Even Tech Has Its Limits: Learning from Students

One of the things I absolutely love about my teaching situation this year is that I have students that are very curious. They ask "what if" questions, make connections between concepts and notice things that I, even after twenty-six years of teaching, haven't noticed.  This type of noticing happened in Advanced Algebra on Friday and reminded me that math tech can have its limits.

Here is what happened. Do students really need to know about end behavior or behavior at x-intercepts for polynomial functions?  They can just plug the function into Desmos and see what it looks like.  But because I let my students discuss ideas and they can share them freely without fear of "being wrong", one of my students commented the function f(x) = x3(x+3)(x-4)  (shown here) seemed to have many

x-intecepts around the origin.  Because we had talked about behavior of x-intercepts of polynomials based on their factored form, she knew that there should only be x-intercepts at 0, -3 and 4.  Yet Desmos was telling her something different.  So, we decided to zoom in to verify that the function really only had one x-intercept at the origin.  And this is what we saw, after zooming in and tracing along the graph. It looked as though my student was right! The graph showed that several of the values close to zero had y-coordinates of 0, but we knew from the factored form of the polynomial that this could not be.  Desmos was lying to us!!! How could we trust it to do our math correctly?  I could tell that several students confidence in Desmos was shook.  At this point, we decided that  Desmos had to round the output to show it on the graph.  And based on what rounding convention Desmos used, it rounded the result to zero.  The students' understanding of polynomial behavior helped us understand what was really happening near the origin.  Incidentally, f(-0.002) is not zero, but it's pretty darn close to it at 9.598 x 10-8. Letting students share what they notice and wonder can be risky.  You don't know what they will say all the time and sometimes they may surprise you or throw you a curve ball, where you don't have a satisfying answer.  But letting students see that you are on a learning journey with them can definitely have unintended positive consequences, where even you as a teacher can learn something new.

Bonus tech tip: If students want to use the Desmos app on their phones, tell them to download the  "Desmos Test Mode" app and use it when doing their math homework.  If the select "Start Test", they can work in a distraction free way - no notifications of any kind will pop up on their phones!