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Appendix D Invited Papers Stating the Obvious: Mathematics Course Taking Matters (W. Tate) Algebra, Technology, and a Remark of I.M. Gelancl (M. Saul) 133
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Stating the Obvious: Mathematics Coarse Taking Maters William F. Tate University of Wisconsin at Madison Madison, Wisconsin Many studies indicate that a disproportionate amount of minority and low SES students receive a mathematics education more closely associated with basic skills (Oakes, 1990; Secada,1992~. In the late 1960s and early 1970s, a mathematics reform movement, "back to basics," emerged, which focused largely on elementary and middle schools (NCTM, 1980~. This movement was partly a product of efforts to achieve equality of educational opportunity through compensatory education. The back-to-basics effort called for teaching a core set of rudimenta- ry mathematics procedures and facts, often to the exclusion of more advanced mathematical ideas. Although the basic skills movement influenced the entire educational system in the United States, it had a particularly significant impact on the mathematics curriculum and pedagogy in low-income urban and rural schools (Strickland and Ascher,1992~. On the positive side, the basic skills movement did result in improved standardized mathematics test scores for students who were traditionally underserved (Secada, 1992~. It illustrated that when teachers and principals agreed on a common standard in mathematics and received adequate institutional support to achieve the standard, students would learn the content. Another reason for the success of the basic skills movement had to do with teachers' beliefs. Much of the basic skills philosophy was consistent with teachers' perceptions of student ability (Zeichner, 1996; Knapp and Woolver- ton,1995~. However, as the vision of what it means to know and understand mathematics has shifted from the basic skills standard to a more demanding goal, the limits of past pedagogical practice have become obvious. Teaching strictly lower level, basic math skills to students throughout their elementary- and middle-school years will not adequately prepare them for the challenges of college preparatory mathematics. Moreover, the practice of tracking students out of college prep mathematics opportunities is akin to restricting them to the basic skills curriculum. SOME RECENT TRENDS: COURSE TAKING MATTERS The impact of restricting students to a basic skills mathematics curriculum is most apparent when studies of course taking and mathematics achievement are examined. Specifically, if the goal is high standards for all students, then a close examination of the influence of course-taking effects is essential. Hoffer and colleagues (1995) reported on the relationship between the number of mathematics courses that high-school students of different racial and SES backgrounds completed and their achievement gain from the end of grade 8 to grade 12. Hoffer et al. indicated that when African American and white students who completed the same courses were compared, the difference in average achievement gains was smaller than in other circumstances, and none was statistically significant. Similarly, Asian and white students' mathematics achievement gains were smaller and also generally reduced among students completing the same number of mathematics courses. More- over, none of the SES comparisons showed significant differences among students taking the same number of courses. These findings can be interpreted to mean that much of the racial and SES differences in mathematics 135
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136 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM achievement in grades 9-12 is the product of the different numbers of mathematics courses that African American and white, Asian and white, and high and low SES students complete during secondary school. Smith (1996) investigated the efforts of early access (eighth grade) to algebra on students' access to advanced mathematics courses and subsequent high-school mathematics achievement. She found that early access to algebra has an effect beyond increased achievement and, in fact, may socialize a student into taking more mathematics. In essence, having credit for a year of algebra at the beginning of high school is a credential, regulating access to more advanced course work in mathematics. Moreover, having this credential increases both the students' and educators' expectations about how much mathematics the student will take in high school, keeping students in the college prep track longer and producing higher achievement as a result. The importance of these two studies is that policy can be used to intervene.) One method of intervention is to mandate specific course requirements at the eighth grade andlor secondary level. To enhance policy intervention, student learning is required at the elementary level. Clearly, mandating more secondary courses without systemic efforts to change elementary-school mathematics experiences and achievement levels is potentially problematic. Further, mandates come with costs avoidance and implementation costs. FINAL REMARKS: IMPLEMENTATION AND AVOIDANCE COSTS Like all policies, mandating mathematics courses has costs associated with it. Space limitations prevent detailed discussion of those costs. However, it is important to note that school systems must examine their practices and goals closely to determine both the social and fiscal costs related to mandating and implementing courses. Many schools have mandated "algebra for all," then constructed multiple tiers of algebra courses. Thus, the policy is implemented on paper, but the intent of the policy is compromised. The costs associated with this kind of practice are avoidance costs. Another tactic that might be employed is to mandate eighth grade "algebra for all," then to create a separate track for a select group of students to take algebra in seventh grade or earlier, thus recreating the credentialing system described by Smith (1996~. I charge teachers and administrators contemplating the mandating of courses to examine implementation and avoidance costs connected to this important policy matter. The future of many underserved students is at stake. REFERENCES Hoffer, T. B., Rasinski, K.A., and Moore, W. (1995.) Social Background Differences in High School Mathematics and Science Course Taking and Achievement (NCES 95-206~. Washington, DC: U.S. Department of Education. Knapp, M.S., and Woolverton, S. (1995.) "Social Class and Schooling." In J.A. Banks and C. Banks (eds.), Handbook of Research on Multicultural Education (pp. 548-569~. New York: Macmillan. National Council of Teachers of Mathematics (NCTM). (1980.) An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, VA: Author. Oakes, J. (1990.) "Opportunities, Achievement, and Choice: Women and Minority Students in Science and Mathematics." In C.B. Cazden (ed.), Review of Research in Education (pp. 153-222~. Washington, DC: American Educational Research Association. Secada, W.G. (1992.) "Race, Ethnicity, Social Class, Language, and Achievement in Mathematics." In D. A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 623-660~. New York: Macmillan. Smith, J.B. (1996.) "Does an Extra Year Make Any Difference? The Impact of Early Access to Algebra on Long-Term Gains in Mathematics Achievement." Educational Evaluation and Policy Analysis, 18~2), 141 - 153. Strickland, D. S., and Ascher, C. (1992.) "Low-Income African American Children and Public Schooling." In P.W. Jackson (ed.), Handbook of Research on Curriculum (pp. 609-625~. New York: Macmillan. Zeichner, K. M. (1996.) "Educating Teachers to Close the Achievement Gap: Issues of Pedagogy, Knowledge, and Teacher Preparation." In B. Williams (ed.), Closing the Achievement Gap: A Vision for Changing Beliefs and Practices (pp.56-76~. Alexandria, VA: Association for Supervision and Curriculum Development. tin contrast, it is nearly impossible to intervene in or change teacher beliefs. If teachers' beliefs are inconsistent with the policy proposed, a significant hurdle is created for the policy-maker interested in implementing a reform e.g., algebra for all.
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Algebra, Technology, and a Remark of I.M. Gelfand Mark Saul Bronxville School Bronxville, New York SOME VIEWS OF ALGEBRAIC THINKING Histoncally, the development of algebra grew from a consideration of the arithmetic of the rational numbers) and particularly from the investigation of methods of solution of polynomial equations. The introduction of variables to make general statements about rational numbers, or to pose and solve problems involving them, was a natural step, taken very early. The conception of the nature of algebra behind these efforts is perhaps most easily seen in Isaac Newton' s book on elementary algebra, which bore the title The Universal Arithmetic.2 Newton himself gives a clear exposition of the philosophy: "Common arithmetic and algebra rest on the same computational foundations and are directed to the same end. But whereas arithmetic treats questions in a definite, particular way, algebra does so in an indefinite, universal manner, with the result that almost all pronouncements which are made in this style of computation and its conclusions especially- may be called theorems. However, algebra most excels, in contrast with arithmetic where questions are solved merely by progressing from given quantities to those sought, in that for the most part it regresses from the sought quantities, treated as given, to those given, as though they were the ones sought, so as...to attain some conclusion that is- equation from which it is permissible to derive the quantities sought.... Yet arithmetic is so instrumental to algebra in all its operations that they seem jointly to constitute but a unique, complete computing science, and for that reason I shall explain both together."3 An essentially different view of algebra emerged in the twentieth century. Morns Kline points out its roots even earlier: "Algebra, for Descartes, precedes the other branches of mathematics.... There is a sketch of a treatise on algebra, known as Le Calcul (1638), written either by Descartes himself or under his direction, that treats algebra as a distinct science. His algebra is devoid of meaning. It is a technique of calculation, or a method, and is part of his general search for method.... Descartes' view of algebra as an extension of logic in treating quantity suggested to him that a broader science of algebra might be created, which would embrace other concepts than quantity and be used to approach all problems."4 ~See, for example, Victor J. Katz, "The Development of Algebra and Algebra Education," in The Algebra Initiative Colloquium, edited by Carole B. Lacampagne, William Blair, and Jim Kaput. (Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement, 1995), 15-30. 2D.T. Whiteside, editor. The Mathematical Papers of Issac Newton, Volume V, 1683-1684 (Cambridge: At the University Press, 1972), 538ff. 3Ibid., 539. 4Morris Kline, Mathematical Thoughtfrom Ancient to Modern Times (New York: Oxford University Press, 1971), 281. 137
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138 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM It was a bit early for this bold step, and Descartes did not complete this work. However, Kline traces echoes of its influence on the work of Leibniz and De Morgan (1187-1189) in developing mathematical logic.5 It was not until the twentieth century that the conception of algebra as the study of manipulation of certain symbols acquired meaning. Katz notes that "it was Emmy Noether who taught the mathematicians in Gottengen in the 1920s that algebra was central to mathematics, that its ideas extended to all areas of the subject, and that an abstract approach was the way to look at algebraic concepts." 6 A REMARK OF I.M. GELFAND I.M. Gelfand offers a twentieth century view of the nature of algebraic thinking that has meaning both for mathematics and for pedagogy.7 His remark: "Arithmetic is essentially the study of the field of rational numbers. Algebra begins with the study of the field of rational expressions." This remark draws an arbitrary but clear line between arithmetic and algebra. It highlights the fact that the essence of algebra is not simply the use of variables. It also distinguishes the study of algebra from the study of functions. Gelfand illustrates his idea with two examples. A common activity for middle-school or upper elementary students is the "guess my number" trick. For example, the teacher asks the student to pick a number, double it, add three, then subtract the original number. If the student gives the answer, the teacher can easily "guess" the number that the student had originally chosen. The learning activity is then for the student to explain the trick, to be able to use it herself, or to make up an original "trick" of the same sort. This activity is an excellent one to get students to generalize statements about arithmetic. In trying to express what is going on, they often are led directly to the use of variables. But there is another level to algebra, the level of computation with rational expressions, which it does not reach. This level is reached when students learn patterns of factoring, for example. Seeing the factorization A2 _ B2 = (A + B)(A - B), a student who has mastered certain algebraic concepts can factor, ~ _ y4, 4X2 _ 1, or cos2 x - sin2 x. Mathematically, the student is then letting the variables A and B stand for other rational expressions (or rational trigonometric expressions). Crudely put, in arithmetic, letters are used to stand for numbers; in algebra they are used to stand for other letters. Gelfand's statement is a type of definition, and, like all definitions, it is arbitrary. It is an attempt to use language to distinguish one set of mathematical concepts from others. We will make the argument that this particular distinction is useful and enlightening, both from a mathematical point of view and from a pedagogical perspective. ALGEBRA AND PEDAGOGY Gelfand's remark implies a view of algebra different from that espoused by several recent writers on learning and teaching. For example, Judah Schwartz has said, "A proposal is made to restructure the post-arithmetic mathematics curriculum around the function (and the entailed concept variable) as the central, and indeed the only necessary mathematical and pedagogical object of the subjects now called algebra, trigonometry, pre-calculus, and calculus."8 This is a pedagogical remark, not a mathematical one, and perhaps Schwartz had in mind the courses labeled "algebra" in many schools and textbooks. A strong argument can be made for using the function concept as a unifying theme in the development of students' mathematical thought. However, we will make the argument that, even pedagogically, it is important to distinguish algebraic thinking from thinking about functions, regardless of the judgment that is then made about the role of either in education. The conception of algebraic thinking implied by Gelfand's remark is closer to that implied by some notes by Kieran and Chalouh: "Most algebra courses begin immediately with the use of letters as mathematical objects and Ibid., 1187-1189. turbid., 30. 7Personal communication to the author, November 1995. Quoted in Algebra for the Twenty-First Century: Proceedings of the August 1992 Conference. (Reston, VA: NCTM, 1992), 26.
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APPENDIX D 139 then proceed to the operations that can be earned out on these objects."9 Kieran and Chalouh go on to describe the experiences that students need to understand algebra: "Links between the use of numbers in arithmetic and the use of letters in algebra are rarely accorded more than a passing nod in high school algebra."~° Implicit in these comments is the idea that algebra is different from its application to anthmetic. We argue below that algebra is a separate entity from either the study of functions or a generalization of arithmetic but that both are important applications of the central ideas of algebra. HOW IS THE STUDY OF ALGEBRA RELATED TO THE STUDY OF ARITHMETIC? The fact that students can "learn" algebra without understanding its application as a generalization of arithmetic poses problems for every teacher of the subject. The following two vignettes reflect the author's own classroom practice in grappling with these problems. Vignette I This scene took place some years ago in an urban high school. The teacher (the present writer) was working on a fairly standard New York State Regents curriculum with a group of well-motivated, high-ability ninth graders. The lesson took place in April, after the class had covered several topics in the factoring of polynomials. The teacher wrote on the board: 4x4= 16 SxS =25 6 x 6 = 36 7 x 7 = 49 3xS= 15 4x6=24 Sx7=35 6x8 =48 15 x 15 = 225 14 x 16 = 224 (This scene took place before calculators were readily available. A more modern practice might be to structure the examples so that the students, using the calculator, would get the answers to the multiplication problems themselves.) The teacher then asked the class two questions: 1) Multiply 149 x 151 mentally. 2) Factor the number 3599. The students in the class had learned all the "factoring patterns" necessary for success in the standard algebra course and, in particular, had some facility with the fact that (x - 1) (x + 1) = x2 - 1. They could factor such expressions as 4a2 _ 1, gx2 _ 4y2, and even (a + b)2 - (a - b)2 with relative ease. Yet many of them were puzzled when faced with this situation. Tom, for example, engaged in the following dialogue with the teacher: Tom: But what are the dots for? Teacher: Well, what comes after 7 x 7 = 49? Tom: 8 x 8. [His voice indicates that he has come to the end of his response.! Teacher: Then? Tom: .equals 64. Teacher: Good. And then? Tom: 9x9 =81. Teacher: Excellent. And what is in the right column? [He points to the space described.] 9Kiernan, Carolyn, and Louise Chalouh, "Prealgebra: The Transition from Arithmetic to Algebra" in Research Ideasfor the Classroom: Middle Grades Mathematics, Douglas T. Owens, editor (Reston, VA: NCTM, 1993), 179. 10Ibid.
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140 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Tom: 8 x 10 = 80. Teacher: Why? Tom: Well, the number before 9 is 8, and the number after 9 is 10. Teacher: OK. Write the next line. Tom: [He fills in both columns of the next line, then makes a statement.] I see. The dots mean "and so on..." So... [He pauses.] 149 x 151 is the same as 150 x 150. Teacher: The same? Tom: Oh. It's one more. Less. [He says this in a loud voice, as if to erase the previous error.] [He was then able to solve both problems posed.] Teacher: Now suppose we wrote x x x here [in the left column] . What could you write here [in the right column] ? Tom: (x-l~and~x+l). Teacher: And what is (x- D(x + 1~? Tom: X2 - 1. Oh, cool. That says that the numbers here [right column] are one less than these numbers [left column]. I knew that. Discussion of Vignette I Tom is physically a large young man who plays baseball with older students. At this point, his frame is trembling with excitement at the fact that he can understand a difficult problem. And the teacher also is excited both about Tom's experience and about the teacher's own discovery that Tom's thorough knowledge of factoring is different, on a cognitive level, from an understanding of the application of factoring to the description of numerical patterns. The teacher's (my) experience with Tom is typical: students who "know" algebra and even who test well on traditional algebraic subject matter may not see how algebra can tell us something about the arithmetic of rational numbers. Can we say that they do not understand algebra? They certainly understand how to work with rational expressions. What they do not yet see is the application of these expressions to the arithmetic of rational numbers. Learning algebra may or may not involve learning a "universal arithmetic." As Kieran and Chalouh point out, the application of algebra to arithmetic requires a different set of experiences than the acquisition of purely algebraic skills. Vignette 2 This scene took place in a small remedial class. The tenth grade students were well motivated but had problems learning the arithmetic of fractions and negative numbers. Marion, for example, subtracts negative numbers by considering eight cases. If she is working on a - b, she considers the signs of a and b and also whether a is larger or smaller than b in absolute value. She has no mental image of what she is doing but simply applies the eight rules she has learned. She succeeds about two-thirds of the time-enough to pass any short-answer standardized test. However, Marion can do well with certain other problems that can be regarded as purely algebraic in nature. An example follows: Teacher: [Writing on the chalkboard, he says the following.] Three apples and 2 bananas cost 13 cents. Two apples and one banana cost 8 cents. What is the price of 5 apples and 3 bananas? Marion: I don't know. How much does an apple cost? Teacher: I'm not sure. But I don't think you need to know. Think of a package of 3 apples and 2 bananas. [He draws on the board.] And then a package of 2 apples and 1 banana. [He draws on the board.] Marion: Oh. You just get one of these and one of those. Teacher: So how much do you spend? Marion: 13 + 8 = 21 cents. Teacher: And suppose you wanted to get 6 apples and 4 bananas? Marion: Easy. You buy 2 of those [she is pointing to the first package] for 26 cents. Teacher: And how about 8 apples and 5 bananas? Marion: [She thinks a bit.] You could buy 4 of those. [She points to the package of two apples and a banana.] No. That would only give you 4 bananas. We need 5. I give up. Teacher: Well, take some of these [he labels the package of 3 apples and 2 bananas as "P"], and some of those [he labels the package of 2 apples and 1 banana as "Q"]. Marion: OK. I could take 2 P's. Then I need 1 Q to make 8 apples. Teacher: But do you have the right number of bananas? Marion: Well... yes! I get 4 bananas from the 2 P's and 1 more from the Q.
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APPENDIX D Discussion of Vignette 2 141 This lesson evolved into a unit on the solution of simultaneous equations. Later on, the students were able to use the letter a to represent the unknown price of an apple and b to represent the price of a banana. They then were able to add and subtract linear expressions in a and b. They also could "scale" them by writing Ma + 2bJ = l5a + lob, for example. Often, they had to return to the drawings of packages of apples and bananas for inspiration. Perhaps more important, the students in this class were able to perform the quintessentially algebraic operation of substitution. That is, their description of Ma + 2bJ + Ma + by as "five of these packages and three of those packages" evolved into the expression SP +3Q, where P stood for 3a + 2b and Q for 2a + b. Marion and her classmates were learning useful algebraic operations without referring to arithmetic. That is, they would have more trouble evaluating 3a + 2b when a = 1/5 and b = 2/3 than they had evaluating SP + 3Q as described above. Conclusion about the Vignettes The two vignettes above are intended to show examples of how algebraic thinking can be distinguished from arithmetic, or generalized arithmetic. HOW IS THE STUDY OF ALGEBRA RELATED TO THE STUDY OF FUNCTIONS? The confounding of a function with its algebraic representation has deep historical roots. It is common in the history of mathematics to find that the idea of a function originated as a name for an algebraic expression. From a more modern viewpoint, it is clear that the study of functions is not the same as the study of algebra. Many functions can be represented graphically more easily than algebraically. For others, we cannot write a closed algebraic expression and so have contrived names, such as sin x or Ax), to remind us of the function of which we are thinking. In still other cases, such as functions of a random variable, the functional values are not related to computations, either arithmetic or algebraic. In each of these cases, however, we use techniques of algebra to handle the functions, once they have been named. It is eauallv clear that algebraic expressions can be studied anart from their use in representing functions. - -~ -- - - ./ - - - - - - - - - - C7 - - - - - - ~- - - - - - - - - - - - - -- -- - -- - - ~- - - - ~ C7 Perhaps the simplest example of this is in studying the integers module N (for some fixed N). Two polynomials with coefficients in such a set may be distinct as polynomials but may define the same function. Having separated algebraic thinking from thinking about functions. we should expect that the corresponding ~ ~ ~ ~ ~ , ~ ~ ~ pedagogy might differ as well. Indeed, while algebra is a useful tool for the study of functions, recent literature is replete with examples of classroom work supporting the function concept in ways that avoid the use of algebraic expressions. i2 AN HYPOTHESIS CONCERNING THE TEACHING OF ALGEBRA To summarize the argument so far: (1) Algebraic thought begins with the study of the field of rational expressions; (2) Algebraic thought is not the same as working with variables nor is it the generalization of arithmetic nor is it the study of functions; and (3) Algebraic techniques find important applications in generalizing arithmetic and in studying functions. What should we make of the experience of so many teachers (including the author) that a direct study of algebra as a set of formal rules for manipulating rational expressions leaves the student unenlightened? Classroom practice has shown that a frontal attack at the ideas of algebra (as defined by Gelfand's remark) will not give the student access to the power and utility of algebraic techniques but, rather, will render it, as one colleague has said, "an intensive study of the last three letters of the alphabet."~3 ~. . . ~. ~. . ~. . . 1lFor example, Euler sometimes used the term function this way. See example, Carl B. Boyer, A History of Mathematics (New York: John Wiley and Sons, 1968), 484ff. 12See, for example, Stephen S. Willoughby, C`Activities to Help in Learning about Functions," in Mathematics Teaching in the Middle School, Vol. 2, No. 4, February 1997. The author of this article makes a conscious and successful attempt not to use algebraic expressions in addressing the function concept with middle-school students.
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142 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM The mathematical statement of Gelfand concerning the relationship of arithmetic to algebra perhaps can be used as a guide here. It has long been recognized that the learning of arithmetic is most effectively approached through its applications. Early learning about numbers must be very concrete: a set of two birds has something in common with a set of two cupcakes, and both are different in a certain way from a set of three cupcakes. Later on, this "something in common" becomes the concept of cardinal number. Similarly, teachers build on concrete applications of the operations of addition and multiplication to develop the concepts they imply, then the algorithms that reflect them. Our hypothesis is that the learning of algebra proceeds in just the same way. We start students on the road to algebra with its concrete applications, such as the description of number patterns or functions. Later on, the algebraic ideas are used to solve polynomial equations (where the algebraic expressions represent rational, real, or complex numbers) and to manipulate functions. Slowly, through years of experiences, the algebraic expressions take on a life of their own, and the student can use them wherever necessary without reference to their interpretation as numbers, for example. The student is then working within the field of rational expressions. A COURSE CALLED ALGEBRA A look at the "algebra" taught in schools today confirms this hypothesis. We have slowly moved from a stiff and formal course in symbolic manipulation towards an approach to algebra through its applications. Indeed, the typical first-year "algebra" course contains very little algebra (in the twentieth-century sense). It has long included such arithmetic topics as signed numbers and numerical radicals (although a case can be made that ~ is neither treated nor perceived as a number, in the sense of quantity, in these courses). For at least 40 years, this course has included extensive work with functions, including graphical and other representations, as well as the use of algebraic techniques. In more recent decades, it has included work in combinatorics and even logic, both with and without the application of algebraic concepts to these areas. This is not a new observation but only a new description of an evolutionary process that we have taken in matching our pedagogy to the achievements of twentieth-century mathematics. It explains and unifies a number of recent observations about the teaching of algebra. For example, Jim Kaput calls for us to "de-coursify Algebra: to weave strands of algebra throughout the grades, so that algebraic ideas grow naturally with the child."~4 This practice has proved effective in the teaching of arithmetic. There is no "course" labeled arithmetic in elementary school. The teaching of arithmetic has been integrated into the development of other mathematical concepts. Perhaps the most emblematic acknowledgment of this change was the renaming of the National Council of Teachers of Mathematics journal from The Arithmetic Teacher to Teaching Children Mathematics. The danger in "de-coursifying" algebra is that we may lose what is valuable in the subject. Identifying the study of the field of rational expressions as the essential beginning of algebra will held us keep algebraic thinking in the curriculum, while we also approach it through other areas of mathematical thought. ALGEBRA FOR ALL? How do these observations about the nature of algebra impact on the issue of who should learn algebra? Is it important for lawyers to learn about the field of rational expressions? Or nurses? Or kindergarten teachers? Why should the study of rational expressions be so important to the work force of tomorrow? These are not simple questions. A deeper consideration, which is out of place here, would quickly lead us close to a central problem of the philosophy of mathematics; namely, the question of why the mental constructs of mathematics (if that is what they are) so unreasonably match our experience of the physical world. The connection Beverly Williams, personal communication with the author, April 1997. whim Kaput. "Long-Term Algebra Reform: Democratizing Access to Big Ideas," in The Algebra Initiative Colloquium, edited by Carole B. Lacampagne, William Blair, and Jim Kaput (Washington: U.S. Department of Education, 1995), 34.
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APPENDIX D 143 between the learning of mathematics any mathematics and physical (i.e., economic) survival in later life is more complex than some recent writings would indicate. A simplistic answer, sometimes given by educators whose expertise is in fields other than mathematics, is a sort of "socialist realism" doctrine of mathematics: that school mathematics must always be related to concrete physical objects or demonstrably useful in earning a living. This solution is certainly not correct. One can think of numerous specific exercises or problems that may seem contrived (from this point of view) but that serve as efficient stepping- stones to a deeper understanding of mathematics. Algebra (in Gelfand's sense) may be seen as one of these stepping-stones. In another sense, a clear understanding (on the part of teachers and writers of curriculum) of algebra and what it means can be seen as an important unifying strand for the teaching of other, more obviously "useful" mathemat- ics. Finally, offering algebra to all students is offering an opportunity to all students. Not many of our students will end up thinking like Emmy Noether, but it is important to allow as many students as possible access to this career path, whether or not they eventually follow it. For all these reasons, twentieth century algebra, like other areas of mathematics, is certainly important for all students whether or not the students themselves reach the point where they can consciously identify the algebraic ideas they are using. ALGEBRA AND TECHNOLOGY We have seen that the focus of algebraic thinking is on the study of the field of rational expressions and operations on them. How have, or how will, the use of new technologies impact the study of algebra? We can do no more than suggest where to look for fruitful applications and see how these opportunities play out in future classrooms. We can distinguish two categories in the use of new technologies. On the one hand, technology can allow us to teach better (or sooner) concepts that we have been teaching all along. On the other, technology can allow us access to new concepts or ideas that we have not been able to address in more traditional environments. But in beginning this search for opportunities, we immediately run into a difficulty: What is "better?" Does sooner mean better? Does easier mean better? Which uses of technology are of value, and which are window dressing? Again, our examination of the nature of algebraic thinking can help out. We already have a certain amount of experience with the use of calculators in teaching arithmetic. Perhaps we can anticipate some of the same experiences in teaching algebra. For example, a judicious use of the calculator can help with the acquisition of mechanical arithmetic skills. By anticipating the calculator' s results, students can deepen their sense of how the operations of arithmetic work. By exploring number patterns with the calculator, students can begin the process of generalizing arithmetic much earlier and much more naturally than they could without calculators. By using the calculator with the ugly numbers of "real" data and comparing their work with a problem involving simple but contrived data, they can strengthen their concept of the arithmetic operations. Let us look at the corresponding situations for algebra. One algebraic analogue of the calculator is symbolic manipulation software, such as "Derive," "Mathematica," or "Maple" (although symbolic manipulation software is closer to algebraic computation than most calculators are to arithmetic computation because the issue of decimal representation and approximation of answers does not occur). On the simplest level, students can certainly check algebraic manipulations against the software. But the possibility of doing algebraic computations quickly and accurately immediately presents another opportunity; of exploring algebraic patterns. The following is an example of a classroom exploration that I have found useful where good use of symbolic manipulation software can be made. This exploration can be used as early as a first-year algebra level or with more advanced students. We will look at how the problem set might be used without, then with, symbolic manipulation software.
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144 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Problem 1: Factorx2- 1 (Students who cannot do this easily are not ready for this exploration.) Problem 2: Factor X3 - 1 (We assume that students are not yet familiar with the factorization of the difference of two cubes. If students are working the examples by hand, it might be best to ask them to show that x - 1 is a factor of X3 - 1.) Problem 3: Factor x4 - 1 Problem 4: Factor X5 - 1 (By this time, students will generally be able to guess that x - 1 is a factor. The form of the other factor is not difficult to guess in this particular case. Or, students can use a division algorithm for polynomials to obtain it.) Problem 5: One of two factors of X)5 - 1 iS X - 1. What is the other factor? (Students frequently are able to guess the form of the second factor, and checking the computation by hand in this case is not really tedious.) From here, the exploration can take a variety of paths. Two examples are the factorization of an _ bn in general and the formula for the sum of a (finite) geometric progression. How does the same sequence of problems play out with symbolic manipulation software at hand? The first four problems are not very different. However, the software will probably give the complete factorization of xn - 1 and not just the two factors that indicate the particular pattern shown above. If students guess at the form of the second factor, they can check it with the software. This guess can be motivated by the form of the second factor when n is pnme. But much more can be done with the software. For example, students might consider the following factorizations, which would probably not be available to them working manually: X5 - 1 = ~X- 1~`X4 +X3 +X2 +X+ 1' XiO - 1 = (X - l)(x - 1~(X4 + X3 + x2 + X + 1~(X4 + X3 + X2 + X + 1' i5 _ 1 ~X- 1~`X2 +X+ 1~`X4 +X3 +X2 +X+ 1~(XS -X7 +X5 -X4+X3-X+ 1) (and so on). A variety of patterns can be observed, the most prominent of which is that certain factors keep coming up. A bit of inquiry will yield the relationship between these recurring factors of the polynomial xn - 1 and the prime factors of the number n. While a complete study of the cyclotomic polynomials is certainly outside the scope of a high- school course, many insights can be gleaned on this level. Some of these can be developed into methods or theorems, especially with the study of complex numbers (for which an entirely different set of software explorations might be developed) or the factor theorem. Work with algebraic patterns is not yet well documented in the literature. Some easy places to look for applications of this idea are in exploring the binomial theorem, teaching factoring patterns, or exploring algebraic symmetry (for example, by using the software to permute the variables). An entirely different set of activities allows for the exploration of algebraic ideas through their applications in studying functions. CONCLUSIONS The current trend in education is to unify the teaching of different areas and to create strands throughout the grades of subjects formerly taught in discrete packages. This trend has helped us to reach past mathematical techniques to mathematical concepts. As we do this, it is important that we not lose track of the nature of the concepts we are addressing. An inquiry into the nature of algebraic thinking and its relation to other areas of mathematics, as well as into the ways in which we encourage it by our students, will help us in our efforts to place this important tool at our students' disposal.
Representative terms from entire chapter: