Nspiring Times, Inquiring Minds: Fostering Critical Examinations of Curriculum with Handheld Graphing Technology

In the examples that follow, this approach will be illustrated in two different activities. The first (referred to as "The Trinomial Factoring Investigation") employs spreadsheets enhanced by CAS (Computer Algebra System). The investigation encourages pre-service teachers to reconsider the relevance of quadratic factoring in the high school algebra curriculum. The second (referred to as "Building Parabolas") exploits the multiple representation capabilities of TI-Nspire™ technology to help teacher candidates consider the many rich connections between high school algebra and geometry.

Introduction to the Investigation
In the article "What Should Not Be in the Algebra and Geometry Curricula of Average College-Bound Students? ", Usiskin (1980) notes that "factoring does not work for the vast majority of trinomials - even those with integral coefficients . . . instead of factoring, the quadratic formula should be used" (p. 71). As mathematics teacher educators, Usiskin's observation intrigued us. We are interested in encouraging pre-service teachers to actively question content, so we ask our teacher candidates to explore Usiskin's recommendations through the following content-oriented task.

Task Details
The built-in, rich features of TI-Nspire CAS technology enable teacher candidates to explore the previous task as a probability-oriented simulation. Using a data analysis approach, pre-service teachers determine an experimental probability associated with randomly generating a perfect square trinomial.

The TI-Nspire functions randpoly( ) and seq( ) provide candidates with a convenient way to generate a list of random trinomials in a spreadsheet page. For instance, randPoly(x,2) generates a random 2nd degree polynomial (such as 6x^2-4x+1). The seq( ) function generates sequences; combining randpoly( ) and seq( ) into one command allows candidates to generate a list of random trinomials. Such an approach is highlighted in Figure 1. The command seq(randpoly(x,2),a,1,500) is used to store 500 random trinomials into Column A of the TI-Nspire spreadsheet. Next, using the familiar factor( ) function, candidates store factored forms of the random trinomials into column B of the spreadsheet.

Figure 1: Column A is defined as a list of 500 random trinomials. Using the factor() function, factored forms of these trinomials are stored in Column B.

While pre-service teachers may scroll through Column B to determine the number of trinomials in Column A that are factorable (counting items without a squared term), we prefer to automate this task with a user-defined function, countft( ). As Figure 2 (left screen) suggests, candidates may use countft( ) to calculate the percentage of factorable trinomials in Column A. Moreover, by re-randomizing the trinomials in Column A, candidates quickly calculate percentages for multiple trials. Figure 2 (right screen) illustrates a plot of percentages generated in such a manner.

Figure 2: (Left) Candidates calculate the percentage of factorable trinomials found in column A; (Right) Dot plot of percentages using TI-Nspire technology's built-in Data & Statistics application.

Discussion of the Investigation
The "Trinomial Factoring Activity" is by no means a mathematically rigorous or exhaustive task.  For instance, pre-service teachers generate only a small subset of possible trinomials and they don't explore theoretical probabilities associated with the task. Nevertheless, the activity is a powerful one. For many candidates, the activity represents the first time they have been asked to seriously question the inclusion of specific topics into the secondary school curriculum. The following is a student response to the second part of the task:

"I found the results of this activity truly amazing. When I was in high school, we must have spent five or six weeks of class studying factoring rules for quadratics. That's nearly a sixth of the school year. Given how few quadratics actually factor, it's clear to me now that we spent a disproportional amount of time on this. I'll think twice before spending this much time covering trinomial factoring in the courses I teach." (EDT 429A candidate, Fall 2007)

Pre-service teachers, at this point in their training, are more comfortable acting as "students" than as "teachers." The task is purposely framed as a probability-oriented task rather than a "teaching methods" activity.

Since few of our pre-service teachers have taught trinomial factoring to high school students, a strictly pedagogical consideration of the topic fails to generate much interest. On the other hand, the content-oriented feel of the calculator-based factoring activity draws students into a setting in which they have far more experience - namely solving mathematics problems. From this initial activity, we build a foundation for more pedagogical-oriented discussions.

Introduction to the Investigation
As NCTM (2000) notes, technology "blurs some of the artificial separations among some topics in algebra, geometry and data analysis by allowing students to use ideas from one area of mathematics to better understand another area of mathematics" (p. 26). Mindful of this, we present the "Building Parabolas" activity as a way to illustrate the benefits of an integrated mathematics curriculum. This activity capitalizes on the multiple representations feature of TI-Nspire technology to blur distinctions between algebraic and geometric definitions of parabolas.

Task Details
The geometric definition of a parabola is the set of points equidistant from a point and a line (the focus and directrix, respectively). As Figure 3 suggests, one can construct a parabola using this definition with the interactive geometry capabilities of TI-Nspire technology.


Figure 3: Geometric construction of a parabola - the locus of points constructed by the perpendicular bisector of FG and the line perpendicular to the directrix through point P.

Typically when algebra students are asked to define a parabola, they respond with x² or (if we are lucky) the standard form of a quadratic function, f(x) = a(x-h)² + k. How does the algebraic equation tie together with the geometric definition? By adding coordinates to the dynamic geometry sketch and confining the directrix to the x-axis, pre-service teachers can begin to make these connections.

After completing a geometric construction of a parabola collaboratively, we ask teacher candidates to confirm that an arbitrary point on the curve (i.e., point P in Figure 3) satisfies the definition of a parabola.  That is, we collect data showing the distances from P to G and from P to F are always congruent (Figure 4).

Figure 4: Collected data to confirm the distances from P to F ('d1' in L2) and from P to G ('d2' in L3) are congruent as we dragged point G along the directrix. The x-coordinate for point G is also collected in L1, as 'xd'.

Next, candidates add the algebraic representation of the parabola to the same window by graphing x² (Figure 5).

Figure 5: (Left) After typing in the function x² in a textbox, teachers use TI-Nspire technology to "grab-and-move" the expression on the x-axis; (Right) A graph of x² is generated.

Utilizing the grab-and-move feature of TI-Nspire technology's interactive geometry environment, teachers transform the algebraically-defined curve to fit the locus of points defining the parabola (Figure 6). The resulting window displays the graph and equation of the parabola in standard form.


Figure 6: Teachers "grab" the graph of the function and "drag" it, resizing as necessary, until it overlaps the geometrically constructed parabola.

Stepping back and assigning general coordinates to points F, G, P and the parabola's vertex, we ask the pre-service teachers to algebraically derive the standard form of a quadratic function as illustrated in Figure 7.


Figure 7: Connecting geometric and algebraic representations of a parabola by equating the distances from P to G and from P to F and solving for y.
We obtain the standard form of a parabola in terms of h, k and the constant distance p, y = .
We further discuss the role and advantages of the algebraic and geometric representations using dynamic TI-Nspire technology, and how this activity connects to both state and national algebra and geometry standards.

Discussion of the Investigation
For many teacher candidates, the notion of connecting geometric and algebraic definitions of parabola is novel. For students who have experienced mathematics as a splintered area of study for most of their lives, the activity is truly an "aha" experience that we believe impacts the way our students plan to teach. Comments such as the following are commonplace: