Algebra for Secondary Teachers: Engaging Students to Develop Symbolic Meaning

As we begin to examine the concept of variable, we will first restrict our discussion to the literature surrounding the use of “letters” as a form of representation. Using letters is probably the most common vantage point regarding the concept of variable for most people. In fact, one could argue that a thumbnail definition of variable for most secondary school students could be characterized by “the advanced study of the last three letters of the alphabet”. For this reason, we first turn our attention to the use of letters to represent variables. Within this context, Trigueros and Ursini (2003) propose three structural categories: unknown number, generalized number, and functional relationship.

Since one of the most useful applications of algebra is the solution of equations, it should come as no surprise that variable as an unknown number is one way to view the concept of variable. A variable may be thought of as an unknown number when we are faced with a statement like \(x^2-5x=-6\) where there are specific value(s) that the variable can take. Recall from our example in the “Guess My Number” game (see Figure 4.1.1) played on computer algebra system. In this game, the concept of variable as an unknown number is one way to look at the use of the letter, \(z\text{.}\) In this case, one player has stored a specific value in the calculator’s memory and linked it to the letter. The second player must then attempt discover the hidden value. Here, there is a single value being used and thus the letter, \(z\text{,}\) represents an unknown number.

Although the “Guess My Number” game can be viewed as using the concept of variable as an unknown number, the continuation of play within the game illustrates our next perspective for interpreting variable. Recall that the game is quite simple. One player stores a number in the calculator as a letter and the opponent’s job is to guess it by using only expressions of inequalities or other expressions using the letter. As play continues in the game, the players repeatedly change the “unknown number” in order to challenge their opponent. It is this “changing of the unknown” that illustrates our second view of variable as a generalized number. In the instance of a generalized number, the variable may have a less specific referent as compared to the unknown number. In mathematics, we often use a variable to express a generalized number while simultaneously allowing the possibility that it can also stand for an unknown number within a specific context. For example, consider \(n\) in the expression \(2n+1\text{.}\) As we use this expression in a general sense to represent an odd integer, the symbol, \(n\text{,}\) is a generalized number generating a set of various odd integers. However, when we decide to place a restriction on the same expression where we state that \(2n+1=7\text{,}\) \(n\) now represents an unknown number rather than a generalized number. So when we ask what \(n\) represents in the expression \(2n+1\text{,}\) the answer is that it all depends on the question we are asking.

As we develop notation, we often need a way to distinguish the difference between a generalized number and a specific number. You may recall from your study of Calculus, that mathematicians often do this by using \(f'\left(x\right)\) to represent the generalized derivative of a function and the notation \(f'\left(x_0\right)\) to represent the derivative at a specific point, \(\left(x_0, f\left(x_0\right)\right)\text{.}\) Although the format of the symbols is the same, there is an implied difference that only a person enculturated into the mathematical community would necessarily see.

The third view of variable given by Trigueros and Ursini is that of variables in a functional relationship. Although this might seem similar to the view of generalized number stated earlier in the form of \(2n+1\text{,}\) this conceptualization is different. Here, Trigueros and Ursini refer to a specific relationship such as Hook’s Law, \(F=kx\text{.}\) In this case, there is an implied variation involved between two variables where the nature of the variation itself is of interest.

Whenever we consider a graph generated by an equation such as \(y=x^2\text{,}\) we are witnessing variables in a functional relationship. We can ask questions such as, how does change in one variable affect change in another? How fast is one variable changing with respect to the other? What happens to \(y\) as \(x\) gets very large in the positive direction? Compared to the view of variable as a generalized number where an expression like \(2n+1\) simply stands for an arbitrary odd integer, the view of variables in a functional relationship carries with it a context aimed at describing relationships.

As we consider the ways in which students use the concept of variable (in this case the use of letters or symbols to represent variables) we may be concerned with which view should come first or which view is the most important. However, we would like to propose that these questions are of lesser concern. If we are interested in helping students build a deeper conceptualization of variable, we must ask ourselves how do students connect all of these views? We posit that the issue on conceptual connection is far more important than which view should come first, although we acknowledge that the concern of order might impact the question of connection. Our argument is that in order for students to develop a concept of variable, they must be able to merge these views of variable and be ably to flexibly move among them. In addition, not only do students need to be able to move among varying views of variable, they must also be able to navigate back and forth between symbols and meaning (Sfard, 2000). Therefore, the question then becomes, how do we develop curricula that encourage students to move back and forth among various views of variable across multiple representations? In this section, we have focused on the symbolic representation of variable, but we will spend more time looking at other representations in a section devoted specifically to this issue.

As far as the symbolic representation is concerned, one resource we have to help students merge various views of the concept of variable is the use of computer algebra systems (CAS). In order to discuss this issue and the use of a technical tool such as CAS we feel it necessary to first disclose the theoretical perspective that we use to approach the development of conceptual understanding. While scholars debate the merits and drawbacks of various theoretical perspectives ranging from radical constructivism to socio-cultural paradigms, we consider ourselves to be social constructivists. What this means is that we believe students develop mathematical understanding from both internal and external influences. Cognitive conflict from within a learner is a powerful mechanism for conceptual change, but the influence of the mathematical community is also a powerful tool. So why do we worry about our theoretical perspective here? Well, the use of CAS carries with it an implied pre-ordained structure set forth by the mathematical community. In order for a student to use a CAS, there is certain syntax and structure that must be followed. In this case, the more open view of radical constructivism as a theory of concept development become problematic while within the social constructivist view students may first develop notation, but later be forced to conform to the community’s notation needed to use CAS as a tool for exploration. In addition, there may be times where the teacher must introduce notation even before the meaning has been explored. Is this a concern? We tend to side with Sfard’s (2000) perspective when she states:

The issue of the symbiotic relationship between notation and concept is important. Often in mathematics, notation drives concept development in a way similar to the reverse relationship where the concept drives the notation. To think that there is a single approach appropriate to all situations, limits creativity and produces a very narrow view to teaching and learning. We would rather let the classroom context guide the order of this relationship. It is often the translation among notation systems that helps deepen the concept and therefore as Sfard posits, we seek more to aid in the translation.

In order to illustrate this notational translation and the deepening of concepts, consider the use of a CAS to work with sequences and series. In standard mathematical notation, we often refer to a sequence or series with notation such as \(a_n=\frac{x^n}{n!}\) and \(\sum\limits^{\infty}_{n=0}{\frac{x^n}{n!}}\text{.}\) The problem is that in order to explore these mathematical objects using CAS we must first translate these into computer or “calculator speak”. The use of function notation helps us to deal with subscripts and variables within one symbol system while encouraging students to focus attention on a dual variation happening within a single expression. For example, in the sequence we have two quantities that can “vary”—the term index and the input variable. In translating this to a useful language in a CAS, we define \(a\left(x,n\right)=\frac{x^n}{n!}\text{.}\) The use of a function of two variables serves us in three ways. The first is the ability to quickly evaluate a specific term of the sequence using a simple input such as \(a\left(0.6,13\right)\) to represent the evaluation of the 13th term using \(x=0.6\text{.}\) This encourages the student to create an object and then operate on it in a very abstract way. The second benefit we gain is the cognitive conflict that is generated by the notational use. As stated earlier, when students are introduced to function notation, they often confuse the use of parentheses to separate the function name from the input variable with the operation of multiplication between integers. The fact that a comma is being used within the parentheses causes the student to pause and re-evaluate his/her understanding of the notation. A byproduct of this re-evaluation is that the student may realize that there are two quantities that can vary within the expression.

The third benefit we gain from the use of CAS is the ability of the student to move from an understanding of the expression as a process to that of an object. This goes beyond the simple evaluation of the sequence and allows the student to explore general behavior. For example, if we wish to explore the convergence of the series \(\sum\limits^{\infty}_{n=0}{\frac{x^n}{n!}}\text{,}\) we can combine previously defined objects to create another sequence describing the \(k^{th}\) partial sum. In this case we might define \(s\left(x,k\right)=\sum\limits^{k}_{n=0}{a\left(x,n\right)}\text{.}\) Here the students are encouraged to make sense of the parts of the definition in order to understand the meaning. It is important to note that the design of the curricula utilizing this approach is crucial so that students will simply try to memorize procedures. Deliberate questions must be present to guide the students to make the connections among notation systems and meaning.

The moral of this story is that technology can serve as a powerful tool for merging different views of variable. The challenge is that the classroom environment must transform to allow discourse so that students feel comfortable with both processes of creation and translation of notation. The teacher must be cognizant of the interplay among the processes involved in this dynamic in order to successfully orchestrate the classroom to allow the formation of algebraic concepts with the depth we desire.