Teaching and learning fundamental mathematics

teaching and learning functional mathematics and teaching and learning mathematical problem solving
GemmaBeven Profile Pic
GemmaBeven,Malaysia,Professional
Published Date:14-07-2017
Your Website URL(Optional)
Comment
Using Research to Shift From the “Yesterday” Mind to the “Tomorrow” Mind Teaching and Learning Mathematics Dr. Terry Bergeson March 2000 State Superintendent of Public InstructionTeaching and Learning Mathematics Using Research to Shift From the “Yesterday” Mind to the “Tomorrow” Mind Dr. Terry Bergeson State Superintendent of Public Instruction Rosemary Fitton Assistant Superintendent Assessment, Research, and Curriculum Pete Bylsma Director, Research and Evaluation Beverly Neitzel and Mary Ann Stine Mathematics Specialists The Office of Superintendent of Public Instruction complies with all federal and state rules and regulations and does not discriminate on the basis of race, color, national origin, sex, disability, age, or marital status. March 2000About the Author This document was written by Dr. Jerry Johnson, Professor of Mathematics at Western Washington University in Bellingham. He has a B.A. from Augsburg College, a M.S. from California Institute of Technology, a M.A. from the University of California, Los Angeles, and a Ph.D. from the University of Washington. Dr. Johnson began teaching at WWU in 1984 and currently teaches classes in both mathematics and mathematics education. He is also part of the WWU faculty team working toward the integration of science, mathematics, and technology curricula. He can be reached by e-mail at Johnsonjcc.wwu.edu. About This Document This document can be found on our Web site (www.k12.wa.us). A free copy of this document can be obtained by placing an order on the website, by writing the Resource Center, Office of Superintendent of Public Instruction, PO Box 47200, Olympia, WA 98504-7200, or by calling the Resource Center toll-free at (888) 595- 3276. If requesting more than one copy, contact the Resource Center to determine the printing and shipping charges. This document is available in alternative format upon request. Contact the Resource Center at (888) 595-3276, TTY (360) 664-3631, or e-mail ericksonospi.wednet.edu. The contents of this document can be reproduced without permission. Reference to this document would be appreciated. Funding for this project was provided by the Excellence in Mathematics Initiative, a state-funded program supporting mathematics education. For questions regarding the content of this document, call (360) 664-3155. Acknowledgements Various staff at the Office of Superintendent of Public Instruction helped prepare this document for publication. Pete Bylsma, Beverly Neitzel, and Mary Ann Stine reviewed the draft document and provided other assistance. Lisa Ireland and Theresa Ellsworth provided editing assistance.CONTENTS Chapter 1 Introduction 1 Chapter 2 Overview of the Research: A Washington State Perspective 5 Number Sense Number and Numeration 5 Computation 9 Estimation 13 Measurement Attributes and Dimensions 15 Approximation and Precision 17 Systems and Tools 18 Geometry Sense Shape and Dimension 18 Relationships and Transformations 19 Probability and Statistics Chance 21 Data Analysis 22 Prediction and Inference 23 Algebraic Sense Relations and Representations 26 Operations 28 Problem Solving 30 Communication 33 Mathematical Reasoning 35 Connections 37 Chapter 3 Mathematics in the Classroom: What Research Tells Educators 39 Constructivism and Its Use 39 Role and Impact of Using Manipulatives 40 How Students Solve Word Problems Involving Mathematics 43 Mastery of Basic Facts and Algorithms 47 Use and Impact of Computing Technologies 49 Culture of the Mathematics Classroom 52 Impact of Ability Grouping 55 Individual Differences and Equity Issues 56 Teacher Attitudes and Student Attitudes 57 Using Performance-Based Assessment 61 Chapter 4 Other Research and Issues 64Professional Development Programs for Mathematics Teachers 64 Changes in How Teachers Teach Math and How Students Learn Math 68 Next Steps: Using Research as Educators 69 Step 1: Mathematics Teachers Accepting Responsibility for Change 69 Step 2: The Reeducation of Mathematics Teachers 70 Step 3: Mathematics Teachers in Their New Roles as Researchers 70 Appendix A Further Explorations 73 Appendix B List of References 75 Index 101 Abbreviations EALRs Essential Academic Learning Requirements WASL Washington Assessment of Student LearningChapter 1 INTRODUCTION Welcome In as friendly and useful manner as possible, our goal is to provide a research-based overview of the potential and challenges of teaching quality mathematics (K–12). Though the primary contexts are the Washington State essential academic learning requirements (EALRs) in mathematics and the correlated Washington Assessment of Student Learning (WASL), each reader must interpret and reflect on the content within his/her own district or classroom situation. Without this important step toward interpretation and reflection by each reader, this publication becomes yet one more resource to be piled on a shelf for reading on that rainy day that never seems to come in Washington. We are fully aware of the ominous nature of the word “research” and its associated baggage. The mere inclusion of the word in the title of articles or workshop offerings often causes teachers and administrators to look for an escape route, whether it is physical or mental. Yet, our intent is to counter this attitude by constructing a research-based perspective that helps both teachers and administrators further the mathematics education reform efforts in Washington at all grade levels. As Charles Kettering, an American engineer and inventor (1876–1958), once said: Research is a high-hat word that scares a lot of people. It needn’t. It is rather simple. Essentially, research is nothing but a state of mind–a friendly, welcoming attitude toward change … going out to look for change instead of waiting for it to come. Research … is an effort to do things better and not to be caught asleep at the switch…. It is the problem solving mind as contrasted with the let-well-enough-alone mind…. It is the “tomorrow” mind instead of the “yesterday” mind. From Kettering’s words, we pull the guiding theme for this book: to use research-based information to support the necessary shift from a “yesterday” mind to a “tomorrow” mind in the making of the many decisions as to how mathematics is taught or learned in Washington. An introductory road map through this text can be useful, especially for those readersreluctant to make the trip. • First, we discuss what research in mathematics education can and cannot do. This section is important because it helps orient the “tomorrow” mind in a positive direction while also ensuring that teachers and administrators are aware of potential misuses of research. Teaching and Learning Mathematics 1Chapter 1 ♦ Introduction • Second, we overview some of the research results related to each of the essential learning academic requirements in mathematics. The key word here is “some,” as the volume of research available in mathematics education is quite large and varied (in both quality and applicability). Though an attempt was made to sort through and select research results in a fair manner, the goal of supporting the “tomorrow” mind was always in full view. If we omitted mention of research results that you have found useful, we apologize for their omission and suggest that you share them with your colleagues. Also, some of the research results mentioned may seem dated but was included because it contributes in some fashion to our current situation and concerns. • Third, we address specific questions in mathematics education as raised by teachers, administrators, or parents. These questions range from the classroom use of calculators or manipulatives, to the role of drill and algorithmic practice, to the best models for the professional development of teachers. In most instances, the research evidence is not sufficient to answer the questions raised in a definitive manner. We suggest that even small insights or understandings are better than teaching in the dark. • And fourth, we outline a plan that a teacher, district, or state can follow to maintain relevancy relative to this document and the issues it addresses. That is, the text should be viewed as another small step forward for Washington State teachers and administrators. Combined with the other efforts of the Office of Superintendent of Public Instruction (OSPI), districts, administrators, teachers, and professional groups, these steps forward help us both gain and maintain momentum in adopting the “tomorrow” mind in mathematics education. Given that road map, we ask you to now join us on this trip through the field of mathematics education research and hope that you find the journey useful. As our ultimate goal is to support teachers and administrators in their efforts to improve student learning in mathematics, we know that an increased awareness of research results is an important form of support. Our apologies are offered if we have either misrepresented or misinterpreted the research results as reported. Also, we apologize in advance for any misleading interpretations or summaries of the research conclusions of others; these lapses were not intentional. RESEARCH IN MATHEMATICS EDUCATION: WHAT IT CAN AND CANNOT DO Think of the many things that can be investigated in mathematics education; it is easy to be overwhelmed. Four key ingredients can be identified: • The students trying to learn mathematics—their maturity, their intellectual ability, their past experiences and performances in mathematics, their preferred learning styles, their attitude toward mathematics, and their social adjustment. • The teachers trying to teach mathematics—their own understanding of mathematics, their beliefs relative to both mathematics itself and how it is Teaching and Learning Mathematics 2Chapter 1 ♦ Introduction learned, their preferred styles of instruction and interaction with students, their views on the role of assessment, their professionalism, and their effectiveness as a teacher of mathematics • The content of mathematics and its organization into a curriculum—its difficulty level, its scope and position in possible sequences, its required prerequisite knowledge, and its separation into skills, concepts, and contextual applications. • The pedagogical models for presenting and experiencing this mathematical content—the use of optimal instructional techniques, the design of instructional materials, the use of multimedia and computing technologies, the use of manipulatives, the use of classroom grouping schemes, the influences of learning psychology, teacher requirements, the role of parents and significant others, and the integration of alternative assessment techniques. All of these ingredients, and their interactions, need to be investigated by careful research. Again, it is easy to be overwhelmed (Begle and Gibb, 1980). Our position is that educational research cannot take into account all of these variables. The result we must live with is acceptance that educational research cannot answer definitively all of the questions we might ask about mathematics education. At best, we can expect research in mathematics education to be helpful in these ways: • It can inform us (e.g., about new pedagogical or assessment techniques). • It can educate us (e.g., about the pros/cons of using different grouping models). • It can answer questions (e.g., about the potential impact of professional development models for teachers). • It can prompt new questions (e.g., about the impact of using the Internet to make real-world connections). • It can create reflection and discussion (e.g., about the beliefs that students and teachers hold toward mathematics). • It can challenge what we currently do as educators (e.g., about our programs for accommodating students with differing ability levels or learning styles). • It can clarify educational situations (e.g., about how assessment can inform instruction). • It can help make educational decisions and educational policy (e.g., about student access to calculators or performance benchmarks). Yet, research in mathematics education can also be counterproductive or fall short of what we would expect in these ways: • It can confuse situations (e.g., about which math curriculum is the best). • It can focus on everything but your situation (e.g., about your classroom, your specific students, and their learning of mathematics). • It can be hidden by its own publication style (e.g., its scholarly vocabulary and overwhelming statistics). • It can be flawed (e.g., about the interpretation of the research data). • It can be boring and obtuse (e.g., its technical jargon, its overuse of statistics and graphs, and its pompous style). Teaching and Learning Mathematics 3Chapter 1 ♦ Introduction Above all, despite the wishes of many teachers and administrators, educational research cannot PROVE anything At best, educational research provides information that the community of educators can use, misuse, or refuse. It is a well-established notion that research results tend not to be used by educators and at times are purposely ignored. For example, Reys and Yeager (1974) determined that while 97.5 percent of the elementary teachers frequently read general education journals, 87.5 percent of these same teachers seldom or never read the research- flavored articles. When asked why, 80 percent of the teachers replied with “lack of time” or “lack of direct classroom implications.” In contrast and on a more positive side, Short and Szabo (1974) found that mathematics teachers at the secondary level were much more knowledgeable about and favorable toward educational research than their colleagues in English and social science. The situation needs to change, as research results must be reflected on and integrated as an important part of the mathematics education plan and process in Washington. The entire education community—mathematics teachers, administrators, legislators, parents, and college mathematics educators—must take and share in the responsibility for this reflection process and integration of research results, whether it occurs at the individual learner level, the classroom level, the district level, the university level, or the state level. This resource text is designed to serve as a catalyst for promoting reflection, discussion, and problem solving within this education community, helping this same community continue to shift from the “yesterday” mind to the “tomorrow” mind in its approach to mathematics education. Teaching and Learning Mathematics 4Chapter 2 OVERVIEW OF THE RESEARCH: A WASHINGTON STATE PERSPECTIVE Each of the EALRs will be considered within the context of some known research results. The search for relevant research results was broad but not exhaustive. The majority of the research results have been omitted (fortunately or unfortunately for the reader), with the few results selected being those that can whet your appetite and illustrate best how research can inform and educate the education community. When it was appropriate, research results that are conflicting or complementary have been juxtaposed to prompt further reflection and discussion. The text’s format will vary from the reporting of interesting research results to the suggesting of research implications that can be adapted for use within a classroom. Without being obtrusive, references are included for those readers who would like to pursue the ideas in more detail. Our primary constraint was to provide summaries of research results in a very concise format. In most instances, this constraint precluded any attempts to describe the actual research that was done. Thus, we often had to omit important factors such as the subjects’ ages, the subjects’ grade levels, the population size, the experimental design, the null hypotheses, the experimental instruments, the data analysis, the levels of statistical significance, or the researchers’ interpretations. These omissions can be dangerous, as it may be misleading to state conclusions based on research involving a few subjects and without replication. Furthermore, no formal effort was made to evaluate the quality of the research efforts as criteria for inclusion in this text. As such a broad review by itself is enormous, we now leave it to you the reader to investigate further each result and evaluate its reasonableness. RESEARCH ON NUMBER SENSE Number and Numeration • The research is inconclusive as to a prerequisite relationship between number conservation and a child’s ability to learn or do mathematics (Hiebert, 1981). Teaching and Learning Mathematics 5Chapter 2 ♦ Overview of the Research: A Washington State Perspective • A child’s acquisition of and fluency with the number-word sequence (e.g., one, two, three …) is a primary prerequisite for the ability to count. A worthy goal is for the student’s fluency to be bidirectional, where the number-words can be produced in sequence in either direction easily (Bergeron and Herscovics, 1990). • Colored chips and money often are used as manipulatives to represent place value concepts and operations, but they prompt increased cognitive complexity. The reason is that the place value notions are not explicitly represented in the color of the chips or the physical sizes of the money (English and Halford, 1995). • Place value is extremely significant in mathematical learning, yet students tend to neither acquire an adequate understanding of place value nor apply their understanding of place value when working with computational algorithms (Fuson, 1990; Jones and Thornton, 1989). • A major reason for place value lapses is the linguistic complexity of our place- value system in English. For example, we do not name “tens” as done in some languages (e.g., “sixty” vs. “six-tens”), arbitrarily reverse the number names between 10 and 20 (e.g., eleven and thirteen), and accept irregularities in our decade names (e.g., “twenty” vs. “sixty”) (Fuson, 1990; English and Halford, 1995). • Students confronted with a new written symbol system such as decimals need to engage in activities (e.g., using base-ten blocks) that help construct meaningful relationships. The key is to build bridges between the new decimal symbols and other representational systems (e.g., whole number place values and fractions) before “searching for patterns within the new symbol system or practicing procedures” such as computations with decimals (Hiebert, 1988; Mason, 1987). • In their extensive study of student understanding of place value, Bednarz and Janvier (1982) concluded that: 1. Students associate the place-value meanings of “hundreds, tens, ones” more in terms of order in placement than in base-ten groupings. 2. Students interpret the meaning of borrowing as “crossing out a digit, taking one away, and adjoining one to the next digit,” not as a means of regrouping. • Students often fail to make the correct interpretation when using base-ten blocks to model place-value or an addition computation. They might not arrange the blocks in accordance with our base-ten positional notation (decreasing value left-to- right) or they might manipulate the blocks in any order (trading whenever necessary or adding left-to-right in place values). Teachers need to be aware that both of these possibilities occur as natural events when students use base-ten blocks (Hiebert, 1992). • Rational number sense differs from whole number sense. The primary difference seems to be that rational number sense is directly connected to students’ understanding of decimal and fraction notations, while whole number sense does not have to be directly connected to the written symbols (Sowder and Schappelle, 1989; Carraher et al., 1985). Teaching and Learning Mathematics 6Chapter 2 ♦ Overview of the Research: A Washington State Perspective • Base-ten blocks are a good physical representation of whole numbers and place value, but prompt increased cognitive complexity when representing decimal numbers. The difference is both a hindrance and an opportunity, as the designation of the unit block may shift as necessary. For example, the base-ten block representation of the number 2.3 will change if the unit block is the flat or the rod (English and Halford, 1995). • Students’ conceptual misunderstandings of decimals lead to the adoption of rote rules and computational procedures that often are incorrect. This adoption occurs despite a natural connection of decimals to whole number, both in notation and computational procedures (English and Halford, 1995). • The place-value connections (or analogs) between whole numbers and decimal numbers are useful for learning, but children often focus directly on the whole number aspects and fail to adjust for the decimal aspects (Hiebert, 1992). For example, a common error is a student’s ordering of decimal numbers as if they were whole numbers, claiming 0.56 is greater than 0.7 because 56 is greater than 7. The reading of decimal numbers seemingly as whole numbers (e.g., “point five six” or “point fifty-six”) contributes to the previous error (Wearne and Hiebert, 1988b; J. Sowder, 1988). • Students with a weak understanding of place value have a difficult time understanding decimals. For example, students will mentally separate a decimal into its whole number part and its pure decimal part, such as rounding 148.26 to 150.3 (Threadgill-Sowder, 1984). Or, students will assume that “more digits” implies that a number is larger, such as 0.1814 being larger than 0.385 and 0.3 (Hiebert and Wearne, 1986). • To construct a good understanding of decimals, students need to focus on connecting the familiar (e.g., written symbols, place value principles, procedural rules for whole number computations and ordering) with the unfamiliar (e.g., decimal notation and the new quantities they represent). Concrete representations of both the symbols and potential actions on these symbols can help make these connections (Hiebert, 1992). • Students who connect the physical representations of decimals with decimal notation are more apt to create their own procedures for new tasks, such as ordering decimals or converting a decimal to its fractional notation (Wearne and Hiebert, 1988a). • When students construct an understanding of the concept of a fraction, the area model (i.e., a continuous attribute) is preferred over the set model (i.e., a discrete attribute) because the total area is a more flexible, visible attribute. Furthermore, the area model allows students to encode almost any fraction whereas the set model (e.g., group of colored chips) has distinct limitations, especially for a part/whole interpretation. For example, try to represent 3/5 using either four cookies or a sheet of paper (English and Halford, 1995; Hope and Owens, 1987). Teaching and Learning Mathematics 7Chapter 2 ♦ Overview of the Research: A Washington State Perspective • Many teachers have a surface level understanding of fractions and decimals, with the result being that students are engaged in learning activities and discussions that are misleading and prompt misconceptions such as “multiplication makes bigger” and “division makes smaller” (Behr et al., 1992). • Students tend to view fractions as isolated digits, treating the numerator and denominator as separate entities that can be operated on independently. The result is an inconsistent knowledge and the adoption of rote algorithms involving these separate digits, usually incorrectly (Behr et al., 1984; Mack, 1990). • Unlike the situation of whole numbers, a major source of difficulty for students learning fractional concepts is the fact that a fraction can have multiple meanings—part/whole, decimals, ratios, quotients, or measures (Kieren, 1988; Ohlsson, 1988). • Student understandings of fractions are very rote, limited, and dependent on the representational form. First, students have greater difficulty associating a proper fraction with a point of a number line than associating a proper fraction with a part-whole model where the unit was either a geometric region or a discrete set. Second, students able to associate a proper fraction on a number line of length one often are not successful when the number line had length two (i.e., they ignore the scaling and treat the available length as the assumed unit) (Novillis, 1976). Finally, though able to form equivalents for a fraction, students often do not associate the fractions 1/3 and 2/6 with the same point on a number line (Novillis, 1980). • As students build some meaning for the symbolic representations of fractions, they overgeneralize their understanding of symbolic representations of whole numbers to fractions and the reverse as well (Mack, 1995). • Students need to work first with the verbal form of fractions (e.g., two-thirds) before they work with the numerical form (e.g., 2/3), as students’ informal language skills can enhance their understanding of fractions. For example, the word “two- thirds” can be associated with the visual of “two” of the “one-thirds” of an object (Payne, 1976). • Students with good understandings of the part/whole interpretation of a fraction still can have difficulty with the concept of fraction equivalence, confuse quantity notions with proportionality, possess limited views of fractions as numbers, and have cognitive difficulty relating fractions to division (Kerslake, 1986). • Students taught the common denominator method for comparing two fractions tend to ignore it and focus on rules associated with ordering whole numbers. Students who correctly compare numerators if the denominators are equal often compare denominators if the numerators are equal (Behr et al., 1984). • Students’ difficulties with ratios are often due to the different referents involved in the ratio situation (Hart, 1984). A ratio can refer to a comparison between two parts (e.g., 1 can of frozen concentrate to 3 cans of water), a comparison between a Teaching and Learning Mathematics 8Chapter 2 ♦ Overview of the Research: A Washington State Perspective part and a whole (e.g., 1 can of frozen concentrate to 4 cans of lemonade), or a comparison between two wholes (e.g., 1 dollar to 4 hours of work). • Students do not make good use of their understandings of rational numbers as a starting point for developing an understanding of ratio and proportion (Heller et al., 1990). • The unit rate method is clearly the most commonly used and perhaps the best method for working with problems involving ratios and proportions. The distinction as “most common” disappears once students are taught then apply by rote the cross-product algorithm for proportions (Post et al., 1985, 1988). Nonetheless, the unit rate method is strongly suggested as “scaffolding” for building proportional reasoning. • The cross-product algorithm for evaluating a proportion is (1) an extremely efficient algorithm but rote and without meaning, (2) usually misunderstood, (3) rarely generated by students independently, and (4) often used as a “means of avoiding proportional reasoning rather than facilitating it” (Cramer and Post, 1993; Post et al., 1988; Hart, 1984; Lesh et al., 1988). • Students see their work with ratios as an additive operation, often replacing the necessary multiplicative concepts with repeated additions (K. Hart, 1981c). • Students’ intuitive understanding of the concept of infinity remains quite stable over the middle grades and is relatively unaffected by mathematics instruction (Fischbein et al., 1979). Computation • Students learning multiplication as a conceptual operation need exposure to a variety of models (e.g., rectangular array, area). Access only to “multiplication as repeated addition” models and the term “times” leads to basic misunderstandings of multiplication that complicate future extensions of multiplication to decimals and fractions (Bell et al., 1989; English and Halford, 1995). • Division situations can be interpreted as either a partition model (i.e., the number of groups is known and the number of members in a group needs to be found) or a measurement model (i.e., the number of members in a group is known and the number of groups needs to be found). Measurement problems are easier for students to model concretely (Brown, 1992), yet partition problems occur more naturally and more frequently in a student’s daily experiences. The partition model also is more representative of the long division algorithm and some fraction division techniques (English and Halford, 1995). • Students learning the processes of addition and subtraction need a “rich problem solving and problem-posing environment” that should include: Teaching and Learning Mathematics 9Chapter 2 ♦ Overview of the Research: A Washington State Perspective 1. Experiences with addition and subtraction in both in-school and out-of-school situations to gain a broad meaning of the symbols +/-. 2. Experiences both posing and solving a broad range of problems. 3. Experiences using their contextual meaning of +/- to solve and interpret arithmetic problems without a context. 4. Experiences using solution procedures that they conceptually understand and can explain (Fuson, 1992a). • When performing arithmetic operations, students who make mistakes “do not make them at random, but rather operate in terms of meaning systems that they hold at a given time.” The teachers feedback should not focus on the student as being “wrong,” but rather identify the student’s misunderstandings which are displayed “rationally and consistently” (Nesher, 1986). • Whole-number computational algorithms have negative effects on the development of number sense and numerical reasoning (Kamii, 1994). • Confronted with decimal computations such as 4.5+0.26=?, students can respond using either a syntactic rule (e.g., line up the decimal points, then add vertically) or semantic analysis (e.g., using an understanding of place values, you need only add the five-tenths to the two-tenths). The first option relies on a student’s ability to recall the proper rules while the second option requires more cognitive understanding on the student’s part. Research offers several insights relevant to this situation. First, students who recall rules experience the destructive interference of many instructional and context factors. Second, when confronted with problems of this nature, most students tend to focus on recalling syntactic rules and rarely use semantic analysis. And third, the syntactic rules help students be successful on test items of the same type but do not transfer well to slightly different or novel problems. However, students using semantic analysis can be successful in both situations (Hiebert and Wearne, 1985; 1988). • The standard computational algorithms for whole numbers are “harmful” for two reasons. First, the algorithms encourage students to abandon their own operational thinking. Second, the algorithms “unteach” place value, which has a subsequent negative impact on the students’ number sense (Kamii and Dominick, 1998). • Students view the multiplication and division algorithms primarily as “rules to be followed,” leading to a persistence that the numbers involved are to be viewed as separate digits and not grouped amounts involving place-values. The result often is an incorrect answer, impacted unfortunately by students’ restricted access to their understanding of estimation, place value, and reasonableness of results (Behr et al., 1983; Fischbein et al., 1985; Lampert, 1992). • Many students never master the standard long-division algoriths. Even less gain a reasonable understanding of either the algorithm or the answers it produces. A major reason underlying this difficulty is the fact that the standard algorithm (as usually taught) asks students to ignore place value understandings (Silver et al., 1993). Teaching and Learning Mathematics 10Chapter 2 ♦ Overview of the Research: A Washington State Perspective • Students have great difficulty “admitting” that the answer to a division of one whole number by another could contain a decimal or a fraction. The cognitive difficulty is compounded if the task involves division of a number by a number larger than itself. The difficulty seems to reflect a dependence on the partition model for division and a preference for using remainders (M. Brown, 1981a, 1981b). • Any approach to performing division, including the long-division algorithm, requires reasonable skills with proportional reasoning, which in turn requires a significant adjustment in a student’s understanding of numbers and the role of using numbers in counting (Lampert, 1992). • Students constructing meanings underlying an operation such as long division need to focus on understanding why each move in an algorithm is appropriate rather than on which moves to make and in which sequence. Also, teachers should encourage students to invent their own personal procedures for the operations but expect them to explain why their inventions are legitimate (Lampert, 1992). • In a “classic” research study, Silver et al. (1993) showed that when students work with division problems involving remainders, their performance is impacted adversely by the students’ dissociation of sense making from the solution of the problem. A second important factor is the students’ inability to write reasonable accounts of their mathematical thinking and reasoning while solving the division problems. • Students’ use of base-ten blocks improves their understanding of place-value, their accuracy while computing multi-digit addition and subtraction problems, and their verbal explanations of the trading/regrouping involved in these problems (Fuson, 1986; Fuson and Briars, 1990). Furthermore, a positive relationship exists between the amount of student verbalizations about their actions while using the base-ten blocks and the students’ level of understanding (Resnick and Omanson, 1987). • Students need a good understanding of the concept of both a fraction and fraction equivalence before being introduced to computation situations and procedures involving fractions (Mack, 1993; Bezuk and Bieck, 1993). • Students learning computational algorithms involving fractions have difficulty connecting their concrete actions with manipulatives with their symbolic procedures. Often, a student’s personal competence with a rote procedure “outstrips” his/her conceptual understanding of fractions; the unfortunate result is that students cannot monitor their work, can check their answers only by repeating the rote procedure, and are unable to judge the reasonableness of their answer (Wearne and Hiebert, 1988b). • Computational algorithms involving fractions prevent students from even trying to reason or make sense of fraction situations. In fact, students tend to not only remember incorrect algorithms but also have more faith in them compared to their own reasoning (Mack, 1990). Teaching and Learning Mathematics 11Chapter 2 ♦ Overview of the Research: A Washington State Perspective • The traditional “invert-and-multiply” algorithm for dividing fractions does not develop naturally from students using manipulatives (Borko et al., 1992). In contrast, the common denominator approach to dividing fractions can be modeled by students using manipulatives and capitalizes on their understanding of the measurement model of whole number division using repeated subtraction (Sharp, 1998). • Students openly not confident when using fractions operate with fractions by adapting or misapplying the computational rules for whole numbers (K. Hart, 1981b). • Many students solve problems involving proportions by using additive strategies which produce incorrect results, not realizing that such problems involve a multiplicative structure (Hart, 1988). • Students gain little value from being taught the cross-multiplication algorithm for evaluating a proportion because of its lack of a conceptual basis (K. Hart, 1981c). • The cross-multiplication algorithm for a proportion is (1) an extremely efficient algorithm but is rote and without meaning, (2) usually misunderstood by students, (3) rarely generated by students independently, and (4) often used as a “means of avoiding proportional reasoning rather than facilitating it” (Cramer and Post, 1993; Post et al., 1988; Hart, 1984; Lesh et al., 1988). • Students begin with useful percent strategies (e.g., using benchmarks, pictorial representations, ratios, and fractions) that are quickly discarded and replaced by their extensive use of school-taught equation strategies. Students’ successes with the earlier conceptual strategies have little impact (Lembke and Reys, 1994). • Students bring informal and self-constructed computational techniques into algebra classrooms where more formal methods are developed. Teachers must (1) recognize students who use such informal methods for a given problem, (2) recognize and value these informal methods, and (3) discuss possible limitations of the informal methods (Booth, 1988). • Young students allowed to develop, use, and discuss personally invented algorithms demonstrate enhanced number sense and operational sense (Kamii et al., 1993; J. Sowder, 1992a). These students also develop efficient reasoning strategies, better communication skills, and richer experiences with a wider range of problem solving strategies (Carroll and Porter, 1997). • The number line is not a good representational model for working with integer operations, except for addition. A discrete model (e.g., where the positive elements can cancel the negative elements) is preferred because it has documented success with students and it is more consistent with the actions involved (Kuchemann, 1981a). Teaching and Learning Mathematics 12Chapter 2 ♦ Overview of the Research: A Washington State Perspective • Students tend to avoid using parentheses when doing arithmetic or algebra, believing that the written sequence of the operations determines the order of computations. Some students even think that changing the order of the computations will not change the value of the original expression (Kieran, 1979; Booth, 1988). • Students tend not to view commutativity and associativity as distinct properties of a number system (numbers and operators), but rather as “permissions” to combine numbers in any order (Resnick, 1992). Estimation • Students need to recognize the difference between estimation and approximation in order to select and use the appropriate tool in a computational or measurement situation. Estimation is an educated guess subject to “ballpark” error constraints while approximation is an attempt to procedurally determine the actual value within small error constraints (J. Sowder, 1992a). • Good estimators tend to have strong self-concepts relative to mathematics, attribute their success in estimation to their ability rather than mere effort, and believe that estimation is an important tool. In contrast, poor estimators tend to have a weak self-concept relative to mathematics, attribute the success of others to effort, and believe that estimation is neither important nor useful (J. Sowder, 1989). • The inability to use estimation skills is a direct consequence of student focus on mechanical manipulations of numbers, ignoring operational meaning, number sense, or concept of quantity/magnitude (Reys, 1984). • The ability to multiply and divide by powers of ten is “fundamental” to the development and use of estimation skills (Rubenstein, 1985). • Three estimation processes are used by “good” estimators in Grades 7 through adult. First, reformulation massages the numbers into a more mentally- friendly form using related skills such as rounding, truncating, and compatible numbers (e.g., using 6+8+4 to estimate 632+879+453 or using 7200 60 to estimate 7431 58). Second, translation alters the mathematical structure into an easier form (e.g., using the multiplication 4x80 to estimate the sum 78+82+77+79). And third, compensation involves adjustments made either before or after a mental calculation to bring the estimate closer to the exact answer. In this study, the less- skilled students “felt bound” to make estimates using the rounding techniques they had been taught even if the result was not optimal for use in a subsequent calculation (e.g., use of compatible numbers) (Reys et al., 1982). • Student improvement in computational estimation depends on several skills and conceptual understandings. Students need to be flexible in their thinking and have a good understanding of place value, basic facts, operation properties, and number comparisons. In contrast, students who do not improve as estimators seem “tied” to the mental replication of their pencil-and-paper algorithms and fail to see any Teaching and Learning Mathematics 13Chapter 2 ♦ Overview of the Research: A Washington State Perspective purpose for doing estimation, often equating it to guessing (Reys et al., 1982; Rubenstein, 1985; J. Sowder, 1992b). Also, good estimators tended to be self- confident, tolerant of errors, and flexible while using a variety of strategies (Reys et al., 1982). • Teacher emphasis on place value concepts, decomposing and recomposing numbers, the invention of appropriate algorithms, and other rational number sense skills have a long-term impact on middle school students’ abilities using computational estimation. Rather than learning new concepts, the students seemed to be reorganizing their number understandings and creating new ways of using their existing knowledge as “intuitive notions of number were called to the surface and new connections were formed” (Markovits and Sowder, 1994). • Students prefer the use of informal mental computational strategies over formal written algorithms and are also more proficient and consistent in their use (Carraher and Schliemann, 1985). • Students’ acquisition of mental computation and estimation skills enhances the related development of number sense; the key seems to be the intervening focus on the search for computational shortcuts based on number properties (J. Sowder, 1988). • Experiences with mental computation improve students’ understanding of number and flexibility as they work with numbers. The instructional key was students’ discussions of potential strategies rather than the presentation and practice of rules (Markovits and Sowder, 1988). • Mental computation becomes efficient when it involves algorithms different from the standard algorithms done using pencil and paper. Also, mental computational strategies are quite personal, being dependent on a student’s creativity, flexibility, and understanding of number concepts and properties. For example, consider the skills and thinking involved in computing the sum 74+29 by mentally representing the problem as 70+(29+1)+3 = 103 (J. Sowder, 1988). • The “heart” of flexible mental computation is the ability to decompose and recompose numbers (Resnick, 1989). Teaching and Learning Mathematics 14Chapter 2 ♦ Overview of the Research: A Washington State Perspective • The use of a context enhances students’ ability to estimate in two ways. First, a context for an estimation helps students overcome difficulties in conceptualizing the operations needed in that context (e.g., the need to multiply by a number less than one producing a “smaller” answer). Second, a context for an estimation helps students bypass an algorithmic response (e.g., being able to truncate digits after a decimal point as being basically insignificant when using decimal numbers) (Morgan, 1988). • Young students tend to use good estimation strategies on addition problems slightly above their ability level. When given more difficult problems in addition, students get discouraged and resort to wild guessing (Dowker, 1989). • Students have a difficult time accepting either the use of more than one estimation strategy or more than one estimation result as being appropriate, perhaps because of an emphasis on the “one right answer” in mathematics classrooms. These difficulties lessened as the students progressed from the elementary grades into the middle school (Sowder and Wheeler, 1989). • Students need to be able to produce reasonable estimates for computations involving decimals or fractions prior to instruction on the standard computational algorithms (Mack, 1988; Owens, 1987). • Students estimating in percent situations need to use benchmarks such as 10 percent, 25 percent, 33 percent, 50 percent, 75 percent, and 100 percent, especially if they can associate a pictorial image. Also, student success seems to depend on a flexible understanding of equivalent representations of percents as decimals or fractions (Lembke and Reys, 1994). RESEARCH ON MEASUREMENT Attributes and Dimensions • The research is inconclusive as to the prerequisite relationship between conservation and a child’s ability to measure attributes. One exception is that conservation seems to be a prerequisite for understanding the inverse relationship between the size of a unit and the number of units involved in a measurement situation. For example, the number expressing the length of an object in centimeters will be greater than its length in inches because an inch is greater than a centimeter (Hiebert, 1981). • Young children lack a basic understanding of the unit of measure concept. They often are unable to recognize that a unit may be broken into parts and not appear as a whole unit (e.g., using two pencils as the unit) (Gal’perin and Georgiev, 1969). Teaching and Learning Mathematics 15

Advise: Why You Wasting Money in Costly SEO Tools, Use World's Best Free SEO Tool Ubersuggest.