As examined in the previous post, spreadsheet software, like Excel or Numbers, has the capacity to go beyond organizing statistical data in the classroom. By making tables and graphs, spreadsheet software can be used to explore properties of sequences and series or solve algebraic equations. The question, then, is whether spreadsheets provide the most effective means by which students can gain conceptual understanding of these topics. One important consideration when answering this question is evaluating if students have existing familiarity with spreadsheet software, as this will determine the types of tasks and level of scaffolding necessary to keep students focused on the math rather than syntax and technique. However, even if students have basic skills working with spreadsheets, other available technology often proves to be more effective.
For example, consider a problem in which a student is asked to solve the following equation:
3x^2-4x-7 = x+4
We will examine two methods by which technology can be used to solve this problem.
First, we will use spreadsheet software to solve for x. In order to do this, we will set up three columns: one for the values of x, one for the left side of the equation, and one for the right side. After generating a sequence of values for x, we can then apply a corresponding formula in each of the remaining two columns, in this way quickly generating many values for the left and right sides of the equation for given x-values. Next, we can analyze the data to determine where the left and right sides of the equation are equal.
One way to do this is by creating line graphs with the coordinates from the left and right sides of the equation, as shown below. By finding approximately where the two graphs intersect, we can narrow down the interval on which the graphs are equal (as we see here, between (-2,-1) and (2,3)). Alternatively, we can examine the table of values and find the x-values at which the value of the left side of the equation becomes less than the value of the right side, and vice versa.
Finally, we can use the same process as before to generate x-values between (-2,-1) and (2,3) in order to find more precise x-values for which the left and right sides are equal. In this problem, the solutions (when rounded to the nearest hundredth) are
x= -1.25 and x= 2.92
Using Desmos is the second method we will examine in order to solve this problem. Using Desmos, a student can graph the left and right sides of the equation and then tap the points of intersection to find the x-values of those points, as shown below.
Unlike the first method, which takes at least 15 minutes and a solid knowledge of spreadsheet basics, using Desmos to find the same solution takes only a couple of minutes, not to mention that Desmos has a much more intuitive interface. Furthermore, by having to manually generate smaller and smaller increments of x in order to find the approximate solution in the spreadsheet, students are distracted from visualizing the basic concept of solving equations: the expressions are equal when they have the same y-values for given x-values. Desmos clearly demonstrates this with easy to graph and easy to find points of intersection.
As seen in this example, spreadsheets are often not the most effective technology choice for illustrating a given concept. When a teacher decides the significance of a given technology, Dick and Hollebrands (2011) propose the following question is most important: “[W]hat questions could [the teacher] ask that [he or she] could not ask before?” With regard to spreadsheets, the answer to this question is very few new questions. Overall, the limited functionality and technical requirements of spreadsheets make them less than ideal for use in the math classroom.
Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to
support reasoning and sense making. Reston, VA: National Council of Teachers of