Conceptions of School Algebra and Uses of Variables

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only convention causes us to use x instead of ? or but you can write “For any numbers a and b, a • b = K or ___ to represent an unknown. The verbal de- b • a.” The specific instance 6 • 12 = 12 • 6 looks scription of a situation, as in the first question, like the algebra and does not look at all like the Conceptions of School “What number, when added to 3, gives 7?” may verbal description. seem to be the least algebraic, but it was the way So you are doing algebra if you discuss generaliza- that many people did algebra before the invention Algebra and tions such as “Add 0 to a number, and the answer of modern symbolism in the 1590s. (The use of x is that number. Add a number to itself, and the re- and y to represent unknowns dates from Descartes sult is the same as two times the number.” But in- in the early 1600s.) Thus, there is a sense that you stead of writing them down in English, you use the Uses of Variables are doing algebra whenever you ask students to language of algebra (0 + n = n; t + t = 2t). find an unknown in a situation. Placeholders Formulas Most people have played Monopoly or other board If we have the formula A = LW for the area of a games in which the following kind of direction is rectangle and we ask students to find A when L = given: “Roll the dice. Whatever number you get, Zalman Usiskin 5 and W = 7, we are doing algebra. If we ask stu- move forward twice the number of spaces.” In al- dents to find n when 5 × 7 = n, whether we are gebraic language it means “If you roll d on the doing algebra is not clear. dice, then move forward 2d.” which have the same form—the product of two If the teacher asks, “What number can I replace n What Is School Algebra? Spreadsheets use algebra. Take the number in one numbers equals a third: by and make this a true statement?” the teacher is cell of an array, subtract it from a number in a sec- lgebra is not easily defined. The algebra treating the statement as algebra. If the teacher ond cell, and put the difference in a third cell. As A taught in school has quite a different cast 1. A = LW asks, “What is the answer?” then the teacher is treat- in the dice situation, we do not need to know from the algebra taught to mathematics majors. 2. 40 = 5x ing the question as arithmetic. The point is that what numbers we have to understand the direc- Two mathematicians whose writings have greatly 3. sin x = cos x • tan x much of the difference between arithmetic and al- tions. If the number in the first cell is x and the influenced algebra instruction at the college level, 4. 1 = n • (1/n) gebra is in the ways questions are couched. It is not number in the second cell is y, then the number in Saunders Mac Lane and Garrett Birkhoff (1967), 5. y = kx hard to do algebra, even with very young students. the third cell is y – x. begin their Algebra with an attempt to bridge school and university algebras: Each of these has a different feel. We usually call Generalized Patterns Consequently, whenever one plays a “pick a num- (1) a formula, (2) an equation (or open sentence) ber” game—pick a number, add 3 to it, subtract 5, Algebra starts as the art of manipulating sums, prod- to solve, (3) an identity, (4) a property, and (5) an My father was a bookkeeper by trade, and he and so on—one is verbally doing algebra, for one ucts, and powers of numbers. The rules for these equation of a function of direct variation (not to be manipulations hold for all numbers, so the manipu- taught me a number of shortcuts for doing arith- is thinking of a number, any number, and dealing solved). These different names reflect different metic. For instance, to multiply a number by 19, I lations may be carried out with letters standing for with it. the numbers. It then appears that the same rules uses to which the idea of variable is put. In (1), A, could multiply the number by 20 and then subtract L, and W stand for the quantities area, length, and the number. The algebraic description is short. If n hold for various different sorts of numbers … and Relationships that the rules even apply to things … which are not width and have the feel of knowns. In (2), we tend is the number, 19n = 20n – n. This special case of numbers at all. An algebraic system, as we will to think of x as unknown. In (3), x is an argument the distributive property of multiplication over sub- Bob is two years older than Marisha. What could study it, is thus a set of elements of any sort on of a function. Equation (4), unlike the others, gen- traction is called just the distributive property for be their ages? If Marisha is 7, then Bob is 9. If Mar- which functions such as addition and multiplication eralizes an arithmetic pattern, and n identifies an short. Notice how much shorter the algebraic de- isha is 4, then Bob is 6. We do not have to know operate, provided only that these operations satisfy instance of the pattern. In (5), x is again an argu- scription is than the description in words. Further- their ages to know how they are related. If Bob’s certain basic rules. (p. 1) ment of a function, y the value, and k a constant more, the algebraic description bears a visual re- age is represented by B and Marisha’s age is repre- If the first sentence in the quote above is thought (or parameter, depending on how it is used). Only semblance to the arithmetic. For instance, if you sented by M, then we could write the following: with (5) is there the feel of “variability,” from buy 19 notebooks at $2.95 each, substitute $2.95 of as arithmetic, then the second sentence is B = M + 2 (Bob is 2 years older than Marisha.) which the term variable arose. Even so, no such for n. school algebra. For the purposes of this article, B – M = 2 (The difference in their ages is 2.) then, school algebra has to do with the under- feel is present if we think of that equation as repre- 19 • $2.95 = 20 • $2.95 – $2.95 M = B – 2 (Marisha is 2 years younger than standing of “letters” (today we usually call them senting the line with slope k containing the origin. variables) and their operations, and we consider Many people can calculate the right side using men- Bob.) Conceptions of variable change over time. In a students to be studying algebra when they first en- tal arithmetic. It equals $59.00 – $2.95, or $56.05. text of the 1950s (Hart 1951a), the word variable is Any of these representations is correct. Although counter variables. The algebraic description just given suggests that there are many ways to write the relationship be- not mentioned until the discussion of systems (p. algebra is the most appropriate language for writ- tween B and M, they are equivalent. This equiva- However, since the concept of variable itself is 168), and then it is described as “a changing num- ing down general properties in arithmetic. You lence is easier to determine in the algebraic de- multifaceted, reducing algebra to the study of vari- ber.” The introduction of what we today call vari- may tell students, “You can multiply two numbers scriptions than in the English descriptions in ables does not answer the question “What is ables comes much earlier (p. 11), through formu- in either order, and the answer will be the same,” parentheses beside them. school algebra?” Consider these equations, all of las, with these cryptic statements: “In each

6 ALGEBRAIC THINKING, GRADES K–12 DEFINING ALGEBRAIC THINKING AND AN ALGEBRA CURRICULUM 7 Usiskin, Zalman. “Conceptions of School Algebra and Uses of Variables.” In Algebraic Thinking, Grades K–12: Readings from NCTM’s School-Based Journals and Other Publications, edited by Barbara Moses, pp. 7–13. Reston, Va.: National Council of Teachers of Mathematics, 1999. formula, the letters represent numbers. Use of let- for a vector; and in higher algebra the variable * It is clear that these two issues relate to the very Bushaw et al. [1980]). The actual record at the end ters to represent numbers is a principal character- may represent an operation. The last of these purposes for teaching and learning algebra, to the of 1985 was 3 minutes 46.31 seconds. istic of algebra” (Hart’s italics). In the second book demonstrates that variables need not be repre- goals of algebra instruction, to the conceptions we The key instructions for the student in this concep- in that series (Hart 1951b), there is a more formal sented by letters. have of this body of subject matter. What is not as tion of algebra are translate and generalize. These definition of variable (p. 91): “A variable is a literal obvious is that they relate to the ways in which Students also tend to believe that a variable is al- are important skills not only for algebra but also number that may have two or more values during variables are used. In this paper, I try to present a ways a letter. This view is supported by many ed- for arithmetic. In a compendium of applications of a particular discussion.” framework for considering these and other issues ucators, for arithmetic (Usiskin and Bell 1984), Max Bell and I relating to the teaching of algebra. My thesis is that Modern texts in the late part of that decade had a concluded that it is impossible to adequately study 3 + x = 7 and 3 + ! = 7 the purposes we have for teaching algebra, the different conception, represented by this quote arithmetic without implicitly or explicitly dealing conceptions we have of the subject, and the uses from May and Van Engen (1959) as part of a care- are usually considered algebra, whereas with variables. Which is easier, “The product of of variables are inextricably related. Purposes for ful analysis of this term: any number and zero is zero” or “For all n, n • 0 = 3 + ___ = 7 and 3 + ? = 7 algebra are determined by, or are related to, dif- 0”? The superiority of algebraic over English lan- Roughly speaking, a variable is a symbol for which ferent conceptions of algebra, which correlate with are not, even though the blank and the question guage descriptions of number relationships is due one substitutes names for some objects, usually a the different relative importance given to various number in algebra. A variable is always associated mark are, in this context of desiring a solution to an to the similarity of the two syntaxes. The algebraic uses of variables. with a set of objects whose names can be substi- equation, logically equivalent to the x and the !. description looks like the numerical description; tuted for it. These objects are called values of the In summary, variables have many possible defini- the English description does not. A reader in doubt variable. (p. 70) tions, referents, and symbols. Trying to fit the Conception 1: Algebra as Generalized of the value of variables should try to describe the Today the tendency is to avoid the “name object” idea of variable into a single conception oversim- Arithmetic rule for multiplying fractions first in English, then distinction and to think of a variable simply as a plifies the idea and in turn distorts the purposes In this conception, it is natural to think of variables in algebra. symbol for which things (most accurately, things of algebra. as pattern generalizers. For instance, 3 + 5.7 = 5.7 Historically, the invention of algebraic notation in from a particular replacement set) can be substituted. + 3 is generalized as a + b = b + a. The pattern 1564 by Francois Viete (1969) had immediate ef- fects. Within fifty years, analytic geometry had The “symbol for an element of a replacement set” Two Fundamental Issues in 3 • 5= 15 been invented and brought to an advanced form. conception of variable seems so natural today that 2 • 5= 10 Algebra Instruction Within a hundred years, there was calculus. Such is it is seldom questioned. However, it is not the only 1 • 5= 5 view possible for variables. In the early part of this Perhaps the major issue surrounding the teaching 0 • 5= 0 the power of algebra as generalized arithmetic. century, the formalist school of mathematics con- of algebra in schools today regards the extent to is extended to give multiplication by negatives sidered variables and all other mathematics sym- which students should be required to be able to Conception 2: Algebra as a Study of bols merely as marks on paper related to each (which, in this conception, is often considered al- do various manipulative skills by hand. (Everyone Procedures for Solving Certain Kinds other by assumed or derived properties that are seems to acknowledge the importance of students gebra, not arithmetic): of Problems also marks on paper (Kramer 1981). having some way of doing the skills.) A 1977 –1 • 5 = –5 Although we might consider such a view tenable NCTM-MAA report detailing what students need to –2 • 5= –10 Consider the following problem: to philosophers but impractical to users of mathe- learn in high school mathematics emphasizes the importance of learning and practicing these skills. This idea is generalized to give properties such as When 3 is added to 5 times a certain number, matics, present-day computer algebras such as the sum is 40. Find the number. MACSYMA and muMath (see Pavelle, Rothstein, Yet more recent reports convey a different tone: –x • y = –xy. and Fitch [1981]) deal with letters without any need The basic thrust in Algebra I and II has been to give At a more advanced level, the notion of variable as The problem is easily translated into the language to refer to numerical values. That is, today’s com- students moderate technical facility.… In the future, pattern generalizer is fundamental in mathematical of algebra: students (and adults) may not have to do much al- puters can operate as both experienced and inex- modeling. We often find relations between num- 5x + 3 = 40 perienced users of algebra do operate, blindly ma- gebraic manipulation.… Some blocks of traditional bers that we wish to describe mathematically, and drill can surely be curtailed. (CBMS 1983, p. 4) nipulating variables without any concern for, or variables are exceedingly useful tools in that de- Under the conception of algebra as a generalizer knowledge of, what they represent. A second issue relating to the algebra curriculum is scription. For instance, the world record T (in sec- of patterns, we do not have unknowns. We generalize known relationships among numbers, and so Many students think all variables are letters that the question of the role of functions and the timing onds) for the mile run in the year Y since 1900 is we do not have even the feeling of unknowns. stand for numbers. Yet the values a variable takes of their introduction. Currently, functions are rather closely described by the equation Under that conception, this problem is finished; are not always numbers, even in high school treated in most first-year algebra books as a rela- T = – 0.4Y + 1020. we have found the general pattern. However, mathematics. In geometry, variables often repre- tively insignificant topic and first become a major under the conception of algebra as a study of pro- sent points, as seen by the use of the variables A, topic in advanced or second-year algebra. Yet in This equation merely generalizes the arithmetic cedures, we have only begun. B, and C when we write “if AB = BC, then !ABC some elementary school curricula (e.g., CSMP values found in many almanacs. In 1974, when the is isosceles.” In logic, the variables p and q often [1975]) function ideas have been introduced as early record was 3 minutes 51.1 seconds and had not We solve with a procedure. Perhaps add –3 to stand for propositions; in analysis, the variable f as first grade, and others have argued that functions changed in seven years, I used this equation to each side: often stands for a function; in linear algebra, the should be used as the major vehicle through which predict that in 1985 the record would be 3 minutes variable A may stand for a matrix or the variable v variables and algebra are introduced. 46 seconds (for graphs, see Usiskin [1976] or 5x + 3 + –3 = 40 + –3

8 ALGEBRAIC THINKING, GRADES K–12 DEFINING ALGEBRAIC THINKING AND AN ALGEBRA CURRICULUM 9 formula, the letters represent numbers. Use of let- for a vector; and in higher algebra the variable * It is clear that these two issues relate to the very Bushaw et al. [1980]). The actual record at the end ters to represent numbers is a principal character- may represent an operation. The last of these purposes for teaching and learning algebra, to the of 1985 was 3 minutes 46.31 seconds. istic of algebra” (Hart’s italics). In the second book demonstrates that variables need not be repre- goals of algebra instruction, to the conceptions we The key instructions for the student in this concep- in that series (Hart 1951b), there is a more formal sented by letters. have of this body of subject matter. What is not as tion of algebra are translate and generalize. These definition of variable (p. 91): “A variable is a literal obvious is that they relate to the ways in which Students also tend to believe that a variable is al- are important skills not only for algebra but also number that may have two or more values during variables are used. In this paper, I try to present a ways a letter. This view is supported by many ed- for arithmetic. In a compendium of applications of a particular discussion.” framework for considering these and other issues ucators, for arithmetic (Usiskin and Bell 1984), Max Bell and I relating to the teaching of algebra. My thesis is that Modern texts in the late part of that decade had a concluded that it is impossible to adequately study 3 + x = 7 and 3 + ! = 7 the purposes we have for teaching algebra, the different conception, represented by this quote arithmetic without implicitly or explicitly dealing conceptions we have of the subject, and the uses from May and Van Engen (1959) as part of a care- are usually considered algebra, whereas with variables. Which is easier, “The product of of variables are inextricably related. Purposes for ful analysis of this term: any number and zero is zero” or “For all n, n • 0 = 3 + ___ = 7 and 3 + ? = 7 algebra are determined by, or are related to, dif- 0”? The superiority of algebraic over English lan- Roughly speaking, a variable is a symbol for which ferent conceptions of algebra, which correlate with are not, even though the blank and the question guage descriptions of number relationships is due one substitutes names for some objects, usually a the different relative importance given to various number in algebra. A variable is always associated mark are, in this context of desiring a solution to an to the similarity of the two syntaxes. The algebraic uses of variables. with a set of objects whose names can be substi- equation, logically equivalent to the x and the !. description looks like the numerical description; tuted for it. These objects are called values of the In summary, variables have many possible defini- the English description does not. A reader in doubt variable. (p. 70) tions, referents, and symbols. Trying to fit the Conception 1: Algebra as Generalized of the value of variables should try to describe the Today the tendency is to avoid the “name object” idea of variable into a single conception oversim- Arithmetic rule for multiplying fractions first in English, then distinction and to think of a variable simply as a plifies the idea and in turn distorts the purposes In this conception, it is natural to think of variables in algebra. symbol for which things (most accurately, things of algebra. as pattern generalizers. For instance, 3 + 5.7 = 5.7 Historically, the invention of algebraic notation in from a particular replacement set) can be substituted. + 3 is generalized as a + b = b + a. The pattern 1564 by Francois Viete (1969) had immediate ef- fects. Within fifty years, analytic geometry had The “symbol for an element of a replacement set” Two Fundamental Issues in 3 • 5= 15 been invented and brought to an advanced form. conception of variable seems so natural today that 2 • 5= 10 Algebra Instruction Within a hundred years, there was calculus. Such is it is seldom questioned. However, it is not the only 1 • 5= 5 view possible for variables. In the early part of this Perhaps the major issue surrounding the teaching 0 • 5= 0 the power of algebra as generalized arithmetic. century, the formalist school of mathematics con- of algebra in schools today regards the extent to is extended to give multiplication by negatives sidered variables and all other mathematics sym- which students should be required to be able to Conception 2: Algebra as a Study of bols merely as marks on paper related to each (which, in this conception, is often considered al- do various manipulative skills by hand. (Everyone Procedures for Solving Certain Kinds other by assumed or derived properties that are seems to acknowledge the importance of students gebra, not arithmetic): of Problems also marks on paper (Kramer 1981). having some way of doing the skills.) A 1977 –1 • 5 = –5 Although we might consider such a view tenable NCTM-MAA report detailing what students need to –2 • 5= –10 Consider the following problem: to philosophers but impractical to users of mathe- learn in high school mathematics emphasizes the importance of learning and practicing these skills. This idea is generalized to give properties such as When 3 is added to 5 times a certain number, matics, present-day computer algebras such as the sum is 40. Find the number. MACSYMA and muMath (see Pavelle, Rothstein, Yet more recent reports convey a different tone: –x • y = –xy. and Fitch [1981]) deal with letters without any need The basic thrust in Algebra I and II has been to give At a more advanced level, the notion of variable as The problem is easily translated into the language to refer to numerical values. That is, today’s com- students moderate technical facility.… In the future, pattern generalizer is fundamental in mathematical of algebra: students (and adults) may not have to do much al- puters can operate as both experienced and inex- modeling. We often find relations between num- 5x + 3 = 40 perienced users of algebra do operate, blindly ma- gebraic manipulation.… Some blocks of traditional bers that we wish to describe mathematically, and drill can surely be curtailed. (CBMS 1983, p. 4) nipulating variables without any concern for, or variables are exceedingly useful tools in that de- Under the conception of algebra as a generalizer knowledge of, what they represent. A second issue relating to the algebra curriculum is scription. For instance, the world record T (in sec- of patterns, we do not have unknowns. We generalize known relationships among numbers, and so Many students think all variables are letters that the question of the role of functions and the timing onds) for the mile run in the year Y since 1900 is we do not have even the feeling of unknowns. stand for numbers. Yet the values a variable takes of their introduction. Currently, functions are rather closely described by the equation Under that conception, this problem is finished; are not always numbers, even in high school treated in most first-year algebra books as a rela- T = – 0.4Y + 1020. we have found the general pattern. However, mathematics. In geometry, variables often repre- tively insignificant topic and first become a major under the conception of algebra as a study of pro- sent points, as seen by the use of the variables A, topic in advanced or second-year algebra. Yet in This equation merely generalizes the arithmetic cedures, we have only begun. B, and C when we write “if AB = BC, then !ABC some elementary school curricula (e.g., CSMP values found in many almanacs. In 1974, when the is isosceles.” In logic, the variables p and q often [1975]) function ideas have been introduced as early record was 3 minutes 51.1 seconds and had not We solve with a procedure. Perhaps add –3 to stand for propositions; in analysis, the variable f as first grade, and others have argued that functions changed in seven years, I used this equation to each side: often stands for a function; in linear algebra, the should be used as the major vehicle through which predict that in 1985 the record would be 3 minutes variable A may stand for a matrix or the variable v variables and algebra are introduced. 46 seconds (for graphs, see Usiskin [1976] or 5x + 3 + –3 = 40 + –3

8 ALGEBRAIC THINKING, GRADES K–12 DEFINING ALGEBRAIC THINKING AND AN ALGEBRA CURRICULUM 9 Then simplify (the number of steps required de- tions is that, here, variables vary. That there is a some students have difficulty with it. Let us analyze arguments may they be considered as dummy vari- pends on the level of student and preference of fundamental difference between the conceptions is the usual solution. We begin by noting that points ables; this special use tends to be not well under- the teacher): evidenced by the usual response of students to the on a line are related by an equation of the form stood by students. following question: 5x = 37 y = mx + b. What happens to the value of 1/x as x gets Conception 4: Algebra as the Study of Now solve this equation in some way, arriving at x This is both a pattern among variables and a for- larger and larger? = 7.4. The “certain number” in the problem is 7.4, mula. In our minds it is a function with domain Structures The question seems simple, but it is enough to baf- and the result is easily checked. variable x and range variable y, but to students it is The study of algebra at the college level involves fle most students. We have not asked for a value of not clear which of m, x, or b is the argument. As a structures such as groups, rings, integral domains, In solving these kinds of problems, many students x, so x is not an unknown. We have not asked the pattern it is easy to understand, but in the context fields, and vector spaces. It seems to bear little re- have difficulty moving from arithmetic to algebra. student to translate. There is a pattern to generalize, of this problem, some things are unknown. All the semblance to the study of algebra at the high Whereas the arithmetic solution (“in your head”) in- but it is not a pattern that looks like arithmetic. (It volves subtracting 3 and dividing by 5, the algebraic letters look like unknowns (particularly the x and school level, although the fields of real numbers is not appropriate to ask what happens to the value y, letters traditionally used for that purpose). form 5x + 3 involves multiplication by 5 and addi- of 1/2 as 2 gets larger and larger!) It is fundamen- and complex numbers and the various rings of tion of 3, the inverse operations. That is, to set up tally an algebraic pattern. Perhaps because of its in- Now to the solution. Since we know m, we substi- polynomials underlie the theory of algebra, and the equation, you must think precisely the opposite trinsic algebraic nature, some mathematics educa- tute for it: properties of integral domains and groups explain of the way you would solve it using arithmetic. tors believe that algebra should first be introduced why certain equations can be solved and others y = 11x + b not. Yet we recognize algebra as the study of struc- In this conception of algebra, variables are either through this use of variable. For instance, Fey and Thus m is here a constant, not a parameter. Now tures by the properties we ascribe to operations on unknowns or constants. Whereas the key instruc- Good (1985, p.48) see the following as the key real numbers and polynomials. Consider the fol- tions in the use of a variable as a pattern general- questions on which to base the study of algebra: we need to find b. Thus b has changed from para- lowing problem: izer are translate and generalize, the key instruc- meter to unknown. But how to find b? We use one For a given function f(x), find— pair of the many pairs in the relationship between tions in this use are simplify and solve. In fact, Factor 3x2 + 4ax – 132a2. 1. f(x) for x = a; “simplify” and “solve” are sometimes two different x and y. That is, we select a value for the argument 2. x so that f(x) = a; The conception of variable represented here is not names for the same idea: For example, we ask stu- x for which we know y. Having to substitute a pair 3. x so that maximum or minimum values of the same as any previously discussed. There is no dents to solve x – 2 = 5 to get the answer x = 7 of values for x and y can be done because y = mx f(x) occur; function or relation; the variable is not an argu- or x = –3. But we could ask students, “Rewrite x – + b describes a general pattern among numbers. 4. the rate of change in f near x = a; ment. There is no equation to be solved, so the 2 = 5 without using absolute value.” We might With substitution, 2 5. the average value of f over the interval variable is not acting as an unknown. There is no then get the answer (x – 2) = 25, which is another 2 = 11 • 6 + b, (a,b). arithmetic pattern to generalize. equivalent sentence. and so b = –64. But we haven’t found x and y Under this conception, a variable is an argument The answer to the factoring question is (3x + Polya (1957) wrote, “If you cannot solve the pro- even though we have values for them, because (i.e., stands for a domain value of a function) or a 22a)(x – 6a). The answer could be checked by posed problem try to solve first some related prob- they were not unknowns. We have only found the parameter (i.e., stands for a number on which substituting values for x and a in the given poly- lem” (p. 31). We follow that dictum literally in solv- unknown b, and we substitute in the appropriate other numbers depend). Only in this conception do nomial and in the factored answer, but this is al- ing most sentences, finding equivalent sentences equation to get the answer the notions of independent variable and dependent most never done. If factoring were checked that with the same solution. We also simplify expres- variable exist. Functions arise rather immediately, sions so that they can more easily be understood y = 11x – 64. way, there would be a wisp of an argument that for we need to have a name for values that depend here we are generalizing arithmetic. But in fact, and used. To repeat: simplifying and solving are Another way to make the distinction between the on the argument or parameter x. Function notation the student is usually asked to check by multiply- more similar than they are usually made out to be. different uses of the variables in this problem is to (as in f(x) = 3x + 5) is a new idea when students ing the binomials, exactly the same procedure that engage quantifiers. We think: For all x and y, there first see it: f(x) = 3x + 5 looks and feels different the student has employed to get the answer in the exist m and b with y = mx + b. We are given the Conception 3: Algebra as the Study of from y = 3x + 5. (In this regard, one reason y = f(x) first place. It is silly to check by repeating the may confuse students is because the function f, value that exists for m, so we find the value that Relationships among Quantities process used to get the answer in the first place, rather than the argument x, has become the para- exists for b by using one of the “for all x and y” but in this kind of problem students tend to treat When we write A = LW, the area formula for a rec- meter. Indeed, the use of f(x) to name a function, pairs, and so on. Or we use the equivalent set lan- the variables as marks on paper, without numbers tangle, we are describing a relationship among as Fey and Good do in the quote above, is seen by guage: We know the line is {(x,y): y = mx + b} and as a referent. In the conception of algebra as the three quantities. There is not the feel of an un- some educators as contributing to that confusion.) we know m and try to find b. In the language of known, because we are not solving for anything. sets or quantifiers, x and y are known as dummy study of structures, the variable is little more than That variables as arguments differ from variables as The feel of formulas such as A = LW is different variables because any symbols could be used in an arbitrary symbol. unknowns is further evidenced by the following from the feel of generalizations such as 1 = n their stead. It is rather hard to convince students There is a subtle quandary here. We want students question: (1/n), even though we can think of a formula as a and even some teachers that {x: 3x = 6} = {y: 3y = to have the referents (usually real numbers) for special type of generalization. Find an equation for the line through (6,2) with 6}, even though each set is {2}. Many people think variables in mind as they use the variables. But we Whereas the conception of algebra as the study of slope 11. that the function f with f(x) = x + 1 is not the same also want students to be able to operate on the relationships may begin with formulas, the crucial The usual solution combines all the uses of vari- as the function g with the same domain as f and variables without always having to go to the level distinction between this and the previous concep- ables discussed so far, perhaps explaining why with g(y) = y + 1. Only when variables are used as of the referent. For instance, when we ask students

10 ALGEBRAIC THINKING, GRADES K–12 DEFINING ALGEBRAIC THINKING AND AN ALGEBRA CURRICULUM 11 Then simplify (the number of steps required de- tions is that, here, variables vary. That there is a some students have difficulty with it. Let us analyze arguments may they be considered as dummy vari- pends on the level of student and preference of fundamental difference between the conceptions is the usual solution. We begin by noting that points ables; this special use tends to be not well under- the teacher): evidenced by the usual response of students to the on a line are related by an equation of the form stood by students. following question: 5x = 37 y = mx + b. What happens to the value of 1/x as x gets Conception 4: Algebra as the Study of Now solve this equation in some way, arriving at x This is both a pattern among variables and a for- larger and larger? = 7.4. The “certain number” in the problem is 7.4, mula. In our minds it is a function with domain Structures The question seems simple, but it is enough to baf- and the result is easily checked. variable x and range variable y, but to students it is The study of algebra at the college level involves fle most students. We have not asked for a value of not clear which of m, x, or b is the argument. As a structures such as groups, rings, integral domains, In solving these kinds of problems, many students x, so x is not an unknown. We have not asked the pattern it is easy to understand, but in the context fields, and vector spaces. It seems to bear little re- have difficulty moving from arithmetic to algebra. student to translate. There is a pattern to generalize, of this problem, some things are unknown. All the semblance to the study of algebra at the high Whereas the arithmetic solution (“in your head”) in- but it is not a pattern that looks like arithmetic. (It volves subtracting 3 and dividing by 5, the algebraic letters look like unknowns (particularly the x and school level, although the fields of real numbers is not appropriate to ask what happens to the value y, letters traditionally used for that purpose). form 5x + 3 involves multiplication by 5 and addi- of 1/2 as 2 gets larger and larger!) It is fundamen- and complex numbers and the various rings of tion of 3, the inverse operations. That is, to set up tally an algebraic pattern. Perhaps because of its in- Now to the solution. Since we know m, we substi- polynomials underlie the theory of algebra, and the equation, you must think precisely the opposite trinsic algebraic nature, some mathematics educa- tute for it: properties of integral domains and groups explain of the way you would solve it using arithmetic. tors believe that algebra should first be introduced why certain equations can be solved and others y = 11x + b not. Yet we recognize algebra as the study of struc- In this conception of algebra, variables are either through this use of variable. For instance, Fey and Thus m is here a constant, not a parameter. Now tures by the properties we ascribe to operations on unknowns or constants. Whereas the key instruc- Good (1985, p.48) see the following as the key real numbers and polynomials. Consider the fol- tions in the use of a variable as a pattern general- questions on which to base the study of algebra: we need to find b. Thus b has changed from para- lowing problem: izer are translate and generalize, the key instruc- meter to unknown. But how to find b? We use one For a given function f(x), find— pair of the many pairs in the relationship between tions in this use are simplify and solve. In fact, Factor 3x2 + 4ax – 132a2. 1. f(x) for x = a; “simplify” and “solve” are sometimes two different x and y. That is, we select a value for the argument 2. x so that f(x) = a; The conception of variable represented here is not names for the same idea: For example, we ask stu- x for which we know y. Having to substitute a pair 3. x so that maximum or minimum values of the same as any previously discussed. There is no dents to solve x – 2 = 5 to get the answer x = 7 of values for x and y can be done because y = mx f(x) occur; function or relation; the variable is not an argu- or x = –3. But we could ask students, “Rewrite x – + b describes a general pattern among numbers. 4. the rate of change in f near x = a; ment. There is no equation to be solved, so the 2 = 5 without using absolute value.” We might With substitution, 2 5. the average value of f over the interval variable is not acting as an unknown. There is no then get the answer (x – 2) = 25, which is another 2 = 11 • 6 + b, (a,b). arithmetic pattern to generalize. equivalent sentence. and so b = –64. But we haven’t found x and y Under this conception, a variable is an argument The answer to the factoring question is (3x + Polya (1957) wrote, “If you cannot solve the pro- even though we have values for them, because (i.e., stands for a domain value of a function) or a 22a)(x – 6a). The answer could be checked by posed problem try to solve first some related prob- they were not unknowns. We have only found the parameter (i.e., stands for a number on which substituting values for x and a in the given poly- lem” (p. 31). We follow that dictum literally in solv- unknown b, and we substitute in the appropriate other numbers depend). Only in this conception do nomial and in the factored answer, but this is al- ing most sentences, finding equivalent sentences equation to get the answer the notions of independent variable and dependent most never done. If factoring were checked that with the same solution. We also simplify expres- variable exist. Functions arise rather immediately, sions so that they can more easily be understood y = 11x – 64. way, there would be a wisp of an argument that for we need to have a name for values that depend here we are generalizing arithmetic. But in fact, and used. To repeat: simplifying and solving are Another way to make the distinction between the on the argument or parameter x. Function notation the student is usually asked to check by multiply- more similar than they are usually made out to be. different uses of the variables in this problem is to (as in f(x) = 3x + 5) is a new idea when students ing the binomials, exactly the same procedure that engage quantifiers. We think: For all x and y, there first see it: f(x) = 3x + 5 looks and feels different the student has employed to get the answer in the exist m and b with y = mx + b. We are given the Conception 3: Algebra as the Study of from y = 3x + 5. (In this regard, one reason y = f(x) first place. It is silly to check by repeating the may confuse students is because the function f, value that exists for m, so we find the value that Relationships among Quantities process used to get the answer in the first place, rather than the argument x, has become the para- exists for b by using one of the “for all x and y” but in this kind of problem students tend to treat When we write A = LW, the area formula for a rec- meter. Indeed, the use of f(x) to name a function, pairs, and so on. Or we use the equivalent set lan- the variables as marks on paper, without numbers tangle, we are describing a relationship among as Fey and Good do in the quote above, is seen by guage: We know the line is {(x,y): y = mx + b} and as a referent. In the conception of algebra as the three quantities. There is not the feel of an un- some educators as contributing to that confusion.) we know m and try to find b. In the language of known, because we are not solving for anything. sets or quantifiers, x and y are known as dummy study of structures, the variable is little more than That variables as arguments differ from variables as The feel of formulas such as A = LW is different variables because any symbols could be used in an arbitrary symbol. unknowns is further evidenced by the following from the feel of generalizations such as 1 = n their stead. It is rather hard to convince students There is a subtle quandary here. We want students question: (1/n), even though we can think of a formula as a and even some teachers that {x: 3x = 6} = {y: 3y = to have the referents (usually real numbers) for special type of generalization. Find an equation for the line through (6,2) with 6}, even though each set is {2}. Many people think variables in mind as they use the variables. But we Whereas the conception of algebra as the study of slope 11. that the function f with f(x) = x + 1 is not the same also want students to be able to operate on the relationships may begin with formulas, the crucial The usual solution combines all the uses of vari- as the function g with the same domain as f and variables without always having to go to the level distinction between this and the previous concep- ables discussed so far, perhaps explaining why with g(y) = y + 1. Only when variables are used as of the referent. For instance, when we ask students

10 ALGEBRAIC THINKING, GRADES K–12 DEFINING ALGEBRAIC THINKING AND AN ALGEBRA CURRICULUM 11 to derive a trigonometric identity such as 2sin2x – 1 = setting for variable. Computer applications tend to For example, consider the question of paper-and- Mathematical Sciences Curriculum K–12: What Is Still sin4x – cos4x, we do not want the student to think involve large numbers of variables that may stand pencil manipulative skills. In the past, one had to Fundamental and What Is Not. Report to the NSB of the sine or cosine of a specific number or even for many different kinds of objects. Also, comput- have such skills in order to solve problems and in Commission on Precollege Education in Mathematics, to think of the sine or cosine functions, and we are ers are programmed to manipulate the variables, order to study functions and other relations. Today, Science, and Technology. Washington, D.C.: CBMS, 1983. not interested in ratios in triangles. We merely so we do not have to abbreviate them for the pur- with computers able to simplify expressions, solve want to manipulate sin x and cos x into a different pose of easing the task of blind manipulation. sentences, and graph functions, what to do with Davis, Robert B., Elizabeth Jockusch, and Curtis McK- form using properties that are just as abstract as manipulative skills becomes a question of the im- night. “Cognitive Processes in Learning Algebra.” In computer science, the uses of variables cover all Journal of Children’s Mathematical Behavior 2 the identity we wish to derive. portance of algebra as a structure, as the study of the uses we have described above for variables. (Spring 1978): 1–320. In these kinds of problems, faith is placed in prop- There is still the generalizing of arithmetic. The arbitrary marks on paper, as the study of arbitrary relationships among symbols. The prevailing view Fey, James T., and Richard A. Good. “Rethinking the Se- erties of the variables, in relationships between x’s study of algorithms is a study of procedures. In quence and Priorities of High School Mathematics today seems to be that this should not be the major and y’s and n’s, be they addends, factors, bases, or fact, there are typical algebra questions that lend Curricula.” In The Secondary School Mathematics Cur- exponents. The variable has become an arbitrary themselves to algorithmic thinking: criterion (and certainly not the only criterion) by riculum, 1985 Yearbook of the National Council of object in a structure related by certain properties. It which algebra content is determined. Teachers of Mathematics, pp. 43–52. Reston, Va.: Begin with a number. Add 3 to it. Multiply it by is the view of variable found in abstract algebra. NCTM, 1985. 2. Subtract 11 from the result.… Consider the question of the role of function ideas in the study of algebra. It is again a question of the Hart, Walter W. A First Course in Algebra. 2d ed. Boston: Much criticism has been leveled against the prac- In programming, one learns to consider the vari- relative importance of the view of algebra as the D. C. Heath & Co., 1951a. tice by which “symbol pushing” dominates early able as an argument far sooner than is customary study of relationships among quantities, in which . A Second Course in Algebra. 2d ed., enlarged. experiences with algebra. We call it “blind” manip- in algebra. In order to set up arrays, for example, the predominant manifestation of variable is as ar- Boston: D. C. Heath & Co., 1951b. ulation when we criticize; “automatic” skills when some sort of function notation is needed. And fi- we praise. Ultimately everyone desires that stu- gument, compared to the other roles of algebra: as Kramer, Edna E. The Nature and Growth of Modern nally, because computers have been programmed dents have enough facility with algebraic symbols generalized arithmetic or as providing a means to Mathematics. Princeton, N.J.: Princeton University to perform manipulations with symbols without to deal with the appropriate skills abstractly. The solve problems. Press, 1981. any referents for them, computer science has be- key question is, What constitutes “enough facility”? Thus some of the important issues in the teaching MacLane, Saunders, and Garrett Birkhoff. Algebra. New come a vehicle through which many students York: Macmillan Co., 1967. It is ironic that the two manifestations of this use learn about variables (Papert 1980). Ultimately, and learning of algebra can be crystallized by cast- May, Kenneth O., and Henry Van Engen. “Relations and of variable—theory and manipulation—are often because of this influence, it is likely that students ing them in the framework of conceptions of alge- Functions.” In The Growth of Mathematical Ideas, viewed as opposite camps in the setting of policy bra and uses of variables, conceptions that have will learn the many uses of variables far earlier Grades K–12, Twenty-fourth Yearbook of the National been changed by the explosion in the uses of toward the algebra curriculum, those who favor than they do today. Council of Teachers of Mathematics, pp. 65-110. mathematics and by the omnipresence of comput- manipulation on one side, those who favor theory Washington, D.C.: NCTM, 1959. on the other. They come from the same view of ers. No longer is it worthwhile to categorize alge- National Council of Teachers of Mathematics and the variable. Summary bra solely as generalized arithmetic, for it is much Mathematical Association of America. Recommendations more than that. Algebra remains a vehicle for solv- The different conceptions of algebra are related to for the Preparation of High School Students for College Variable in Computer Science different uses of variables. Here is an oversimpli- ing certain problems but it is more than that as Mathematics Courses. Reston, Va.: NCTM; Washington, fied summary of those relationships: well. It provides the means by which to describe D.C.: MAA, 1977. Algebra has a slightly different cast in computer and analyze relationships. And it is the key to the Conception of algebra Use of variables Papert, Seymour. Mindstorms: Children, Computers, science from what it has in mathematics. There is characterization and understanding of mathemati- and Powerful Ideas. New York: Basic Books, 1980. often a different syntax. Whereas in ordinary alge- cal structures. Given these assets and the increased Generalized arithmetic Pattern generalizers Pavelle, Richard, Michael Rothstein, and John Fitch. mathematization of society, it is no surprise that al- bra, x = x + 2 suggests an equation with no solu- (translate, generalize) “Computer Algebra.” Scientific American, December tion, in BASIC the same sentence conveys the re- gebra is today the key area of study in secondary 1981, pp. 136–52. placement of a particular storage location in a Means to solve certain Unknowns, constants school mathematics and that this preeminence is Polya, George, How to Solve It. 2d ed. Princeton, N.J.: computer by a number two greater. This use of problems (solve, simplify) likely to be with us for a long time. Princeton University Press, 1957. variable has been identified by Davis, Jockusch, Study of relationships Arguments, parameters and McKnight (1978, p. 33): Usiskin, Zalman. Algebra through Applications. Chicago: (relate, graph) Bibliography Department of Education, University of Chicago, 1976. Computers give us another view of the basic mathe- Structure Arbitrary marks on paper Bushaw, Donald, Max Bell, Henry Pollak, Maynard Usiskin, Zalman, and Max Bell. Applying Arithmetic. matical concept of variable. From a computer point (manipulate, justify) Thompson, and Zalman Usiskin. A Sourcebook of Ap- Preliminary ed. Chicago: Department of Education, of view, the name of a variable can be thought of plications of School Mathematics. Reston, Va.: National University of Chicago, 1984. as the address of some specific memory register, Earlier in this article, two issues concerning in- Council of Teachers of Mathematics, 1980. Viete, Francois. “The New Algebra.” In A Source Book and the value of the variable can be thought of as struction in algebra were mentioned. Given the Comprehensive School Mathematics Program. CSMP the contents of this memory register. on Mathematics, 1200–1800, edited by D. J. Struik, pp. discussion above, it is now possible to interpret Overview. St. Louis: CEMREL, 1975. 74–81. Cambridge, Mass.: Harvard University Press, In computer science, variables are often identified these issues as a question of the relative impor- Conference Board of the Mathematical Sciences. The 1969. strings of letters and numbers. This conveys a dif- tance to be given at various levels of study to the ferent feel and is the natural result of a different various conceptions.

12 ALGEBRAIC THINKING, GRADES K–12 DEFINING ALGEBRAIC THINKING AND AN ALGEBRA CURRICULUM 13 to derive a trigonometric identity such as 2sin2x – 1 = setting for variable. Computer applications tend to For example, consider the question of paper-and- Mathematical Sciences Curriculum K–12: What Is Still sin4x – cos4x, we do not want the student to think involve large numbers of variables that may stand pencil manipulative skills. In the past, one had to Fundamental and What Is Not. Report to the NSB of the sine or cosine of a specific number or even for many different kinds of objects. Also, comput- have such skills in order to solve problems and in Commission on Precollege Education in Mathematics, to think of the sine or cosine functions, and we are ers are programmed to manipulate the variables, order to study functions and other relations. Today, Science, and Technology. Washington, D.C.: CBMS, 1983. not interested in ratios in triangles. We merely so we do not have to abbreviate them for the pur- with computers able to simplify expressions, solve want to manipulate sin x and cos x into a different pose of easing the task of blind manipulation. sentences, and graph functions, what to do with Davis, Robert B., Elizabeth Jockusch, and Curtis McK- form using properties that are just as abstract as manipulative skills becomes a question of the im- night. “Cognitive Processes in Learning Algebra.” In computer science, the uses of variables cover all Journal of Children’s Mathematical Behavior 2 the identity we wish to derive. portance of algebra as a structure, as the study of the uses we have described above for variables. (Spring 1978): 1–320. In these kinds of problems, faith is placed in prop- There is still the generalizing of arithmetic. The arbitrary marks on paper, as the study of arbitrary relationships among symbols. The prevailing view Fey, James T., and Richard A. Good. “Rethinking the Se- erties of the variables, in relationships between x’s study of algorithms is a study of procedures. In quence and Priorities of High School Mathematics today seems to be that this should not be the major and y’s and n’s, be they addends, factors, bases, or fact, there are typical algebra questions that lend Curricula.” In The Secondary School Mathematics Cur- exponents. The variable has become an arbitrary themselves to algorithmic thinking: criterion (and certainly not the only criterion) by riculum, 1985 Yearbook of the National Council of object in a structure related by certain properties. It which algebra content is determined. Teachers of Mathematics, pp. 43–52. Reston, Va.: Begin with a number. Add 3 to it. Multiply it by is the view of variable found in abstract algebra. NCTM, 1985. 2. Subtract 11 from the result.… Consider the question of the role of function ideas in the study of algebra. It is again a question of the Hart, Walter W. A First Course in Algebra. 2d ed. Boston: Much criticism has been leveled against the prac- In programming, one learns to consider the vari- relative importance of the view of algebra as the D. C. Heath & Co., 1951a. tice by which “symbol pushing” dominates early able as an argument far sooner than is customary study of relationships among quantities, in which . A Second Course in Algebra. 2d ed., enlarged. experiences with algebra. We call it “blind” manip- in algebra. In order to set up arrays, for example, the predominant manifestation of variable is as ar- Boston: D. C. Heath & Co., 1951b. ulation when we criticize; “automatic” skills when some sort of function notation is needed. And fi- we praise. Ultimately everyone desires that stu- gument, compared to the other roles of algebra: as Kramer, Edna E. The Nature and Growth of Modern nally, because computers have been programmed dents have enough facility with algebraic symbols generalized arithmetic or as providing a means to Mathematics. Princeton, N.J.: Princeton University to perform manipulations with symbols without to deal with the appropriate skills abstractly. The solve problems. Press, 1981. any referents for them, computer science has be- key question is, What constitutes “enough facility”? Thus some of the important issues in the teaching MacLane, Saunders, and Garrett Birkhoff. Algebra. New come a vehicle through which many students York: Macmillan Co., 1967. It is ironic that the two manifestations of this use learn about variables (Papert 1980). Ultimately, and learning of algebra can be crystallized by cast- May, Kenneth O., and Henry Van Engen. “Relations and of variable—theory and manipulation—are often because of this influence, it is likely that students ing them in the framework of conceptions of alge- Functions.” In The Growth of Mathematical Ideas, viewed as opposite camps in the setting of policy bra and uses of variables, conceptions that have will learn the many uses of variables far earlier Grades K–12, Twenty-fourth Yearbook of the National been changed by the explosion in the uses of toward the algebra curriculum, those who favor than they do today. Council of Teachers of Mathematics, pp. 65-110. mathematics and by the omnipresence of comput- manipulation on one side, those who favor theory Washington, D.C.: NCTM, 1959. on the other. They come from the same view of ers. No longer is it worthwhile to categorize alge- National Council of Teachers of Mathematics and the variable. Summary bra solely as generalized arithmetic, for it is much Mathematical Association of America. Recommendations more than that. Algebra remains a vehicle for solv- The different conceptions of algebra are related to for the Preparation of High School Students for College Variable in Computer Science different uses of variables. Here is an oversimpli- ing certain problems but it is more than that as Mathematics Courses. Reston, Va.: NCTM; Washington, fied summary of those relationships: well. It provides the means by which to describe D.C.: MAA, 1977. Algebra has a slightly different cast in computer and analyze relationships. And it is the key to the Conception of algebra Use of variables Papert, Seymour. Mindstorms: Children, Computers, science from what it has in mathematics. There is characterization and understanding of mathemati- and Powerful Ideas. New York: Basic Books, 1980. often a different syntax. Whereas in ordinary alge- cal structures. Given these assets and the increased Generalized arithmetic Pattern generalizers Pavelle, Richard, Michael Rothstein, and John Fitch. mathematization of society, it is no surprise that al- bra, x = x + 2 suggests an equation with no solu- (translate, generalize) “Computer Algebra.” Scientific American, December tion, in BASIC the same sentence conveys the re- gebra is today the key area of study in secondary 1981, pp. 136–52. placement of a particular storage location in a Means to solve certain Unknowns, constants school mathematics and that this preeminence is Polya, George, How to Solve It. 2d ed. Princeton, N.J.: computer by a number two greater. This use of problems (solve, simplify) likely to be with us for a long time. Princeton University Press, 1957. variable has been identified by Davis, Jockusch, Study of relationships Arguments, parameters and McKnight (1978, p. 33): Usiskin, Zalman. Algebra through Applications. Chicago: (relate, graph) Bibliography Department of Education, University of Chicago, 1976. Computers give us another view of the basic mathe- Structure Arbitrary marks on paper Bushaw, Donald, Max Bell, Henry Pollak, Maynard Usiskin, Zalman, and Max Bell. Applying Arithmetic. matical concept of variable. From a computer point (manipulate, justify) Thompson, and Zalman Usiskin. A Sourcebook of Ap- Preliminary ed. Chicago: Department of Education, of view, the name of a variable can be thought of plications of School Mathematics. Reston, Va.: National University of Chicago, 1984. as the address of some specific memory register, Earlier in this article, two issues concerning in- Council of Teachers of Mathematics, 1980. Viete, Francois. “The New Algebra.” In A Source Book and the value of the variable can be thought of as struction in algebra were mentioned. Given the Comprehensive School Mathematics Program. CSMP the contents of this memory register. on Mathematics, 1200–1800, edited by D. J. Struik, pp. discussion above, it is now possible to interpret Overview. St. Louis: CEMREL, 1975. 74–81. Cambridge, Mass.: Harvard University Press, In computer science, variables are often identified these issues as a question of the relative impor- Conference Board of the Mathematical Sciences. The 1969. strings of letters and numbers. This conveys a dif- tance to be given at various levels of study to the ferent feel and is the natural result of a different various conceptions.

12 ALGEBRAIC THINKING, GRADES K–12 DEFINING ALGEBRAIC THINKING AND AN ALGEBRA CURRICULUM 13