INTASC Principle #1

Teachers responsible for mathematics instruction
at any level understand ... the interaction
between technology and the discipline.

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     In my own high school experience, technology was rarely used or, when it was, it was generally used in a way that did not take full advantage of the technology. My experience with computers in mathematics classes was minimal, and my use of calculators was usually only for performing simple calculations or creating graphs, rather than exploring mathematical concepts. Because I myself had not often experienced effective technology-based learning, I chose to spend the year investigating INTASC Standard #1:

Teachers responsible for mathematics instruction at any level understand the key concepts and procedures of mathematics and have a broad understanding of the K-12 mathematics curriculum. They approach mathematics and the learning of mathematics as more than procedural knowledge. They understand the structures within the discipline, the past and the future of mathematics, and the interaction between technology and the discipline.
Specifically, I focused on the following: “Teachers responsible for mathematics instruction at any level understand…the interaction between technology and the discipline.” In my coursework at the University of Georgia, I not only learned more about various forms of technology but also investigated ways in which technology can be used to truly deepen students’ mathematical understanding. Then, as a student teacher, I was able to apply and build on this knowledge.

     I was most interested in learning about how to use technology in a way that made students’ learning experience not only different from traditional instructional methods, but richer. In one of my courses, I wrote a research paper about the differences between appropriate and inappropriate calculator use—for example, calculating 6 x 7 versus analyzing large data sets. Despite my early opinion of calculators as nothing more than a crutch to avoid mental math, my research showed me that calculators can serve as a tool to help students investigate more complex problems than pencil-and-paper methods allow. One lesson that particularly stood out to me was an “Average Game,” wherein students were asked to add pieces of data to a given set to bring the data’s mean to a given target number. This game would have been tedious if students were expected to use only pencil-and-paper methods, but calculators eliminated the need for these basic calculations and allowed students to focus on mathematics deeper than simply calculating the mean of a set of data. I began to realize that, when used correctly, calculators could actually give students the opportunity to explore mathematics in a way that traditional approaches did not.

     I considered ways in which this research applied to other forms of technology, such as graphing calculators, spreadsheets, and GSP—what special qualities do various types of technology have to help students learn more advanced mathematics? In investigating a problem related to the Simson Line, I used Geometer’s Sketchpad to help me form and test conjectures. Technology gave me a different perspective on the problem than traditional methods would have: the dynamic nature of GSP allowed me to approach the problem in a way that related the solution to many triangles, rather than just the few I could draw by hand. My write-up of the problem shows both my solution and the methods I used to find it. By being in the position of a learner using technology to investigate a problem, I was able to better understand how technology can be used to further develop mathematical knowledge. I would not have been able to see this problem, or many others I investigated throughout the year, in the way I did without GSP’s immediate feedback about my what-ifs—“What if it were an obtuse triangle?”, “What if the pedal point were at a vertex?”, “What if the original triangle were degenerate?”, etc. The ability to easily adjust the many variables in such a problem showed me first-hand that technology’s unique capabilities can give learners new insights into a problem.

     During my student teaching, I used my experiences as a student to try to be a better teacher. For one of the first days of the semester, I developed an Algebra II lesson to review linear equations. Students used paper thermometers to find equivalent Celsius and Fahrenheit temperatures, and we used a graphing calculator to graph these (F, C) pairs quickly and find a best-fit line for the data. Although the lesson gave students experience with the school’s graphing calculators, I felt when I reflected on the day that the project had been very algorithmic rather than exploratory. I remembered more about what I had told students to do—which buttons to push, what to look at, how to proceed—than what they had said. Disappointed, I felt this lesson was too similar to those I myself had experienced in high school. I thought about the assignments I had investigated with technology throughout the year, and realized that the use of technology in this lesson had not given students the unique perspective on the problem that it had given me, for example, in the Simson Line project.

     In the same class several weeks later, we spent a day using the Calculator-Based Laboratory to see a real-life example of a parabola, as an introduction to quadratic functions. A volunteer rolled a tennis ball up an inclined pipe toward a motion sensor, which both recorded the ball’s distance from the sensor as it rolled up and then back down the pipe and instantly graphed this data on the attached graphing calculator, which was projected onto the board. Students’ comments indicated to me that the technology we used in the activity made the lesson less abstract than it might have been without the CBL. For instance, when I asked if the parabola were truly symmetric, students were able to use what they had learned in their Physical Science class to answer that the ball must have taken the same amount of time to roll down the pipe as it did to roll up it, and thus the parabola was indeed symmetric. Without the technology to translate from the physical scenario to the graph in real-time, I do not feel students would have recognized as many interesting properties of the parabola and its relationship to the ball rolling as they did. Additionally, unlike the Fahrenheit-Celsius lesson, students responded to this activity by offering their own observations, conjectures, and questions. For example, one student asked how we could ever get shuttles into space, if parabolas always “come back down.” Students’ enthusiastic participation, rather than my own pre-set script, led the discussion. Unfortunately, the lesson did not proceed ideally—at one point I failed to record the scale of the graph from the CBL, so data for future calculations had to be fabricated—but such glitches gave me a better awareness of the potential difficulties of technology use and how to avoid them.

     This year, I learned more about technology from the perspective of both a student and a teacher. Having rarely been in a classroom where I felt technology was used effectively, I struggled to develop a philosophy about what “effective technology use” really is and how to achieve it. However, as I spent more time in technology-centered classrooms and researched the subject more, I was able to use my new experiences as a learner to cultivate my own teaching style. I learned that technology can be more than just a gadget to play with to make a day’s lesson different; it can actually shed new light on a topic. Throughout the year, I saw my research on the subject come to life, both in my own experience as a learner (as with the Simson Line project) and as a teacher (for instance, during the parabola lesson). My opinion of technology in the classroom changed from somewhat skeptical to very enthusiastic as I experienced and learned from both disappointment and success.