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From page 29... ... Instead of connecting with, building on, and refining The madhematical understandings, intuitions, and resourcefulness dhat students 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 mathematics fluently around a set of core mathematical concepts (Pnnciple 2) Read the entire page →
From page 30... ... 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 Pnnciple 2 argues for a foundation of factual knowledge (procedural fluency) , tied to a conceptual framework (conceptual understanding) Read the entire page →
From page 31... ... The fret principle of How People Learn emphasizes bodh the need to build on existing knowledge and The need to engage students preconceptions particularly when they interfere with learning. In mathematics, certain preconceptions dhat are often fostered early on in school settings are in fact counterproductive. Read the entire page →
From page 32... ... If one needs to find The exact answer to The above problem, computation 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 correctness of a computation. Read the entire page →
From page 33... ... 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. Read the entire page →
From page 34... ... 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' sensemaking 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. Read the entire page →
From page 35... ... 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) Read the entire page →
From page 36... ... If, on the odher hand, students believe That for each kind of madh situation 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. Read the entire page →
From page 37... ... 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 secondgrade students can explain their methods rather than simply performing steps in a memorized procedure. It also shows how to make student thinking visible. Read the entire page →
From page 38... ... Manuel,don'teraseyourproblem. I knowyou think it is probably wrong because you got a different answer, but remember how making a mistake helps everyone learn�because other Rafael Peter Ca rmen Peter Ca rmen Peter Ca rmen Peter Teacher Read the entire page →
From page 39... ... But students were eventually helped to understand both the strengths and weaknesses of their existing methods and to find ways of improving their approaches. SOURCE: Karen Fuson, Children's Math Worlds Project. Read the entire page →
From page 40... ... Students need to feel comfortable expressing dheir ideas and revising their thinking when feedback suggests The need to do so. Madh talk allows teachers to draw out and work widh The preconceptions students bring widh Them to The classroom and Then helps students learn how to do tills sort of work for themselves and for odhers. Read the entire page →
From page 41... ... More-advanced students also helped less-advanced students learn by modeling, asking questions, and helping odhers form more complete descriptions. Initially in The Children's Madh Worlds Project, all students made conceptual support drawings such as Those in Figure 5-1. Read the entire page →
From page 43... ... Identifying real-world contexts whose features help direct students' attention and thinking in mathematically productive ways is particularly helpful in building conceptual bridges between students' informal experiences and The new formal madhematics they are learning Examples of such bndging contexts are a key feature of each of The Three chapters Flat follow. PRINCIPLE #2: UNDERSTANDING REQUlRES FACTUAL KNOWLEDGE AND CONCEPTUAL } RAMEWORKS The second principle of How People Learn suggests The importance of bodh conceptual understanding and procedural fluency, as well as an effective organization of knowledge�in This case one that facilitates strategy development and adaptive reasoning. Read the entire page →
From page 44... ... Teachers can help students move at least to intemmediate "good-enough" methods that can be understood and explained. Box 5-4 describes such a learning path for single-digit addition and subtraction that is seen worldwide. Read the entire page →
From page 45... ... Teachers in some countries do tills while also facilitating problem solving with alternative n~etllods. Overall, knowing about student learning padhs and knowledge networks helps teachers direct madh talk along productive lines toward valued knowledge networks Research in madhematics learning has uncovered important information on a number of typical learning padhs and knowledge networks involved in acquiring knowledge about a variety of concepts in madhematics (see the next three chapters for examples) Read the entire page →
From page 46... ... At this level, students can chunk The approaches in the three chapters that follow identify the central conceptual structures in several areas of mathematics. The areas of focus whole number, rational number, and functions were identified by Case and his colleagues as requiring major conceptual shifts. Read the entire page →
From page 47... ... In each area of focus, instructional approaches were developed that enable teachers to help children move Trough learning padhs in productive ways. In doing so, teachers often find That They also build a more extensive knowledge network. Read the entire page →
From page 48... ... 236 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM BOXS-S Accessible Algorithms I In over a decade of working with a range of urban and suburban classrooms in the Children's Math Worlds Project, we found that one multidigit addition method and one multidigit subtraction method were accessible to all students. The students easily learned, understood, and remembered these methods and learned to draw quantities for and explain them. Read the entire page →
From page 49... ... The final level is "math-talk." Here students share responsibility for discourse with the teacher, justifying their own ideas and asking questions of and helping other students. Key shifts in teacher practice that support a class moving through these levels include asking questions that focus on mathematical thinking rather than just on answers, probing extensively for student thinking, modeling and expanding on explanations when necessary, fading physically from the center of the classroom discourse (e.g., moving to the back of the classroom) Read the entire page →
From page 50... ... On the other hand, helping students experience their own abilities to find patterns and problems, invent solutions (even if they are not quite as good as expert solutions) , and contribute to and learn from discussions with others provides the kinds of experiences that can help them learn with understanding, as well as change their views about the subject matter and themselves.'9 However, research on metacognition suggests that an additional instructional step is needed for optimal learning�one that involves helping students reflect on their experiences and begin to see their ideas as instances of larger categories of ideas. Read the entire page →
From page 51... ... They also can offer teachers windows into students' thinking and thus provide information about how better to help students along a learning path to efficient problem-solving methods. An Emphasis on Debugging Metacognitive functioning is also facilitated by shifting from a focus on answers as just right or wrong to a more detailed focus on "debugging" a wrong answer, that is, finding where the error is, why it is an error, and correcting it. Read the entire page →
From page 52... ... For example, it will be difficult for students to develop a good conceptual understanding for functions and the ways in which their representations are interconnected because the graph of y = 2x + I is a straight line rather than the parabolic curve of y = xz + 1. He does know, however, how to make a table of values and to graph resulting pairs of values. Read the entire page →
From page 53... ... The vignette in Box 5-3 illustrates such communication about mathematical thinking after it has been developed in a classroom. Expenence in the Children's Math Worlds Project indicates that students from all backgrounds can learn to think critically and ask thoughtful questions, reflect on and evaluate their own achievement, justify their points of view, and understand the perspectives of others Even first-grade students can learn to interact in these ways. Read the entire page →
From page 54... ... The instruction described is learner-centered in that it draws out and builds on student thinking. It is also knowledgecentered in that it focuses simultaneously on the conceptual understanding and the procedural knowledge of a topic, which students must master to be proficient, and the learning paths that can lead from existing to more advanced understanding. Read the entire page →
From page 55... ... Two such teacher communities, involving video clubs and lesson study, respectively, are sur manzed in Boxes 5-7 and 5-8. A third approach to a teacher learning community is to organize teacher discussions around issues that anse from teaching a curriculum that supports conceptual approaches. Read the entire page →
From page 56... ... NOTE: Resources, including a handbook, videotapes, listserve, and protocols for teachers who wish to engage in lesson study, can be found at the websites of the Lesson Study Research Group at Teachers College, Columbia University: (http:// www.tc.columbia.edu/lessonstudyp and the Mills College Lesson Study Group (www.lessonresearch.net) Read the entire page →
From page 57... ... taking responsive action that is open to ongoing examination and adjustment. They suggest that teaching with curriculum guides can be improved as teachers recognize and embrace their role while navigating openings in the curriculum to determine learning paths for students. Read the entire page →
From page 58... ... Ceci and Liker, 1986; Ceci, 1996. National Council of Teachers of Mathematics, 2000. Read the entire page →
From page 59... ... . The acquisition of addition and subt action concepts in grades one through three. Read the entire page →
From page 60... ... . Multidigit addition and subtraction methods invented in small groups and teacher support of problem solving and ref ection. Read the entire page →
From page 61... ... (1986a) Roles of representation and verbalization in the teaching of mu udigit addition and subt action. Read the entire page →
From page 62... ... . Children's conceptual st octanes for mu udigit numbe s and methods of mu udigit addition and subtraction. Read the entire page →
From page 63... ... . Conceptua/ a nd procedural knowledge Abe case of nzatbenzatics Mahwah, NJ: Lawrence Erlbaum Associates. Read the entire page →
From page 64... ... New York: Teachers College Press. Lampert, M Read the entire page →
From page 65... ... Reston, VA: National Council of Teachets of Mathematics. National Council of Teachets of Mathematics. Read the entire page →
From page 66... ... . Mahwah, NJ: Lawrence Erlbaum Associates. Read the entire page →
From page 67... ... . Mahwah, NJ: Lawrence Erlbaum Associates. Read the entire page →
From page 68... ... New York : Teachers College Press. Leinhardt, G., Putnam, R.T, and Hattrup, R.A. Read the entire page →

From page 29...

... Instead of connecting with, building on, and refining The madhematical understandings, intuitions, and resourcefulness dhat students 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 mathematics fluently around a set of core mathematical concepts (Pnnciple 2)