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I(eynote Addresses Algebra with Integrity and Reality (H. Bass) Making Algebra Dynamic and Motivating: A National Challenge hi. Dossey)
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Algebra with Intensity and Reality Hyman Bass Columbia University New York, New York I come to you as a university mathematician who does research in algebra and who is interested in improving mathematics education in the schools. While I have lived professionally with the subject we call algebra, I am not an expert on what algebra means and looks like in the school curricula. Indeed, there does not seem to be much consensus about this. Much of the discussion here seems to envisage broad areas of mathematics, such as functions or modeling, and to identify within them certain algebraic ideas and techniques. I choose instead to focus on some cognitive issues in algebra and on our sense of the number systems that are at the root of algebra. This focus is germane to teaching in the elementary grades on up. In preparing this talk, I have benefited greatly from communications with Deborah Ball, Joan Ferrini-Mundy, and Mark Saul, to whom I extend my thanks and appreciation. WHAT IS ALGEBRA? To begin, I shall adopt a working definition of algebra that may strike you as traditional. In some sense, the subject grows from its efforts to solve equations. The starting point, rooted in our experience with counting, is the natural numbers: O. 1, 2, 3, 4, 5,....i Then, trying to solve equations such as 8 + x = 5, involving only natural numbers, leads us to introduce negative numbers and, so, to arrive at the integers: ..., -2, - 1, O. 1, 2, 3, 4, 5,.... Then, in order to solve equations such as 2x = 3, with integer coefficients, we are led to introduce fractions, i.e., the rational numbers: numbers of the form p/q with p and q (i O) integers. HI am following Bourbaki in including "O" in the natural numbers, and, in fact, there are some mathematical reasons for doing this. There is divergent usage, however. 9
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10 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Many natural equations, such as x2 = 2, cannot be rationally solved. We could continue enlarging our number system accordingly. However, even if we adjoin the solutions of all polynomial equations with rational coeffi cients, we still won't capture all of the real numbers. The passage from the rational numbers to the real numbers is not via solutions to equations but, rather, by a very different process of "geometric completion," a sort of "filling in the holes" left by the rational numbers in the "real line." This process is not discussed explicitly in the schools and not even in most college mathematics courses. One of the main issues I want to address here is, What sense of the real numbers is appropriate and useful for school mathematics ? I want to advocate the following "definition," about which I'll say more later: real numbers are the points on the real number line, as shown here: o 1 With real numbers, we select an origin, 0, and a unit, 1, to the right of the origin, and then each point is measured by its (oriented) distance from the origin, counted negatively if it is to the left of 0. In particular, by counting off multiples of the unit distance, we are able to identify integers with (evenly spaced) points on this real number line. Squares of real numbers are positive, so equations like x2 + 1 = 0, have no solutions in the real numbers. To solve them, we are led to introduce the complex numbers. These are numbers of the form z = x + iy, where x and y are real numbers, and i denotes a square root of -1. We often identify z with the point (x,y) in the coordinate plane and so identify the set of complex numbers with the plane, the real numbers then forming the x-axis. The so-called "Fundamental Theorem of Algebra" says that we can now stop here, in the following sense: Any algebraic (i.e. polynomial) equation with complex coefficients can be solved with complex numbers. 2 This is not, of course, the end of the story of equations. The complex numbers simply give us a stable environment in which solutions live. We can still ask how deeply into the complex numbers we have to go to find solutions for particular types of equations and how we can describe them. For example, we know the familiar quadratic formula for the solutions of quadratic polynomial equations. Is there a similar formula, involving only rational operations and extraction of roots (square roots, cube roots, and so on) that describes solutions of higher degree polynomial equations? After much searching, the answer turns out to be "no," starting with degree five. The methods invented to prove this gave rise to a major branch of algebra called "group theory." The most important recent successful analysis of algebraic equations came with Andrew Wiles' celebrated proof of Fermat's Last Theorem. The complex plane is a very attractive mathematical object. The arithmetic of the complex numbers gives a natural and integrated setting in which to introduce some linear algebra and in which to study the Euclidean transformations of plane geometry, an attractive possibility for the high-school curriculum. This topic deserves a paper of its own. But I want to focus here on some more fundamental, cognitive issues. OUR SENSE OF THE NUMBER SYSTEMS Suppose that I ask you, "What is a number?" The answer might be an instance of it, a visual or symbolic representation, a mental image .... What I want to emphasize also are major cognitive distinctions between thinking of individual numbers and thinking of "number systems." For natural numbers and integers, the answer to my question is pretty clear and natural, and it is well reflected in early school mathematics. Counting activities and discovery of number patterns with integers prevail. Numbers are represented as counts of collections of objects (apples, rods, tiles, for example). These settings provide models of addition and multiplication, and the integers are listed in order on an integer number line. The mental sense of the number 5 might progress from images of 5 apples or 5 children or 5 circles or 5 marks to a place labeled "5" on the integer number line. For very large numbers, the 2Despite the use of the words "theorem" and "algebra," this is not really a theorem in algebra, since its proof has to invoke non-algebraic ideas from either calculus or topology.
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KEYNOTE ADDRESSES 11 concrete images are no longer so intuitive, but the compressive power of place-value notation and the indefinite geometric extension of the integer number line can still bring them, at least implicitly, into one's mental universe. What is the corresponding story for real numbers, and how are the two developments related? I asked myself, "What, in the minds of school teachers and students, is a real number?" This is hard to know, but one can at least look at textbooks and curricula. One of the explicit answers given is that "real numbers are the rationale and the irrationals." For this answer to be sensible, there has to be an implied understanding of some "number universe" in which the "irrationals are the complementary set to the rationale." According to the above definition, the real numbers would constitute this implied number universe. But has this universe been explicitly described? Until it is, the question of what the real numbers are has not really been answered. If one looks at curricula for an operational definition of real numbers, they seem to appear ultimately, and only tacitly, as numbers represented by decimals as a kind of by-product of the division algorithm for integers. This definition has some mathematical body, but it is technical, not intrinsic, and it poses some computational problems, as we shall see. The answer that I want to propose as appropriate for school mathematics is that a real number is a point on the real number line. You may well ask, "What do you mean by the real number line?" My answer is that I view this as one of the primordial objects of mathematical experience, on which we can build our early mathematical learning. Just as whole numbers are rooted in our primitive experience of counting and enumeration of collections of discrete objects, so too slightly later do we have qualitative experience with (continuous) geometric measurement of height, distance, speed, for example and in such activities as cutting paper models, sewing, simple carpentry, and so on. This gives us a natural intuitive access to and geometric sense of the real line as a scale of measurement, even before we have a developed notational system for designating individual real numbers. Our geometric intuition is a sound and coherent foundation on which to build a geometric discussion of the basic arithmetic operations with real numbers, as I shall try to illustrate. And I suggest, also, that there are significant advantages to giving this geometric model of the real numbers an earlier and more prominent appearance in our school curricula. First let us look at how the American curriculum develops the number systems. As far as I can tell, early mathematics instruction begins with counting numbers. Addition is modeled by aggregating collections of objects: 5 + 7 is represented by a basket of 5 apples, combined with a basket of 7 apples. Multiplication is modeled by iterated addition: 5 x 7 is represented by 5 baskets, each with 7 apples. In the course of this, one learns the base- 10 place-value notation for numbers and develops the conventional algorithms first for addition and then multiplica- tion of such numbers first by memorizing single digit multiplication and then by learning an algorithm that reduces general products to a sequence of steps, each being a single digit multiplication or an addition. Fractions are then introduced and also division and decimal notation. The rules for arithmetic are extended to fractions and finite decimals, in each case as though these were new situations. When students are taught to divide integers, they encounter infinite decimals. In this way, finally, the real numbers, as arbitrary decimals, appear unannounced from the wings, almost as a kind of by-product of the algorithm for division in place-value notation. If this is how real numbers emerge in the American curriculum, it presents some mathematical difficulties. For example, the conventional algorithms for both addition and multiplication begin with operations on the rightmost digits in decimal form. What can this mean for infinite decimals? This can be addressed mathematically, of course, but it doesn't seem to be. In fact, the very existence and properties of multiplication of real numbers appear to be unspoken assumptions, given as a part of some canon of unexplained origin or perhaps simply provided as a calculator command. To illustrate the mathematical problem, contrast the understanding of arithmetic with fractions with that of real numbers. For example, multiplying by 1/2 is understood to represent cutting, or dividing, something in half. But how is one to interpret multiplying by ~ or by it? Yet students will comfortably discuss expressions involving xy, where x and y are tacitly allowed to vary over all real numbers. It seems to me that the most satisfactory way to define multiplication for all real numbers is in terms of areas of rectangles, using the geometric model of the real numbers. The present curricular approach, almost exclusively tied to base-10 place-value and decimal notation for numbers, does not support a conceptually well-grounded sense of the real number system. Missing is an early and systematic emphasis on the geometric representation of the (continuous) real line and, correspondingly, on the geometric representation of the operations of arithmetic. The classical Greek geometers did not have the benefit of our place-value notation but did have a deep sense of the real number system and focused a great deal of attention on number theoretic questions, e.g. the existence and
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-. 1 1 1 1 1 1 1 1 12 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM irrationality of certain square roots and the nature of the number it. For the ancient Greeks, real numbers were measures of geometric quantities (length, area, and so on). It is in that same spirit that I propose that we found our sense of the real numbers on the geometric real line model (see below), in which we first name the integer points and eventually name all of the points with the aid of decimal notation. Of course, there is the serious question about whether what I am advocating can be made accessible to young students. -4 -3 -2 -1 0 2 1 2 3 4 1 , 1 ' 1 1, 1 I ~r 1 17r What's the point? What does one gain by this? First of all, one has an intuitively accessible and authentic model of the real numbers that can be apprehended at a very early age Second, the model's geometric representation is completely intrinsic and not dependent on the structure of place-value or any other arbitrary notation. Third, the basic operations of +, -, and x have very natural aeon Metric interpretations, making some properties transparent when they are not at all in the combinatorial or counting model. For example, addition can be done by juxtaposing intervals; i.e., a + b is obtained by placing the left end of an interval of length b at the right end of one of length a, as shown by the first figure below. Subtraction, a - b, can be modeled by making the right ends of the intervals coincide and then locating the left end of the lo-interval, as shown by the second figure below. a 1 ' b Cab a -- 1 a-b b Multiplication, in contrast, is two dimensional; i.e., a x b is the area of a rectangle with sides a and b. Commutativity is made transparent, a fact not obvious in the combinatorial model: compare 5 baskets of 7 apples with 7 baskets of 5 apples. The rectangle model of multiplication also makes the distributive law transparent. Notice further that these geometric constructions directly, and intrinsically, define the operations for all real numbers, whether or not they have been named (e.g. by place-value notation).3 a I axb b =: a b ad + ~ = a x (bl+ c) a b C b+c The process of adding or multiplying a fixed number with a variable number can be geometrically interpreted as a transformation of the real line. Adding 2 is a translation distance 2 to the right. Adding -5 is a translation- distance 5 to the left. Multiplying by 3 is a tripling of scale. Multiplying by 1/4 is a reduction of scale to one-fourth size. Multiplication by -2 is a doubling of scale, combined with a (mirror) reflection through the origin. It is this reflection interpretation of multiplication by negative numbers which demystifies the infamous formula, (-a)~-b) = ab, because double reflection brings you back to where you started. Such illuminating discussion of arithmetic operations as geometric transformations of the line seems to be absent from the current curriculum until one investigates what happens to the graph of a function, y Infix), when you change x to ax, or x + a, or change y to ay, or y + a. But there is hardly need to wait for the discussion of function graphs to introduce this transformational point of view. 3There is a serious underlying problem here: explaining how one can convert a measure of area into a real number (i.e., a linear measure). This may require reversion to the repeated addition model but with grids of finer and finer units and then a passage to a limit. In the early grades, one could give only the most heuristic account of this.
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KEYNOTE ADDRESSES 13 If one reaches the complex numbers, then the same transformational interpretation of addition and multiplica- tion (plus complex conjugation) leads to all of the fundamental Euclidean (rigid) motions of plane geometry, on which the notion of geometric congruence and similarity of figures is based. This discussion has suggested how one could approach some of the early learning of arithmetic using the geometric model of the real numbers. Next we will look at some of the features of this model at a more advanced level of study. EUCLIDEAN DIVISION First, I am not arguing for a de-emphasis of place-value notation. After all, place-value is the basis for our most effective algorithms for arithmetic. Thus, if the above geometric point of view is to receive emphasis, it must be reconciled with place-value notation. This reconciliation is itself a very instructive undertaking and can be based on some of the fundamental properties of real numbers, which themselves perhaps deserve increased attention. Place-value flows from a basic result, sometimes called "Euclidean Division," which I believe deserves some emphasis, since, as I shall illustrate below, it has many other significant applications as well. The idea is as follows: Given a number b > 0, we mark off the real line with "fence posts" at distance b apart, with one post at 0. These posts are then located at the set of integer multiples of b and look on the real line just like the integers but for a change of scale. -4b -3b -2b -b O b 1 1 1 1 2b 3b 4b 1 1 1 1 - 1 1 1 1 ' ' ' ' -~ Now any given number will lie between two consecutive posts. If qb (with q being an integer) is the nearest post to the left of a, then we have this picture: qb < a < (q + lab 1 1 1 1- 1 1 - qb a (q+l)b Subtracting qb as represented on the previous page, we have O < a - qb < b Now, when we put r = a - qb, we have proved (heuristically) Euclidean Division, which is as follows: · Fix a real number b > 0. Then for any real number a, there exist a unique integer q ("quotient") and real number r ("remainder"), 0 < r < b, such that a = qb + r. Note that if a and b are integers, then so also is r = a - qb. OBSERVATIONS Let me make some observations to explain the significance of this result. 1. This "proof" was purely geometric (and intuitive) and not dependent on any special notation for numbers, other than their geometric definition on the real line. 2. The result is related to ordinary division as follows: Dividing by b transforms the equation a = qb + r to a/b = q + r/b, with q an integer and 0 < r/b < 1. Here, q called the "integer part" of a/b is what appears to the left of the decimal point when we divide a by b, while r/b called the "fractional part" of a/b is, when written in decimal form, what appears to the right of the decimal point when dividing a by b.
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14 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM For example, with b = 123 and a = 1997, we have a/b = 1997/123 = 16.235772357723577..., so q = 16 and r/b (= 29/123) = 0.235772357723577.... 3. Consider what happens for b = 10. For example, taking a = 5,297, we have 5,297 = 529 x 10 + 7, so q = 529 and r = 7, the units digit of a. Repeat this with q = 529 to get 529 = 52 x 10 + 9. Substituting this in the first equation, we get 5,297 = (52 x 10 + 9~10 + 7 = 52 x 102 + 9 x 10 + 7. Finally, writing 52 = 5 x 10 + 2, and substituting in the last equation gives 5,297 = (5 x 10 + 2) 102 + 9x 10+7=Sx 103+2x 102+9x 10+7. Thus, repeated application of Euclidean Division to the successive quotients yields the expansion of a as a sum of powers of Jo, with base-lO digits (the numbers O. l, 2, 3,..., 9J as coefficients. This is exactly equivalent to the base-lO place-value representation of a. The significance here is that we obtained it intrinsically from repeated application of Euclidean Division. Moreover, by exactly the same process, for any integer b > 1, we can so obtain the base-lo place-value representation of any integer a > 0 . Further, if a > 0 is not an integer, then we can apply the above to the integer part of a, and then apply a related but modified procedure to the fractional part of a to derive the place-value representation of a to the right of the "b-ecimal" point. While this is all a bit technical, I simply wish to emphasize that Euclidean Division contains the genesis of the place-value representation of all numbers, to an arbitrary integer base b > 1. This is one reason for its mathematical importance. 4. Given integers a and b, suppose that we want to calculate their greatest common divisor, gcd~a,b). A natural way to try to do this is first to calculate the prime factorization of a and b and then, for each prime p, to take the highest power of p dividing both of them, and, finally, to take the product of these prime powers. This turns out to be a computationally burdensome method. The task of factoring very large integers is extremely laborious, even with powerful computers. Indeed, the difficulty of this problem is the source of security in some of the commonly used cryptosystems. On the other hand, one can quite simply calculate gcd~a,b) without knowing the prime factorizations, as follows. Say that b < a. Then we can (Euclidean) divide a by b to get a remainder r < b. If r = 0, then b divides a, and b = gcd~a,b). If r > 0, we then divide b by r and get a new remainder, r' < r. If r' = 0, then r = gcd~a,b). If r' > 0, we then can continue in the same fashion. This process must terminate in a finite number of steps, as the successive remainders keep getting smaller. More formally, we may define the (decreasing) sequence of integers al, at, as, a4,... by al = a, a2= b, and, for n > 1, am is the remainder obtained when dividing ant by an. This decreasing sequence of non-negative integers strictly decreases until it reaches 0 in a finite number of steps, and the last non-zero term is gcd~a,b). Thus, Euclidean Division gives us an algorithm for calculating the greatest common divisor, gcd~a,b), for any two integers a and b. A further important consequence of this algorithm is an expression of the form, gcd~a,b) = ra + sb, for suitable integers r and s. It is this latter result that underlies the proof of unique prime factorization for integers, i.e., the so-called "Fundamental Theorem of Arithmetic." 5. One of the most interesting and far-reaching analogies in mathematics is the similarity of the arithmetic of the integers and that of the "ring" of polynomials in one variable x, say with real or complex coefficients. The starting point of this analogy is Euclidean division of polynomials, where now the "size" of a polynomial is measured by its degree. Again, we have the following: ~7~
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KEYNOTE ADDRESSES 15 Given polynomials b and a, with b ~ 0, there exist unique polynomials q and r, so that degree(r) < degree(b), and a = qb + r. The idea is to keep subtracting suitable multiples of b from a to keep lowering the degree, until one obtains a remainder of degree smaller than that of b. Formally this argument is carried out by mathematical induction on degree(aj. Now, just as for the integers, this leads to an algorithm for calculating the greatest common divisor of two polynomials and, as above, to a proof of unique factorization for polynomials. 6. There is a further interesting, formal similarity between place-value notation for integers and for polynomi- als. As noted above, base-10 place-value representation corresponds to expression of a number as a sum of powers of 10, with digit coefficients. For example, 5,297 = 5 x 103 + 2 x 102 + 9 x 10 + 7. If we substitute "x" for "10" here, then this becomes a polynomial. So we can more or less think of place- value notation as expressing numbers as "polynomials in 10" instead of "polynomials in a variable x." But the analogy is not so simple, and it breaks down when one analyzes the rules for arithmetic. The way that this breaks down in fact illuminates some of the complications of our algorithms for arithmetic. Consider first two polynomials, for example, a = aO + Ax + a2x2, and b = be + box. Then we have, from commutativity of addition and distributivity of multiplication, a + b = (aO + be) + (a + b.)x + a,x2~ and a x b = aObO + (albO + aObl)x + (a2b0 + albl)X + a2blX For polynomials, this is the end of the story. The reason is that the coefficients of polynomials, the scalars (integer, rational, real, or complex), are closed under addition and multiplication. Now, in contrast, imagine in the equations above that we replace x by 10 and assume that the coefficients aO, al, a2, bo, b1 are base-10 digits. Then exactly the same formulas for addition and multiplication apply, since they are the result simply of commutativity and distributivity of addition and multiplication. The complication arises from the fact that new coefficients, such as al + b1 or a2b0 + albl, need no longer be base-10 digits; they may be larger than 10. Hence, to express the answers in standard place-value form, we have to remove some lO's, and "carry" a 1 to the adjacent coefficient on the next higher power of 10. This is the source of the "carries" in our standard algorithms for addition and multiplication. SUMMARY I have argued that it is both natural and advantageous to give an early emphasis to the geometric real line model of the real numbers, in which basic arithmetic operations are interpreted geometrically and developed alongside the more algorithmic development rooted in base-10 place-value representation. I also have used Euclidean Division to illustrate several important algebraic phenomena. First, I have shown how it can be used to derive place-value representation from the geometric model of the real line. Second, you can see that, when applied to integers, Euclidean Division furnishes an effective algorithm for calculating greatest common divisors and the basis for proving the "Fundamental Theorem of Arithmetic" unique prime factoriza- tion. Then, an analogue of Euclidean division for polynomials can be used to lead to analogous results, thus showing a "mathematically similar structure," so to speak, for the arithmetic of integers and the arithmetic of polynomials. (I want to note that searching for this kind of similarity of mathematical structure in different looking settings is characteristic of research in algebra and, indeed, in all mathematics in general.) Finally, I have offered some remarks on the similarities and differences between integer and polynomial addition and multiplication.
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Making, Al~,~ebra Dynamic and Motivating,: A National Challenge ,iahn A. Dossey Illinois State University Normal, Illinois I have been given the task of discussing what high-school algebra is and what it should be. To lessen the suspense, I will tell you that I am not going to give you the answer as to what it should be. However, this is a question that deserves to be addressed in a place as august as the National Academy of Sciences. I cannot think of a more important time in the history of our country than now to focus efforts on the teaching and learning of algebra in our schools. As Hy Bass has shown, algebra has a long and beautiful history, but the translation of that into a school subject has lost some dynamism. For many students and teachers, algebra has become a mere passing on of information from generation to generation. To prepare my remarks, I went back and looked at school books for algebra from more than a century ago. I found that they are not very different from my high-school algebra book and, in fact, not very different from some books that have 1996 copyrights. Charles Davies' algebra text (1846), which is a translation of the French version by M. Bourdon, begins with some definitions that focus on number systems. Next, it looks at polynomial expressions and solutions of first-degree equations. The focus then shifts to exponents and radicals, at least to the point you can talk about quadratic equations and what square roots might be. Davies then does a little factoring and solutions of quadratic equations and systems of linear equations. Does all that sound familiar to you? As my text analysis moved toward the present, I noticed some differences. In some cases, there has been a permutation of topics. In some cases, the solution of first-degree equations appears before the discussion of polynomial expressions. These differences serve to highlight the shift in the treatment of the subject from meaning to manipulation. Overall, the text analysis showed the very static nature of the curriculum in secondary school algebra. Discussing the role of algebra in the broader realm of the K-14 curriculum is an even more daunting task. After all, we are products of the system of algebra teaching that I have just outlined. When we talk about making changes in algebra teaching and learning, we are talking about changing the system of concepts, skills, and knowledge that provided part of the basis for our being here today. We are the survivors of that system and the sifting process it helped define. The very skills and knowledge that gave us our identifications as mathematicians and mathematics educators those knowledgeable about mathematics and its teaching were a result of our performance as we moved through "algebra." Now, one might say, "We have the Curriculum and Evaluation Standards for School Mathematics now, and, so, things are changing." But what do we see when we look at the National Assessment of Educational Progress (NAEP) results and the Third International Mathematics and Science Study (TIMSS) reports from teachers about what they are doing in the algebra classroom? The data show a little bit of movement toward doing projects. They show a little bit of movement toward using technology. Involved mathematics educators immediately react to this stating that they know many teachers who really are changing. However, if you triangulate from the NAEP 17
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18 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM assessments, the TIMSS findings, and from dissertation research on teacher beliefs, you see a movement toward change that can be measured in terms of geologic time scales. There has been change as a result of the Standards. It is of an undeniable and growing magnitude. But, it is not clear that it has taken a uniformly accepted direction. This growing vector of change may yet lead the way to a new algebra for school mathematics. We see the beginnings of change in algebraic related pedagogy. It is not clear that the underlying content to which this pedagogy is applied has really received due attention yet, however. Referring to M. Fullan's book (1993) on educational change, we probably are on course. We are now nearly 10 years past the release of the Standards and that is an appropriate time frame for the initiation of ideas. We now need to shift to the beginning of the implementation phase. The question is, what do we implement? We must take into consideration that beyond the stage of implementation is institutionalization. That is what we are suffering from right now. We are suffering from the institutionalization of a certain type of school algebra that is at least 200 years old. What changes do we see in new programs and new books? In programs, we see a lot of nice problems building from patterns to projects involving expressions. In new books and other materials, we see a shift in the way variables are addressed and introduced (but in many cases that shift has only been a pagination shift from pg. 55 to pg. 3~. In projects, algebra seems to have lost some of its structural core and the cohesiveness that makes it a field of study as described by Hy Bass. In the texts, we still see hesitation to break free of the idea of a variable being an unknown and the relentless march from integers to quadratic equations via the solution of linear equations. Algebra is beginning to be seen as the gatekeeper to quantitative literacy, in which quantitative literacy is viewed as comprising more than statistics. If we think of calculus as being either a pump or a filter, I think we have to look at algebra as being the faucet. What do things like the NAEP results, the Conference Board of the Mathematical Sciences' (CBMS) report on undergraduate programs, and so on, tell us about the algebra enterprise? First of all, such information tells us that about 24 percent of our students today are taking a course called "algebra" in 8th grade or earlier. Further, we know that 94 percent of students have taken some sort of a course with "algebra" in its name by the time they reach the 12th grade. Of that 94 percent, one-half will have taken a second course in algebra called "algebra II" or something like "intermediate algebra" or "algebra/trigonometry." Beyond this point, we know that enrollment drops precipi- tously, with about 13 percent of all students going on to take a college preparatory mathematics class before high- school graduation. Now, with that kind of a background, what do these students carry away from their algebra course or courses? I think the movie "Peggy Sue Got Married" captures it well. In it, actress Kathleen Turner is drifting back like a ghost into her high-school algebra class. Suddenly, she says that her algebra teacher lied because she never did need algebra in the world beyond that classroom. We teaching professionals indeed most educated adults have never successfully refuted her claim, at least not with students and not on a regular basis. This symposium provides us with an opportunity to focus on what school algebra is, how it can be structured, and to note the roles that algebra plays in the lives of citizens. MODELS FOR UNDERSTANDING SCHOOL ALGEBRA The 1988 NCTM Yearbook, The Ideas of Algebra, K-12, and the recently released Approaches to Algebra: Perspectives for Research and Teaching (Bednarz, Kiernan, Lee, 1996) point to several models we might consider in understanding school algebra. One of these is structure. How do we take a set of objects and impose an operation or a set of operations on them? How do we bring this beauty and structure to students and at what age? One might think of these objects as symbols. One might think of them as more concrete entities. With objects and operations, we can begin to look at properties and at statements of generality that we can put with them. Much of the history of our doing this algebra has evolved from our actual work with computation and our ability to abstract from the structures surrounding computation. We all know about sequences of decreasing exponents on integers and using them to decide what a negative integral exponent means. We have taken a look at various patterns that result from operating with matrices, small matrices, two by two, and then trying to deduce from a set of these what an inverse might look like and if and when inverses might exist. We have developed properties, and, from these properties, we have created structures. It is here that we begin to speak of algebra. Representations are developed for ideas that we have learned to manipulate verbally. There has been recent discussion about the rule of three: graphical, tabular, and symbolic representations. I think we have to have a rule of four, including verbal representation, and think of verbal and the
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KEYNOTE ADDRESSES 19 other three representations as located at the vertices of a tetrahedron. The edges of the tetrahedron then serve as models of the shifts in representations that we hope students can engage in for a given concept. How do we take all these words and translate them? How do we actually assess students' development of linguistic representations? How do we look at students' growth in making shifts along the edges of this tetra- hedron? Is it the students' ability to make such shifts that shows us best the beginning of the students' depth of understanding? Many concepts in algebra are tied up in language. Hy Bass just spoke about the geometric representations, but there are hierarchies within that representation. There is length, measured in centimeters. There is area, and it is represented in square centimeters. That is a bit more difficult to think about. Then there is volume, which comes in cubic centimeters. That is even more difficult for many students to conceptualize. And there are rates, which are comparisons of objects that have different units. Students can handle rates when doing comparisons of such things as price per soda can. However, if I start to talk about velocity or density, I hit a wall with students because these rates are not rooted in their concrete conceptions of rate. Part of the wall is the students' inability to think of objects, to tie them to a concrete model, and to develop the linguistic ability to function with that model. This is part of learning and thinking about algebra, an algebra of representations. There also is quantification: "each," "all," and "some." These words, along with "and," "or," and "not," play major roles in the translation of problems to the real world. Another model for algebra is built around the study of functions and relations. There appears to be a great tension between the more structural, historical, polynomial-based approach and the functional representation approach. Some would say that the function approach is really more analysis than algebra. If so, where do we cross the line, and does it really matter? I think that these issues are not as important as developing a clear understanding by students of the mathematics that underlie what we want them to know and be able to do. This is especially true in situations involving simple systems and numerical relationships but also functions and operations that represent the relationships. Finally, in terms of models, there is the modeling version of what algebra might be, which involves looking at and building models to represent real-world contexts and problems within those contexts. This approach often draws on statistics, analysis, geometry, and many other forms of representation. What is most important as we take a look at algebra across the curriculum is for us to begin to think about the translations and the transitions that are necessary as students move from understanding arithmetic to beginning to understand divisive or rate-based structures and how to operate in them. Moving on to generalize examples in these situations is algebraic. This type of modeling approach relates back to the approach that builds relational models from the data. Reflecting on the four representations of algebra that I just outlined structural, linguistic, functional, and modeling none of them can survive or prosper alone. In fact, what we really have to do is think of how to merge them to support a coherent program of study in algebra. That means developing a program of study that enables students to meet four goals of algebra in the context of the school mathematics program. FOUR GOALS FOR SCHOOL ALGEBRA The following four goals for school algebra programs are to help students see and use algebra as a way of · representing quantity and relationships among quantities; · predicting what happens in quantitative settings; · controlling, where possible, the outcomes to quantitative processes; · extending the applications and establishing the validity of new relationships in the structure of algebra. These goals may look a bit different than the customary goals one would list for students of algebra. For example. we normally don't think about using algebra to Predict. but that is exactly what we do when we build models and apply them. We use algebra in this fashion because it allows us to look at how the manipulation of objects could suggest what might happen. We use algebra to control when we establish a structure of statements that runs a computerized assembly line. We use algebra to control the paths of action we might want to follow in running a process.
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20 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Beyond these goals, we need to look at how an algebra program provides motivation for students; i.e., How does it address why they are studying algebra? How does it relate to their future and to their daily lives? How does it provide a basis for their continued learning? We can do many interesting things with algebra at several points in the K-14 curriculum, but what we do may not provide the foundation for the next steps students themselves may have to take or will want to take sometime in the future. We constantly need to be looking at how we can provide a base for student growth in algebraic thinking. Along these lines, we also must work to prevent Peggy Sue's lack of connection between her algebra class and the outside world. I, too, have never had to package pecans, peanuts, and cashews to make mixtures in my life! BIG IDEAS There are a few big ideas that all students need to understand. The most important is variable. There are many roles for variables, and Zal Usiskin has listed and characterized them well in his lead article for the NCTM Yearbook (1988~. Among the uses listed by Usiskin, I would place a great deal of stress on expression of functional relationships, when variable is a symbol expressing a quantity but, further, for expressing a quantity that varies. From whence does algebra grow? It grows from the study of growth itself. One of the first places students see growth is when they look at patterns and patterns of numbers. When they arrange centimeter cubes to model a n~tt`~rn of nilmb`~r~ Allah ~ ~ ~ Q 1 n 1 o - ~ - - - ~ - - - - - -, - - - - c' - - - HI Jo KIEV ~11 ~ ~ Vat ~ ~ JO ~ ~ ~ ~ a... ~ they are beginning to study growth. This pattern becomes apparent in something that they are doing with pattern blocks, such as putting them together and looking at the perimeters as they extend a pattern of consecutive squares. Or it may come as a result of a calculation. What is central, however, is the pattern of arithmetic growth. Now, when we look at arithmetic growth at a later stage in our learning, we might look at this growth a little differently. We might look at it as an application of difference equations where the next value is the previous value plus some addition. In our pattern block example, four was my initial value, and then each next value was the previous value plus two; i.e., Pn+1 = Pn + 2 and where P1 = 4. As elementary-school children begin to build out from their study of patterns, they are developing the foundation of the next big idea linearity. Many children approach linearity recursively and then move to an understanding that approximates seeing linearity like a function of a real variable. However, this transition is really hidden from them, hidden, in fact, even from many undergraduate students. When we come to examine change in linear situations, we first see it as adding on unitary change, the idea behind slope. Eventually, we shift to developing algebraic formulas, written something liken = 2n + 4, for the perimeter example we examined earlier. The second form of growth students encounter is geometric growth. This is evident in situations having a constant multiplier. They may start with 2, and then each successive value comes from multiplying that previous value by 2. The result is the pattern of 2, 4, 8, 16, 32, 64,.... Another big idea is exponential growth. We want to be able to capitalize on student actions to develop other ways of looking at growth. Situations may have some additive parts and some multiplicative parts. Some situations may have some additive parts, some geometric parts, and some may even have a strange function tacked on that might be polynomial, logarithmic, exponential, trigonometric, or who knows what? We want students to be able to take a look at a growth situation, ferret out the patterns embedded in it, and develop algebraic expressions that allow them to predict, control, and understand what makes it tick. This is one of the lessons that Peggy Sue missed. Algebra is the key to understanding change in the world about us. Consider the typical annual report graph from a Fortune 500 company. Such graphs commonly show how the Standard & Poor's index is moving, how the stock price of the company is moving, and how the volume of shares of the company sold in a given day has varied. Now, if you want to talk about things being a function of other things, all the little points along this graph are revealing of significant happenings, in the company or in the market itself, for example. If one wanted to place a value on that company in order to sell it, all of these data points would be important for constructing a helpful mathematical model. Some of the models that investment bankers build are systems of difference equations. If you look at derivatives on Wall Street, you are looking at systems of partial differential equations. Mathematics and algebra play a very strong role in modeling financial things in our world. Peggy Sue, take note. I want to step back into the student's world. Suppose you had the task of answering the following market research question. It is a common question involving which TV shows capture which portions of the audience:
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KEYNOTE ADDRESSES Background Consider the problem of market research as sociated with monitoring the audience share watching a given late night show. Each night the surveys show that 25 percent of yuppies watching the "Tonight Show" with Jay Leno watch it again the next night. Likewise, 50 percent of the yuppies watching the "Late Show" with David Letterman watch again the next night. The remaining audience shifts back and forth. Problem a) Model the number of individuals watching the shows over time with a system of difference equations. b) Find the general solution for the system of difference equations. c) Given that Tom Cruise and Cindy Crawford are on the season premiere of the "Tonight Show" and all yuppies watch that show, find the particular solution for the system of difference equations. d) For the long term, determine what portion of the yuppies watch the "Late Show." 21 You can probably see the system of equations in the situation outlined above, but the system actually happens to be a system of difference equations. How would you solve it? Well, perhaps there is a relationship between how we would approach this and how we would approach a system of differential equations, equations that are algebraic in form and that also account for change. In the past, we used systems of linear equations with algorithms like Cramer's Rule because they appeared to give us quick and efficient solutions. It was a paper-and-pencil method that was fairly efficient for very low- dimension systems. But the introduction of technology into the classroom has permitted us to start to use the power of algebraic structure to consider what we mean by efficiency. We quickly see that Cramer's Rule is not very efficient once the size of systems increases. Further, it requires us to add additional baggage to the school curriculum determinants. In fact, once we start to take a look at solving systems of linear equations, we really want to take a look at what analogies we can build for our students. We can solve ax = b as a polynomial equation x = a-ib. But if we use the same type of structure, and we want to solve a system of equations, then we have coefficient matrix A multiplied by X, the variable matrix, equaling the constant matrix B. And this can be expressed by AX = B. Its solution, where it exists, is given by X = A-iB. Are there some similarities here? What have we gained and what have we given up in making such changes in the content and approaches to our study of linear systems in a technological age? Algebra is very much part of our world as representations and as models. Our ability to take and pull structure from problems, our ability to look at the objects involved, and our ability to act on these objects as algebraic objects constitute the essence of what we are about in school algebra. How do we manipulate? How do we actually take and interpret the results of those manipulations? How do we look at the linguistic parts? How do we look at the relationships in these situations? Is this not the essence of what we are talking about at this symposium? And how we can make it happen for our students at all levels in grades K-14? Today, the ways in which we represent things are changing. Bar codes and public key cryptography codes are now used to structure and represent number. And these structures have all types of uses. The numbers built into the ISBN codes on the back of a textbook or the grids of bars on your most recent FedEx package provide you with security. The digits on the back of the book you want have a check digit built into to them to make sure that the typist does not transpose them while entering your order in the computer. (The check digit is an algebraic property of such numbers making application of a structural property of numbers based on divisibility by 11.) A similar form of algebraic representation allows a can of Campbell_' s soup to be identified and accurately priced by the scanner in your local grocery store, as well as allows the grocer to make almost instant pricing changes. Bar codes are also used to identify individuals in security settings. What I have tried to do is talk about some of the different models of how we represent algebra. How might we look at beginning to put them together in a program across grades K-14 for students? One of the first things we want to do is involve students with number and with data, and I think we want to do this at all levels. The variety of representations that number may take from integers to matrices of complex numbers will require a great deal of flexibility from students. They must learn to see that data can come in many forms. We want students to look for and to begin to describe pattern within its context. Pattern within and among numbers calls on students to start to draw on their representational and linguistic skills. We want them to start to recognize the quantity that is embedded in the data they confront. What is that which can be measured? What is the quantity, that is, the essence of the particular setting? We might want students to move to expressions, to take that variable and be able then to put it into a form that makes use of and begins to look like what we call algebra. What
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22 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM we have done at this point if not before is to begin to solve equations. The essential difference is that we already have moved through the study of functional representation. We have moved to function when we represent the pattern as 4 + 2x, but we haven't earned all the trappings of functional notation with us. Interestingly, students are thinking functionally. They are looking at direct and joint vanation. I think that we can look at an expression and that we can evaluate it; i.e., we can use formulas. I think students can begin to ask, does the form 2x + 4 ever take on the value 100? If so, when? This leads to the building of real equations from expression forms to answer the question of whether a given number will ever appear in the list of potential outcomes. Finally, the student moves to modeling and building models. The sequence of steps involves rich data, recognition of pattern, developing quantity within that pattern, representing that quantity through the use of vanable, developing a function-like expression to represent typical values within the pattern, using equations to study the pattern, and using the models developed to represent, manipulate, predict, and control with algebra. How do we mold these visions of algebra into a coherent program for students in grades K to 14? How do we build the concepts of vanable, table, graph, expression, function, equation, and others into a solid program a program that deals with properties, pnnciples, procedures, and proof and has the kinds of understandings I talked about earlier? How do we develop articulation in these programs? We no longer can afford high-school programs that don't accept 8th-grade programs or college programs that do not recognize good work done at the secondary school levels. We have to have school system stakeholders talking to one another across important articulation points. We have to know, to some degree, more about the foci at particular points in the curriculum because, without that kind of information about the curnculum, there is no way to monitor opportunities afforded to our children, our students, our future citizens. When no one has the responsibility for seeing that students have the opportunity to receive, expenence, and move through quality instruction, or to build carefully a program that includes these important steps, we are in trouble. We have to think about how we assess progress in these important stages to learning and using algebra. This is essential because, in many cases in the past, assessment systems have been the biggest deterrents to changing school algebra. Perhaps, at least, I have given you some food for thought about the history of school algebra and its status. The future is in our hands in the hands of all who walk with and work with children in mathematical situations. I hope that the algebra programs we move to are more motivating for students lead students to see the need for algebra and the dynamic roles it can play in their lives. REFERENCES Bednarz, N., Kieran, K., and Lee, L. (1996.) Approaches to Algebra: Perspectives for Research and Teaching. Dordrecht, The Netherlands: Kluwer Academic Publishers. Coxford, A. F., and Shulte, A. P. (Eds.) (1988.) The Ideas of Algebra, K-12. 1988 Yearbook of the National Council of Teachers of Mathematics (NCTM). Reston, VA: NCTM. Davies, C. (1846.) Elements of Algebra. New York: A. S. Barnes & Co. Fullan, M. (1993.) Change Forces: Probing the Depths of Educational Reform. London: The Falmer Press. National Council of Teachers of Mathematics (NCTM.) ( 1989.) Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.
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