“I’d like you to envision your ideal classroom. What are you seeing? What are students doing?”

At Twitter Math Camp this summer, I participated in a 3-day workshop for math teachers, coaches, and administrators. Our facilitator had us begin by working in small groups to write a vision statement for math class. After a minute of quiet jotting, my partners and I turned to one another to share our ideas.

I want to hear students talking to one another, one said.
Another: I want to see students using math to solve challenging problems.
I’d like students to share their strategies and not be embarrassed if their answers are wrong.

These are lovely visions. And given the stasis that pervades math education — students still sitting in rows; students still solving rote problems; teachers still prioritizing answers over reasoning — realizing them would be a major win.

But I don’t think it would be a complete win. I think we can win bigger. I think we can envision more. From NCTM to Twitter Math Camp, @MTBoS to #ObserveMe, we as a math education community possess an insatiable appetite for improving our craft, a fervent desire to help students love math. At the same time, I think we maintain an overly narrow conception of what math is: what it means to teach math, what’s possible in math class, and what it means to provide students with a complete mathematical experience.

Nowhere have I observed this more than in how we approach — or really, in how we overlook — the distinction between math as an object of inquiry and math as an object for inquiry. As obtuse as this may sound, I believe it’s one of the most important issues we face as a community, one that goes not just to the nature of math but to the purpose of school itself.

Conceptual Understanding vs. Application

As described in the Common Core Standards and elsewhere, a “rigorous” math education is one that includes an equal emphasis on three areas: procedural fluency, conceptual understanding, and application.

When it comes to developing procedural fluency, educators typically have an easy time identifying resources: Khan Academy, iXL, the myriad worksheets and rote practice systems designed to help students strengthen their mathematical muscle memory. Whatever we may think of the emphasis on such activities, we tend to have little trouble recognizing them.

When it comes to distinguishing between conceptual understanding and application, though, I find there’s much less clarity. For instance, consider the following activity and ask yourself: Is the purpose to help students understand a concept, or is it to help students apply math?

“How long will the apple peel be?” Source: Graham Fletcher

I’ve shared this task with teachers around the country, and invariably around half describe it as an application. Their reasoning? “It has to do with apples.” If an activity involves a real-world context, the thinking goes, it’s an application. Otherwise it’s a conceptual understanding task. This criteria seems reasonable, and it’s certainly easy to understand.

However, I think it’s too simplistic and leads to weird results. To see why, imagine two other tasks:

1. Start with the number 1. If you double it thirty times, what number will you end up with?
2. Start with a penny. If you double it every day, how much will you have at the end of a month?

Based on the context-or-not criteria, we’d classify the first activity as a conceptual understanding task and the second as an application. But that’s absurd! The tasks are fundamentally the same; the only difference is the frame.

An Alternative Criteria

If we’re to provide students with a complete math experience, then instead of focusing on whether a task involves a context, I find it more helpful to focus on the role the context serves. Put another way, to ask, “What is the task about?”

While the second doubling task (above) involves pennies, it’s still about exponential growth. While the peeling task involves an apple, I’d argue the apple is really a prism for looking at an underlying mathematical concept: the relationship between the surface area and volume of a sphere. When students are shown a larger apple, they discover that doubling the radius causes the skin to quadruple and the meat to octuple. They do some great reasoning about πr² vs. πr³ and scaling in different dimensions. What they don’t do, though, is reason about fruit. Should juice companies use organic or genetically modified crops? What about old folks with fiber deficiency? These aren’t conversations students have because apples aren’t what the lesson’s about.

From Three Acts to Mathalicious, Yummy Math to Desmos, there are plenty of resources that incorporate “real world” situations. So when evaluating an activity, how can we discern whether its purpose is for students to understand a concept or to apply a concept?

The way I see it, if students are primarily using the world to look at math (world → math), the task should be classified as conceptual understanding. On the other hand, if students are using math to look at the world (math → world), the task is an application.

Using this criteria, consider the tasks below. In “Will It Hit the Hoop,” students are presented with a series of basketball images and use a parabola to determine whether the shot will result in a basket. In “Out of Left Field,” students create an equation for the trajectory of the average Major League home run, then compare it to the left field wall of every MLB stadium. Both lessons involve sports. Both lessons involve parabolas. So are they both applications?

“Will It Hit the Hoop?” (Source: Desmos)	“Out of Left Field” (Source: Mathalicious)

In “Out of Left Field,” students use a parabola to explore why it’s easier to hit home runs in some Major League stadiums than in others, then debate the implications of this. For instance, the average home run would clear Busch Stadium by 23 feet but Wrigley Field by less than three. So was the 1998 home run race between Mark McGuire (Cardinals) and Sammy Sosa (Cubs) fair, and should outfield walls be standardized? While the lesson involves quadratics, it’s about baseball. Since math → world, I’d classify it as an application.

Contrast this with “Will It Hit the Hoop?” While this task involves basketball, students don’t emerge with a deeper understanding of the game. Instead, they use the context to reason about parabolas: how their graphs are symmetrical, and how to use an equation to find the height for a given distance. Though the lesson involves a foul shot, it seems fundamentally about quadratics. Here world → math, and I’d classify the task as conceptual understanding.

Math as Telescope

Of course, many (maybe even most) educators might say, “The purpose of math class isn’t to discuss fruit! We’re not here to talk about baseball!” I get that perspective. I also think it’s wrong. While math class is a place to talk about concepts, I believe it’s also a place to apply those concepts to discuss issues in the world. What are the odds of finding life on other planets? How should police departments address excessive use of force? If I get 19 in blackjack, should I double down?

In many ways, mathematics is like a telescope. Just as a telescope is an intricate interplay of metal and glass, math is an elegant structure of logic and reason. In this sense, math is a beautiful structure to look at. At the same time, it’s powerful prism to look with. While a telescope helps us see distant galaxies, math can help us resolve more clearly the myriad questions we might ask.

In my experience, most people agree with this dual-nature of mathematics. And yet…that’s rarely how it’s taught. Instead of prioritizing them equally, we often evaluate applications exclusively through the lens of conceptual understanding.

• Should we increase the minimum wage? Great way to introduce linear systems!
• Do JCPenney coupons really save you money? I can use that to introduce proportions!
• Should basketball players foul at the buzzer? Woo-hoo. Compound probabilities!!

To be sure, there are plenty of educators who artfully use application lessons to help develop students’ conceptual understanding; the process of applying a tool can certainly deepen our understanding of it. Even if this weren’t the case, though, applications would still be valuable in and of themselves. And every time I hear a teacher praise “how is wealth distributed in the United States?” as a “great context for box-and-whisker plots,” I can’t help but imagine an astronomy teacher shouting, “Thank god for Saturn. I’ve been looking for a better way to teach refraction!”

Why I Think This Matters

If you would have told me ten years ago that one day I’d care about the distinction between conceptual understanding tasks and applications, I would have rolled my eyes. The reason I’m so passionate about it, though — and the reason I hope we as a community will take it seriously — is because (1) school represents a unique opportunity for students to analyze and understand the world we’ve created, and (2) math is a powerful tool for doing that.

In order for students to understand the world, of course, they first have to look at it. And for that to happen, I think we as a math community must do a better job distinguishing between conceptual understanding and application, and prioritizing them equally.

Unfortunately, I don’t think we’re there yet. In the Publishers’ Criteria for Mathematics, applications are described as “activities centered on application scenarios.” Not only is this circular — which may be inevitable; it’s hard to avoid words like “apply” or “use” — but the description fails to distinguish between the existence of a scenario (involves) and the purpose of the scenario (about). As a result, the “how many pennies” task from before could reasonably be classified as an authentic application, as could the following activities from Illustrative Math’s new OpenUp curriculum.

When EdReports evaluated the Grade 8 EngageNY curriculum, it concluded that “the instructional material meets the expectations for the criterion for rigor and balance with a perfect rating,” which is to say: EngageNY achieves a perfect balance of procedures, concepts, and applications. And what does types of problems does this best-in-show curriculum include? From a section specifically titled “Applications of Linear Equations:”

• Marvin paid an entrance fee of $5 plus an additional $1.25 per game at a local arcade. Altogether, he spent $26.25. Write and solve an equation to determine how many games Marvin played.
• A book has x pages. How many pages are in the book if Maria read 45 pages of a book on Monday, 1/2 the book on Tuesday, and the remaining 72 pages on Wednesday?
• Shonna skateboarded for some number of minutes on Monday. On Tuesday, she skateboarded for twice as many minutes as she did on Monday, and on Wednesday, she skateboarded for half the sum of minutes from Monday and Tuesday. Altogether, she skateboarded for a total of three hours. How many minutes did she skateboard each day?

These may be fine contexts for exploring linear functions. But authentic applications? I don’t think they’re even close.

Completing the Square

“I’d like you to envision your ideal classroom,” the facilitator asked. “What are you seeing? What are students doing?”

There’s a lot that I love about this question, but what strikes me most is how rarely we ask it. For a whole bunch of reasons — testing pressures, frenetic schedules, new curricula, the anesthetizing effect of inertia — I find that we as educators often forget to consider why we do what we do. What is our purpose? What are our goals? 

Where our goal as educators is to provide students with a better mathematical experience, I agree with my table-mates that we must encourage them to talk to one another, to share their solution strategies, and to value wrong answers as opportunities for growth. If bridges aren’t a good way to contextualize parabolas, let’s use foul shots. If textbooks provide too much information too quickly, let’s use video to scaffold them more gradually. By all means, let’s do all of that.

But let’s also recognize that framing a conversation differently is not the same as having a different conversation, and providing students with a better experience is not the same as providing them with a complete experience. If our goal is to do that — if our mission is to help students not just understand math but also apply math to think more critically about the world — we must do more than create better ways for them to look at the telescope. We also need to give them consistent opportunities to look with the telescope: to peer through the eyepiece, train it on the world, and discuss what they see.

Teaching applications is difficult. It involves a lot of moving parts and is different than teaching procedures or concepts. But it’s doable and gets easier over time. In my experience, the hardest part about prioritizing applications may actually be giving ourselves permission; allowing ourselves to walk into class thinking, “Today we’re going to talk about the world, because that’s a critical part of what we’re here to do.”

It’s worth it, though. You know why? Because it’s invigorating. Because the world is incredibly interesting. And because Saturn’s fucking beautiful.