Algebra for Secondary Teachers: Engaging Students to Develop Symbolic Meaning

To begin, consider The Border Problem: Part 1 found in Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning (2005) by Jo Boaler and Cathy Humphreys. In this video case, students are exploring the pattern for finding the number of small shaded squares along the outer edge of a larger \(10 \times 10\) square as seen in Figure 2.3.2.

Now that we have speculated about Cathy’s plan for her class, let’s see where she takes the discussion on the next day. In the next activity, pay attention to the way Cathy tries to draw connections between the physical representations of border squares and the way the computations are taking place.

The classroom exchange provides a context to explore this type of sequence. Understanding how a sequence behaves can give us, as teachers, a way to allow students to explore their own observations while facilitating a fruitful discussion. Cathy does a masterful job in allowing students to express their ideas and connecting them during the whole-class interaction. So, let’s look a little deeper at this type of sequence by exploring the question Cathy posed when she asked the students to image shrinking the square to a \(6\times6\) and using various methods to compute the total number of small squares on the border. In this case, we will extend it more to consider the \(n \times n\) case.

Some sequences like arithmetic sequences have properites like a common difference between adjacent terms. There are many types of phenomena that can be explained by this behavior; however, there are other types of behavior that do no have a common difference (or constant rate of change). In the next activity, we will explore what can happen when the change between terms is not so easily described. To begin, consider the bouncing of a racquet ball as described in Figure 2.3.4.

The relationship from term to term you have just explored is called geometric. Just like with arithmetic sequences where we can determine its arithmetic nature by subtracting adjacent terms to see if we always get the same result, with geometric sequences we can do the same, but using division instead of subtraction. In a sense, the structure of the behavior is the same, but instead of observing the relationship between addition and subtraction, we notice the analogous relationship between multiplication and division.

You will notice in the previous definition, if we take successive ratios as you did in the Ball Bounce activity, we get \(\frac{ar}{a}=\frac{ar^2}{ar}=\frac{ar^3}{ar^2}=\ldots =r\text{.}\) We use the notation, \(r\text{,}\) here since we are referring to the common ratio.

In order to make connections with what is taught in the secondary curriculum, we begin by examining some activities from the Interactive Mathematics Program (IMP). This activity will set the stage for reflecting on both the mathematics and the teaching and learning practices involved in the exploration. In particular, we will consider the relevance of the depth of content knowledge on classroom decisions made by the teacher.

Now that you have started to reflect on the mathematics and pedagogy present in the exploration, consider the following classroom exchange as a teacher (Jill) interacts with her students. Following the vignette, you will be asked to respond to questions surrounding the discussion.

At this point we will take some time to explore some mathematical ideas related to the previous classroom vignette using an activity called, Evaluate Me Again Sam. In your groups, complete . The material needed for the activity is only a computer algebra system.

Before we continue our discussion of the mathematics involved here, reread the vignette of the classroom discussion in the Case Study: Over and Over and Out found in Activity 2.3.8.

One of the difficulties common to pre-service teacher preparation is the ability to see how the undergraduate mathematics curriculum is connected to the secondary curriculum. To address this, we will again examine a secondary mathematics curriculum, The Interactive Mathematics Program and reflect on how this type of curriculum is related to the classroom discussion from our previous vignette and to the undergraduate mathematics explored in Evaluate Me Again, Sam (Activity 2.3.9).

The previous explorations have given us some insight into predicting when an iterative sequence will converge and when it might diverge. So how can we be certain of the behavior and then, as teachers, help our students discover the properties behind the behavior? Here we will take a deeper look into the observations made in our previous investigations and see how iterative sequences can play a role in practical applications.

The concept of iteration is an important part of the mathematical landscape. It is used in real-world applications to find solutions that have no direct algebraic solution as well as for approximating values of mathematical constants such as \(e\) and \(\pi\text{.}\) Having taken Calculus, you are already familiar with one such iteration process, Newton’s Method. Recall that Newton’s Method allows the approximation of roots by using the projection along a tangent line to the x-axis as a new approximation of the root. Then the new value from the x-axis is used as the new point of tangency and the process continues until the desired level of accuracy for the root is achieved.

\(\mathbf{Newton's}\) \(\mathbf{Method:}\) \(\mathbf{A}\) \(\mathbf{Refresher}\)

We know that some equations of the form \(f\left(x\right)=0\) can be solved exactly (e.g. using the quadratic formula). However, the general equations that can be solved exactly are very few within the realm of mathematical endeavor. To be able to approximate a solution, Isaac Newton noticed that as a point on a curve approaches the root of the function, its tangent line at that point begins to have an x-intercept that is approximately the root desired (see Figure 2.3.8).

Newton’s idea was to “surf” the tangent line. In other words, pick an x-coordinate of a point you know is close to the actual root, and then find where the tangent line to the curve at that point intersects the x-axis. Now use the x-intercept of the tangent line as the new x-coordinate as stated earlier and repeat the process. With each iteration, the approximation gets closer to the root of the function.

For example, suppose you wish to find the solution for \(f\left(x\right)=0\text{,}\) where \(f\left(x\right)=x^2-3\text{.}\) As you can see, one solution is approximately 2 as can be seen from the graph in Figure 2.3.9.

In order to find a better approximation, we can notice that if \(x_0\) is our initial guess, then the next choice for an approximation of the root will be \(x_1=x_0-\frac{f\left(x\right)}{f'\left(x\right)}\text{,}\) since this is where the tangent line at \(x=x_0\) intersects the x-axis. Repeating this process will give better approximations.

In practice, we can approximate using Newton’s Method on our CAS by entering the original function in one location and the formula for the next approximation in another function location referring to the original function. In the figure below, the original function is entered in the function \(f\left(x\right)\) and the iteration formula is in \(nm\left(x\right)\) for “Newton’s Method” (see Figure 2.3.10).

We now simply evaluate the \(nm\left(x\right)\) function at our initial guess and then successively using our result as the new input into \(nm\left(x\right)\text{.}\) You can see that with just a few iterations, we get a solution to 11-decimal place accuracy. Here we have actually approximated \(\sqrt{3}\) since the function we are using is \(f\left(x\right)=x^2-3\text{.}\)

From a graphical standpoint, suppose we begin with an initial value of 3. The x-intercept of the tangent line at \(x=3\) is 2 and thus a new tangent is constructed at \(x=2\text{.}\) As can be seen from the sketch in Figure 2.3.11, the new x-intercept is much closer to the root. It does not take many iterations of this process to get extremely accurate approximations.

\(\mathbf{Iterations}\) \(\mathbf{and}\) \(\mathbf{Convergence}\)

As we have seen from exploration and classroom vignettes, not all self-iterating functions yield a convergence. So how can we tell if a particular function will converge to a value at all or whether or not it may have multiple values that are possible for convergence? How do we predict the value to which they will converge? Are there any criteria that can be applied to predict convergence?

Let us begin by examining the familiar function from the activity, Evaluate Me Again Sam, Activity 2.3.9, with \(f\left(x\right)=\sqrt{x}\text{.}\) Below is a graph of \(f\) along with the graph of \(y=x\text{.}\) The reason for \(y=x\) will become evident shortly. Assume that \(f\) is continuous and differentiable on the desired interval. Also let us define the iteration by stating \(x_{n+1}=f\left(x_n\right)\text{.}\) All this says is that we choose the previous output of \(f\) as the new input of \(f\) each time we iterate the function with itself. Here is the reason for graphing \(y=x\) along with \(f\text{.}\) Since the outputs are found along the y-axis, we get \(x_{n+1}=y_n\text{.}\) This process is shown by the arrows marking the progress of the iteration on the graph in Figure 2.3.12. We begin with \(x_0\) and find its corresponding output (\(y_0\)) on the function. We then move horizontally until we meet the line \(y=x\) since this is where x and y are the same. We now move vertically to the x-axis so that we stop on the “new input” (\(x_1=y_0\)). We repeat the process by moving vertically to the function’s new output. As you can see by following the arrows in the graph, we keep moving closer and closer to the point of intersection between the graph of \(f\) and \(y=x\text{.}\) But why is the intersection with \(y=x\) important?

The point where a function intersects the line \(y=x\) is important and is called a fixed point. The reason it plays a vital role in the convergence of iterated functions is that it has the property of the function’s output being equal to the function’s input.

Since the convergence of an iterated function relies on the fact that the \(\lim_{n\rightarrow \infty}f\left(x_n\right)=p\) when \(\lim_{n\rightarrow \infty}x_{n+1}=p\) and thus \(f\left(p\right)=p\) assuming \(f\) continuous, the intersection of \(f\) and \(y=x\) gives us possible values of convergence for our iterated function. We can gain insight into some general principles for convergence by looking at the earlier investigation found in the IMP materials, Over, and Over, and Over, and…. Recall that in this exploration, the students began with linear functions. For functions such as \(f\left(x\right)=3x+2\text{,}\) the iterated function diverged no matter what initial seed value was chosen. Consider the function \(g\left(x\right)=\frac{1}{2}x+1\) and examine the progression of the inputs as shown in Figure 2.3.14 using arrows along the graph. Notice that no matter which side of the fixed point you begin with, the result converges on the fixed point.

Now consider the function, \(h\left(x\right)=2x-1\text{.}\)

As can be seen here, no matter which side of the fixed point we choose, the iteration diverges to either \(\pm \infty\text{.}\) In this case, the fact that \(h\) is steeper than \(y=x\) means as we choose points near the fixed point to use for iteration, the mapping onto the line \(y=x\) prior to the re-evaluation of \(h\) moves away from the fixed point. As we saw with the function \(g\left(x\right)=\frac{1}{2}x+1\) earlier, the mapping back to the line \(y=x\) prior to re-evaluation of \(g\) causes the \(x_n\) to approach the fixed point. The behavior near these types of fixed points has given rise to the categorizations of repelling and attracting fixed points.

So if sometimes we are repelled from a fixed point and sometimes we are attracted to one, how can we tell what will happen? For example, when using \(f\left(x\right)=\sqrt{x}\text{,}\) we saw any seed we used be attracted to 1, but with \(f\left(x\right)=x^2\text{,}\) it was repelled. Why?

To look deeper into this let’s reconsider our earlier activity from IMP, Over, and Over, and Over, and..., we discussed in Activity 2.3.7.

As you have observed in your investigation in Activity 2.3.12, for a linear function, the iteration’s convergence depends on the value of the slope, \(m\text{.}\) This leads to the following theorem where the proof consists of your observations from the activity.

Theorem 2.3.17.

Let \(f\left(x\right)=mx+b\) be a linear function and let \(f^n\left(x\right)=\left(\underbrace{f\circ \cdots \circ f}_{\text{n times}}\right)\left(x\right)\text{.}\) The sequence, \(\left\{f^n\right\}\text{,}\) converges if \(\left|m\right|\lt 1\) and diverges if \(\left|m\right|\geq 1\text{.}\)

Proof.

Consider the iterations for \(f^2\left(x\right)=m^2x+b\left(m+1\right)\) and \(f^3\left(x\right)=m^3x+b\left(m^2+m+1\right)\text{.}\) In general, we could conclude that \(f^n\left(x\right)=m^nx+b\left(m^{n-1}+\cdots m^2+m+1\right)\text{.}\) We can use proof by induction to show this holds. Assume \(f^k\left(x\right)=m^kx+b\left(m^{k-1}+\cdots m^2+m+1\right)\text{.}\) We get \(f^{k+1}\left(x\right)=m^k\left(mx+b\right)+b\left(m^{k-1}+\cdots m^2+m+1\right)\text{.}\) Expanding and collecting like terms gives \(f^{k+1}\left(x\right)=m^{k+1}x+m^kb+b\left(m^{k-1}+\cdots m^2+m+1\right)\text{,}\) and therefore \(f^{k+1}\left(x\right)=m^{k+1}x+b\left(m^k+m^{k-1}+\cdots m^2+m+1\right)\text{.}\) By induction, this holds. In order to converge, the geometric series \(m^n+m^{n-1}+\cdots m^2+m+1\) needs \(\left|m\right|\lt 1\text{.}\) If \(\left|m\right|\lt 1\text{,}\) then \(m^n \rightarrow 0\) as \(n \rightarrow \infty\) and thus \(\left\{f^n\right\}\) converges. If \(\left|m\right|\geq 1\) then the series diverges and the \(m^nx\) term will either diverge or simply be \(x\) in the case of \(m=1\text{.}\) In either case, \(\left\{f^n\right\}\) will diverge.

The use of linear functions to analyze the convergence or divergence of the iterated function is helpful since as with any continuous and differentiable function, we consider them locally linear as we zoom in on a fixed point. If we have a curved function with a fixed point and we start to get close to the fixed point, it will start to behave like the line in Theorem 2.3.17. Therefore, this leads us to the following argument regarding convergence of iterated functions with fixed points.

Theorem 2.3.18.

Let \(f\) be a continuous and differentiable function with a fixed point, \(z=f\left(z\right)\text{.}\) Further, let \(x_n=f\left(x_{n-1}\right)\text{.}\) If \(x_0\) lies on an interval, \(I\text{,}\) containing \(z\) such that \(\left| f'\left(x\right) \right| \lt 1\) for all \(x \in I\text{,}\) then \(x_n\) converges to \(z\text{.}\)

Proof.

Suppose \(x_0 \in I\) such that \(\left| f'\left(x\right) \right| \lt 1\) for all \(x \in I\text{.}\) Since \(x_1=f\left(x_0\right)\) and \(f\left(z\right)=z \Rightarrow\) \(x_1-z=f\left(x_0\right)-f\left(z\right)\text{.}\) Now by the Mean Value Theorem, there exists some \(c\) between \(x_0\) and \(z\) such that \(f'\left(c\right)=\frac{f\left(x_0\right)-f\left(z\right)}{x_0-z}\) and thus \(f\left(x_0\right)-f\left(z\right)=\left(x_0-z\right) \cdot f'\left(c\right)\text{.}\) Since \(c \in I\) and by assumption \(\left| f'\left(x\right) \right| \lt 1\) for all \(x \in I\) \(\Rightarrow\) \(\left|x_1-z\right| \leq k\left|x_0-z\right|\) for some \(k\) with \(\left|k\right| \lt 1\text{.}\) Therefore, \(x_1 \in I\text{.}\) Using a similar argument, \(\left|x_2-z\right| \leq k\left|x_1-z\right|\leq k^2\left|x_0-z\right|\text{.}\) In general, we can state that \(\left|x_n-z\right|\leq k^n\left|x_0-z\right|\text{.}\) Since \(\lim\limits_{n\rightarrow \infty}k^n=0\text{,}\) we get \(\lim\limits_{n\rightarrow \infty} \left|x_n-z\right|=0\) and thus \(\lim\limits_{n\rightarrow \infty}x_n=z=f\left(z\right)\text{.}\)

So what does this theorem say about convergence to fixed points? Basically, the theorem says that if the fixed point, \(z\text{,}\) is in an interval where \(\left|f'\left(x\right)\right| \gt 1\text{,}\) then the mapping that transforms the outputs to inputs back into the function is expanding and thus the iteration process diverges.

\(\mathbf{Fixed}\) \(\mathbf{Point}\) \(\mathbf{Iteration}\) \(\mathbf{and}\) \(\mathbf{Root}\) \(\mathbf{Finding}\)

Although Newton’s Method is very efficient for finding zeros of a function, a more simplistic method involves only the iteration of a function with itself using only a small alteration. Suppose you are looking for the root of a polynomial, \(p\left(x\right)\text{.}\) The process boils down to solving the equation, \(p\left(x\right)=0\text{.}\) Now consider the transformed equation formed by adding \(x\) to both sides of the original equation, \(p\left(x\right)+x=x\text{.}\) If we let \(q\left(x\right)=p\left(x\right)+x\text{,}\) we are essentially finding a solution to \(q\left(x\right)=x\text{.}\) We know from our earlier discussion that this solution is simply a fixed point for \(q\left(x\right)\text{.}\) Based on the fact that repeated iteration will converge to a fixed point (if convergence does occur), we can simply take the original function, add \(x\text{,}\) and then iterate it with itself as a method to find its root. The down side of this process is that not all fixed points are accessible through iteration as we have seen in earlier examples. However, this does remain a very simple algorithm for finding roots although not as efficient as Newton’s Method.

Since we are using the Over and Over and Over and… activity from the Interactive Mathematics Program (IMP) it is fitting to examine the previously mentioned study by Weigand (1991) where he examined 11th graders’ learning of the same iterated sequence \(a_1,f\left(a_1 \right),f\left(f\left(a_1 \right)\right), \ldots\) found in the IMP text. In Weigand’s study, 79 students and 22 secondary mathematics teachers were engaged in exploring sequences generated by iterating functions. One of the goals was to investigate the influence representations have on students’ abilities to discover properties and solve problems. Weigand chose iterated sequences as a context for the study because it is often difficult to express sequences in explicit algebraic terms. Therefore, the use of a variety of representations becomes necessary in order for students to explore the behavior of these sequences. Weigand examined six different non-algebraic representations: Beginning of Sequence, Table, Cobweb Diagram, Graph, Arrow Chain, and Arrow Diagram. These representations can be seen in Figure 2.3.21, Figure 2.3.22, Figure 2.3.23, and Figure 2.3.24.

In these representations, the numerical view of sequence was embodied primarily in the Beginning of Sequence and Table format while the geometric view was primarily embodied within the Cobweb Diagram, Graph, and Arrow Chain diagrams. What is interesting is that from a geometric perspective, the cobweb and arrow representations were not effective in relating the representation to the sequence. These were viewed primarily as pictorial descriptions rather than relationships. For someone already familiar with the concept of iterative sequences, the cobweb diagram makes sense and can lend additional meaning to the user, but to the newcomer, it adds little and in fact, can lead to incorrect mathematical interpretations (Weigand, 1991). Although Weigand’s results cannot definitively explain why the geometric representations apart from the graphs were not effective in connecting the representation to the relations involved in the sequence, one might speculate that it may be related to the lack of exposure to such representations. Weigand suggests that in order for students to deepen their understanding of a concept such as iterative sequences, “[they] must also be given the opportunity to use this variety of representations. When teaching using these elements, the computer can be an indispensable aid and tool” (p. 433).

In order to help students develop reasoning skills related to a concept, it is also important that the teacher is aware of how students make use of representations. As Weigand found, the teachers and students applied different methods of thinking. The teachers tended to use more formal mathematical thinking strategies while the students made use of informal approaches involving different representations. It is crucial that teachers not ignore the ways in which students think as they plan instruction. Weigand suggests that his results support the demand typically given to symbolic representations in contrast to iconic ones within the classroom since, to the teacher, the goal is to move in the direction of symbolic forms.

So what are the advantages and disadvantages of different representations? While the numerical representations found in Weigand’s beginning of sequence and table formats offer views of the actual value to which a sequence is converging, these representations do little to offer insight into an overall behavior of a sequence. Weigand found that unsuccessful students tended to use the numerical forms; beginning of sequence and table representations, and did not use any graphical representations at all. He found that for most solutions strategies, students needed to deal with comparisons of sequences with different initial values. The numerical representations force students to do a comparison of terms on a one-by-one basis. In contrast, the graphical representations provide comparison of many terms in a single view allowing broader patterns to be detected.

Even though the graphical representation allows the students to see the sequence behavior from a broader perspective, Weigand also reports that students still had difficulty connecting the properties of the function used to generate the iterative sequence and the iterative sequence itself. In this case, students did see the iterative sequence as a functional relationship, but described the properties of the sequence dynamically with no connection to the recursive nature of the sequence’s origin or the properties of the generating function. The result of this disconnected view of sequence iteration is that students “hardly used arguments which could be applied to all iteration sequences” (Weigand, 1991, p. 434).

So how do we help students view commonalities that might apply to iteration sequences in general? One of the challenges is to help students make broader observations with respect to sequence behavior. Just as the graphical representation gave a broader perspective of the sequence behavior in comparison to the numerical representation in Weigand’s study, we need to find a way to allow a “bigger view” of multiple graphical representations as parameters in the situation change in order to give a broader perspective of change across situations. Lapp and St. John (2009) describe just such an environment where students notice patterns across multiple graphical representations as parameters in the situation change. The key to describing the change of changing graphs lies in dynamically linked representations where the change in one triggers real-time change in the others. In this case, students have the graph and algebraic representation of the generating function appearing on one side of a viewing window and the graph of the scatter plot of the iterative sequence on the other side of the same viewing window (see Figure 2.3.25). Here the graph of the generating function can be grabbed and pulled resulting in changes in both the graphical and algebraic representations of the function. As the graph is manipulated, the scatter plot of the iterative sequence dynamically changes in the same field of view. This use of technology allows students to not only see the sequence behavior, but also the corresponding behavior of the function that creates the sequence, thus addressing a problem found with students in Weigand’s study.

While the graphical representations can help students notice patterns, in mathematics, we do rely on algebraic reasoning for proving properties in a general sense. A concern for the lack of connection to algebraic representations is voiced by Alcock and Simpson (2004). In this study, some students who used graphical representations maintained a strong sense that they understood the concepts. The problem was that some of these students did not feel a need to examine the more formal representations used for proof. Alcock and Simpson did find that some students who relied on visual or graphical representations made connections to algebraic properties, but these students tended to be students whose “beliefs about learning mathematics prompt them to seek overall integration of both visual and algebraic/verbal representations of the material” (p. 29). The students who seem to be motivated to seek such integration tended to have a stronger sense of internal authority with respect to mathematics. The students who did not seek integration tended to have an external sense of authority where mathematics is viewed as a set of conventions and procedures to be learned.

The lesson to be learned from this is that we need to help students develop a sense of internal authority so that they will more naturally seek integration of informal and formal representations. Accomplishing this task of motivating a sense of internal authority can be a challenge. One suggestion is to transition from guided discovery to inquiry teaching by asking appropriate questions until students begin to ask those questions themselves. Connecting algebraic reasoning to graphical behavior is also described by Lapp and St. John (2009). Here students are asked specifically to explain the graphical behavior they have observed in terms of algebraic representations. The students in this case used the computer algebra system to extend the patterns to algebraic representations and then reason from those representations (see Figure 2.3.26). Alcock and Simpson (2004) suggest this same approach when they state, “in order to help other visualizing students build on their existing strengths, we need to think about ways to have them engage in work that will help them to build such links” (p. 30).

In summary, we suggest a need to use dynamically connected representations in conjunction with a move toward inquiry approaches to teaching. The literature suggests a need to engage students in “doing” mathematics while using many representations. Our perspective is that when students can view the technology as a tool for investigation, autonomy in the problem-solving process will be more likely. If the students can easily move between representations and the teacher models such movement, we can empower the students to become self-learners relying on an internal sense of authority.