I got a call yesterday from a friend asking for help with this problem their kid was assigned for homework:
It's an ok problem. Not the best, not the worst. I wish it wasn't homework.1
I tried to talk through it. Ok, so when we make a rectangle for 3/4, that's really 3 copies of 1/4. What would that look like? If 3/4 is 3 copies of 1/4, how many 1/4s would be in 4 * 3/4?
I don't think I did much good. I mean, the kid got the answer, sure, but I doubt they learned anything. I talked them through each step and they weren’t doing much thinking for themselves. I kept thinking about the experience afterward and there's a mistake here that I make a lot with my students.
Two paragraphs above, where I described how I tried to walk the kid through the problem, what were you thinking? I bet a lot of the math teachers reading this were thinking, "oh I have a different way of explaining that, what if you thought about it this way?" or something along those lines. That's what I often think when I see other people's explanations in situations like that. I think that instinct is counterproductive. That instinct is focused on helping the kid with the problem. Most of the time, the real goal is to help the kid learn something.
If I'm working with a kid like that and they show me their work on that problem, the best thing to do is push the problem to the side, grab a piece of paper or a whiteboard or something, and give the kid some other problems. I might start by asking the student to visualize a few fractions, maybe 1/8, 3/8, 5/8, etc. Then, using the visual, what is 1/8 * 3? What is 1/8 * 5? Then work our way up - what is 3/10 * 2? What is 2/5 * 4? Then some word problems, your recipe needs 2/3 cups of flour and you want to make 2 batches, how much flour do you need? We could debate about improper fractions, mixed numbers, and answers that work out to whole numbers. The sequence of problems you would choose probably looks a bit different. But stepping back, my goal is to give the student some problems building gradually toward the problem we want to solve. I can figure out where they're having a hard time, give them some help, then give them some more problems. Ideally we build up so that when we get back to that original problem the student has the tools to solve it independently. That's actual learning. Getting stuck on a complex problem and having someone help you with it often doesn't result in much learning.
There are two big obstacles here. First, this approach takes a lot of time. It's hard to do in real classrooms full of squirmy kids. But the bigger problem is that problems can kindof stand between the student and the learning. Sure, in plenty of situations problems are great. But a typical human sees a problem and becomes laser focused on solving it, not learning something from it. If the student doesn't know how to solve the problem and the teacher walks them through it they're likely to lose the forest for the trees. In those moments it's smart to step back and ask, "what are some problems this student knows how to solve that can prepare them for the bigger problem?" Those problems become opportunities for more learning than being walked through something they barely understand. This is hard! It’s not easy to step back and do that type of thinking. When there’s a problem in front of us we want to solve it. When students have a problem to solve for homework or classwork they want to solve it, they don’t want someone to give them a bunch more problems. But if the goal is to learn things, stepping back and finding some other problems they can solve is the way to go.
I’ve noticed myself doing this type of thing more and more with my curriculum. A curriculum is mostly a whole bunch of problems that hopefully fit together in a good way. If I put every problem in the curriculum in front of kids I’ll end up in this situation a lot: a group of students doesn’t know what to do, I need to walk them through the problem, and they don’t learn very much. I don’t want to dumb down the curriculum and give students easier problems instead. A better choice is to say, “what are some problems I can give students that they know how to do, and that build toward the problem the curriculum says they should do today?”
Seriously, don’t do this. This is a good problem to do in class, where the teacher can support students who are having a hard time. Keep homework simple and straightforward.
So true! It's not about solving the problems, it's about how the activity becomes a door for learning. There is so much insight in your idea here! But students are so used to solving, getting points and credit and pats on the back for what they solve. The whole system revolves around getting the solution, and it's a hard mindset to break. Sometimes, I like to put the answer to the problem on the board before they start to solve it. I want the students to know that it's not about having the answer, it's about finding it, that process. Anyway, thanks for a good post, reminding us what the real work is that we are doing.