A useful quote

At lunch today I spent a few minutes reading a recent edition of the AWM Newsletter. One article, written by Jackie Dewar, is called “Situated Studies of Teaching and Learning: The New Mainstream.” In it, she gave a great quote that I want to keep handy for later. The quote is from the keynote address at the 2013 ISSOTL Conference, given by Dr. Lee Shulman:

What advice did [Dr. Shulman] offer [Scholarship of Teaching and Learning] investigators? “Do not look for generalizations. Try to figure out what to do tomorrow because it matters.” (emphasis mine)

Whenever I think about my teaching approach and philosophy, I always stumble across the following problem: I have the tendency to think about what I want my classroom to look like, say, five or ten years down the road. I think about what I want the student experience to be and about big, radical changes I’d like to fully implement to get things there. Usually what happens at this point is I am jolted back to reality. I have so many different things pulling me in different directions that, in the end, I never feel like I’ve got momentum in the direction I’d like to go.

This is why I wanted to keep Dr. Shulman’s quote handy. Instead of thinking about big, long-term changes and projects, I really should spend my energy figuring out how to make class better tomorrow, or this week, or this semester. Hopefully small changes over time will have an additive result.

The range of a function

A recent Tweet led me to discover an old blog post written by Christopher Danielson (@Trianglemancsd) back in 2013. His post was titled, “College algebra teachers! Please try this and report back!” Although I don’t teach College Algebra, my current semester includes two sections of our “PreCalculus” course. I tried a modified version of his activity and I’ll describe my version below.

I gave each student a small (roughly 3″x5″) piece of green paper and a small piece of orange paper. I wrote on the board that green meant YES and orange meant NO. Then I passed out a one-page quiz that had six “Yes/No” style questions. The questions were about the function $f(x)=x^2 - 1$. Each question asked, “Is (number) in the range of the function?” I displayed each question on the projector and then asked the students to vote on the answer by holding their green or orange paper above their head.

The first question was, “Is 8 in the range of the function?” In both sections, all of the students answered this question correctly. I asked for a volunteer to explain their reasoning. In the case the reasoning was too short or wasn’t quite right, I asked for another volunteer to explain the answer in a different way.

Next, I asked, “Is -3 in the range of the function?” In both sections, all of the students answered this question correctly. Most explanations cited the fact the graph of the function would show a parabola with a minimum y-value of -1, so -3 was “too low” to be on the parabola.

With the next three questions, most of the students answered correctly and gave appropriate and correct reasoning. The questions asked if 1/4, -1/2, and π were in the range of the function. The last question, though, had an interesting result.

“Is ∞ in the range of the function?”