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.