Image created by Kim Holman. 2022.
Developing my literacy in Desmos has been fascinating and engaging. I have used Desmos before but in a very limited capacity when compared to its capabilities.
Equivalent equations are straightforward in Desmos. Using the sliders we can see, in real-time, how the different variables in an equation impact the shape and location of a graph. We can use these variables, sliders, and inputs to model an equation to see if it is the same as another equation written in a different format - how cool is that?! One of my favorite functions to explore is the quadratic function in vertex form. The impact of the scalar, a, the x-coordinate of the vertex, h, and the y-coordinate, k, are explicit and obvious. When I learned quadratic functions in grade-school we began with standard form, where the variables a, b, and c don't have a discernible, straightforward impact. I believe standard form is generally the first experience most students still have with quadratic equations, but using a mathematical action technology (MAT) like Desmos makes vertex form and factored form more sensible for first exposure (Dick & Hollebrands, 2011).
Standards for teaching mathematics are rigorous, on the one hand, but flexible on the other. We have skills and concepts we are required to teach at each grade level (and moving into high school, specific standards for each course) (Alabama State Department of Education [ALSDE] 2019). Sure, I love spreadsheets - you'll never convince me otherwise! However, there are tools in Desmos that aren't available in Excel, just as there are tools in Excel that aren't available in Desmos. My favorite feature in Desmos is the interactive slider. You can do similar things in Excel, if you know what you're doing, but the inputs are manual and the interface for this particular application of a MAT is clunky, to say the least. For now, we will compare three forms of the quadratic equation: vertex form, factored form, and standard form. Adjust the sliders in the linked Desmos graph; see if you can make the other two equations match after you've moved the sliders on the third without computations or other assistance!
Gif by Giphy.
Mathematics, and teaching mathematics, are my passions. My time as a Graduate Teaching Assistant in the Department of Mathematics and Statistics at Auburn University was lackluster when it comes to MATs. Few of the faculty use them, and in fact the majority of the service courses (courses offered for non-mathematics majors) prohibit the use of technology in testing situations, which in turn provides little room for instructors to even consider using MATs to teach material. Meyer (2020, p. 249) claims, and I agree, that "the field of mathematics education technology has no shortage of people who love technology. What it needs is people who love mathematics and education." My fellow graduate students used technology when they tutored, often using sites such as WolframAlpha, Photomath, or Symbolab to verify solutions or Khan Academy to direct students to supplemental learning resources. Students frequently use Photomath, Symbolab, and Chegg for walkthroughs of homework problems; some use the "watch it" feature in Webassign to follow an algorithm for a particular problem and memorize these algorithms to prepare for tests. While the student will likely get the correct answer for the Webassign assignment, they haven't learned anything. Skemp (1978, p. 153) calls this is "instrumental understanding." Using technology in this manner is ineffective, as little to no learning is taking place and there is no student engagement.
Observation of the ways we teach math at the college level, both at Auburn and my undergraduate institution, Huntingdon College, as well as conversations I've had with my network of mentor-peers in academia indicate that the ways technology is used above has been the standard for some time. Many of my mentor-peers are progressive in their use of MATs but are often subject to pushback, even hostility at times, from their departments who've "always done it this way," meaning, acknowledging that students are going to cheat*, while refusing to change their pedagogy to adapt to the changing needs of the student population.
Strategies for teaching mathematics must change. We need to embrace MATs and engage our students.
*This could mean: violating the proscribed academic honesty policy; skirting the nuances of academic dishonesty without outright committing it; or other real or perceived cheating based on the instructor's perspective.
References
Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school Mathematics: Technology to support reasoning and sense making. National Council of Teachers of Mathematics.
Alabama course of study: Mathematics. (2019). Alabama State Department of Education.
Meyer, D. D. (2020). Social and creative classrooms. The Mathematics Teacher, 113(3), 249-250.
Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15.