Algebra for Secondary Teachers: Engaging Students to Develop Symbolic Meaning

For years, the typical approach for dealing with algebraic expressions consisted of memorizing the rules (e.g. distributive property, associative property, etc.) and then practicing these rules to manipulate one expression until it took the form of another expression that gave some new insight into a possible solution or property not easily seen from the old form. For example, consider the solution to a simple equation like \(5x-3=2x+9\text{.}\) Here students learn that they can manipulate this equation into an equivalent equation by adding \(-2x\) and \(3\) to both sides yielding \(5x+\left(-2x\right)-3+3=2x+\left(-2x\right)+9+3\text{.}\) By simplifying this expression, we obtain \(3x=12\) and thus \(\frac{1}{3}\cdot 3x=\frac{1}{3}\cdot 12 \Rightarrow x=4\text{.}\) Now this solution is not new or special. In fact, most of us can relate to this approach and the experiences that lead to it. We may have experienced the use of a balance as a physical interpretation of an equation applying the physical analogues of arithmetic operations to each side of the balance yielding new expressions. While this use of both symbolic and concrete representations helps students make connections among different views of the same mathematical concept, there are still other representations that can help deepen this understanding and also aid students in approaching other concepts that do not lend themselves as easily to symbolic approaches.

Consider an approach to equivalent expressions that has its origin in the graphical representation. Kieran and Sfard (1999) investigated how students dealt with equivalent expressions first introduced through the perspective of functions and their graphs. The argument used was that without having another representational meaning for a linear expression, it is difficult for the student to tie “rules” to an understanding of the concept. In fact, Kieren and Sfard (1999) define understanding as the ability to inspect, talk about, and move between representations of a concept. By this definition, it is not possible to have an understanding without multiple representations.

In order to examine students’ understanding of basic algebraic properties (e.g. distributive), the students were asked to graph linear functions and “add” them in a graphical representation aided by a technological tool in a similar way we have explored in the exercise from Section 4.4 where Liam asks Quinn about quadratics. For example, if one function was stored as \(f_1\left(x\right)=250+50x\) and another as \(f_2\left(x\right)=50+100x\text{,}\) then the student can simply graph \(f_3\left(x\right)=f_1\left(x\right)+f_2\left(x\right)\) (see Figure 4.5.1) and examine the resulting graph and its properties. When comparing the slopes of \(f_1\text{,}\) \(f_2\text{,}\) and \(f_3\) the students deal with the “rule” for the distributive property from a graphical perspective. In an instance cited, a student is able to manipulate the symbolic representation through the use of graphical visualization in his head viewing the various algebraic properties as transformations of points on the graphs.

The example of Kieren and Sfard’s teaching experiment serves to illustrate the depth of understanding that can happen through the use of multiple representations. While they do not claim that this is a quick fix for all students, they make the argument that for some students it does serve as a tool for making sense of the abstract properties that make up algebra. The challenge is for teachers and curriculum developers to create engaging materials that encourage students to take an active role in sense-making where the movement between various representational forms is required. Some students may have certain representational preferences, but it is still the ability to move flexibly among these representations that defines a depth of understanding.

What is a root and how is it related to various representations? What does it mean to be a solution to an equation? These are standard questions we expect our students to be able to explain and apply. The problem is that while many students may be able to solve equations, far too many have only procedural knowledge of the equation-solving process. Consider the following explanation posed by Aaron. In question 8 from an interview (Lapp, Ermete, Brackett, & Powell, 2013), when asked to solve the equation \(\left(x-2\right)\left(x+3\right)=6\) (with no graph given), he began by setting each factor equal to 6 (see Figure 4.5.2).

Aaron: What I would do is have \(x-2=6\) and then \(x+3=6\) (writes \(-3\) under the \(x+3\) and the 6), \(x=3\text{,}\) (writes \(+2\) under the \(x-2\text{,}\) see Figure 4.5.2), \(x=8\text{,}\) so then I have the two solutions…like that...I mean, I don’t know if those approaches are right...I’d have to know exactly where we are at.

Interviewer: What do you mean, where you’re at?

Aaron: Um, what we’re trying to get out of it…out of the equation. What we’re trying to do with it. What we’re trying to do with the numbers we’re trying to find...if we’re trying to find where it’s going to cross on the x-axis...you know.

Interviewer: As opposed to just the solutions?

Aaron: Yeah.

Interviewer: So you are thinking whether or not you are simply trying to find the solutions as an algebraic method versus what does it mean graphically?

Aaron: Yeah. So basically like when I looked at that, it’s like this is just an algebraic expression...you know, just find the answer. As opposed to like over here (referring to question 7, see Figure 4.5.3), I would have tried to figure out what the solution is as opposed to where it is on the graph.

Interviewer: So in question 7, the graph being there might have affected your interpretation of what in the problem you were supposed to do?

Aaron: It wouldn’t have been, solve this algebraically. It would have been solve this algebraically for the graph instead of just solving it. Um, so you know if I had finished this out, I would have got something...I would have done something more as break it down as what I just did as opposed to actually solving it out like on 8 (referring to his work from question 8, see Figure 4.5.2).

Interviewer: So what would you have done different on like 7?

Aaron: On 7, I would have put them both equal to 4. So like on 7, it would be \(x-1=4\text{...}\) (continues to write \(x+2=4\) in silence, then writes \(+1\) under one equation and two 2s under the other, see Figure 4.5.4). And then I would’ve been looking at this and like uh oh, something’s not right.

Interviewer: What would clue you in that something’s not right?

Aaron: The solutions weren’t...the solutions didn’t match.

Interviewer: So what would you expect the solutions to match? For example, the numbers you got were what, 5 and...

Aaron: 5 and 2, and I would expect them to match -2 and 1 (pointing to the x-intercepts on the graph). Um, just because that’s where they cross.

Interviewer: Where they cross the x-axis?

Aaron: Yeah. So looking at this, I would have been like something’s not right. That’s when I would go back and be like try to look through and try to find where that equation looks like that and then apply it to whatever the example is in the book and try to figure out a new answer.

Interviewer: So you are looking for some sort of consistency between the answer you get from algebraic manipulation and the interpretation you would have from looking at the graph?

Aaron: Yep.

Having given some initial thought to Aaron’s approach and understanding, it is time to take a deeper dive into his misconceptions and what the research says about multiple representations. Aaron continued to try and find resolution between his algebraic solutions and his expectations that these solutions should correspond to the x-intercepts from the graph, but he was unsuccessful in resolving the issue and gave up. Why was Aaron unable to see the error in his solution process even though he did attempt to use multiple representations when a graph was provided? Since the instructor regularly made it a point to use multiple representations in solving equations during class, it is not surprising that Aaron would seek confirmation when the graph is present; however, it is not just the use of multiple representations that helps students make connections. It really has more to do with how these representations are interfaced.

It is important to note that during this particular semester, the students used a basic graphing calculator (TI-84) that did not have dynamically connected representations and thus Aaron did not have the ability to see a “real-time” linkage between the key features of the graph and their corresponding algebraic counterparts from the symbolic world. Through research on the use of microcomputer-based laboratory (MBL) technology (also calculator-based data collection devices), we have seen that real-time changes in linked representations simultaneously visible in the same field of view, allows students to make connections among corresponding features of the various representations (Lapp & Cyrus, 2000; Beichner, 1990; Brasell, 1987). While Aaron had access to the use of multiple representations, these representations were not dynamically linked and thus he did not make a connection between the meaning of the graphical representation of x-intercepts and the zero property of multiplication as a technique for equation solving. To discuss the specifics of the experiences we can give students to foster this transfer, we first must consider the dynamics of how symbolic meaning comes to be.

One of the powerful aspects of mathematics is its use of symbolization. On the other hand, from the aspect of the learner, one of its worst obstacles is its symbolization. Since the early development of mathematics, power has come from the development of symbol systems used to describe the concepts that are fundamental to furthering mathematical knowledge. The encapsulation of big ideas, that before took pages and pages to communicate, into smaller collections of symbols allows the reader to see various stages of arguments within one field of view. However, from a learner’s perspective, it is the unpacking of the symbolization that poses a significant hurdle in the development of conceptual understanding.

If we think of this in terms of concept image and concept definition (Tall & Vinner, 1981), a small collection of symbols typically elicits much more information than could be contained in paragraphs of written prose. For example, suppose we say the word “dog”. What comes to mind? It might likely be a mental image of a beloved pet from your childhood or an image of a traumatic event you had with a strange animal while walking last week. These images are far more complex than the simple three letters “dog” or a formal definition from Webster’s dictionary. It is our experiences that are connected to the compact system of symbols “dog” that form our concept image of a dog. Similarly, it is the experiences that a student has between mathematical ideas and the symbols that represent them that determine her/his concept image.

So how does the use of symbol systems influence our learning, and even more, our development of new ideas? Recall that Kaput, Blanton, and Moreno (2008) propose a model Figure 1.1.5 that describes how we develop both our understanding of existing ideas as well as the creation new concepts. The key to this model is the communication and analysis between two worlds. One is the "real world" (i.e. the world of either broader mathematical or physical experiences) and the other is the world of the symbols that we use to represent these "real world" experiences. In this model, the learner creates raw representations of ideas and in turn these representations provide the learner with a tool to reason about these ideas. As the learner uses these raw representations for reasoning, they begin to view the "real world" experiences differently as the representations serve as a lens through which reality is filtered. As this process progresses, new, more refined representations and symbols are created influencing both the tools used for reasoning as well as the learner’s view of the real world itself.

It is important to note that this process, while focusing on representations, is consistent with Sfard’s (1991) process of interiorization, condensation, and reification where concepts become internalized through interaction with the various natures of the mathematical ideas. In some ways, the use of technology here reduces the length of time for Sfard’s interiorization and facilitates the condensation process. Alleviating the computational load can also keep the lack of procedural fluency from becoming a cognitive obstacle to reaching condensation or reification as observed by Lapp, Nyman, and Berry (2010).

Given this theoretical model for the development of symbolic meaning, the next question is how do we create experiences that allow our students to move fluidly among various representations while maintaining an isomorphic mapping (see what I did there — wink) of meaning between various worlds used to represent the ideas involved? One way is to utilize technology that allows for linked representations. In other words, representations that dynamically update in real time as each is manipulated.

In the same study as Aaron, in contrast to the Fall semester’s class using only basic graphing technology, the students interviewed in the Spring semester showed an ability to move among representations as a means for justifying their reasoning. To illustrate this process, we examine Jon’s response to the same question posed to Aaron, namely, solving the equation \(\left(x-2\right)\left(x+3\right)=6\text{.}\) Unlike students from the previous semester, Jon realized that he must get the equation set equal to zero by subtracting 6 from both sides. In this exchange, Jon refers to his solutions to other problems referring to problems 6 and 7 (Figure 4.5.5).

Interviewer: So what do you think is different about this? Why is it important that you get the six to the other side?

Jon: Because I think if you were to graph it, you’re trying to find a completely different line, if it’s a six or zero and if it’s zero... You’re trying to find \(y\text{.}\) And then if \(y\) was zero... I don’t know.

Interviewer: So you’re thinking of it from a graphical standpoint?

Jon: Yeah, I’m thinking about it from a graph. If it’s equal to six it’s going to be up higher and it’s gonna be... You’re not looking for the zeros [points to problems 6 and 7 from the earlier questions].

Interviewer: So kind of like the difference in those earlier problems?

Jon: Basically I took number seven and tried to turn it more into like number six. Like I tried... I just moved it down so that way I would be able to solve for it. I visualized it as more like bringing it down and I just solved for the vertex [here he said vertex, but was referring to the zeros], which I know how to do. When it comes to that, it’s more complicated to get to that point.

Interviewer: Oh, the x-intercepts you mean, as opposed to the vertex?

Jon: Yeah cause when it’s at equals six, it’s like the whole thing got raised six. So I just made it...I brought it down to the vertex...and then I just solved for it there because that’s easier for me to do algebraically.

In this case, Jon had first been exposed to dynamically linked representations through technologically rich investigations in class. In one investigation the manipulation of a graph gave real time change in the algebraic representation of the quadratic function as well as the factored form of the function. Jon was asked to label the x-intercepts on the graph and as the graph was manipulated, he noticed a connection between the numerical values of the x-coordinates of the x-intercepts and the numerical values, \(r_1\) and \(r_2\) found in the factored form of the function \(a\left(x-r_1\right)\left(x-r_2\right)\) (see Figure 4.5.6).

The mind of the student naturally seeks out aspects of a situation that are invariant across representations. In this case, the student noticed that no matter how the graph was manipulated the numerical values in both the labeled x-intercepts and the factored form of the function remained related. The classroom investigation also asked questions designed to focus the student’s attention on the effects on the function’s output for entering each x-coordinate of the zeros into the factored form of the function as well as the values of each factor. It was this combination (communication and analysis between symbolic worlds) that enabled the student to articulate the reason for use of factoring as a technique for equation solving and the reason that it is imperative the equation be set equal to zero before factoring.

In another investigation, the students developed a connection between algebraic and graphical representations for the process of completing the square and observed relationships among various parameters found in the algebraic expressions (see ). Here the students further experienced the connection between both algebraic and graphical transformations. This led to Jon’s strategy of transforming the graph of a parabola and its intersection with a horizontal line above the x-axis into one where the parabola was shifted down until the horizontal line was superimposed with the x-axis. In his desire to transform an equation by subtracting 6 from both sides, he expressed a graphical understanding linked to the algebraic transformation of subtracting 6. He stated that he was essentially moving the dotted line (see Figure 4.5.5) down to coincide with the x-axis so that he could use his technique of factoring that required the equation to be set equal to zero. In this instance, he was able to justify his process and not just procedurally execute it.

The influence of the Completing the Square (see Figure 4.5.7) investigation can also be seen in Jon’s reference to the movement of the parabola’s vertex during his verbal description of his graphical understanding. In the Completing the Square activity, the students were specifically asked to follow the movement of the vertex.

Jon clearly demonstrated an understanding of the concept of root and its connection to factoring as an equation solving technique. In explaining why he could not simply set each factor equal to zero in the equation \(\left(x-2\right)\left(x+3\right)=6\) he states,

The fact that Jon could articulate an understanding of the root solving process along with his verbal reference to the vertex movement indicates an influence of both of these investigations on his mathematical understanding and its relationship to the symbolic world that describes these ideas.

The bottom line of these contrasting examples is that we should not have to only expect procedural fluency, but rather that conceptual understanding can go hand-in-hand with procedural ability. This example challenges the conventional wisdom that students must first become procedurally fluent before they come to understand the concepts that we teach. Heid (1988) more than two decades ago challenged this position and argued that we should rethink the sequencing of skills and concepts in Calculus by using computer algebra systems to develop concepts prior to teaching procedures. Here we see this same principle applied to high school algebra concepts. The difference between this study and Heid’s is that we suggest it is not just the computer algebra system that influences student connection of concepts, but rather the dynamic linking of various representations. The students in the first semester of this study used the TI-84 and did have the development of concepts prior to skills; however, they did not have the ability to manipulate various representations and see real time effects among representations.

A second aspect of concept development that we noticed involves student control of the environment. As Lapp & Cyrus (2000) from research on the use of data collection devices as well as Lapp, Ermete, Brackett, & Powell (2013) experience within a CAS microworld suggest, the student’s ability to manipulate the environment plays a significant role in making connections. During the first semester of this study, the instructor merely demonstrated some of the dynamically linked representations using a computer during class since the students did not have use of this technology individually. Results of our interviews showed that none of these students could articulate connections among various representations. However, during the second semester, each student had a TI-Nspire CAS device and used it during student-centered investigations. We argue that it was this ability to communicate and analyze via dynamically connected representations between the symbolic world and the world of mathematical ideas that allowed the students make connections between concepts and procedures

As technology that links representations has become readily available, there is no reason we should not take advantage of it to better develop students’ mathematical understanding; however, technology alone cannot make these connections for students. Kaput Blanton, and Moreno (2008) as well as Sfard (2008) suggest that it is the students’ negotiation or discourse between the symbolic world and the world of mathematical ideas that plays a key role in the development of symbolic meaning. Therefore, as teachers, we need to use appropriate technology to engage students in investigations that allow them to make mathematically meaningful observations and justify them.