It’s been 20 years since I bought my first car with an in-dash navigation system. It was magic. Type in a destination, and it’ll show you a map with turn-by-turn directions telling you how to get there. “This will make you stupid,” one of my colleagues exclaimed at the time. “You don’t need to know anything.” And it’s true in the sense that I don’t need to get directions anymore. I don’t have a paper map in the car. In the early days, when people would ask me if I know how to get somewhere, I’d reply “no, but my car does.” There are now holes in my memory. I know that I’ve attended events, and I’ve driven to them, but I have no idea where they were or how I got there. My car figured it out.

But the most amazing part was the navigation system’s ability to adapt. We were driving through Danbury, Connecticut on our way to Boston one year, and it looked like the traffic jam was going to extend for miles. So we got off the highway. It didn’t matter where we were. The car recalculated. It knew where we were trying to go, and adapted its instructions based on where we were. And no matter how many wrong turns we made or instructions we ignored, it always patiently recalculated to give us the best route to our destination given our current location and trajectory.

Those systems are a lot better now. They adapt to traffic and weather conditions. They know about construction and temporary closures. They can optimize for distance, time, or fuel economy.

I was reminded of this recently when this passage was shared with me from Tracy Johnston Zager’s Becoming the Math Teacher You Wish You’d Had. She begins by quoting Richard Skemp (page 185):

A person with a set of fixed plans can find his way from a certain set of starting points to a certain set of goals. The characteristics of a plan is that it tells him what to do at each choice point: turn right out of the door, go straight on past the church, and so on. But if at any stage he makes a mistake, he will be lost; and he will stay lost if he is not able to retrace his steps and get back on the right path.

In contrast, a person with a mental map of the town has something from which he can produce, when needed, an almost infinite number of plans by which he can guide his steps from any starting point to any finishing point, provided only that both can be imagined on his mental map. And if he does take a wrong turn, he will still know where he is, and thereby be able to correct his mistake without getting lost; even perhaps to learn from it.

Zager continues by connecting this to math instruction:

A look through typical math textbooks reveals that we focus almost entirely on point-A-to-point-B instrumental understanding: can students subtract two-digit numbers with regrouping, divide three-digit numbers by one-digit numbers, multiply fractions, find the least common denominator? Each chapter is separate from the ones that come before and after. We’re teaching that limited number of “fixed plans” to get from particular starts to particular goals. We can generate a lot of right answers and even high scores on standardized tests that way, but when our students get lost, they are lost because they have no mental map of the terrain. If the problems look at all unfamiliar, learners have no idea where they are.

When I was in college learning to be a math teacher, our methods professor called this “what kind is this math”. Every day, we teach students how to solve a different kind of math problem. Then, all they need to do is figure out which kind of problem they’re trying to solve, and plug in the numbers to get the answer. There are two issues with this approach. One is that we spend very little time teaching students how to recognize what kind of problem they’re trying to solve. That’s why you hated story problems so much. The point of story problems was to get you to figure out what kind of problem you were dealing with. But all of the story problems were contrived to fit the kind of problem you were solving that day. So really, the work was done for you based on the problem of the day.

The second issue is that we often encounter problems that we don’t have a formula for. This isn’t any of the kinds. What do I do now? That depends. Do you have a fundamental understanding of mathematics, or do you just know how to plug numbers into formulas? Can you figure out how to solve new problems?

Back in 1990, we called this problem solving and talked a lot about developing a conceptual understanding of the structure of mathematics. That approach created a whole lot of angst in the decades that followed, because parents who don’t understand math struggle with helping kids who are being taught to understand mathematical concepts rather than rote processes. To a large extent, we’re still fighting that battle.

Here’s another example. Several years ago, I had a garden full of tomatoes and peppers, and I wanted to make salsa. After some web searching for recipes, I stumbled upon the one that has become my favorite:

Salsa requires four things: Peppers, aromatics, acidic fruits, and salt. Peppers are usually chiles, but they can be other kinds. Aromatics are onions or garlic (or both). Acidic fruits are generally tomatoes, but the can be lots of things. Salt is salt.

Put it all together. Add lime juice or cilantro or whatever you want. Then adjust. If it’s too tomatoey or not hot enough, add peppers. If it’s too hot, you need more fruit. If it’s too acidic, add more aromatics. If it’s too blah, it needs salt.

We don’t need to get into amounts or specifics. It’s not a set of directions. It’s a map of the city. Develop an understanding of it, and you can go wherever you want. Last summer I used cucumbers and Hungarian wax peppers, because that’s what I had. It wasn’t a traditional Mexican salsa, but it certainly worked.

I want our students to be able to solve the problems they don’t know they have. How can we foster that kind of understanding?