The kind of task we ask our learners to engage in algebra communicates particular notions of algebra and with it particular use of variable. There are at least four conceptions of algebra embedded in the curriculum. These are reflected in the tasks in textbooks and in our lessons. Zalman Usiskin proposed the following conceptions of algebra in school mathematics. These are present in the curriculum in varying degrees.
Conception 1: Algebra as Generalized Arithmetic
Usually the task here involves extending numbers patterns to include other types of numbers such as the negative numbers. It also include some statements about number relationships and the task is to state it in symbolic form, usually with a variable. The key instructions for the student in this conception of algebra are translate and generalize. These are important skills not only for algebra but also for arithmetic. The aim of the teacher for these tasks should be to highlight to the learners and make them see that in algebra what they are doing is to be able to make a general statement about number relationships that is already known to them. For example, a generalization for a number pattern that relates the product of a number and its reciprocal is n x (1/n) = 1.
You may want to read my post on When is it algebra, when is it arithmetic?
Conception 2: Algebra as a Study of Procedures for Solving Certain Kinds of Problems
This may be considered the traditional way algebra was conceived in the curriculum. This is the era when textbooks will have classifications of problems such as mixture problems, distance problems, work problems, age problems etc. For each type, the learners are expected to study the procedures for setting up the equation and then solving them. For example,
When 3 is added to 5 times a certain number, the sum is 40. Find the number.
The problem is translated as 5x+3 = 40. According to Usiskin, as far as conception #1 is concerned, this is already algebra. The problem has been translated to its symbolic form. However, conception #2 is about procedures and solving hence one needs to find the unknown number symbolized by x using a procedure – adding -3 to both sides and then dividing both sides of the equal sign by 5. In solving these kinds of problems, many students have difficulty moving from arithmetic to algebra, accordinng to Usiskin. Whereas the arithmetic solution (“in your head”) involves subtracting 3 and dividing by 5, the algebraic form 5x + 3 involves multiplication by 5 and addition of 3, the inverse operations. That is, to set up the equation, you must think precisely the opposite of the way you would solve it using arithmetic.
Note the difference between the meaning of the letter symbol here and the one in conception #1. Here, the letter symbol stands for either unknowns or constants. Whereas the key instructions in the use of a variable as a pattern generalizer are translate and generalize, the key instructions in this use are simplify and solve.
Conception 3: Algebra as the Study of Relationships among Quantities
The focus here is the notion of function and the idea of variable that does vary. Usiskin argued that when we write A = LW, the area formula for a rectangle, we are describing a relationship among three quantities. There is no feel of an unknown, because we are not solving for anything. The feel of formulas such as A = LW is different from the feel of generalizations such as 1 = n (1/n), even though we can think of a formula as a special type of generalization.
In addition, whereas the conception of algebra as the study of relationships may begin with formulas, the crucial distinction between this and the previous conceptions is that, here, variables vary. That there is a fundamental difference between the conceptions is evidenced by the usual response of students to the following question:
What happens to the value of 1/x as x gets larger and larger?
The question seems simple, but it is enough to baffle most students. It does not ask for a value of x, so x is not an unknown. It does not ask the student to translate. There is a pattern to generalize, yes, but it is not a pattern that looks like arithmetic.
When function is the focus of the study of algebra, Fey and Good (1985, p.48) proposed following as the key questions on which to base the study of algebra:
For a given function f(x), find—
- f(x) for x = a;
- x so that f(x) = a;
- x so that maximum or minimum values of f(x) occur;
- the rate of change in f near x = a;
- the average value of f over the interval (a,b).
Usiskin further clarified that under this conception, a variable is an argument (i.e., stands for a domain value of a function) or a parameter (i.e., stands for a number on which other numbers depend). Only in this conception do the notions of independent variable and dependent variable exist.
Conception 4: Algebra as the Study of Structures
We do not speak here of algebraic structures in college mathematics. Here we recognize algebra as the study of structures in terms of the properties we ascribe to operations on real numbers and polynomials. Consider the following problem:
Factor 3x^2 + 4ax – 132a^2.
The conception of variable represented here is not the same as any previously discussed. There is no function or relation; the variable is not an argument. There is no equation to be solved, so the variable is not acting as an unknown. There is no arithmetic pattern to generalize.
The answer to the factoring question is (3x + 22a)(x – 6a). The answer could be checked by substituting values for x and a in the given polynomial and in the factored answer, but this is almost never done. If factoring were checked that way, there would be a wisp of an argument that here we are generalizing arithmetic. But in fact, the student is usually asked to check by multiplying the binomials, exactly the same procedure that the student has employed to get the answer in the first place. It is silly to check by repeating the process used to get the answer in the first place, but in this kind of problem students tend to treat the variables as marks on paper, without numbers as a referent. In the conception of algebra as the study of structures, the variable is little more than an arbitrary symbol.
If we are aware of these different conceptions of algebra and with it the particular meaning of the variable in each, then perhaps we can plan our lessons better knowing what the learners need to contend with in learning algebra.