Here is the oldest kind of math problem that exists:

Some of you knew what kind of problem this was before you had finished the first sentence. You could blur your eyes and without reading the words you saw that there were two unknown quantities and two facts about them and you knew this was a problem about solving a system of equations.

Whoever wrote this problem knows that students struggle to learn how to solve systems and struggle to remain awake while solving systems. I presume that’s why they added a context to the system and it’s why they scaffolded the problem all the way to the finish line.

How could we improve this problem — and other problems like this problem?

I asked that question on Twitter and I received responses from, roughly speaking, two camps.

One group recommended we change the adjectives and nouns. That we make the problem more real or more relevant by changing the objects in the problem. For example, instead of analyzing an animated movie, we could first survey our classes for the movie genres they like most and use those in the problem.

This makeover is common, in my experience. I don’t doubt it’s effective for some students, particularly those students already adept at the formal, operational work of solving a system of equations through elimination. The work is already easy for those students, so they’re happy to see a more familiar context. But I question how much that strategy interests students who aren’t already adept at that work.

Another strategy is to ignore the adjectives and nouns and change the verbs, to change the work students do, to ask students to do informal, relational work first, and use it as a resource for the formal, operational work later.

This makeover is hard, in my experience. It’s especially hard if you long ago became adept at the formal, operational work of solving a system of equations through elimination. This makeover requires asking yourself, “What is the core concept here and what are early ways of understanding it?”

No adjectives or nouns were harmed during this makeover. Only verbs.

The theater you run charges $4 for child tickets and $12 for adult tickets.

1. What’s a large amount of money you could make?
2. What’s a small amount of money you could make?
3. Okay, your no-good kid brother is working the cash register. He told you he made:
   • $2,550 on Friday
   • $2,126 on Saturday
   • $1,968 on Sunday

   He’s lying about at least one of those. Which ones? How do you know?

This makeover claims that the core concept of systems is that they’re about relationships between quantities. Sometimes we know so many relationships between those quantities that they’re only satisfied and solved by one set of those quantities. Other times, lots of sets solve those relationships and other times those relationships are so constrained that they’re never solved.

So we’ve deleted one of the relationships here. Then we’ve ask students to find solutions to the remaining relationship by asking them for a small and large amount of money. There are lots of possible solutions. Then we’ve asked students to encounter the fact that not every amount of money can be a solution to the relationship. (See: Kristin Gray, Kevin Hall, and Julie Reulbach for more on this approach.)

From there, I’m inclined to take Sunday’s sum (one he wasn’t lying about) and ask students how they know it might be legitimate. They’ll offer different pairs of child and adult tickets. “My no-good kid brother says he sold 342 tickets. Can you tell me if that’s possible?”

Slowly they’ll systematize their guessing-and-checking. It might be appropriate here to visualize their guessing-and-checking on a graph, and later to help students understand how they could have used algebraic notation to form that visualization quickly, at which point the relationships start to make even more sense.

If we only understand math as formal, operational work, then our only hope for helping a student learn that work is lots and lots of scaffolding and our only hope for helping her remain awake through that work is a desperate search for a context that will send a strong enough jolt of familiarity through her cerebral cortex.

That path is wide. The narrow path asks us to understand that formal, operational ideas exists first as informal, relational ideas in the mind of the student, that our job is devise experiences that help students access those ideas and build on them.

BTW. Shout out to Marian Small and other elementary educators for helping me see the value in questions that ask about “big” and “small” answers. The question is purposefully imprecise and invites students to start poking at the edges of the relationship.