J Kilpatrick, J. Swafford, and B. Findell (Eds.). Many people in the United States consider procedural fluency to be the heart of the elementary school mathematics curriculum. New York: Columbia University Press. Computing 3 Applying 5 Big Ideas in Beginning Reading 1. Carpenter, Franke, Jacobs, Fennema, and Empson, 1998. 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. Click here to buy this book in print or download it as a free PDF, if available. All rights reserved. Adding It Up (National Research Council, 2001), an influential report on how students learn mathematics describes five strands involved in being mathematically proficient: (1) conceptual . Stereotype threat and the intellectual test performance of African-Americans. The connection might be made explicit as follows: Let each level in the stack of blocks denote a particular topping (e.g., 1, catsup; 2, onions; 3, pickles; 4, lettuce; 5, tomato) and let the color signify whether the topping is to be included (e.g., green, include; red, exclude). Maher, C.A., & Martino, A.M. (1996). Examples from each strand illustrate the current situation.54. Classroom Data Analysis with the Five Strands of Mathematical Proficiency. NAEP findings regarding race/ethnicity: Students performance, school experiences, and attitudes and beliefs. English, 1997a, p. 4. Students need enough time to engage in activities around a specific mathematical topic if they are to become proficient with it. Research findings on what children know about numbers by the time they arrive in pre-K and the implications for mathematics instruction. Flexibility of approach is the major cognitive requirement for solving nonroutine problems. In terms of the five strands, the two that are most closely related to mathematical practices are strategic competence and adaptive reasoning. - proficiency in mathematics - mathematical processes - computation, algorithms and the use of digital tools in mathematics - protocols for engaging First Nations Australians - m eeting the needs of diverse learners ; Key connections new section addressing It is particularly important that students represent the quantities mentally, distinguishing what is known from what is to be found. the knowledge of procedures, and the knowledge of when and how to use them appropriately. Steele, 1997; and Steele and Aronson, 1995, show the effect of stereotype threat in regard to subsets of the GRE (Graduate Record Examination) verbal exam, and it seems this phenomenon may carry across disciplines. Without sufficient procedural fluency, students have trouble deepening their understanding of mathematical ideas or solving mathematics problems. Shannon, A. Behavioral and Social Sciences and Education. Alexander, White, and Daugherty, 1997, p. 122. If necessary, however, the cluster can be unpacked if the student needs to explain a principle, wants to reflect on a concept, or is learning new ideas. The NAEP data reported on the five strands are drawn from chapters in Silver and Kenney, 2000. ED 332 054). 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. For example, as students build strategic competence in solving nonroutine problems, their attitudes and beliefs about themselves as mathematics learners become more positive. Brownell, W.A. Kouba, Carpenter, and Swafford, 1989, p. 83. Kilpatrick, Swafford and Findell (2001) define mathematical proficiency as having five intertwining strands: conceptual understandingan understanding of concepts, operations and relations. Making sense: Teaching and learning mathematics with understanding. J Kilpatrick, J. Swafford, and B. Findell (Eds.). Available: http://books.nap.edu/catalog/6160.html. Journal for Research in Mathematics Education, 28, 652679. Registration confirmation will be emailed to you. p. 116). For views about learning in general, see Bransford, Brown, and Cocking, 1999; Donovan, Bransford, and Pellegrino, 1999. This practice leads to a compartmentalization of procedures that can become quite extreme, so that students believe that even slightly different problems require different procedures. It is not sufficient to justify a procedure just once. In D.Grouws (Ed. By renaming the fractions so that they have the same denominator, the students might arrive at a common measure for the fractions, determine the sum, and see its magnitude on the number line. Academic challenge in high-poverty classrooms. SOURCE: 1996 NAEP assessment. Much more important is the drop in U.S. educational standards and outcomes. Nunes, T. (1992a). Other views of mathematics learning have tended to emphasize How the strands of mathematical proficiency interweave and support one another can be seen in the case of conceptual understanding and procedural fluency. As students learn to execute a multidigit multiplication procedure such as this one, they should develop a deeper understanding of multiplication and its properties. Terms in this set (5) conceptual understanding. 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. (1989). For example. Goals for mathematics instruction like those outlined in our discussion of mathematical proficiency need to be set in full recognition of the differential access students have to high-quality mathematics teaching and the differential performance they show. On the other hand, as they deepen their conceptual understanding, they should become more fluent in computation. It is not as critical as it once was, for example, that students develop speed or efficiency in calculating with large numbers by hand, and there appears to be little value in drilling students to achieve such a goal. Yet our various backgrounds have led us to formulate, in a way that we hope others can and will accept, the goals toward which mathematics learning should be aimed. Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency to capture what we believe is necessary for anyone to learn mathematics successfully. 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 high-quality mathematics teaching.71, One concern has been that too few girls, relative to boys, are developing mathematical proficiency and continuing their study of mathematics. is defined as Mayer, R.E., & Wittrock, M.C. Number and operations. Those goals should never be set low, however, in the mistaken belief that some students do not need or cannot achieve proficiency. There is a precedent for this term: Students come to think of themselves as capable of engaging in independent thinking and of exercising control over their learning process [contributing] to what can best be called the disposition to higher order. Pose purposeful questions. Strategic Competence. Model with mathematics. A significant indicator of conceptual understanding is being able to represent mathematical situations in different ways and knowing how different representations can be useful for different purposes. 53187). This could allow us to address unequal acquisition of mathematical proficiency in school. All reality is composed of atoms in a void. Druckman, D., & Bjork, R.A. Steele, C.M. For example, finding the product of 567 and 46 is a routine problem for most adults because they know what to do and how to do it. (1987). 193 224). Students need to see that procedures can be developed that will solve entire classes of problems, not just individual problems. ), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. They need to be able to learn new concepts and skills. Hagarty, M., Mayer, R.E., & Monk, C.A. (1995). New York: Columbia University, Teachers College, Bureau of Publications. This separation limits childrens ability to apply what they learn in school to solve real problems. Use appropriate tools strategically. This. As they go from pre-kindergarten to eighth grade, all students should become increasingly proficient in mathematics. New York: Macmillan. Upper Saddle River, NJ: Prentice Hall . Journal of Educational Psychology, 87, 1832. Some of the most important consequences of students failure to develop a productive disposition toward mathematics occur in high school, when they have the opportunity to avoid challenging mathematics courses. Dweck, C. (1986). Students develop procedural fluency as they use their strategic competence to choose among effective procedures. Hence, our view of mathematical proficiency goes beyond being able to understand, compute, solve, and reason. Fifth graders solving problems about getting from home to school might describe verbally the route they take or draw a scale map of the neighborhood. Less successful problem solvers tend to focus on specific numbers and keywords such as $1.13, 5 cents, less, and 5 gallons rather than the relationships among the quantities.25. It also refers to knowing when and how to use. Bransford, Brown, and Cocking, 1999, pp. Strategic competence - ability to formulate, represent, and solve . These strands are not independent; they represent different aspects of a complex whole. McLeod, D.B. Washington, DC: National Center for Education Statistics. Mathematics Learning Study Committee, Center for Education, Division of If students do understand, they are less likely to forget critical steps and are more likely to be able to reconstruct them when they do. Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Analyses of students eye fixations reveal that successful solvers of the two-step problem above are likely to focus on terms such as ARCO, Chevron, and this, the principal known and unknown quantities in the problem. Chicago: University of Chicago Press. Fennema, E., & Romberg, T.A. 5. Only 1% of eighth graders in 1996 provided a satisfactory response for both claims, and only another 21% provided a partially correct response. Adaptive reasoning interacts with the other strands of proficiency, particularly during problem solving. Instruction, understanding, and skill in multidigit addition and subtraction. (1992). Carpenter, Corbitt, Kepner, Lindquist, and Reys, 1981. Mathematical proficiency is not a one-dimensional trait, and it cannot be achieved by focusing on just one or two of these strands. Washington, DC: National Center for Education Statistics. Rittle-Johnson, B., & Siegler, R.S. See, for example, Stevenson and Stigler, 1992. To represent a problem accurately, students must first understand the situation, including its key features. are 5 WIDA standards (Social/instructional, Language Arts, Math, Science, and Social Studies). In general, U.S. boys have more positive attitudes toward mathematics than U.S. girls do, even though differences in achievement between boys and girls are, in general, not as pronounced today as they were some decades ago.64 Girls attitudes toward mathematics also decline more sharply through the grades than those of boys.65 Differences in mathematics achievement remain larger across groups that differ in such factors as race, ethnicity, and social class, but differences in attitudes toward mathematics across these groups are not clearly associated with achievement differences.66, The complex relationship between attitudes and achievement is well illustrated in recent international studies. Available: http://books.nap.edu/catalog/1580.html. American Psychologist, 52, 613629. ), Handbook of educational psychology (pp. So there are 368=28 bikes. Cited in Wearne and Kouba, 2000, p. 186. Reasoning about operations: Early algebraic thinking in grades K-6. The continuing failure of some groups to master mathematicsincluding disproportionate numbers of minorities and poor studentshas served to confirm that assumption. /* ]]> */, The Five Strands of Mathematics Proficiency, http://books.nap.edu/catalog.php?record_id=10434, Promoting Social Justice and Environmental Justice, Developing Myself as an Antiracist Math Educator -Promoting Social Justice, Reflecting on Culturally Sustaining Pedagogy, Math Modeling at the Core of Equitable Teaching, Learning More about Math Modeling as a lever for Social Justice, Learning how to teach synchronously online- My PD. ), Assessment in transition: Monitoring the nations educational progress (Background Studies, pp. to the integrated and functional grasp of mathematical ideas, which Students conceptual understanding of number can be assessed in part by asking them about properties of the number systems. Arithmetic Teacher, 34(8), 1825. ), Results from the fourth mathematics assessment of the National Assessment of Educational Progress (pp. . Consider, for instance, the multiplication of multidigit whole numbers. Learning and teaching with understanding. In R. Glaser (Ed. 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 problem-solving operations and techniques, and aids to learning and transfer.31, Some researchers have concluded that childrens 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 representation-building experiences, children can demonstrate sophisticated reasoning abilities. As we indicated earlier and as the preceding discussion illustrates, the five strands are interconnected and must work together if students are to learn successfully. 2 THE STATE OF SCHOOL MATHEMATICS IN THE UNITED STATES, 4 THE STRANDS OF MATHEMATICAL PROFICIENCY, 6 DEVELOPING PROFICIENCY WITH WHOLE NUMBERS, 7 DEVELOPING PROFICIENCY WITH OTHER NUMBERS, 8 DEVELOPING MATHEMATICAL PROFICIENCY BEYOND NUMBER, 10 DEVELOPING PROFICIENCY IN TEACHING MATHEMATICS. Saxe, G. (1990). (Eds.). Schoenfeld, A.H. (1992). To search the entire text of this book, type in your search term here and press Enter. Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. Carpenter, T.P., & Levi, L. (1999, April). Over the same period, African American and Hispanic students recorded increases at grades 4 and 12, but not at grade 8.74 Scores for African American, Hispanic, and American Indian students remained below scale scores for white students. Such reasoning is correct and valid, stems from careful consideration of alternatives, and includes knowledge of how to justify the conclusions. PROCEDURAL FLUENCY. In addition to providing tools for computing, some algorithms are important as concepts in their own right, which again illustrates the link between conceptual understanding and procedural fluency. Adding It Up: adaptive reasoning, strategic . Proficiency in mathematics is therefore an important foundation for further instruction in mathematics as well as for further education in fields that require mathematical competence. http://books.nap.edu/catalog.php?record_id=10434, http://books.nap.edu/catalog.php?record_id=9822. How children change their minds: Strategy change can be gradual or abrupt. 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. (1994). Adding it up: Helping children learn mathematics. The mathematics achievement gaps between average scores for these subgroups did not decrease in 1996.75 The gap appears to be widening for African American students, particularly among students of the best-educated parents, which suggests that the problem is not one solely of poverty and disadvantage.76, Students identified as being of middle and high socioeconomic status (SES) enter school with higher achievement levels in mathematics than low-SES students, and students reporting higher levels of parental education tend to have higher average scores on NAEP assessments. Mathematics, gender and research. Terms in this set (6) Conceptual Understanding. Then the keywords how much and 5 gallons suggest that 5 should be multiplied by the result, yielding $5.40. Washington, DC: Author. These findings indicate that teacher educators should be aware of Senior High School students across different strands' attitudes and seek to improve them in order to positively influence students' proficiency in mathematics. (1998). Pesek, D.D., & Kirshner, D. (2000). Analogical reasoning and early mathematics learning. Math Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Plya, 1945, defined such problems as follows: In general, a problem is called a routine problem if it can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some well-worn conspicuous example (p. 171). By studying algorithms as general procedures, students can gain insight into the fact that mathematics is well structured (highly organized, filled with patterns, predictable) and that a carefully developed procedure can be a powerful tool for completing routine tasks. National Assessment Governing Board. Rational numbers. Washington, DC: National Academy Press. In D.A.Grouws (Ed. Reston, VA: National Council of Teachers of Mathematics. Do you enjoy reading reports from the Academies online for free? ; J.Teller, Trans.). those procedures. During the twentieth century, the meaning of successful mathematics learning underwent several shifts in response to changes in both society and schooling. In L.D.English (Ed. (1999). The attention they devote to working out results they should recall or compute easily prevents them from seeing important relationships. The five strands provide a framework for discussing the knowledge, skills, abilities, and beliefs . 8292). A compact procedure involves applying a written numerical algorithm that carries out the same steps without the bundles of sticks. procedural fluency. Take away 5 of the bundles (corresponding to subtracting 50), and take away 9 individual sticks (corresponding to subtracting 9). Ansell, E., & Doerr, H.M. (2000). TIMSS 1999 international mathematics report: Findings from IEAs repeat of the Third International Mathematics and Science Study at the eighth grade . Berdasarkan hasil penelitian di atas terlihat bahwa mathematical proficiency dapat dikembangkan dalam diri siswa. The Five Strands of Mathematics back to Dr. Suh's home The Five Strands of Mathematics Proficiency DEVELOPING MATHEMATICIANS Click on each strand for classroom structures that promote this strand: Elicit and use evidence of student thinking. [July 10, 2001]. Atoms come, For Plato, the Forms are the __________ foundation of reality, which means that knowledge of reality is grounded in knowledge of the Forms. (NRC, 2001, p. 116), is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.Mental gymnastics- Flexibility with numbers, is the ability to formulate, represent, and solve mathematical problems. Students need well-timed practice of the skills they are learning so that they are not handicapped in developing the other strands of proficiency. This frame-, Box 41 Intertwined Strands of Proficiency, work has some similarities with the one used in recent mathematics assessments by the National Assessment of Educational Progress (NAEP), which features three mathematical abilities (conceptual understanding, procedural knowledge, and problem solving) 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. Reston, VA, National Council of Teachers of Mathematics. (1997). 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. ), Handbook of research on mathematics teaching and learning (pp. (1) Conceptual understanding refers to the integrated and functional grasp of mathematical ideas , which enables them [students] to learn new ideas by connecting those ideas to what they already know. Such a scheme establishes a correspondence between the 22222=32 stacks of blocks and the 32 kinds of hamburgers. ), Handbook of research on mathematics teaching and learning (pp. following five factors. The authors of Principles and Standards for School Mathematics (NCTM, 2000)summarize it best 2: "Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.". Representing a problem situation requires, first, that the student build a mental image of its essential components. View our suggested citation for this chapter. 597622). New York: Macmillan. Developing conceptions of algebraic reasoning in the primary grades. Understanding makes learning skills easier, less susceptible to common errors, and less prone to forgetting. Students who view their mathematical ability as fixed and test questions as measuring their ability rather than providing opportunities to learn are likely to avoid challenging problems and be easily dis-, couraged by failure.42 Students who view ability as expandable in response to experience and training are more likely to seek out challenging situations and learn from them. 93 115). "What will ultimately determine the standard of living of this country is the skill . The reform movement of the 1980s and 1990s pushed the emphasis toward what was called the development of mathematical power, which involved reasoning, solving problems, connecting mathematical ideas, and communicating mathematics to others. A longitudinal study of invention and understanding in childrens multidigit addition and subtraction. National Assessment Governing Board, 2000. (2000). Because young children tend to learn the doubles fairly early, they can use them to produce closely related sums.10 For example, they may see that 6+7 is just one more than 6+6. For a more general discussion of classroom norms, see Cobb and Bauersfeld, 1995; and Fennema and Romberg, 1999. Wearne, D., & Hiebert, J. Procedural fluency and conceptual understanding are often seen as competing for attention in school mathematics. Now that we have looked at each strand separately, let us consider mathematical proficiency as a whole. ability to formulate, represent, and solve mathematical problems. They know that 108 is only 80, so multiplying two numbers less than 10 and 8 must give a product less than 80. Proofs (both formal and informal) must be logically complete, but a justification may be more telegraphic, merely suggesting the source of the reasoning. Beaton, A.E., Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Kelly, D.L., & Smith, T.A. 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The proficiency strands describe the . Davis, R.B., & Maher, C.A. The solution to this system of equations also yields 28 bikes and 8 tricycles. this error.15 Further, when students learn a procedure without understanding, they need extensive practice so as not to forget the steps. This variation allows students to discuss the similarities and differences of the representations, the advantages of each, and how they must be connected if they are to yield the same answer. Cobb, P., Yackel, E., & Wood, T. (1995). This frame- Page 117 Suggested Citation: "4 THE STRANDS OF MATHEMATICAL PROFICIENCY." Relevant findings from NAEP can be found in Silver, Strutchens, and Zawojewski, 1997; and Strutchens and Silver, 2000. For example, even seemingly simple concepts such as even and odd require an integration of several ways of thinking: choosing alternate points on the number line, grouping items by twos, grouping items into two groups, and looking at only the last digit of the number. Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Gregory, K.D., Garden, R.A., OConnor, K. M., Chrostowski, S.J., & Smith, T.A. Connecting students to a changing world: A technology strategy for improving mathematics and science education: A statement. 7475; Hiebert and Wearne, 1996. Only 61% of 13-year-olds chose the right answer, which again is considerably lower than the percentage of students who can actually compute the result. In a position page on procedural fluency, the National Council of Teachers of Mathematics ( NCTM . sary nor efficient. There is reason to believe that the conditions apply more generally. ), The teaching of arithmetic (Tenth Yearbook of the National Council of Teachers of Mathematics, pp. It is an intertwining combination of the. Mathematics Learning Study Committee, J Kilpatrick and J. Swafford, Editors. Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently. These methods include, in addition to written procedures, mental methods for finding certain sums, differences, products, or quotients, as well as methods that use calculators, computers, or manipulative materials such as blocks, counters, or beads. Phi Delta Kappan, 76, 770776. Still others might ask how the layout of the cafeteria might be improved. NAEP findings regarding gender: Achievement, affect, and instructional experiences. [July 10, 2001]. In L.V.Stiff (Ed. (1) Conceptual Understanding (Understanding): Comprehending mathematical concepts, operations, and relations knowing what mathematical symbols, diagrams, and procedures mean. 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 applies equally to all domains of mathematics. 117148). How children discover new strategies. 103-109. [July 10, 2001]. 5. Basic math facts: Guidelines for teaching and learning. Donovan, M.S., Bransford, J.D., & Pellegrino, J.W. NAEP 1996 mathematics report card for the nation and the states. Thus, learning with understanding is more powerful than simply memorizing because the organization improves retention, promotes fluency, and facilitates learning related material. 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