Editor’s note: This is the seventh piece in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder” by Barry Garelick, a second-career math teacher in California. He has written articles on math education that have appeared in The Atlantic, Education Next, Education News and AMS Notices. He is also the author of three books on math education. Says Mr. Garelick: “At its completion, this series will be published in book form by John Catt Educational, Ltd.” The previous chapters can be found here:
Ch 7: A Gnarly Problem, Critical Thinking, and Authentic Struggle
My meetings with my parole office/mentor Diane occurred once a week in the early morning at a local coffee house a few blocks from my previous school. At one particular meeting she showed me her notes from an observation she had made of a lesson I gave my seventh grade class. Her notes were typical “hunting for problems” comments, such as my not noticing a particular boy who was unfocused, or another student who was talking, and so on.
“Any comments?” she asked.
I resisted the urge to say that it sounded like she was hunting for problems to enter on her online checklist. I talked instead about the lesson itself. It had been about taking a situation like “A bowling alley charges $5 for shoes and $3 per game bowled” and writing an algebraic expression for the cost of x games. (5 + 3x). Knowing that Diane wanted to see me extend JUMP’s scaffolded approach to more “gnarly” problems, I told her about a problem I gave the seventh graders on one of the Warm-Up questions the day after the lesson she observed.
They were to write an expression representing the cost for n hours, if a babysitter charges a flat fee of $10 and $15 per hour, but with the first hour free. I was met with the usual questions of “How do you do this?”
In my highly scientific approach I helped the first person who asked “How do you do this?” which happened to be Kyle. He was a talkative boy who was quite good at problems when he put his mind to it.
“How many hours does the babysitter charge for 6 hours of work if the first hour is free?” I asked.
“Five,” he answered.
“Right: 6 -1 = 5. So for five hours of work what does he charge?”
“Five minus one,” he said. I gave him a few more numbers and then asked “How much for n hours work?”
“Oh! n-1,” he said. He was then able to see it was 10 + 15(n-1), though he and others needed help with the parentheses.
“Yes, that’s a good problem,” Diane said. “But it wasn’t really critical thinking.”
“Why not?” I asked.
“You led them there.”
I said nothing, hoping for an awkward silence and got my wish.
“There’s nothing wrong with what you did,” she said. “But true critical thinking would involve them struggling to come up with a solution.”
I recognized this immediately as the “struggle is good” philosophy which holds that if students aren’t struggling they aren’t learning. There are nuances to this philosophy including “productive struggle”, “desirable difficulties” and “students should be able to use prior knowledge in new situations without scaffolding because otherwise it is inauthentic.” I’ve read variations of this thinking in books that I’ve thrown across the room.
“Let me give you a problem that I want you to solve,” I said. “Two cars head towards each other on the same highway. One car starts from the north heading south, at 80 mph. The other car starts from the south heading north at 70 mph. They meet somewhere on the highway. How far apart are they one hour before they meet?”
She took a gulp of coffee and tried to smile.
“You do not need to know the distance they are apart to solve it”, I said.
She looked perplexed and gave me a look I see on my students’ faces when they ask “How do you do this?”
“Tell me this,” I said. “How far does a car going 80 mph travel in one hour?”
“Eighty miles,” she said.
I drew a line on a napkin and marked a point near the middle with an X. “Where was the 80 mph car 1 hour before he got here?”
“Well, that would be 80 miles north of that point.”
” What about the 70 mph car?”
“Uh, 70 miles south of the point?”
“Good. Can you put that together somehow?”
She suddenly saw it. “Oh, I see! They’re 150 miles apart one hour before they meet.”
“Good work. Now let me ask you something. I gave you some hints. Would you say that you used those hints in thinking about the problem and coming up with a solution?”
She smiled knowingly. “Ah, I see. Critical thinking.”
“So would you say that what you did qualifies as critical thinking?” She agreed.
“Then why would you say that what I did with the baby sitting question did not qualify as critical thinking.”
“I’ll have to think about what I mean by critical thinking,” she said. “I think applying an algorithm repeatedly does not entail critical thinking.”
“Even if it leads to a conclusion? And in essence that was what I had you do when you think about it. And you put it together like my students did. Why would you not call that critical thinking? In your mind is there no difference between thinking and critical thinking?”
“I guess I might have to look up the definition of critical thinking.”
“I’ll send you a definition tonight by email,” I said. “My concern is this. If your goal is to look for examples of critical thinking in my classes using the definition you’ve presented, you will probably never see critical thinking in my classes. I use worked examples and scaffolding and problems that ramp up. That’s how I teach. You’ll see this more in my algebra class, and I hope you observe one of those.”
She said she definitely would. I thanked her for having the discussion with me. “I felt it was important that we understand the language we’re speaking and what I’m about.”
“Yes,” she said. The conversation then shifted to lighter topics. I felt a bit bad for putting her on the spot with my math problem. But then again, her struggle with critical thinking was productive if not authentic.