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217 MATHEMATICAL UNDERSTANDING: AN INTRODUCTION 5 Mathematical Understanding: An Introduction Karen C. Fuson, Mindy Kalchman, and John D. Bransford For many people, free association with the word “mathematics” would produce strong, negative images. Gary Larson published a cartoon entitled “Hell’s Library” that consisted of nothing but book after book of math 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 mathematics 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 that makes use of the three principles that are the focus of this volume. Instead of connecting with, building on, and refining the mathematical understandings, intuitions, and resourcefulness that stu- dents bring to the classroom (Principle 1), mathematics 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 (Principle 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 math- ematics problems. Box 5-1 provides a vignette involving a student who gives an answer to a problem that is quite obviously impossible. When quizzed, he can see that 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 whether his answer made sense, but he believes that sense making is irrelevant.
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218 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM BOX 5-1 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 need?” His answer was 18 pints. I said, “How much in each jug?” “Two- thirds of a pint.” I said, “Is that more or less that a pint?” “Less.” I said, “How many jugs are there?” “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.”1 A recent report of the National Research Council,2 Adding It Up, reviews a broad research base on the teaching and learning of elementary school mathematics. The report argues for an instructional goal of “mathematical proficiency,” a much broader outcome than mastery of procedures. The report argues that five intertwining strands constitute mathematical profi- ciency: 1. Conceptual understanding—comprehension of mathematical con- cepts, operations, and relations 2. Procedural fluency—skill in carrying out procedures flexibly, accu- rately, efficiently, and appropriately 3. Strategic competence—ability to formulate, represent, and solve math- ematical problems 4. Adaptive reasoning—capacity for logical thought, reflection, expla- nation, and justification 5. Productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy These strands map directly to the principles of How People Learn. Prin- 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 both describe the phenomenon of ongoing sense making, reflection, and explanation to oneself and others. And, as we argue below, the preconceptions students bring to the study of mathematics affect more than their understanding and problem solving; those preconceptions also play a major role in whether students have a productive
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219 MATHEMATICAL UNDERSTANDING: AN INTRODUCTION 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 this introduction, we explore how those principles ap- ply to the subject of mathematics more generally. We draw on examples from the Children’s Math 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 with language, that awareness appears to be universal in nor- mally developing children, though the rate of development varies at least in part because of environmental influences.5 But it is not only the awareness of quantity that develops without formal training. Both children and adults engage in mathematical 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 mathematics 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 mathemati- cal problems when comparison shopping, even though the women could not solve problems presented abstractly in a classroom that required the same mathematics.7 A similar result was found in a study of a group of Weight Watchers, who used strategies for solving mathematical measure- ment problems related to dieting that they could not solve when the prob- lems were presented more abstractly.8 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 mathematics. But they also suggest that the link is not automatic. If there is no bridge between informal and formal mathematics, the two often remain disconnected. The first principle of How People Learn emphasizes both 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 that are often fostered early on in school settings are in fact counterproductive. Students who believe them can easily conclude that the study of mathematics 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: MATHEMATICS IN THE CLASSROOM Some Common Preconceptions About Mathematics Preconception #1: Mathematics is about learning to compute. Many of us who attended school in the United States had mathematics instruction that 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 that is the procedure they learned in school. Finding the lowest common denominator is not easy in this instance, and the problem seems difficult. A few people take a con- ceptual rather than a procedural (computational) approach and realize that 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 that mathematics is about problem solving and that 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 that some aspect of problem solving has gone awry. If one believes that mathematics is about computation, however, then sense making may never take place. Preconception #2: Mathematics is about “following rules” to guarantee correct answers. Related to the conception of mathematics as computation is that of math- ematics as a cut-and-dried discipline that specifies rules for finding the right 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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221 MATHEMATICAL UNDERSTANDING: AN INTRODUCTION view use of these procedures only as following the rules. But the “rules” should not be confused with the game itself. The authors of the chapters in this part of the book provide important suggestions about the much broader nature of mathematical proficiency and about ways to make the involving nature of mathematical inquiry visible to students. Groups such as the National Council of Teachers of Mathematics10 and the National Research Council11 have provided important guidelines for the kinds of mathematics instruction that accord with what is currently known about the principles of How People Learn. The authors of the following chapters have paid careful attention to this work and illustrate some of its important aspects. In reality, mathematics 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 mathematics 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 methods of calculating with numerals and more recently by methods that involve pushing buttons on a calculator. If mathematics 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. Preconception #3: Some people have the ability to “do math” and some don’t. This is a serious preconception that is widespread in the United States, but not necessarily in other countries. It can easily become a self-fulfilling prophesy. In many countries, the ability to “do math” is assumed to be attributable to the amount of effort people put into learning it.12 Of course,
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222 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM some people in these countries do progress further than others, and some appear to have an easier time learning mathematics than others. But effort is still considered to be the key variable in success. In contrast, in the United States we are more likely to assume that ability is much more important than effort, and it is socially acceptable, and often even desirable, not to put forth effort in learning mathematics. 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.13 This preconception likely shares a common root with the others. If mathematics 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), they can easily be forgotten. Without a conceptual understanding of the nature of problems and strategies for solving them, failure to retrieve learned procedures can leave a student completely at a loss. Yet students can feel lost not only when they have forgotten, but also when they fail to “get it” from the start. Many of the conventions of math- 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 efficiency, they can be baffled by things that are left unexplained. If students never understand that x and y have no intrinsic meaning, but are conven- tional notations for labeling unknowns, they will be baffled when a z ap- pears. When an m precedes an x in the equation of a line, students may wonder, Why m? Why not s for slope? If there is no m, then is there no slope? To someone with a secure mathematics understanding, the missing m is simply an unstated m = 1. But to a student who does not understand that 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 they are couched in meaningful contexts, there are few clues to help a student who is lost in mathematics. Providing a secure conceptual understanding of the mathematics enterprise that is linked to students’ sense- making capacities is critical so that students can puzzle productively over new material, identify the source of their confusion, and ask questions when they do not understand.
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223 MATHEMATICAL UNDERSTANDING: AN INTRODUCTION 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 mathematics so students come to appreciate that it is not about computation and following rules, but about solving important and relevant quantitative problems? This perspective in- cludes an understanding that the rules for computation and solution are a set of clever human inventions that in many cases allow us to solve complex problems more easily, and to communicate about those problems with each other effectively and efficiently. Second, how can we link formal mathemat- ics training with students’ informal knowledge and problem-solving capaci- ties? Many recent research and curriculum development efforts, including those of the authors 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 informal problem-solving strate- gies, at least initially, and then guiding their mathematical thinking toward more effective strategies and advanced understandings. • Encouraging math talk so that students can clarify their strategies to themselves and others, and compare the benefits and limitations of alternate approaches. • Designing instructional activities that can effectively bridge commonly held conceptions and targeted mathematical understandings. Allowing Multiple Strategies To illustrate how instruction can be connected to students’ existing knowl- edge, consider three subtraction methods encountered frequently in urban second-grade classrooms involved in the Children’s Math 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 method. Two of the methods are correct, and one is mostly 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 that 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 math 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 math” are sown. An answer is either wrong or
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224 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM Three Subtraction Methods BOX 5-2 Maria’s add-equal- Peter’s ungrouping Manuel’s mixed quantities method method method 11 14 11 14 1 2 14 12 4 12 4 – 15 6 – 15 –56 6 68 68 5 8 right, 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 with 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 other hand, students believe that for each kind of math situa- tion or problem there can be several correct methods, 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 reflects 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 methods 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- scription in the vignette of how the class together figured out over time how Maria’s method worked (this method is commonly taught in Latin America and Europe). Understanding a new method can be a worthwhile mathemati- cal project for the class, and others can be involved in trying to figure out why a method works. This illustrates one way in which a classroom commu- nity can function. If one relates a calculation method to the quantities in- volved, one can usually puzzle out what the method 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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225 MATHEMATICAL UNDERSTANDING: AN INTRODUCTION Engaging Students’ Problem-Solving Strategies BOX 5-3 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. Teacher Maria, can you please explain to your friends in the class how you solved the problem? Maria Six is bigger than 4, so I can’t subtract here [pointing] in the ones. So I have to get more ones. But I have to be fair when I get more ones, so I add ten to both my numbers. I add a ten here in the top of the ones place [pointing] to change the 4 to a 14, and I add a ten here in the bottom in the tens place, so I write another ten by my 5. So now I count up from 6 to 14, and I get 8 ones [demonstrating by counting “6, 7, 8, 9, 10, 11, 12, 13, 14” while raising a finger for each word from 7 to 14]. 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 + ? = 12 tens. Oh, I know 6 + 6 = 12, so my answer is 6 tens.”] Jorge I don’t see the other 6 in your tens. I only see one 6 in your answer. Maria The other 6 is from adding my 1 ten to the 5 tens to get 6 tens. I didn’t write it down. Andy But you’re changing the problem. How do you get the right answer? Maria 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. Michelle Why did you count up? Maria Counting down is too hard, and my mother taught me to count up to subtract in first grade.
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226 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM Continued BOX 5-3 Teacher How many of you remember how confused we were when we first saw Maria’s method last week? Some of us could not figure out what she was doing even though Elena and Juan and Elba did it the same way. What did we do? Rafael We made drawings with our ten-sticks and dots to see what those numbers meant. And we figured out they were both tens. Even though the 5 looked like a 15, it was really just 6. And we went home to see if any of our parents could explain it to us, but we had to figure it out ourselves and it took us 2 days. Teacher Yes, I was asking other teachers, too. We worked on other methods too, but we kept trying to understand what this method was and why it worked. And Elena and Juan decided it was clearer if they crossed out the 5 and wrote a 6, but Elba and Maria liked to do it the way they learned at home. Any other questions or comments for Maria? No? Ok, Peter, can you explain your method? Peter Yes, I like to ungroup my top number when I don’t have enough to subtract everywhere. So here I ungrouped 1 ten and gave it to the 4 ones to make 14 ones, so I had 1 ten left here. So 6 up to 10 is 4 and 4 more up to 14 is 8, so 14 minus 6 is 8 ones. And 5 tens up to 11 tens is 6 tens. So my answer is 68. Carmen How did you know it was 11 tens? Peter Because it is 1 hundred and 1 ten and that is 11 tens. Carmen I don’t get it. Peter Because 1 hundred is 10 tens. Carmen Oh, so why didn’t you cross out the 1 hundred and put it with the tens to make 11 tens like Manuel? Peter I don’t need to. I just know it is 11 tens by looking at it. Teacher Manuel, don’t erase your problem. I know you think it is probably wrong because you got a different answer, but remember how making a mistake helps everyone learn—because other
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227 MATHEMATICAL UNDERSTANDING: AN INTRODUCTION students make that same mistake and you helped us talk about it. Do you want to draw a picture and think about your method while we do the next problem, or do you want someone to help you? Manuel Can Rafael help me? Teacher Yes, but what kind of helping should Rafael do? Manuel He should just help me with what I need help on and not do it for me. Teacher Ok, Rafael, go up and help Manuel that way while we go on to the next problem. I think it would help you to draw quick-tens and ones to see what your numbers mean. [These draw- ings are explained later.] But leave your first solution so we can all see where the problem is. That helps us all get good at debugging— finding our mistakes. Do we all make mis- takes? Class Yes. Teacher Can we all get help from each other? Class Yes. Teacher So mistakes are just a part of learning. We learn from our mistakes. Manuel is going to be brave and share his mistake with us so we can all learn from it. Manuel’s method combined Maria’s add-equal-quantities method, which he had learned at home, and Peter’s ungrouping method, which he had learned at school. It increases the ones once and decreases the tens twice by subtracting a ten from the top number and adding a ten to the bottom subtracted number. In the Children’s Math Worlds Project, we rarely found children forming such a meaningless combination of meth- ods if they understood tens and ones and had a method of drawing them so they could think about the quantities in a problem (a point discussed more later). Students who transferred into our classes did sometimes initially use Manuel’s mixed approach. But students were eventually helped to understand both the strengths and weaknesses of their existing meth- ods and to find ways of improving their approaches. SOURCE: Karen Fuson, Children’s Math Worlds Project.
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246 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM this work indicates that we have begun the crucial journey into mathemati- cal proficiency for all and that the principles of How People Learn can guide us on this journey. NOTES 1. Holt, 1964, pp. 143-144. 2. National Research Council, 2001. 3. See Fuson, 1986a, 1986b, 1990; Fuson and Briars, 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; Ron, 1998. 4. Carey, 2001; Gelman, 1990; Starkey et al., 1990; Wynn, 1996; Canfield and Smith, 1996. 5. Case et al., 1999; Ginsburg, 1984; Saxe, 1982. 6. Carraher, 1986; Carraher et al., 1985. 7. Lave, 1988; Sternberg, 1999. 8. De la Rocha, 1986. 9. Ceci and Liker, 1986; Ceci, 1996. 10. National Council of Teachers of Mathematics, 2000. 11. National Research Council, 2001. 12. See, e.g., Hatano and Inagaki, 1996; Resnick, 1987; Stigler and Heibert, 1997. 13. Stigler and Heibert, 1999. 14. National Research Council, 2004. 15. See, e.g., Tobias, 1978. 16. Hufferd-Ackles et al., 2004. 17. Sherin, 2000a, 2002. 18. See, e.g., Bransford et al., 1989. 19. See, e.g., Schwartz and Moore, 1998. 20. Sherin, 2000b, 2001. 21. Lewis, 2002, p. 1. 22. Fernandez, 2002; Lewis, 2002; Stigler and Heibert, 1999. 23. Remillard, 1999, 2000. 24. Remillard and Geist, 2002. 25. Remillard, 2000. REFERENCES Anghileri, J. (1989). An investigation of young children’s understanding of multiplica- tion. Educational Studies in Mathematics, 20, 367-385. Ashlock, R.B. (1998). Error patterns in computation. Upper Saddle River, NJ: Prentice- Hall. Baek, J.-M. (1998). Children’s invented algorithms for multidigit multiplication prob- lems. In L.J. Morrow and M.J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics. Reston, VA: National Council of Teachers of Mathematics.
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247 MATHEMATICAL UNDERSTANDING: AN INTRODUCTION Baroody, A.J., and Coslick, R.T. (1998). Fostering children’s mathematical power: An investigative approach to k-8 mathematics instruction. Mahwah, NJ: Lawrence Erlbaum Associates. Baroody, A.J., and Ginsburg, H.P. (1986). The relationship between initial meaning- ful and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 75-112). Mahwah, NJ: Lawrence Erlbaum Associates. Beishuizen, M. (1993). Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in Math- ematics Education, 24, 294-323. Beishuizen, M., Gravemeijer, K.P.E., and van Lieshout, E.C.D.M. (Eds.). (1997). The role of contexts and models in the development of mathematical strategies and procedures. Utretch, The Netherlands: CD-B Press/The Freudenthal Institute. Bergeron, J.C., and Herscovics, N. (1990). Psychological aspects of learning early arithmetic. In P. Nesher and J. Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education. Cambridge, England: Cambridge University Press. Bransford, J.D., Franks, J.J., Vye, N.J., and Sherwood, R.D. (1989). New approaches to instruction: Because wisdom can’t be told. In S. Vasniadou and A. Ortony (Eds.), Similarity and analogical reasoning (pp. 470-497). New York: Cambridge University Press. Brophy, J. (1997). Effective instruction. In H.J. Walberg and G.D. Haertel (Eds.), Psychology and educational practice (pp. 212-232). Berkeley, CA: McCutchan. Brownell, W.A. (1987). AT Classic: Meaning and skill—maintaining the balance. Arith- metic 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). Evolutionary and ontogenetic foundations of arithmetic. Mind and Language, 16(1), 37-55. Carpenter, T.P., and Moser, J.M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15(3), 179-202. Carpenter, T.P., Fennema, E., Peterson, P.L., Chiang, C.P., and Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experi- mental study. American Educational Research Journal, 26(4), 499-531. Carpenter, T.P., Franke, M.L., Jacobs, V., and Fennema, E. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and sub- traction. Journal for Research in Mathematics Education, 29, 3-20. Carraher, T.N. (1986). From drawings to buildings: Mathematical scales at work. In- ternational Journal of Behavioural Development, 9, 527-544. Carraher, T.N., Carraher, D.W., and Schliemann, A.D. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3, 21-29. Carroll, W.M. (2001). A longitudinal study of children using the reform curriculum everyday mathematics. Available: http://everydaymath.uchicago.edu/educators/ references.shtml [accessed September 2004].
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248 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM Carroll, W.M., and Fuson, K.C. (1999). Achievement results for fourth graders using the standards-based curriculum everyday mathematics. Unpublished document, University of Chicago, Illinois. Carroll, W.M., and Porter, D. (1998). Alternative algorithms for whole-number opera- tions. In L.J. Morrow and M.J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (pp. 106-114). Reston, VA: National Council of Teachers of Mathematics. Case, R. (1985). Intellectual development: Birth to adulthood. New York: Academic Press. Case, R. (1992). The mind’s staircase: Exploring the conceptual underpinnings of children’s thought and knowledge. Mahwah, NJ: Lawrence Erlbaum Associates. Case, R. (1998). A psychological model of number sense and its development. Paper presented at the annual meeting of the American Educational Research Associa- tion, April, San Diego, CA. Case, R., and Sandieson, R. (1988). A developmental approach to the identification and teaching of central conceptual structures in mathematics and science in the middle grades. In M. Behr and J. Hiebert (Eds.), Research agenda in mathemat- ics education: Number concepts and in the middle grades (pp. 136-270). Mahwah, NJ: Lawrence Erlbaum Associates. Case, R., Griffin, S., and Kelly, W.M. (1999). Socioeconomic gradients in mathemati- cal ability and their responsiveness to intervention during early childhood. In D.P. Keating and C. Hertzman (Eds.), Developmental health and the wealth of nations: Social, biological, and educational dynamics (pp. 125-149). New York: Guilford Press. Ceci, S.J. (1996). On intelligence: A bioecological treatise on intellectual development. Cambridge, MA: Harvard University Press. Ceci, S.J., and Liker, J.K. (1986). A day at the races: A study of IQ, expertise, and cognitive complexity. Journal of Experimental Psychology, 115(3), 255-266. Cotton, K. (1995). Effective schooling practices: A research synthesis. Portland, OR: Northwest Regional Lab. Davis, R.B. (1984). Learning mathematics: The cognitive science approach to math- ematics education. Norwood, NJ: Ablex. De la Rocha, O.L. (1986). The reorganization of arithmetic practice in the kitchen. Anthropology and Education Quarterly, 16(3), 193-198. Dixon, R.C., Carnine, S.W., Kameenui, E.J., Simmons, D.C., Lee, D.S., Wallin, J., and Chard, D. (1998). Executive summary. Report to the California State Board of Education, review of high-quality experimental research. Eugene, OR: National Center to Improve the Tools of Educators. 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 reflection. In A. Baroody and A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ: Lawrence Erlbaum Associates. Fernandez, C. (2002). Learning from Japanese approaches to professional develop- ment. The case of lesson study. Journal of Teacher Education, 53(5), 393-405.
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249 MATHEMATICAL UNDERSTANDING: AN INTRODUCTION Fraivillig, J.L., Murphy, L.A., and Fuson, K.C. (1999). Advancing children’s math- ematical thinking in everyday mathematics reform classrooms. Journal for Re- search in Mathematics Education, 30, 148-170. Fuson, K.C. (1986a). Roles of representation and verbalization in the teaching of multidigit addition and subtraction. European Journal of Psychology of Educa- tion, 1, 35-56. Fuson, K.C. (1986b). Teaching children to subtract by counting up. Journal for Re- search in Mathematics Education, 17, 172-189. Fuson, K.C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cogni- tion and Instruction, 7, 343-403. Fuson, K.C. (1992a). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, R.T. Putnam, and R.A. Hattrup (Eds.), The analysis of arithmetic for mathematics teaching (pp. 53-187). Mahwah, NJ: Lawrence Erlbaum Associates. Fuson, K.C. (1992b). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243-275). New York: Macmillan. Fuson, K.C. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, W.G. Martin, and D. Schifter (Eds.), A r esearch companion to prin- ciples and standards for school mathematics (pp. 68-94). Reston, VA: National Council of Teachers of Mathematics. Fuson, K.C., and Briars, D.J. (1990). Base-ten blocks as a first- and second-grade learning/teaching approach for multidigit addition and subtraction and place- value concepts. Journal for Research in Mathematics Education, 21, 180-206. Fuson, K.C., and Burghardt, B.H. (1993). Group case studies of second graders in- venting multidigit addition procedures for base-ten blocks and written marks. In J.R. Becker and B.J. Pence (Eds.), Proceedings of the fifteenth annual meeting of the North American chapter of the international group for the psychology of mathematics education (pp. 240-246). San Jose, CA: The Center for Mathematics and Computer Science Education, San Jose State University. Fuson, K.C., and Burghardt, B.H. (1997). Group case studies of second graders in- venting multidigit subtraction methods. In Proceedings of the 19th annual meet- ing of the North American chapter of the international group for the psychology of mathematics education (pp. 291-298). San Jose, CA: The Center for Math- ematics and Computer Science Education, San Jose State University. Fuson, K.C., and Fuson, A.M. (1992). Instruction to support children’s counting on for addition and counting up for subtraction. Journal for Research in Mathemat- ics Education, 23, 72-78. Fuson, K.C., and Kwon, Y. (1992). Korean children’s understanding of multidigit addition and subtraction. Child Development, 63(2), 491-506. Fuson, K.C., and Secada, W.G. (1986). Teaching children to add by counting with finger patterns. Cognition and Instruction, 3, 229-260.
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256 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM Zucker, A.A. (1995). Emphasizing conceptual understanding and breadth of study in mathematics instruction. In M.S. Knapp (Ed.), Teaching for meaning in high- poverty classrooms. New York: Teachers College Press. SUGGESTED READING LIST FOR TEACHERS Carpenter, T.P. Fennema, E., Franke, M.L., Empson, S.B., and Levi, L.W. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Fuson, K.C. (1988). Subtracting by counting up with finger patterns. (Invited paper for the Research into Practice Series.) 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: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann. Jensen, R.J. (Ed.). (1993). Research ideas for the classroom: Early childhood math- ematics. New York: Macmillan. Knapp, M.S. (1995). Teaching for meaning in high-poverty classrooms. New York: Teachers College Press. Leinhardt, G., Putnam, R.T., and Hattrup, R.A. (Eds.). (1992). The analysis of arith- metic for mathematics teaching. Mahwah, NJ: Lawrence Erlbaum Associates. Lo Cicero, A., De La Cruz, Y., and Fuson, K.C. (1999). Teaching and learning cre- atively with the Children’s Math Worlds Curriculum: Using children’s narratives and explanations to co-create understandings. Teaching Children Mathematics, 5(9), 544-547. Owens, D.T. (Ed.). (1993). Research ideas for the classroom: Middle grades math- ematics. New York: Macmillan. Schifter, D. (Ed.). (1996). What’s happening in math class? Envisioning new practices through teacher narratives. New York: Teachers College Press. Wagner, S. (Ed.). (1993). Research ideas for the classroom: High school mathematics. New York: Macmillan.