solve pervasive problems of everyday life. In Europe in the Middle Ages, for example, people used calculating cloths marked with ver- tical columns and carried out procedures with counters to perform calcula- tions. Other cultures fastened Their counters on a rod to make an abacus. Both of these physical means were at least partially replaced by written medhods of calculating with numerals and more recently by medhods That involve pushing buttons on a calculator if madhematics procedures are un- derstood as inventions designed to make common problems more easily solvable, and to facilitate communications involving quantity, those proce- dures take on a new meaning. Different procedures can be compared for Their advantages and disadvantages. Such discussions in the classroom can deepen students understanding and skill, Preeoreeptior #3. Some people have tlbe ability to "do math" and some don't. This is a serious preconception That is widespread in The United States, but not necessarily in o he- countries. It can easily become a self-fulfilling prophesy. In many countries, The ability to "do madh" is assumed to be attributable to The amount of effort people put into learning it.'3 Of course,

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222 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM some people in These countries do progress further Than odhers, and some appear to have an easier time learning madhematics Than odhers. But effort is still considered to be The key variable in success. In contrast, in The United States we are more likely to assume dhat ability is much more important Than effort, and it is socially acceptable, and often even desirable, not to put forth effort in learning madhematics. This difference is also related to cultural differences in the value attributed to struggle. Teachers in some countries believe it is desirable for students to struggle for a while with problems, whereas teachers in the United States simplify Things so That students need not struggle at all.'3 This preconception likely shares a common root widh The odhers. If madhematics learning is not grounded in an understanding of The nature of The problem to be solved and does not build on a student's own reasoning and strategy development, Then solving problems successfully will depend on The ability to recall memorized rules. If a student has not reviewed Those rules recently (as is The case when a summer has passed), dhey can easily be forgotten. Widhout a conceptual understanding of The nature of problems and strategies for solving Them, failure to retr eve learned procedures can leave a student completely at a loss. Yet students can feel lost not only when dhey have forgotten, but also when dhey fail to "get it" from The start. Many of The conventions of madh- ematics have been adopted for The convenience of communicating efficiently in a shared language. If students learn to memorize procedures but do not understand that The procedures are full of such conventions adopted for off ciencv, dhey can be baffled by Things dhat are left unexplained. If students never understand that x and y have no intrinsic meaning, but are conven- tional notations for labeling unknowns, dhey will be baffled when a z ap- pears. When an m precedes an x in The equation of a line, students may wonder, Why me Why not s for slope? If There is no m, Then is There no slope? To someone widh a secure madhematics understanding, The missing m is simply an unstated m = 1. But to a student who does not understand dhat The point is to write The equation efficiently. The missing m can be baffling. Unlike language learning, in which new expressions can often be figured out because dhey are couched in meaningful contexts, There are few clues to help a student who is lost in madhematics Providing a secure conceptual understanding of The madhenlarics enterprise dhat is linked to students' sense- making capacities is critical so dhat students can puzzle productively over new material, identify The source of Their confusion, and ask 4uesoons when dhey do not understand.

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MATHEMAT CAE UNDERSTAND NO: AN INTFDDUCT ON 223 Engaging Students' Preconceptions and Building on Existing Knowledge Engaging and building on student preconceptions, then, poses two in- structional challenges. First, how can we teach madhematics so students come to appreciate dhat it is not about computation and following rules, but about solving important and relevant quantitative problems? This perspective in- cludes an understanding dhat The rules for computation and solution are a set of clever human inventions dhat in many cases allow us to solve complex problems more easily, and to communicate about those problems widh each odher effectively and efficiently. Second, how can we link formal madhemat- ics training with students' informal knowledge and problem-solving capaci- ties? Many recent research and curriculum development efforts, including those of the audhors of The chapters that follow, have addressed These ques- tions. While There is surely no single best instructional approach, it is pos- sible to identify certain features of instruction that support the above goals: . Allowing students to use their own infommal problem-solving strate- gies, at least initially, and Then guiding Their mathematical thinking toward more effective strategies and advanced understandings. . Encouraging madh talk so that students can clalify Their strategies to themselves and odhers, and compare The benefits and limitations of alternate approaches. · t)esigning instructional activities dhat can effectively blidge commonly held conceptions and targeted mathematical understandings. Allowing Multiple Strategies To illustrate how instruction can be connected to students' existing know]- edge, consider three subtraction methods encountered frequently in urban second-grade classrooms involved in the Children's Madh Worlds Project (see Box 5-2). Maria, Peter, and Manuel's teacher has invited Them to share Their methods for solving a problem, and each of Them has displayed a different medhod. Two of The methods are correct, and one is mosey correct but has one error What the teacher does depends on her conception of what math- ematics is. One approach is to show The students The "right" way to subtract and have Them and everyone else practice dhat procedure. A very different ap- proach is to help students explore Their methods and see what is easy and difficult about each. If students are taught that for each kind of madh situa- tion or problem, There is one correct method that needs to be taught and learned, the seeds of the disconnection between Their reasoning and strat- egy development and "doing madh" are sown. An answer is either wrong or

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224 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM BOXS-2 Three Subtraction Methods Maria's add-equal- quantities method 1 2'4 -'5 6 6 8 Peter's ungrouping method Manuel's mixed method ~ ~ ~ 4 -'5 6 5 8 nght, and one does not need to look at wrong answers more deeply—one needs to look at how to get The right answer The problem is not that stu- dents will fail to solve The problem accurately widh This instructional ap- proach; indeed, they may solve it more accurately. But when The nature of the problem changes slightly, or students have not used the taught approach for a while, they may feel completely lost when confronting a novel prob- lem because The approach of developing strategies to grapple with a prob- lem situation has been short-circuited. If, on the odher hand, students believe That for each kind of madh situa- tion or problem There can be several correct medhods, their engagement in strategy development is kept alive. This does not mean that all strategies are equally good. But students can learn to evaluate different strategies for Their advantages and disadvantages. What is more, a wrong answer is usually partially correct and redects some understanding; finding The part That is wrong and understanding why it is wrong can be a powerful aid to under- standing and promotes metacognitive competencies. A vignette of students engaged in the kind of mathematical reasoning That supports active strategy development and evaluation appears in Box 5-3. It can be initially unsettling for a teacher to open up the classroom to calculation medhods That are new to The teacher But a teacher does not have to understand a new method immediately or alone, as indicated in The de- scnption in the vignette of how the class togedher figured out over time how Maria's medhod worked (this method is commonly taught in Latin America and Europe). Understanding a new medhod can be a worthwhile madhemati- cal project for the class, and odhers can be involved in trying to figure out why a medhod works. This illustrates one way in which a classroom commu- nity can function. If one relates a calculation medhod to the quantities in- volved, one can usually puzzle out what The medhod is and why it works. This also demonstrates That not all mathematical issues are solved or under- stood immediately; sometimes sustained work is necessary,

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MATHEMAT CAE UNDERSTAND FIG: AN INTFDDUCT ON 225 BOX 5-3 Engaging Students' Problem-Solving Strategies The following example of a classroom discussion shows how second- grade students can explain their methods rather than simply performing steps in a memorized procedure. It also shows how to make student thinking visible. After several months of teaching and learning, the stu- dents reached the point illustrated below. The students' methods are shown in Box 5-2. Ma ria Jo rge Ma ria Andy Maria Michelle Ma ria Teacher Maria, can you please explain to your friends in the class how you solved the problems Six is bigger than 4, so l can't subtract here [pointingl in the ones. So I have to get more ones. But I have to be fairwhen I get more ones, so l add ten to both my numbers. I add a ten here in the top of the ones place IPointingl to change the 4 to a 14, and I add a ten here in the bottom in the tens place, so l write another ten by my5. So now I count up from 6 to 14, and I get 8 ones I demonstrating by counting "6, 7, 8, 9, 10,11,12,13,14" while raising a finger for each word from 7 to 141. And I know my doubles, so 6 plus 6 is 12, so I have 6 tens left. [She thought, "1 + 5 = 6 tens and 6 + 7 = 12 tens. Oh, I know 6 + 6 = 12, so my answer is 6 tens. ~ | I don't see the other 6 in your tens. I only see one 6 in your answer The other 6 is from adding my 1 ten to the 5 tens to get 6 tens. I didn't write it down. But you're changing the problem. How do you get the right answers If I make both numbers bigger by the same amount, the difference will stay the same. Remember we looked at that on drawings last week and on the meter stick. Why did you count up7 Counting down is too hard, and my mother taught me to count up to subtract in first g fade.

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246 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM this work indicates that we have begun the crucial journey into mathemati- cal proficiency for all and that the principles of How Pt ople Learn can guide us on this journey. NOTES Holt, 1964, pp. 143-144. National Hesearch Counci, 2001. ,. See Fuson, 1986a, 1986b, 1990; Fuson and Bnats, 1990; Fuson and Burghardt, 1993, 1997; Fuson et al., 1994, 2000; Fuson and Smith, 1997; Fuson, Smith, and Lott, 1977; Fuson, Wearne et al., 1997; Fuson, Lo Cicero et al., 1997; Lo Cicero et al., 1999; Fuson et al., 2000; Hon. 1998. =. Carey, 2001; Ge man, 1990; Starkey et al., 1990; Wynn, 1996; Can told and Smith, 1996. Case et a ., 1999; Ginsburg, 1984; Saxe, 1982. Cam her, 1986; Cartaher et a ., 1985. Lave, 1988; Sternberg, 1999. De la Rocha, 1986. Ceci and Liker, 1986; Ceci, 1996. National Council of Teachers of Mathematics, 2000. National Hesearch Counci, 2001. See, e.g., Hatano and Inagaki, 1996; Resnick, 1987; stig er and Heibert, 1997. stig er and Heibert, 1999. National Hesearch Counci, 2004. See, e.g., Tobias, 1978. Hufferd-Ackles et al . 200 i Shelin, 2000a, 2002. See, e.g., Bransford et al., 1989. See, e.g., Schwartz and Moore, 1998. Sherin, 2000b, 2001. Lewis, 2002, p. 1. Fernandez, 2002; Lewis, 2002; Stiger md Heibert, 1999. Remillard, 1999, 2000. Remillard and Geist, 2002. Remillard, 2000. REFERENCES Anghi en, J. (1989). An investigation of young chi d en's understanding of multiplica- thm Educational Studies in Mathematics, 20, 367-385. Ashlock, R. B. (1998). Errorpatterns in computation. Upper Sadd e River, NJ: Prentice- Hall. Back, J.-M. (1998). Chi d en's invented algonthms for multidigit multiplication ptob- lems. In LJ. Morrow and MJ. Kenney (Eds.), The leaching and learning of algontlhms in school mathematics Reston, VA: National Counci of Teachers of Mathematics.

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MATHEMAT CAE UNDERSTAND NO: AN INTFDDUCT ON 247 Baroody, A J., and Coslick, R. T. (1998). Fostering children's matbematScalpouer An investigative approach to k{S mathematics instruction. Mahwah, NJ: Lawrence Erlbaum Associates. Baroody, AJ., and Ginsburg, H P. (1986). The relationship between initial meaning- ful and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural krrorrledge Tbe case of mathematics (pp. 75-112). Mahwah, NJ: Lawrence Erlbaum Associates. Beishuizen, M. (1993). Mental strategies and materials or mode s for addition and subt action up to 100 in Dutch second grades. Jourruzlfor Research in Matb- ematics Education, 24, 294 323. Beishuizen, M., Gravemerjer, K P.E., and van Lieshout, E C.D.M. (Eds.). (1997). Tbe role of contexts and models in the dot elopr,rent of mathematical strategies and procedures. Utretch, The Netherlands: CD B Press/The Freudentha institute. Bergeron, J.C., and Herscovics, N. (1990) Psychologica aspects of learning early anthmetic. In P. Nesher and J. Ki Patrick (Eds.), Matbematics and cognition A research synthesis by the International Groupior the Psychology of Mathematics Education. Cambndge, Eng and: Cambndge University PĒss. Btansford, J.D., Franks, JJ., Vye, NJ., and Sherwood, R. D. (1989). New approaches to inst Action: Because wisdom can't be told. In s. Vasniadou and A. Ortony (Eds.), SimiSanty and analogical reasoning (pp. 470 497). New York: Cambndge University PĒss. Brophy, J. (1997) Effective inst Action. In HJ. Walberg and G.D. Hae tel (Eds.), Psychology and educationalpractice(pp. 212-232). Berkeley, CA: McCutchan. Browned, WA. (1987). AT C assic: Meaning and ski—maintaining the balance. Aritb- met~c Teacher, 34(8), 18-25. Canfield, R. L., and Smith, E G. (1996). Number-based expectations and sequential enumeration by 5-month-old infants. Developmental Psychology, 32, 269-279. Carey, s. (2001) Evolutiona y and ontogenetic foundations of anthmetic. Mind and language, 16~1), 37-55. Carpenter, T. P. and Moser, J.M. (1984). The acquisition of addition and subt action concepts in grades one through three. Journal for Research in Mathematics Education, 15~3), 179-202. Carpenter, TP, Fennem I, E, Peterson, P. L., Chiang, C P. and Loef, M. (1989). Using knowledge of chi den s mathematics thinking in class oom teaching: An experi- mental study. American Educational Research Journal, 26 4), 499-531. Carpenter, TP, Franke, M.L., Jacobs, V., and Fennema, E (1998). A longitudinal study of invention and unde standing in children's multidigit addition and sub- traction. Journal for Reserzrci in . Mati emetics Ed ucatSon, 29, 3-20. Car aher, TN. (1986). From drawings to buildings: Mathematical scales at work. In- ternatSonalJournal of Bebavioural Development, 9, 527-544. Car aher, T.N., Carraher, D.W., and Schiemann, A.D. (1985). Mathematics in the streets and in schoo s. BntSsb Journal of Developmental Psychology, 3, 21 -29. Carroll, W.M. (2001). A SongStudSnal study of cbSsdren usSng tbe reform cumculum everyday matbemancs. Available: http://everydaymath.uchicago.edu/educato s/ references.shtml [accessed September 2004].

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248 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM Car all, W.M., and Fuson, EC. (1999). Ackietement resultsforfourtb graders using the standards-based curriculum et eryday matbematics. Unpublished document, University of Chicago, Illinois. Carroll, W.M., and Porter, D. (1998). Alternative algorithms for whole-number opeta- tions. In LJ. Morrow and MJ. Kenney (Eds.), The teaching and learning of algonthnzs in school matbematics(pp. 106 114). Reston, VA: National Counci of Teachers of Mathematics. Case, R. (1985). Intellectual detelopment Birth to tztbdthood New York: Academic Press. Case, R. (1992). The mind's staircase Exploring the conceptual underpinnings of ci ildren's thought and knouledge Mahwah, NJ: Lawence Erlbaum Associates. Case, R. (1998). A psychological model of nunzhersense and its development. Paper presented at the annual meeting of the American Educational Research Associa- tion, Apnl, San Diego, CA. Case, R., and Sandieson, R. (1988). A developmental approach to the identification and teaching of cent al conceptual st uctures in mathematics and science in the midd e g ades. In M. Behr and J. Hiebert (Eds.), Research agenda in matbemat- ics education Number concepts and in the middle grades (p p. 136-270). Mahwah, NJ: Lawence Erlbaum Associates. Case, R., Griffun, S., and Kelly, WM. (1999). Socioeconomic g adients in mathemati- cal abi ity and their responsiveness to intervention during early chi dhood. In D.P lieating and C. Hertzman (Eds.), Developmental health and the toeallb of nations Social, biological, andeducationaldynamics(pp. 125-149). NewYork: Gui ford Press Ceci, S J. (1996). On intelligence A bioecological treatise on intellectual Bet elopment Cambndge, MA: Harvard University Press. Ceci, SJ., and Liker, J.K. (1986). A day at the races: A study of IQ, expertise, and cogmitive complexity. Journal of Experimental Psyckology, 115(3), 255-266. Cotton, K. (1995). Effectite sckooling practices A researek syntbesrs. Port and, OR: Northwest Regional Lab. Davis, R.B. (1984). Learning matbematics Tl~e cognitive science approacb to matb- ematics education. Norwood, NJ: Ablex. De la Rocha, O.L. (198Q. The eorganization of anthmetic ptactice in the kitchen. Antbropology and Ed ucation Quarterly, 16 3), 193-198. Dixon, R.C., Carnine, S.W., Kameenui, EJ., Simmons, D.C., Lee, D.S., Wallin,J., md Chard, D. (1998). Executite summary. Report to tbe California State Board of Education, revieu of bigb~uabty expenmental researek. Eugene, OR: Nationa Center to Imptove the Tools of Educatots. Dossey, J.A., Swafford, J.O., Parmantie, M., and Dossey, A.E. (Eds.). (2003). Multidigit addition and subtraction methods invented in small groups and teacher support of problem solving and ref ection. In A. Batoody and A. Dowker (Eds.), Tl~e development of anthmetic concepts and skills Constructing adaptite expertise. Mahwah, NJ: Lawence Erlbaum Associates. Fernandez, C. (2002). Learning ftom Japanese approaches to professional develop- ment. The case of lesson study. Journal of Teacber Education, 53(5), 393-405.

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MATHEMAT CAE UNDERSTAND NO: AN INTFDDUCT ON 249 Fraivillig, J.L., Murphy, L.A., and Fuson, K.C. (1999). Advancing children's math- ematical thinking in everyday mathematics reform classrooms. Journa/ for Re- searek in.l1ather,zahc3 Education, 30, 148-170. Fuson, KC. (1986a) Roles of representation and verbalization in the teaching of mu udigit addition and subt action. European Journal of Psychology of Educa- tion, 1, 35-56. Fuson, K.C. (1986b). Teaching children to subtract by counting up. Journal for Re- searek in.l1ather,zahc3 Education, 17, 172-189. Fuson, K.C. (1990). Conceptual st uctures for mu uunit numbers: implications for learning and teaching mu udigit addition, subt action, and place value. Cogni- tion and Instruction, 7, 343 403. Fuson, KC. (1992a) Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, RT Putnam, and R A. Hattrup (Eds.), Me analysis of arithmetic for r,zather,zatics teaching (pp. 53-187). Mahwah, NJ: Lawrence Erlbaum Associates. Fuson, KC. (1992b) Research on whole number addition and subt action. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp 243 275) New York Macmillan Fuson, EC. (2003). Developing mathematical power in whole number operations. In J. Kilpatnck, W.G. Martin, and D. Schifter (Eds.), A research companion to pnn- ciplesand.standardsforsekool nzathenzati OCR for page 62
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254 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM Nesher, P. (1992). Solving multiplication word problems. In G. Leinhardt, R.T. Putnam, and R.A. Hatt up (Eds.), The analysis of arithmetic for matbematScs teaching (pp. 189-220). Mahwah, NJ: Lawrence Erlbaum Associates. Peak, L. (1996). Pursuing excellence A study of the US eighlb-grade matbematScs and science teaching learning, curriculum, and achieue.7zent in an Snterna- tSonal context. Washington, DC: National Center for Education Statistics. Rem lard, J.T. (1999). Cumculum matena s in mathematics education reform: A frame- work for examin ng teachers' cumculum development. CumcuSunz Inquiry, 2~ 3), 315-342. Kemi lard, J.T (2000). Can curriculum materials support teachers' learning? Elemen- tary SchoolJournal, 100(4), 331-350. Remillard,J.T., and Geist, P. (2002). Supporting teachers' professional lea m i rig though navigating openings in the curnculum. Journal of Mathematics TeacberEduca- tSon, 5(1), 7-34. Resnick, L.B. (1987). Education and learning to think. Committee on Mathematics, Science, and Technology Education, Commission on Behaviora and Socia Sci- ences and Education. Washington, DC: National Academy PĒss. Resnick, L.B. (1992). From protoquantities to operators: Building mathematical com- petence on a foundation of everyday knowledge. In G. Leinhardt, R.T Putnam, and R.A. Hattnup (Eds.), Tbe analysis of arithmetic for nuzthenzati OCR for page 67
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256 HOW STUDENTS LEAFN: MATHEMAT CS N THE CEASSFOOM bucker, A.A. (1995). Emphasizing conceptual understanding and breadth of study in mathematics inst uction. In M.S. Knapp (Ed.), Teackingfor meaning Sn kSgk- pouerty classrooms. New York: Teachers College PĒss. SUGGESTED READING LIST FOR TEACHERS Carpenter, T.P. Fennema, E., Ftanke, M.L., Empson, S.B., and Levi, L.W. (1999). CIhSldren's nrathenrtlhcs: CognStSuely guided instruction. Portsmouth, NH: Heinemann. Fuson, K.C. (1988). Subtracting by counting up with finger patterns. (Invited paper for the Research into Practice Senes.) Arithmetic Teacher, 35(5), 29 31. Hiebert, J., Carpenter, T. Fennema, E., Fuson, K.C., Wearne, D., Murray, H., Olivier, A., and Human, P. (1997). Making sense detaching and learning nzathenzahcs uith understanding. Portsmouth, NH: Heinemann. Jensen, RJ. (Ed.). (1993). Research Sdeasfor the classroom: Early childhood matl~- ematScs. New York: Macmi an. Knapp , M. S . (1995) . Teach Sng for meaning in k Sgk poverty classrooms. New York : Teachers College Press. Leinhardt, G., Putnam, R.T, and Hattrup, R.A. (Eds.). (1992). The analysis of antl~- metScior nuzti emetics react Sng Mahwah, NJ: Lawrence Erlbaum Associates. Lo Cicero, A., De La Cruz, Y. and Fuson, K.C. (1999). Teaching and learning cĒ- atively with the Chi dĒn's Math Worlds Curriculum: Using childĒn's narratives md expl Nations to co create understandings. Teaching ChiSdretz Maths noetics 5(9), 544 547. Owens, D.T (Ed.). (1993). Research ideate for the classroom: Middle grades matl~- ematScs. New York: Macmillan. Schifter, D. (Ed.). (1996). That's h~:zp~n in g in math class?EnuisSonSng new practices through teacher narratives New York: Teachers College Press. Wagner, S. (Ed.). (1993). Research Sdeasfor the classroom: High school mathematics New York: Macmillan.

Representative terms from entire chapter:

cae understand