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From page 115... ...
These various emphases have reflected different goals for school mathematics held by different groups of people at different times. Our analyses of the mathematics to be learned, our reading of the research in cognitive psychology and mathematics education, our experience as learners and teachers of mathematics, and our judgment as to the mathematical knowledge, understanding, and skill people need today have led us to adopt a

From page 116... ...
conceptual understanding comprehension of mathematical concepts, operations, and relations · proceduralfluency skill in carrying out procedures flexibly, accurately, efficiently, and appropriately · strategic competence ability to formulate, represent, and solve mathematical problems adaptive reasoning capacity for logical thought, reflection, explana.

From page 117... ...
and includes additional specifications for reasoning, connections, and communication.2 The strands also echo components of mathematics learning that have been identified in materials for teachers. At the same time, research and theory in cognitive science provide general support for the ideas contributing to these five strands.

From page 118... ...
. · ~ proc uct1ve c 1spos1t1on.~ Although there is not a perfect fit between the strands of mathematical proficiency and the kinds of knowledge and processes identified by cognitive scientists, mathematics educators, and others investigating learning, we see the strands as reflecting a firm, sizable body of scholarly literature both in and outside mathematics education.

From page 119... ...
Mnemonic techniques learned by rote may provide connections among ideas that make it easier to perform mathematical operations, but they also may not lead to understanding.7 These are not the kinds of connections that best promote the acquisition of mathematical proficiency. Knowledge that has been learned with understanding provides the basis for generating new knowledge and for solving new and unfamiliar problems.8 When students have acquired conceptual understanding in an area of mathematics, they see the connections among concepts and procedures and can give arguments to explain why some facts are consequences of others.

From page 120... ...
Often, the structure of students' understanding is hierarchical, with simpler clusters of ideas packed into larger, more complex ones. A good example of a knowledge cluster for mathematically proficient older students is the number line.

From page 121... ...
Such understanding also supports simplified but accurate mental arithmetic and more flexible ways of dealing with numbers than many students ultimately achieve. Connected with procedural fluency is knowledge of ways to estimate the result of a procedure.

From page 122... ...
For example, it is difficult for students to understand multidigit calculations if they have not attained some reasonable level of skill in singledigit calculations. On the other hand, once students have learned procedures without understanding, it can be difficult to get them to engage in activities to help them understand the reasons underlying the procedure.~3 In an experimental study, fifthgrade students who first received instruction on procedures for calculating area and perimeter followed by instruction on understanding those procedures did not perform as well as students who received instruction focused only on understanding.~4 Without sufficient procedural fluency, students have trouble deepening their understanding of mathematical ideas or solving mathematics problems.

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This separation limits children's ability to apply what they learn in school to solve real problems. Also, students who learn procedures without understanding can typically do no more than apply the learned procedures, whereas students who learn

From page 124... ...
Students with more understanding would recognize that 598 is only 2 less than 600, so they might add 600 and 647 and then subtract 2 from that sum.20 Strategic Competence Strategic Strategic competence refers to the ability to formulate mathematical probcompetence lems, represent them, and solve them. This strand is similar to what has abi~itYto been called problem solving and problem formulation in the literature of formulate mathematics education and cognitive science, and mathematical problem mathematical solving, in particular, has been studied extensively.

From page 125... ...
In a common superficial method for representing this problem, students focus on the numbers in the problem and use socalled keywords to cue appropriate arithmetic operations.24 For example, the quantities $1.83 and 5 cents are followed by the keyword less, suggesting that the student should subtract 5 cents from $1.13 to get $1.08. Then the keywords low much and S gallons suggest that 5 should be multiplied by the result, yielding $5.40.

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A fundamental characteristic needed throughout the problemsolving process is flexibility. Flexibility develops through the broadening of knowledge required for solving nonroutine problems rather than just routine problems.

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There are There are mutually supportive relations between strategic competence mutually and both conceptual understanding and procedural fluency, as the various supportive approaches to the cycle shop problem illustrate. The development of strata relations · r ~ · · '' ~ ~ ~ ~.

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1Z8 ADDING IT UP Box 43 Subtraction Using Sticks: Modeling Ha 59 = 7 1 1 1_______________ Remove 50 1~ PRRRRR Break apart a bundle Remove 9 27 remain 86=80+6 LLLL LLLL .

From page 129... ...
Our notion of adaptive reasoning is much broader, including not only informal explanation and justification but also intuitive and inductive reasoning based on pattern, analogy, and metaphor. As one researcher put it, "The human ability to find analogical correspondences is a powerful reasoning mechanism."30 Analogical reasoning, metaphors, and mental and physical representations are "tools to think with," often serving as sources of hypotheses, sources of problemsolving operations and techniques, and aids to learning and transfer.3~ Some researchers have concluded that children's reasoning ability is quite limited until they are about 12 years old.32 Yet when asked to talk about how they arrived at their solutions to problems, children as young as 4 and 5 display evidence of encoding and inference and are resistant to counter suggestion.33 With the help of representationbuilding experiences, children can demonstrate sophisticated reasoning abilities.

From page 130... ...
Learners draw on their strategic competence to formulate and represent a problem, using heuristic approaches that may provide a solution strategy, but adaptive reasoning must take over when

From page 131... ...
Productive Disposition Productive disposition refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics.40 If students are to develop conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning abilities, they must believe that mathematics is understandable, not arbitrary; that, with diligent effort, it can be learned and used; and that they are capable of figuring it out. Developing a productive disposition requires frequent opportunities to make sense of mathematics, to recognize the benefits of perseverance, and to experience the rewards of sense making in mathematics.

From page 132... ...
How a teacher views mathematics and its learning affects that teacher's teaching practice,46 which ultimately affects not only what the students learn but how they view themselves as mathematics learners. Teachers and students inevitably negotiate among themselves the norms of conduct in the class, and when those norms allow students to be comfortable in doing mathematics and sharing their ideas with others, they see themselves as capable of understanding.47 In chapter 9 we discuss some of the ways in which teachers' expectations and the teaching strategies they use can help students maintain a positive attitude toward mathematics, and in chapter 10 we discuss some programs of teacher development that may help teachers in that endeavor.

From page 133... ...
Mathematically proficient people believe that mathematics should make sense, that they can figure it out, that they can solve mathematical problems by working hard on them, and that becoming mathematically proficient is worth the effort. Properties of Mathematical Proficiency Now that we have looked at each strand separately, let us consider mathematical proficiency as a whole.

From page 134... ...
It requires knowing that 40 x 268 is 4 x 10 x 268; knowing that in the product of 268 and 10, each digit of 268 is one place to the left; having enough fluency with basic multiplication combinations to find 7 x8, 7 x 60, 7 x200, and 4 x8, 4x60, 4x 200; and having enough fluency with multidigit addition to add the partial products. As students learn to execute a multidigit multiplication procedure such as this one, they should develop a deeper understanding of multiplication and its properties.

From page 135... ...
It is still reasonable, however, to talk about a first grader as being proficient with singledigit addition, as long as the student's thinking in that realm incorporates all five strands of proficiency. Students should not be thought of as having proficiency when one or more strands are undeveloped.

From page 136... ...
In general, the performance of 13yearolds over the past 25 years tells the following story: Given traditional curricula and methods of instruction, students develop proficiency among the five strands in a very uneven way. They are most proficient in aspects of procedural fluency and less proficient in conceptual understanding, strategic competence, adaptive reasoning, and productive disposition.

From page 137... ...
Procedural Fluency An overall picture of procedural fluency is provided by the NAEP longterm trend mathematics assessment,58 which indicates that U.S. students' performance has remained quite steady over the past 25 years (see Box 44~.

From page 138... ...
However, students, especially those in the fourth and eighth grades, had difficulty with more complex problemsolving situations. For example, asked to add or subtract two and threedigit numbers, 73% of fourth graders and 86% of eighth graders gave correct answers.

From page 139... ...
This level of performance is especially striking because this kind of reasoning does not require procedural fluency plus additional proficiency. In many ways it is less demanding than the computational task and requires only that basic understanding and reasoning be connected.

From page 140... ...
Although within most countries, positive attitudes toward mathematics are associated with high achievement, eighth graders in some East Asian countries, whose average achievement in mathematics is among the highest in the world, have tended to have, on average, among the most negative attitudes toward mathematics.

From page 141... ...
Although students appear to think mathematics is useful for everyday problems or important to society in general, it is not clear that they think it is important for them as individuals to know a lot of mathematics.69 Proficiency in Other Domains of Mathematics Although our discussion of mathematical proficiency in this report is focused on the domain of number, the five strands apply equally well to other domains of mathematics such as geometry, measurement, probability, and statistics. Regardless of the domain of mathematics, conceptual understanding refers to an integrated and functional grasp of the mathematical ideas.

From page 142... ...
The continuing failure of some groups to master mathematics including disproportionate numbers of minorities and poor students has served to confirm that assumption. More recently, mathematics educators have highlighted the universal aspects of mathematics and have insisted on mathematics for all students, but with little attention to the differential access that some students have to highquality mathematics teaching.71 One concern has been that too few girls, relative to boys, are developing mathematical proficiency and continuing their study of mathematics.

From page 143... ...
The racial/ethnic diversity of the United States is much greater now than at any previous period in history and promises to become progressively more so for some time to come. The strong connection between economic advantage, school funding, and achievement in the United States has meant that groups of students whose mathematics achievement is low have tended to be disproportionately African American, Hispanic, Native American, students acquiring English, or students located in urban or rural school districts.73 In the NAEP assessments from 1990 to 1996, white students recorded increases in their average mathematics scores at all grades.

From page 144... ...
We also raise the standard for success in learning mathematics and being able to use it. In a significant and fortuitous twist, raising the standard by requiring development across all five strands of mathematical proficiency makes the development of any one strand more feasible.

From page 145... ...
It has developed some procedural fluency, but it clearly has not helped students develop the other strands very far, nor has it helped them connect the strands. Consequently, all strands have suffered.

From page 146... ...
18. Researchers have shown clear disconnections between students' "street mathematics" and school mathematics, implying that skills learned without understanding are learned as isolated bits of knowledge.

From page 147... ...
54. The NAEP data reported on the five strands are drawn from chapters in Silver and Kenney, 2000.

From page 148... ...
. NAEP findings regarding gender: Achievement, affect, and instructional experiences.

From page 149... ...
(1996~. Mathematics achievement in the middle school years: IEA's Third International Mathematics and Science Study.

From page 150... ...
Journal for Research in Mathematics Education 21, 180206.

From page 151... ...
(1993~. Group case studies of second graders inventing multidigit addition procedures for baseten blocks and written marks.

From page 152... ...
, Res? 'ltsfrom the seventh mathematics assessment of the National Assessment of Educational Progress (pp.

From page 153... ...
, The teaching and assessing of mathematicalproblem solving (Research Agenda for Mathematics Education, vol.3, pp.8292~. Reston, VA: National Council of Teachers of Mathematics.

From page 154... ...
, Results from the sixth mathematics assessment of the National Assessment of Educational Progress (pp.

From page 155... ...
, Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp.

From page 156... ...
Chapter 4: The Strands of Mathematical Proficiency 442

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