How Students Learn History, Mathematics, and Science in the Classroom (2005) / Chapter Skim
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Part II MATHEMATICS: 5 Mathematical Understanding: An Introduction
Part II MATHEMATICS: 5 Mathematical Understanding: An Introduction
Pages 215-256
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Part II MATHEMATICS
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From page 217... ...
Instead of connecting with, building on, and refining the mathematical understandings, intuitions, and resourcefulness that students 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.
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From page 218... ...
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.
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The first principle of How People Learn emphasizes both the need to build on existing knowledge and the need to engage students' preconceptions -- particularly when they interfere with learning. In mathematics, certain preconceptions that are often fostered early on in school settings are in fact counterproductive.
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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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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.
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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.
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Allowing Multiple Strategies To illustrate how instruction can be connected to students' existing knowledge, consider three subtraction methods encountered frequently in urban second-grade classrooms involved in the Children's Math Worlds Project (see Box 5-2)
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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.
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After several months of teaching and learning, the stu dents reached the point illustrated below. The students' methods are shown in Box 5-2.
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226 HOW STUDENTS LEARN: MATHEMATICS IN THECLASSROOM BOX 5-3 Continued 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.
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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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Students need an effective bridge between their developing understandings and formal mathematics. Teachers need to use carefully designed visual, linguistic, and situational conceptual supports to help students connect their experiences to formal mathematical words, notations, and methods.
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More-advanced students also helped less-advanced students learn by modeling, asking questions, and helping others form more complete descriptions. Initially in the Children's Math Worlds Project, all students made con ceptual support drawings such as those in Figure 5-1.
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PRINCIPLE #2: UNDERSTANDING REQUIRES FACTUAL KNOWLEDGE AND CONCEPTUAL FRAMEWORKS The second principle of How People Learn suggests the importance of both conceptual understanding and procedural fluency, as well as an effective organization of knowledge -- in this case one that facilitates strategy development and adaptive reasoning. It would be difficult to name a discipline in which the approach to achieving this goal is more hotly debated than mathematics.
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Developing Mathematical Proficiency Developing mathematical proficiency requires that students master both the concepts and procedural skills needed to reason and solve problems effectively in a particular domain. Deciding which advanced methods all students should learn to attain proficiency is a policy matter involving judg ments about how to use scarce instructional time.
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Overall, knowing about student learning paths and knowledge networks helps teachers direct math talk along productive lines toward valued knowledge networks. Research in mathematics learning has uncovered important information on a number of typical learning paths and knowledge networks involved in acquiring knowledge about a variety of concepts in mathematics (see the next three chapters for examples)
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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.
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, plus the 4 in 14, so the answer is 6. These make-a-ten methods demonstrate the learning paths and network of knowledge required for advanced solution meth ods.
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. In the area of mathemat ics, as noted earlier, many people who take mathematics courses "learn" that "they are not mathematical." This is an unintended, highly unfortunate, con sequence of some approaches to teaching mathematics.
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Key shifts in teacher practice that support a class moving through these lev els 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) , and coaching students in their participatory roles in the discourse ("Everyone have a thinker question ready.")
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, 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.19 However, research on metacognition suggests that an additional instruc tional step is needed for optimal learning -- one that involves helping stu dents reflect on their experiences and begin to see their ideas as instances of larger categories of ideas. For example, students might begin to see their way of showing more ones when subtracting as one of several ways to demonstrate this same important mathematical idea.
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Drawings and questions are a means of self-monitoring. 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.
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For example, it will be difficult for students to develop a good conceptual understanding for functions and the ways in which their representations are inter connected because the graph of y = 2x + 1 is a straight line rather than the parabolic curve of y = x2 + 1. He does know, however, how to make a table of values and to graph resulting pairs of values.
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Of course, teachers must help students learn to interact fruitfully. To this end, teachers can model clear descriptions and supportive questioning or helping techniques.
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The instruction described is learner-centered in that it draws out and builds on student thinking. It is also knowledge centered 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 ad vanced understanding.
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This is particularly true if they can examine their own teaching practices, supported by a teachinglearning community of likeminded colleagues. Such a community can help teachers create learning paths for themselves that can move them from their present teaching practices to practices that conform more fully to the principles of How People Learn and thereby create more effective classrooms.
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BOX 5-8 Lesson Study: Learning Together How to Build on Student Knowledge Lesson study is "a cycle in which teachers work together to consider their long term goals for students, bring those goals to life in actual `research lessons,' and collaboratively observe, discuss, and refine the lessons."21 Lesson study has been a major form of teacher professional development in Japan for many decades, and in recent years has attracted the attention of U.S. teachers, school administrators, and educational researchers.22 It is a simple idea.
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They suggest that teaching with curriculum guides can be improved as teachers recognize and embrace their role while navi gating openings in the curriculum to determine learning paths for students. Similarly, Remillard25 found that teachers came to reflect on their beliefs and understandings related to their teaching and its content while involved in the very work of deciding what to do next by interpreting students' understanding with respect to their goals for the students and particular instructional tasks.
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Kenney (Eds.) , The teaching and learning of algorithms in school mathematics.
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. A longitudinal study of invention and understanding in children's multidigit addition and sub traction.
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. Learning mathematics: The cognitive science approach to math ematics education.
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. San Jose, CA: The Center for Math ematics and Computer Science Education, San Jose State University.
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. Supporting multiple 2-digit conceptual structures and calculation methods in the classroom: Issues of conceptual supports, in structional design, and language.
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. Social class and racial influences on early mathematical thinking.
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. A research companion to principles and standards for school mathematics.
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science and math ematics education: Where we stand, where we want to go. Journal of Science Education and Technology, 7(1)
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Kenney (Eds.) , The teaching and learning of algorithms in school mathematics (pp.
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Warfield (Eds.) , Beyond classical pedagogy: Teaching elementary school mathematics (pp.
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. Research ideas for the classroom: High school mathematics.
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