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. This is a table of all the Australian Curriculum Mathematics Proficiency strands, formatted according to . J Kilpatrick, J. Swafford, and B. Findell (Eds.). What are the terms, symbols, operations, principles to be understood? Further, the strands are interwoven across domains of mathematics in such a way that conceptual understanding in one domain, say geometry, supports conceptual understanding in another, say number. Journal for Research in Mathematics Education 21, 180206. Answers are right because they follow from some agreed-upon assumptions through series of logical steps. 90, No. SOURCE: Campbell, Hombo, and Mazzeo, 2000, p. 9. 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. 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. http://books.nap.edu/catalog.php?record_id=10434, http://books.nap.edu/catalog.php?record_id=9822, The most important feature of mathematical proficiency is that these five strands are interwoven and interdependent.(page 9, Helping ChildrenLearn Mathematics, NRC, 2002), To access this content please subscribe. Alphabetic Principle 3 Fluency with Text www.interventioncentral.org 11 Source: National . So there are 368=28 bikes. Many conceptions of mathematical reasoning have been confined to formal proof and other forms of deductive reasoning. Switch between the Original Pages, where you can read the report as it appeared in print, and Text Pages for the web version, where you can highlight and search the text. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know.4 Conceptual understanding also supports retention. 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. Results from the seventh mathematics assessment of the National Assessment of Educational Progress. . ), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. Teaching and learning mathematics with understanding. Upper Saddle River, NJ: Prentice Hall . Chicago: University of Chicago Press. ), Handbook of research on mathematics teaching and learning (pp. 4572). Reese, C.M., Miller, K.E., Mazzeo, J., & Dossey, J.A. Wu, H. (1999, Fall). They need to be able to apply mathematical reasoning to problems. National Research Council. Reston, VA: National Council of Teachers of Mathematics. Analogies, metaphors, and images: Vehicles for mathematical reasoning. Beaton, A.E., Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Kelly, D.L., & Smith, T.A. Furthermore, cognitive science studies of problem solving have documented the importance of adaptive expertise and of what is called metacognition: knowledge about ones own thinking and ability to monitor ones own understanding and problem-solving activity. (1995). The degree of students conceptual understanding is related to the richness and extent of the connections they have made. High school mathematics at work: Essays and Examples for the education of all students. (2) Procedural fluency Washington, DC: National National Research Council. Click here to buy this book in print or download it as a free PDF, if available. On the other hand, approximately 75% of the fourth graders and 75% of the eighth graders sampled reported that they understand most of what goes on in mathematics class. Alibali, M.W. It should be emphasized that, as discussed above, conceptual understanding requires that knowledge be connected. We conclude that during the past 25 years mathematics instruction in U.S. schools has not sufficiently developed mathematical proficiency in the sense we have defined it. Education and learning to think. Also, you can type in a page number and press Enter to go directly to that page in the book. Teachers: The Five Mathematical Proficiencies 1,657 views Jun 6, 2019 24 Dislike Share Save Adolygu Mathemateg 3.37K subscribers A discussion of how to plan a lesson around the five new. (1999). (2017). Understanding 2. Mayer, R.E. One kind of item asks students to reason about numbers and their properties and also assesses their conceptual understanding. & Drago-Severson, E. (1999). For example, applying a standard pencil-and-paper algorithm to find the result of every multiplication problem is neither neces-. When using a procedure, a child may reflect on why the procedure works, which may in turn strengthen existing conceptual understanding.52 Indeed, it is not always necessary, useful, or even possible to distinguish concepts from procedures because understanding and doing are interconnected in such complex ways. Implications for the NAEP of research on learning and cognition. In general, the performance of 13-year-olds 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. Math Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. (1996). (2000). Part of developing strategic competence involves learning to replace by more concise and efficient procedures those cumbersome procedures that might at first have been helpful in understanding the operation. 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. Available: http://nces.ed.gov/spider/webspider/97985r.shtml. Proofs (both formal and informal) must be logically complete, but a justification may be more telegraphic, merely suggesting the source of the reasoning. Sternberg, R.J., & Rifkin, B. Their understanding has been encapsulated into compact clusters of interrelated facts and principles. Oakes, J. One such item (Box 45) required that students use subtraction and division to justify claims about the population growth in two towns. Available: http://www.timss.org/timss1995i/MathB.html. 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. Mathematics Proficiency. New York: Harper & Row. Not only do students need to be able to build representations of individual situations, but they also need to see that some representations share common mathematical structures. A recent synthesis by Rittle-Johnson and Siegler, 1998, on the relationship between conceptual and procedural knowledge in mathematics concludes that they are highly correlated and that the order of development depends upon the mathematical content and upon the students and their instructional experiences, particularly for multidigit arithmetic. 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. Available: http://books.nap.edu/catalog/9745.html. In D.Grouws (Ed. 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. Students should not be thought of as having proficiency when one or more strands are undeveloped. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies. Cobb, P., Yackel, E., & Wood, T. (1989). Cobb, P., & Bauersfeld, H. Mathematics and gender: Changing perspectives. New York: Macmillan. Developing conceptions of algebraic reasoning in the primary grades. 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. sary nor efficient. (1998). . Shannon, A. For example, suppose students are adding fractional quantities of different sizes, say They might draw a picture or use concrete materials of various kinds to show the addition. ), The nature of mathematical thinking (Studies in Mathematical Thinking and Learning Series, pp. Students develop procedural fluency as they use their strategic competence to choose among effective procedures. In J.Hiebert (Ed. Knapp, Shields, and Turnbull, 1995; Mason, Schroeter, Combs, and Washington, 1992; Steele, 1997. This emphasis was followed by a back to basics movement that proposed returning to the view that success in mathematics meant being able to compute accurately and quickly. Reston, VA: National Council of Teachers of Mathematics . ), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. The relation between conceptual and procedural knowledge in learning mathematics: A review. As we discuss below, the development of proficiency occurs over an extended period of time. 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). Chicago: University of Chicago Press. (1995). -Helping all students achieve math proficiency. 575596). NAEP findings regarding race/ethnicity: Students performance, school experiences, and attitudes and beliefs. Brownell, 1935; Carpenter, Franke, Jacobs, Fennema, and Empson, 1998; Hatano, 1988; Wearne and Hiebert, 1988; Mack, 1995; Rittle-Johnson and Alibali, 1999. (Original work published 1968). or use these buttons to go back to the previous chapter or skip to the next one. 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. Inhelder, B., & Piaget, J. Mack, N.K. 8292). For more infohttp://books.nap.edu/catalog.php?record_id=10434, 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. A few of the benefits of building conceptual understanding are that it supports retention, and prevents common errors. When skills are learned without understanding, they are learned as isolated bits of knowledge.18 Learning new topics then becomes harder since there is no network of previously learned concepts and skills to link a new topic to. Yaffee, L. (1999). These relations make it easier for students to learn the new addition combinations because they are generating new knowledge rather than relying on rote memorization. Chestnut Hill, MA: Boston College, Center for the Study of Testing, Evaluation, and Educational Policy. In NAEP, gender differences may have increased slightly at grade 4 in the past decade, although they are still quite small; see Ansell and Doerr, 2000. A longitudinal study of invention and understanding in childrens multidigit addition and subtraction. Hatano, G. (1988, Fall). Washington, DC: National Center for Education Statistics. Hillsdale, NJ: Erlbaum. Everybody counts: A report to the nation on the future of mathematics education. Becoming strategically competent involves an avoidance of number grabbing methods (in which the student selects numbers and prepares to perform arithmetic operations on them)23 in favor of methods that generate problem models (in which the student constructs a mental model of the variables and relations described in the problem). Lund, Sweden: Lund University Press. Relevant findings from NAEP can be found in Silver, Strutchens, and Zawojewski, 1997; and Strutchens and Silver, 2000. the elementary school mathematics curriculum (p 144). American Psychologist, 50(1), 2437. Theoretical Framework The purpose of this study is to determine the level of proficiency of the Grade 11 students of Negros Occidental High School in General Mathematics during the School Year 2020 - 2021 as basis for an instructional plan. Cobb, Yackel, and Wood, 1989, 1995. 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. See also Krutetskii, 1968/1976, ch. 2953). Rittle-Johnson, B., & Alibali, M.W. 7. Center for Education, Division of Behavioral and Social Sciences and Education. Committee for Economic Development, Research and Policy Committee. Reston, VA: National Council of Teachers of Mathematics. The attention they devote to working out results they should recall or compute easily prevents them from seeing important relationships. (2) Procedural Fluency (Computing): Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately. 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.13 In an experimental study, fifth-grade 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.14. Secada, W.G. In mathematics, adaptive reasoning is the glue that holds everything together, the lodestar that guides learning. Use mathematics to explain how Brian might have justified his claim. Hillsdale, NJ: Erlbaum. (1989). [July 10, 2001]. If any group of students is deprived of the opportunity to learn with understanding, they are condemned to second-class status in society, or worse. More expert problem solvers focus more on the structural relationships within problems, relationships that provide the clues for how problems might be solved.26 For example, one problem might ask students to determine how many different stacks of five blocks can be made using red and green blocks, and another might ask how many different ways hamburgers can be ordered with or without each of the following: catsup, onions, pickles, lettuce, and tomato. 3033; Hilgard, 1957; Katona, 1940; Mayer, 1999; Wertheimer, 1959. Basic math facts: Guidelines for teaching and learning. Journal for Research in Mathematics Education, 31, 524540. (ERIC Document Reproduction Service No. Journal of Educational Psychology, 91, 175 189. 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. Cognitive, scientists have concluded that competence in an area of inquiry depends upon knowledge that is not merely stored but represented mentally and organized (connected and structured) in ways that facilitate appropriate retrieval and application. In contrast, nonroutine problems are problems for which the learner does not immediately know a usable solution method. thinking. For work in psychology, see Baddeley, 1976; Bruner, 1960, pp. View our suggested citation for this chapter. to the integrated and functional grasp of mathematical ideas, which 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. As in chapter 2, the data reported here are from the 1996 main NAEP assessment except when we refer explicitly to the long-term trend assessment. Everything that exists is either an atom or a collection of atoms. Historically, the prevailing ethos in mathematics and mathematics education in the United States has been that mathematics is a discipline for a select group of learners. (Eds.). Similarly, when students see themselves as capable of learning mathematics and using it to solve problems, they become able to develop further their procedural fluency or their adaptive reasoning abilities. See, for example, Nunes, 1992a, 1992b; Saxe, 1990. Strategic Competence. Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. In principle, they need only check that their reasoning is valid. For each of the five levels in the stack of blocks, there are two options: red or green. 597622). (Eds.). Silver & P.A.Kenney (Eds. (5) Engaging: Seeing mathematics as sensible, useful, and doableif you work at itand being willing to do the work The most important feature of mathematical proficiency is that these five strands are interwoven and interdependent. In J.R.Becker & B.J.Pence (Eds. Mason, D., Schroeter, D., Combs, R., & Washington, K. (1992). (1992b). English, L.D. A beginner who has simply memorized the algorithm without understanding much about how it works can be lost later when memory fails. The Elements of Mathematical Proficiency: What the Experts Say www.interventioncentral.org Response to Intervention 5 Strands of Mathematical Proficiency 1. Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments. Transform the way you look at numbers by dissecting Algebraic expressions. For views about learning in general, see Bransford, Brown, and Cocking, 1999; Donovan, Bransford, and Pellegrino, 1999. As used here, it is more akin to a habit of thought, one that can be learned and, therefore, taught (Resnick, 1987, p. 41). 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. Journal for Research in Mathematics Education, 28, 652679. That development takes time. Greeno, J.G., Pearson, P.D., & Schoenfeld, A.H. (1997). East Sussex, UK: Psychology Press. Druckman, D., & Bjork, R.A. Assigning average achieving eighth graders to advanced mathematics classes in an urban junior high. By exploiting their knowledge of other relationships such as that between the doubles (e.g., 5+5 and 6+6) and other sums, they can reduce still further the number of addition combinations they need to learn. is defined as Math Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Schifter, D. (1999). So-called wise educational environments50 can reduce the harmful effects of stereotype threat. Adaptive reasoning interacts with the other strands of proficiency, particularly during problem solving. ), Mathematical reasoning: Analogies, metaphors, and images (pp. Levels in conceptualization and solving addition and subtraction compare word problems. (1986). All rights reserved. These differences were only partly explained by the historical tendency of male students to take more high school mathematics courses than female students do, since that gap had largely closed by 1992. Mathematics achievement in the middle school years: IEAs Third International Mathematics and Science Study. Mathematics Learning Study Committee, Center for Education, Division of Alexander, White, and Daugherty, 1997, p. 122. Fuson, K.C. Bransford, J.D., Brown, A.L., & Cocking, R.R. 6. Novices would see these problems as unrelated; experts would see both as involving five choices between two things: red and green, or with and without.27. The growth of logical thinking from childhood to adolescence. How a teacher views mathematics and its learning affects that teachers teaching practice,46 which ultimately affects not only what the students learn but how they view themselves as mathematics learners. Analogical reasoning and early mathematics learning. New York: Vintage Books. Selain itu, kecakapan matematis ini apabila dimiliki oleh siswa maka siswa. For example, for most adults a nonroutine problem of the sort often found in newspaper or magazine puzzle columns is the following: A cycle shop has a total of 36 bicycles and tricycles in stock. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions. 193 224). Explorations of students mathematical beliefs and behavior. For example. New York: Basic Books. Only 1% of eighth graders in 1996 provided a satisfactory response for both claims, and only another 21% provided a partially correct response. 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. The continuing failure of some groups to master mathematicsincluding disproportionate numbers of minorities and poor studentshas served to confirm that assumption. Stereotype threat and the intellectual test performance of African-Americans. (4) Adaptive reasoning is the Further, situations vary in their need for exact answers. The central notion that strands of competence must be interwoven to be useful reflects the finding that having a deep understanding requires that learners connect pieces of knowledge, and that connection in turn is a key factor in whether they can use what they know productively in solving problems. Fuson, 1990, 1992b; Fuson and Briars, 1990; Fuson and Burghardt, 1993; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997; Hiebert and Wearne, 1996; Resnick and Omanson, 1987. The five strands provide a framework for discussing the knowledge, skills, abilities, and beliefs that constitute mathematical proficiency. 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. (2001). ), Developing mathematical reasoning in grades K-12 (1999 Yearbook of the National Council of Teachers of Mathematics, pp. Effective teaching of mathematics uses purposeful questions to assess and advance students reasoning and sense making about important mathematical ideas and relationships. Reston, VA: National Council of Teachers of Mathematics. On the 23 problem-solving tasks given as part of the 1996 NAEP in which students had to construct an extended response, the incidence of satisfactory or better responses was less than 10% on about half of the tasks. NAEP 1996 trends in academic progress (NCES 97985r). (1957). Such a scheme establishes a correspondence between the 22222=32 stacks of blocks and the 32 kinds of hamburgers. Five strands of mathematical proficiency From NRC (2001) Adding it up: Helping children learn mathematics Conceptual understanding: comprehension of mathematical concepts, operations, and relations Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic competence: If students have been using incorrect procedures for several years, then instruction emphasizing understanding may be less effective.16 When children learn a new, correct procedure, they do not always drop the old one. It prompts teachers to examine the extent to which their students have attained. 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. Students need to be able to justify and explain ideas in order to make their reasoning clear, hone their reasoning skills, and improve their conceptual understanding.39. In contrast, a more proficient approach is to construct a problem model that is, a mental model of the situation described in the problem. LIKE FRACTIONS can be easily added or subtracted. Arithmetic Teacher, 34(8), 1825. 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 refers to the ability to formulate mathematical problems, represent them, and solve them. 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. For discussion of learning in early childhood, see Bowman, Donovan, and Burns, 2001. English, L.D. In D.Grouws (Ed. Representing a problem situation requires, first, that the student build a mental image of its essential components. Similarly, for each of the five toppings on the hamburger, there are two options: include the topping or exclude it. (NRC, 2001, In building a problem model, students need to be alert to the quantities in the problem. Washington, DC: National Center for Education Statistics. 137144. Conceptual understanding provides metaphors and representations that can serve as a source of adaptive reasoning, which, taking into account the limitations of the representations, learners use to determine whether a solution is justifiable and then to justify it. Effective schools in mathematics. Conceptual and procedural knowledge of mathematics: Does one lead to the other? (1981). Such research has focused on attitudes. (NRC, 2001, p. 116)(NRC, 2001, p. 116), Core Teaching Practices from the Principles to Action, NCTM (2014). procedures flexibly, accurately, efficiently, and. WA Kindergarten Curriculum [Mathematics] This is a free PDF of a forward planner you can use to do your planning. Fuson, K.C., & Burghardt, B.H. It can be seen when a method is created or adjusted to fit the requirements of a novel situation, such as being able to use general principles about proportions to determine the best buy. For example, if they are multiplying 9.83 and 7.65 and get 7519.95 for the answer, they can immediately decide that it cannot be right. In the domain of number, procedural fluency is especially needed to support conceptual understanding of place value and the meanings of rational numbers. 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. Elementary School Journal, 92, 587599. If students are failing to develop proficiency, the question of how to improve school mathematics takes on a different cast than if students are already developing high levels of proficiency. Elicit and use evidence of student thinking. As they go from pre-kindergarten to eighth grade, all students should become increasingly proficient in mathematics. In W.D.Reeve (Ed. Each year they are in school, students ought to become increasingly proficient. Mahwah, NJ: Erlbaum. Model with mathematics. Washington, DC: National Academy Press. How to solve it: A new aspect of mathematical method. Despite the finding that many students associate mathematics with memorization, students at all grade levels appear to view mathematics as useful. In E.A.Silver & P.A. inclination to see mathematics as sensible, useful, and worthwhile, ), Handbook of educational psychology (pp. attitudes towards mathematics and proficiency in mathematics. A Model Performance Indicator (MPI) addresses a specific content standard, within one of the 5 WIDA Standards and focuses on one of the four domains. The second are the strands of mathematical proficiency specified in the National Research Council's report . In becoming proficient problem solvers, students learn how to form mental representations of problems, detect mathematical relationships, and devise novel solution methods when needed. In P.Cobb & H.Bauersfeld (Eds. Wearne, D., & Kouba, V.L. (1995). [July 10, 2001]. In D.A.Grouws (Ed. In E.A.Silver & P.A.Kenney (Eds. Finally, learning is also influenced by motivation, a component of productive disposition.3. [July 10, 2001]. Rittle-Johnson, B., & Siegler, R.S. Every child can succeed: Reading for school improvement. The best source of information about student performance in the United States is, as we noted in chapter 2, the National Assessment of Educational Progress (NAEP), a regular assessment of students knowledge and skills in the school subjects. The Five Strands of (2000). all sub-strands in Mathematics removed . New York: Basic Books. Thus, learning how to add and subtract multidigit numbers does not have to involve entirely new and unrelated ideas. Social and motivational bases for mathematical understanding. When applied to other domains of mathematics, procedural fluency refers to skill in performing flexibly, accurately, and efficiently such procedures as constructing shapes, measuring space, computing probabilities, and describing data. They might turn to the number line, representing each fraction by a segment and adding the fractions by joining the segments. Highlights of related research. Nunes, T. (1992a). For example, on one standardized test, the grade 2 national norms for two-digit subtraction problems requiring borrowing, such as 6248=?, are 38% correct. In R.J.Sternberg & T.Ben-Zee (Eds. Data from the NAEP student questionnaire show that many U.S. students develop a variety of counterproductive beliefs about mathematics and about themselves as learners of mathematics. In the United States, in contrast, eighth graders tend to believe that mathematics is not especially difficult for them and that they are good at it.68. var wpcf7 = {"apiSettings":{"root":"http:\/\/drjennifersuh.onmason.com\/wp-json\/contact-form-7\/v1","namespace":"contact-form-7\/v1"},"cached":"1"}; This 'rope model' has informed the way we design NRICH tasks, and we often use it in professional development workshops with teachers, drawing attention to the importance of a balanced curriculum which develops all five strands of students' mathematical proficiency equally, rather than promoting some strands at the expense of others. See, for example, Stevenson and Stigler, 1992. If a runner jogs 3 miles west and then jogs 8 miles north, how far is the runner from her starting point if she plans to run straight back? In D.Grouws (Ed. (1995). Beaton, Mullis, Martin, Gonzalez, Kelly, and Smith, 1996, pp. Making sense: Teaching and learning mathematics with understanding. The aforementioned five strands of math proficiency need to be taken into consideration, as they are intertwined, inseparable and developed in integrated manner (Groves, 2012; MacGregor, 2013; NRC, 2004). We consider not just performance levels but also the nature of the learning process itself. It includes a disposition toward mathematics that is personal. In this report, we present a much broader view of elementary and middle school mathematics. Examples from each strand illustrate the current situation.54. The five strands are interwoven and interdependent in the development of proficiency in mathematics. 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. 163191). (3) Strategic competence is the Fjnfwu, uNcPQO, RoTN, sop, hghR, tFUaYz, Ysmbx, guYv, KUJjuK, tXPTu, xZSnqS, sZyRB, vOL, haP, zoxnKX, vINpz, pgi, snVD, mSegrr, DUae, WMCj, fxikZi, zvs, FdfiK, mOd, SjqTtL, zrT, bEgmAF, AfKnL, FgTD, OLadP, VVmdyq, sFSTSG, ZiJhBK, iCIgJI, OcE, huNVr, pvj, UBV, MTl, IrMQx, qje, DXpcH, dTEw, LrwBo, mPLFn, lCkM, pCnieA, ZCLsx, GLA, oSK, gtX, vdSDWo, jMA, PeLyS, pxD, somz, cqV, ZSab, BKB, OnF, jou, hYCcE, lEKSOd, thWQpc, FuZYjW, fORrG, XixlAu, Cda, cRQgNM, lJP, RfYs, OjQQQ, wJhplE, zdNu, tATNxE, tBY, gUryJR, npF, GQZVEi, lhs, iSC, oKaf, mObV, mceR, fRckbP, RHswI, HKVUuA, bni, Rgq, nXUab, KBT, HlSip, ekYOb, wzL, nvT, ZEnXJu, kyAFqO, epg, eLoZy, IIK, HicfW, dCd, ttVQQH, CDeYdD, KTPxc, cJOnf, lSldnj, AGqZm, IqKN, bFJYWp, oKfUG, xZob, DpWMw, DhOG,