Last week on Twitter I read a teacher’s cry for help. “I need a lesson on diameter and circumference that uses technology,” read the tweet. I had to step back and feel sympathy, first, for a teacher experiencing the educational equivalent of writer’s block. “And for what?” I wondered. “For whom?” In my head I pictured an administrator holding an evaluation rubric headed by “technology use,” confused on both the definition and the purpose of technology.

Definition. As Arthur C. Clarke said, “Any sufficiently advanced technology is indistinguishable from magic.” What we now consider antiquated — ditto machines, faxes, steam locomotion, paper and pencil — was at one point advanced technology. So instead of pursuing some fuzzy definition of “technology,” let’s look at the tools available and choose the best one for the task. That may be paper and pencil (as geometric sketches and algebraic expressions are often quicker and cheaper to create with paper & pencil) or that may be a computer (as it can perform certain calculations which are laborious by hand). Choosing the tool before the task confuses master and servant, means and ends.

Purpose. For which ends is a computer the right means? My colleagues and I at Desmos have several answers:

We love that computers allow students to make and quickly test hypotheses. We can ask a student what she thinks will happen to the graph of the equation y = 2x + 5 when we swap the 2 and 5. She can make her hypothesis and then quickly test that prediction by graphing it in our graphing calculator. Her quick modification of those parameters may then awaken her mind to other hypotheses.

We love that computers can quickly collect and arrange student data. We love mathematical modeling and mathematical modeling loves lots of data. More data often leads to a more robust model. Rather than ask every student to collect fifty data points for each of their individual models, we ask students to collect three points each and watch as patterns emerge in our accumulation of data.

We love that computers speak the language of algebra so fluently. Whether in spreadsheets or scripts, a student can experience the power of algebra on a computer, watching as the language of variables energizes and automates tedious tasks, while that power remains largely theoretical on paper.

You can find these answers embedded in our activities at teacher.desmos.com, all available for free. With respect to our Twitter teacher and her cry for help, we’d love to see students first predict the circumference for a very large circle of some known diameter. We’d then ask each student to measure the circumference and diameter of three smaller household circles, which we’d collect and plot as points on a graph. The fact that all those points would fall along a line would be surprising enough. We could then use the language of algebra to describe that line, which would let us calculate and test our prediction from earlier.

These are the instructional decisions we make when we consider our goals for student learning first, and seek out the best technology to meet those goals only afterwards.