It’s very hard to fake it when you don’t know the answer — or have any idea of how to find the answer — to a math problem. So I had to admit, when a student signed on to my office hours yesterday, that I didn’t believe I’d be able to help her find the cosecant of 11pi/2.
I’m what we call the Math Therapist at a small public school in Oakland, California. We made up the title and we made up the job, both of which I love. My job description changes every year, but right now, in the year of covid, it’s this: I work with small groups of students who have flunked Algebra and Geometry. I help them understand the material enough to get credit, and — my hidden agenda here — love math a little bit.
I also have office hours and anyone can visit me online for math tutoring at those times. That’s how I ended up facing one of our talented math students, who is taking a college-level Trigonometry class at the nearby community college, and had to figure out (before tomorrow’s midterm!) the cosecant of 11pi/2.
I might mention here two facts about myself. One is that I was a math major at the University of Michigan, so it’s likely that at one time in my life I was very comfortable with cosecants. The second fact is that I turned 75 last month, so really, give me a break.
Anyway, I did remember one thing: Cosecant is the inverse of sine, but I kept that to myself and asked my student to tell me what she knew. She knew that the cosecant is the inverse of sine, so there went my advantage.
Okay, great, I said. Where do you want to begin?
I think we need to use the unit circle, she said.
Okay, great, I said again.
I gave her the ability to share screen so we could look at the unit circle together. Then it seemed like something clicked for her, because she knew to go counterclockwise in her search.
When she got to the 11pi point on the circle, I suppose I said another Okay, great, and then there were about three more steps until she got to 1/-1, which became -1, the exact right answer.
“Thank you SO much!” she said.
I asked her to turn on her camera so I could see her because I had something important to say. When she did, I looked at her sweet face and said, “I did nothing. You figured it out completely on your own. Excellent thinking.”
“Not really,” she said.
“Really,” I said. “I didn’t teach you anything. All I did was say was ‘Okay, great’ several times.”
“But I wasn’t able to do it before I came here,” she insisted. “I couldn’t figure it out when I was alone.”
I know learning is an interpersonal, social activity, and I know we learn better when we’re listened to, but I still disagreed. We’re talking about mathematical skill, I told her. You’ve got this one.
But she was right of course. There are plenty of things in my life I find nearly impossible to do alone. But give me a trusted friend sitting with me on the floor of my attic, for example, and I can sort through decades of papers, photos, and other treasured items, finally able to decide what to throw away and what to keep. Give me a motivated student to work with and I can help find the cosecant of 11pi/2.