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Adding + It Up: Helping Children Learn Mathematics 8 DEVELOPING MATHEMATICAL PROFICIENCY BEYOND NUMBER In this chapter, we go beyond number to examine other domains of school mathematics in grades pre-K to 8. Because a great deal of the curriculum dealing with number leads naturally to algebra and because whether and how to teach algebra to all children is a hotly debated topic in many schools, we devote the bulk of the chapter to issues of beginning algebra. The first section is organized according to the algebraic activities of representing, transforming, and generalizing and justifying, which allows us to survey the literature relevant to learning algebra in grades pre-K to 8. We close the chapter with two briefer sections: one on measurement and geometry, the other on statistics and probability. As we noted in Chapters 1 and 3, these domains are intimately related to number. Measurement is one of the most common uses of number, and the geometry studied in elementary and middle school uses lengths, areas, and volumes usually expressed as numerical quantities. Statistics and probability involve the quantification of phenomena dealing with data and chance. Throughout the last two sections we emphasize the strands of conceptual understanding and adaptive reasoning because these have been the focus of much recent research and because traditional instruction has tended to emphasize the development of procedural fluency instead. Beginning Algebra For most students, school algebra—with its symbolism, equation solving, and emphasis on relationships among quantities—seems in many ways to signal a break with number and arithmetic. In fact, algebra builds on the proficiency that students have been developing in arithmetic and develops it Algebra builds on the proficiency that students have been developing in arithmetic and develops it further.
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Adding + It Up: Helping Children Learn Mathematics further. In particular, the place-value numeration system used for arithmetic implicitly incorporates some of the basic concepts of algebra, and the algorithms of arithmetic rely heavily on the “laws of algebra.” Nevertheless, for many students, learning algebra is an entirely different experience from learning arithmetic, and they find the transition difficult. The difficulties associated with the transition from the activities typically associated with school arithmetic to those typically associated with school algebra have been extensively studied.1 In this chapter, we review in some detail the research that examines these difficulties and describe new lines of research and development on ways that concepts and symbol use in elementary school mathematics can be made to support the development of algebraic reasoning. These recent efforts have been prompted in part by the difficulties exposed by prior research and in part by widespread dissatisfaction with student learning of mathematics in secondary school and beyond. The efforts attempt to avoid the difficulties many students now experience and to lay the foundation for a deeper set of mathematical experiences in secondary school. Before reviewing the research, we first describe and illustrate the main activities of school algebra. Previous chapters have shown how the five strands of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition are interwoven in achieving mathematical proficiency with number and its operations. These components of proficiency are equally important and similarly entwined in successful approaches to school algebra. The Main Activities of Algebra What is school algebra? Various authors have given different definitions, including, with “tongue in cheek, the study of the 24th letter of the alphabet [x].”2 To understand more fully the connections between elementary school mathematics and algebra, it is useful to distinguish two aspects of algebra that underlie all others: (a) algebra as a systematic way of expressing generality and abstraction, including algebra as generalized arithmetic; and (b) algebra as syntactically guided transformations of symbols.3 These two main aspects of algebra have led to various activities in school algebra, including representational activities, transformational (rule-based) activities, and generalizing and justifying activities.4 The representational activities of algebra involve translating verbal information into symbolic expressions and equations that often, but not always, involve functions. Typical examples include generating (a) equations that represent quantitative problem situations in which one or more of the quan-
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Adding + It Up: Helping Children Learn Mathematics tities are unknown, (b) functions describing geometric patterns or numerical sequences, and (c) expressions of the rules governing numerical relationships (see Box 8–1 for an example of each). Proficiency with representational activities involves conceptual understanding of the mathematical concepts, operations, and relations expressed in the verbal information, and it involves strategic competence to formulate and represent that information with algebraic equations and expressions. Hence, facility with generating expressions and equations combines two of the strands of mathematics proficiency. The second kind of algebraic activities—the transformational or rule-based activities—includes, for instance, collecting like terms, factoring, expanding, substituting, solving equations, and simplifying expressions. These activities are largely concerned with changing the form of an expression or equation to an equivalent one using the rules for manipulating algebraic symbols. For example, in solving the equation 4(x+3)=2x+19, you can replace the expression 4(x+3) by the equivalent expression 4x+12. Subsequently, by subtracting 2x and then 12 from both sides, the equation 4x+12=2x+19 can be replaced by the equivalent equation 2x=7; finally, dividing both sides by Box 8–1 Representational Activities of Algebra There are 3 piles of stones; the first has 5 less than the third, and the second has 15 more than the third. There are 31 altogether. Find the number in each pile. Say to yourself what you see in the picture sequence. T hen state a rule for extending the sequence of pictures indefinitely. The sum of two consecutive numbers is always an odd number. Can you show why, using algebra? SOURCES: Bell, 1995, p. 61; Lee and Wheeler, 1987, p. 160; Mason, 1996, p. 84. Used by permission of Elsevier Science and of Kluwer Academic Publishers.
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Adding + It Up: Helping Children Learn Mathematics 2 yields the solution Facility with symbolic computation in algebra has an obvious parallel with, and indeed draws upon, procedural fluency in the domain of number. Just as in arithmetic, aspects of conceptual understanding and strategic competence interact with each other and with procedural fluency in transformational activities in algebra. Lastly, there are the generalizing and justifying activities. These include problem solving, modeling, noting structure, justifying, proving, and predicting. These activities are not exclusive to algebra, but they often use its language and tools. For example, the consecutive numbers problem (show that the sum of two consecutive numbers is always an odd number) illustrates how algebra is used to generalize and justify.5 Arithmetic can be used to generate many instances to show that the sum of two consecutive numbers is odd: 3+4=7, 12+13=25, and so on. But the representational and transformational aspects of algebra make it possible to justify that the sum is always odd. The sum of two consecutive integers can be represented with algebra as x+(x+1), where the key is the recognition that x represents any whole number. This expression can be transformed into the equivalent expression 2x+1, which is the general form of any odd number. This example illustrates the power of algebra, as against arithmetic, as a tool for making generalizations and providing justifications, at least for those learners who understand how statements using variables express generality. Generalizing and justifying activities typically involve examining and interpreting representations that have previously been generated or manipulated. Such activities can provide insight into, for example, the underlying mathematical structure of a situation, or they can yield answers to specific questions or conjectures. They encourage students to develop an awareness of the role that algebra can play in mathematical thinking. All of the strands of algebraic proficiency come together in these activities, especially adaptive reasoning. One of the great strengths of algebra is that, for experts, a great deal of its transformational activity can be carried out in what appears to be a rather automated manner. Once a student makes the transformation rules his or her own, the algorithms of algebra can be executed, in a sense, without thinking. The student needs to be thinking, for example, not of what the letters in the expressions refer to or of the operations he or she is carrying out, but only that the actions on the symbolic objects are allowable. In fact, once an expression or equation has been generated (or provided) and the goal is known, it seems to be treated in an almost mindless fashion. But is that possible?
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Adding + It Up: Helping Children Learn Mathematics Every algebraic manipulation involves an anticipatory element, a sense of the direction in which you want to be going and of what the desired expression will look like once you get there.6 The development of this sense of anticipation provides an alternative to the “blind” manipulation that is so often performed by beginning algebra students.7 Research suggests, however, that such anticipatory thinking is not acquired without effort. Even students with extensive algebra experience can make poor strategic decisions that leave them “going round in circles” because they cannot seem to “see” the right thing in algebraic expressions.8 The transformational aspects of algebra have traditionally been emphasized in U.S. textbooks, which have tended to pay more attention to the rules to be followed in manipulating symbolic expressions and equations than to the concepts that support those rules or give meaning to the expressions or equations being manipulated. Although few experimental comparisons have been conducted, research has shown that rule-based instructional approaches that do not give students opportunities to create meaning for the rules or to learn when to use them can lead to forgetting,9 unsystematic errors,10 reliance on visual clues,11 and poor strategic decisions.12 For example, experienced algebra students were found to choose inappropriate strategies when deciding what to do next in the simplification of an algebraic expression and would often end up with an expression that was more difficult to deal with, even though they had performed legal transformations.13 Beginning algebra students were found to be quite haphazard in their approach; they might simplify 4(6x–3y)+5x as 4(6x–3y–5x) on one occasion and do something else on another.14 When the consecutive numbers problem was given to 113 high school students who had studied algebra, only 8 worked the problem correctly.15 The rest made a variety of errors, including substituting a few values for x to show the sum’s “oddness,” using different letters for each number (x and y), representing the consecutive numbers as 1x and 2x, and setting the expression x+(x+1) equal to a fixed odd number and then solving for x. In one of the few experimental studies of rule-based instruction, students who were taught an estimate-and-test sense-making strategy performed better in solving systems of equalities and inequalities than students taught rule-based equation solving.16 Data from the National Assessment of Educational Progress (NAEP) further reveal the shortcomings of traditional school algebra. For example, one of the NAEP tasks from the second mathematics assessment involved completing the table shown in Box 8–2. Most of the students with one or two years of algebra could recognize the pattern—adding 7—from the given nu-
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Adding + It Up: Helping Children Learn Mathematics Box 8-2 Table Completion Task from NAEP Give the values of y when x=3 and when x=n. SOURCE: Carpenter, Corbitt, Kepner, Lindquist, and Reys, 1981. Used by permission of the National Council of Teachers of Mathematics. merical values and use it when x=3 (with success rates of 69% and 81% for the two groups of students, respectively). They were less successful, however, when asked to derive from the same table the value of y when x=n (correct response: y=n+7; success rates: 41% and 58%, respectively). The next three sections of the chapter present representative findings from the large body of research on algebra learning and teaching for the three types of algebraic activity sketched above. Since much of this research has been carried out with students making the transition from arithmetic to algebra, it casts light on the kinds of thinking that students bring with them to algebra from the traditional arithmetic curriculum centered on algorithmic computation that has been predominant in U.S. schools.17 Indeed, many studies have been oriented toward either developing approaches to teaching algebra that take this arithmetic thinking into account or, more recently, developing approaches to elementary school mathematics that build foundations of algebraic reasoning earlier. Much research also has focused on linear relations and linear functions, perhaps because these are considered the easiest and are the first ones encountered by students making the transition from arithmetic to algebra. Although the domain of algebra is far richer than linear relations, much of the research at the cusp of arithmetic and algebra focuses on them.18 Some of the newer curriculum programs, however, introduce nonlinear relations along with linear relations in the middle grades. In particular, exponential growth relations (e.g., doubling) have been shown to be an accessible topic for middle school students.19
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Adding + It Up: Helping Children Learn Mathematics Several of the teaching approaches discussed in the following sections have profitably used computer technologies, especially graphics, as a means of making algebraic symbolism more meaningful. These studies provide evidence of the positive role that computer-supported approaches can play in the learning of algebra, as well as suggesting that technology can be a means for making algebra accessible to all students, including those who, for whatever reason, lack skill in pencil-and-paper computation.20 Thus, these examples suggest that some version of “algebra for all” may be viable. The Representational Activities of Algebra What the Number-Proficient Student Brings Traditional representational activities of algebra center on the formation of algebraic expressions and equations. Creating these expressions and equations involves understanding the mathematical operations and relations and representing them through the use of letters and—for equations—the equal sign. It also requires thinking that proceeds in rather different ways from the thinking that develops in traditional arithmetic. In the transition from arithmetic to algebra, students need to make many adjustments, even those students who are quite proficient in arithmetic. At present, for example, elementary school arithmetic tends to be heavily answer oriented and does not focus on the representation of relations.21 Students beginning algebra, for whom a sum such as 8+5 is a signal to compute, will typically want to evaluate it and then, for example, write 13 for the box in the equation 8+5=□+9 instead of the correct value of 4. When an equal sign is present, they treat it as a separator between the problem and the solution, taking it as a signal to write the result of performing the operations indicated to the left of the sign.22 Or, when doing a sequence of computations, students often treat the equal sign as a left-to-right directional signal. For example, consider the following problem: Daniel went to visit his grandmother, who gave him $1.50. Then he bought a book for $3.20. If he has $2.30 left, how much money did he have before visiting his grandmother? In solving this problem, sixth graders will often write 2.30+3.20=5.50–1.50 =4.00, tacking the second computation onto the result of the first.23 Since 2.30+3.20 equals 5.50, not 5.50–1.50, the string of equations they have written violates the definition of equality. To modify their interpretation of the equality sign in algebra, students must come to respect the true meaning
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Adding + It Up: Helping Children Learn Mathematics of equality as a statement that the two sides of an equation are equal to each other. Students oriented toward computation are also perplexed by an expression such as x+3; they think they should be able to do something with it, but are unsure as to what that might be. They are not disposed to think about the expression itself as being the subject of attention. Similarly, they need to rethink their approach to problems. In solving a problem such as “When 3 is added to 5 times a certain number, the sum is 38; find the number,” students emerging from arithmetic will subtract 3 from 38 and then divide by 5— undoing in reverse order, as they have been taught, the operations stated in the problem text. In contrast, they will be taught in algebra classes first to represent the relationships in the situation by using those operations and not reversing them: 3+5x=38. Although most students beginning algebra have had some experience with the use of letters in arithmetic, such as finding the number n such that n+12=37, rarely have they worked with more general problem situations in which the letter can take on any of an infinite set of values. In a third-grade class,24 the students were presented with the problem, “Who can make up a number sentence that equals 10 but has more than two numbers adding up to 10?” Most students started with examples like 5+2+3=10 and 8+1+1=10, but the class went on to generate a variety of equations, including 200–200+10=10 and 1,000,000–1,000,000+10=10. With the teacher’s help, they soon were able to formulate the equation x–x+10=10, for any number x. This use of a letter as variable, where the letter can take on a range of values, is seldom seen in typical elementary school mathematics. More often, the letter, or some placeholder, represents an unknown, and only one numerical value will make the equation true. In algebra, both of these conceptions of literal terms (or letters) are important. A number of recent intervention studies have shown how selected modifications of elementary school mathematics might support the development of algebraic reasoning. One approach infuses elementary mathematics with a systematic use of problems requiring students to generalize, to determine values of a literal term that satisfy quantitative constraints (with or without equations), or to treat numbers in algebraic ways. For example, students might be asked to determine how many ways the number 4 can be written using a given number of 1s and the four basic operations. Since each expression must equal 4, students must distinguish among the different possibilities on the basis of their symbolic form rather than their value when evaluated.25
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Adding + It Up: Helping Children Learn Mathematics Another approach is to assist elementary school teachers in modifying their instructional materials and classroom practices to emphasize generalizing and expressing generality in elementary mathematics, particularly using patterns, functions, and the notions of variable. Third graders whose teachers were given such assistance showed substantial increases in their understanding of variable and equality compared with traditionally instructed students in the same grade and school. Further, these third graders outperformed fourth graders on items testing number sense from a mandated statewide assessment.26 A third approach to modifying elementary school mathematics focuses on helping teachers understand their students’ thinking when the students are asked to generalize operations and properties from arithmetic. In one combination first-and-second-grade class, the teacher focused on number sentences twice a week during the school year. Instruction started with true and false number sentences and progressed to increasingly complex forms of open sentences. Number sentences were also used to help the children articulate and represent conjectures about properties of numbers and operations. By the end of the year most of the students (13 of 17) developed a relational concept of equality and operations, along with an ability to form and express general relations among number sentences.27 In particular, the majority of these students no longer made mistakes like writing 13 for the box in 8+5=□+9. Much of the difficulty that students experience when they first encounter algebra is symptomatic of the cognitive challenges inherent in moving from one mode of thinking to another, from arithmetic reasoning to algebraic reasoning. Research on algebra learning has sought to uncover the ways in which beginning algebra students think, thus helping ease their transition into algebra. In the examples cited above of research on more algebraic approaches to elementary school mathematics that are intended to avoid transition problems, the approaches are in their early stages. Although the long-term impact of these approaches is still unknown, they offer considerable promise for avoiding the difficulties many students now experience. Developing Meaning Much of the algebra research in the 1970s and early 1980s yielded evidence that incoming algebra students have trouble interpreting letters as variables.28 Building on these findings, recent work has focused on how students learn to use algebraic letters to represent a range of values.
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Adding + It Up: Helping Children Learn Mathematics One investigator studied an approach designed to address students’ difficulties with thinking about and symbolizing algebraic expressions.29 Students were asked to give instructions to an “idealized mathematics machine”: for example, “I want the machine to add 5 to any number I give it; how will I write the instructions?” or “I want the machine to add any two numbers I give it” or “Have the machine find the area of any square, given a side.” The students easily made sense of the idea of employing letters to write rules that would enable the machine to solve whole classes of problems. In the examples above, the rules would be expressed using (x+5), (x+y), and x2, respectively. This approach addresses two issues related to the introduction of algebra: the usefulness or purpose of learning algebra, and the difficulty of new algebraic concepts. The investigator emphasized that “children who are not persuaded on the former point will make little effort to try and come to terms with the latter” and added that “certainly the evidence…clearly indicated this to be the case.”30 The majority of the students in the study made significant gains in thinking about the letters in algebraic expressions as taking on multiple values (from 23% correct on the pretest to 85% correct on the delayed posttest) and in improving their attitude toward algebra (at the beginning of the study, they “hated algebra, didn’t understand it” and complained that “letters are stupid; they don’t mean anything”).31 Later research in which students used actual computers confirmed these results, both with respect to increasing the students’ motivation and developing their understanding of algebraic expressions as general computational procedures.32 Representational activities of algebra can interact with well-established natural-language-based habits. Representational activities of algebra can interact with well-established natural-language-based habits. These interactions are particularly clear in the well-studied class of tasks exemplified by the so-called students-and-professors problem:33 At a certain university, there are six times as many students as professors. Using S for the number of students and P for the number of professors, write an equation that gives the relation between the number of students and the number of professors. A robust reversal error is committed by a majority of students, ranging from first-year algebra students to college freshmen, who write “6S=P” and treat the “6” as an adjective modifying the “S” as if it were a noun.34 This error occurs across different versions of the problem and is resistant to easy correction.35 The error, while of intrinsic interest, has an especially important connection to the instruction that students receive prior to studying algebra. In particular, detailed correlational analyses have shown that the error’s robust-
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Adding + It Up: Helping Children Learn Mathematics ness is strongly associated with students’ understanding of rates and ratios— the worse their understanding, the more robust the error.36 Such findings could signal the connections between building proficiency in using algebra as a representational tool and building conceptual understanding of number ideas—in this case, multiplicative ideas. Interestingly, related findings show that a procedural perspective that treats the variables in the equation as input-output pairs leads to improved equation-writing performance,37 which is consistent with the results described above using the idealized machine and the computer. A series of teaching experiments conducted over three years during the late 1980s in Mexico and the United Kingdom demonstrated the potential of computer spreadsheets to help students grasp the meaning of variables and algebraic expressions, including students who had been having difficulty with traditional approaches to algebraic symbolism.38 Further, spreadsheets can provide a vehicle for introducing students to formal symbolism.39 For an example of how a student can profit from the use of a spreadsheet, see Box 8–3. This student was a tenth grader in a low mathematics track of a school in England who had little previous experience with algebra. Experimental studies involving spreadsheets have also shown enhanced student learning relative to traditional instruction.40 Studies of the use of spreadsheets have found that it is relatively easy for students to pass from a mixture of spreadsheet and algebraic notation to traditional algebraic symbolism.41 It should be noted that the spreadsheet approach involves creating a range of values for the expressions that represent the various relationships in the problem statement. Thus, a spreadsheet column of the values that are generated provides an explicit representation of sample values of each variable. Moreover, the particular value of X that solves the problem is often found in one line of the spreadsheet array (if the situation is linear). In the spreadsheet approach, therefore, the unknown is viewed simply as that particular value that satisfies the constraints of the problem. In general, the use of spreadsheets has been found to be an effective way to develop several notions involved in the representational activities of algebra. It encourages discussion of the role of a letter as both a variable and an unknown; it provides meaningful experience in creating algebraic expressions; and it puts the focus squarely on the representation of quantitative relationships. Research from both small-group instruction42 and broad-based implementations involving several schools43 provides support for these claims. Closely related to spreadsheets are intelligent tutors in which students label spreadsheet-like worksheets and fill in calculated results for specific
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Adding + It Up: Helping Children Learn Mathematics M.Koyama (Eds.), Proceedings of the Twenty-fourth International Conference for the Psychology of Mathematics Education (vol. 2, pp. 145–152). Hiroshima, Japan: PME Program Committee. Carry, L.R., Lewis, C., & Bernard, J. (1980). Psychology of equation solving: An information processing study (Final Technical Report). Austin: University of Texas at Austin, Department of Curriculum and Instruction. (ERIC Document Reproduction Service No. ED 186 243). Chaiklin, S., & Lesgold, S. (1984, April). Prealgebra students’ knowledge of algebraic tasks with arithmetic expressions. Paper presented at the meeting of the American Educational Research Association, New Orleans. (ERIC Document Reproduction Service No. ED 247 147). Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13, 16–30. Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly, 88, 286–290. Clements, D.H. (2000). Geometric and spatial thinking in early childhood education. Paper presented at the meeting of the National Council of Teachers of Mathematics, San Francisco. Clements, D.H., & Battista, M.T. (in press). Logo and geometry (Journal for Research in Mathematics Education Monograph Series). Reston, VA: National Council of Teachers of Mathematics. Clements, D.H, & Battista, M.T. (1990). The effects of Logo on children’s conceptualizations of angle and polygons. Journal for Research in Mathematics Education, 21, 356–371. Clements, D.H., & Battista, M.T. (1989). Learning of geometric concepts in a Logo environment. Journal for Research in Mathematics Education, 20, 450–467. Clements, D.H., Battista, M.T., & Sarama, J. (1998). Development of geometric and measurement ideas. In R.Lehrer & D.Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 201–225). Mahwah, NJ: Erlbaum. Clements, D.H., Battista, M.T., Sarama, J., & Swaminathan, S. (1996). Development of turn and turn measurement concepts in a computer-based instructional unit. Educational Studies in Mathematics, 30, 313–337. Clements, D.H., Battista, M.T., Sarama, J., & Swaminathan, S. (1997). Development of students’ spatial thinking in a unit on geometric motions and area. Elementary School Journal, 98, 171–186. Clements, D.H., Swaminathan, S., Hannibal, M.A.Z., & Sarama, J. (1999). Young children’s concepts of shape. Journal for Research in Mathematics Education, 30, 192– 212. Collis, K.F. (1975). The development of formal reasoning. Newcastle, Australia: University of Newcastle. Confrey, J. (1992, April). First graders’ understanding of similarity. Paper presented at the meeting of the American Educational Research Association, San Francisco. Confrey, J. (1994). Splitting, similarity, and the rate of change: New approaches to multiplication and exponential functions. In G.Harel & J.Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 291–330). Albany: State University of New York Press.
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Representative terms from entire chapter: