A youth group with 26 members is going to the beach. There will also be 5 chaperones that will each drive a van or a car. Each van seats 7 persons, including the driver. Each car seats 5 persons, including the driver. How many vans and cars will be needed?
Tuesday was an all-school professional development day. The math departments joined from two campuses to learn about the Gradual Release of Responsibility from a couple of math coaches from the next county over.
One coach modeled a GRR lesson and opened with the problem above.
I leaned into another teacher and whispered, "I'm trying to decide which would be more socially acceptable right now, letting out a loud fart or saying what I really think about this problem."
We broke for lunch and came back to debrief. No one had commented on the problem by the end so I did.
"I see problems like this and I feel myself becoming less of a human and more of a math teacher. And I feel very lucky to teach our neediest students, students who punish me daily for problems like this one, students who are often very hard on me but who in return have helped me hold onto some of that humanity.
"I have three questions about this problem and we can discuss any of them or none of them."
"One, is the problem realistic? Would a real person need to solve this problem?
"Two, is the solution realistic? Would a real person solve the problem using a system of two equations?
"Three, in what ways does this problem help our students become better problem solvers?"
I didn't elaborate. I thought my questions were self-evident and their answers self-explanatory. I was wrong. The coach shrugged me off, saying, "Well, it's in your textbook." and I couldn't disagree. None of my colleagues seemed disturbed by the opening exercise of this quote model lesson unquote, so I didn't belabor the point. In hindsight, I wish I had soapboxed:
- This is a problem you will only find in a math textbook. It's bizarre to me how many different ways just fifty words can fail to square with reality. Why does each chaperone have to drive? Why can't we take five vans? Why do our vehicles have to seat the exact number of people in our group and no more?
- No youth group leader would ever solve this problem with a system of equations. I'd wager that no math teacher, if somehow faced with this completely fantastic scenario, would solve this problem with a system of equations. With 31 people, we'd just shuffle them around until they fit. Even if we insisted on the contrivances in #1, there are only [0, 5] possibilities for the vans so we'd use a table or just guess and check. ¶ I asked the coach why we were forcing the issue of systems when the easiest solution by a long shot was tables. She replied that we learned tables last class and this is the new skill we're learning.
- This kind of algebra makes our students dumb, unimaginative, and scared of real problems. At the end of the model lesson, the coach put up our homework, which was a carbon copy of the original problem, new numbers swapped in for the old. ¶ I can't describe my contempt for this arrangement. ¶ This is how we make kids stupid and impatient with irresolution, eager for contrived problems that look just like the last contrived problem, completely lost if we so much as switch around the order of a few words. "We don't teach them problem solving skills anymore," my department head said to me. "We teach them problem types."
Algebra teachers sell students a cheap distortion of the real world while insisting at the same time that it really is the real world. The cognitive dissonance is obvious and terrible. Students know the difference. It cheapens my relationship to them and their relationship to mathematics when you ask me to lie to them.
It's like offering someone lust or manipulation while insisting that it's love. Not only are the short-term consequences devastating but it makes that person distrustful or wary of the real thing. Make no mistake. We are making an alien of algebra. We are doing real damage here.