# The 3 Vehicles of Conceptual Math

Are you looking for better ways to bring conceptual math to class to your classroom? Conceptual learning is incredibly important for your students. But teaching conceptually also presents significant challenges.

When I was in school, no one seemed interested in conceptual math. I still remember the times tables quizzes we would take every Friday in 3rd grade. The first week was the zeros, next was the ones, and so on.

Now, it felt great to get a 100% just for writing 0 or 1 all the way down the page. But how much thinking was involved?

And by the time we got to the nines, most of us had forgotten half of the 3’s, 6’s and 7’s. And once we got through 12, we all just went on to the next chapter, regardless.

Even in high school, most of the math involved memorizing formulas and procedures. In Calculus, I plugged equation after equation into my graphing calculator. But I never really understood what they meant.

I knew that the integral of sin x was equal to – cos x + C. But did I appreciate the magic of it all?

Today, I think about the fact that you can graph a sine wave, and somehow use a cosine wave to find the area under the curve. Don’t sine and cosine have something to do with triangles? Does anyone else find it ridiculous and beautiful that these things are connected?

## How Much Has Changed?

But back then, it was just another formula to enter into the calculator. And most adults, including myself, would struggle to perform the calculations, even while appreciating the beauty of the mathematics. Does it make you wonder why students need to do so many of these in high school?

We can’t just teach math as a subject of remembering. If we do, our students will quickly forget the procedures and “math facts” that they learn in our classes.

And they’ll fail to see the purpose of what they’re learning. They’ll wonder, “What does this have to do with what I learned last year?” Or “When will I ever use this?”

And despite a lot of attention being paid to the importance of conceptual math, most students are still learning a procedural and memory-based style of mathematics.

I remember sitting in quite a few inservice days, watching a slideshow and listening to a lecture about procedural vs conceptual math. I would think, “Ok, I get it! Conceptual math good, procedural math bad.”

But these slideshows never really changed my teaching. I went back to my classroom, opened my textbook to the next chapter, and assigned exercises 1 through 29. (Odds only).

The reason this type of PD didn’t work was because it addressed the wrong problem. It wasn’t that I didn’t like conceptual math or didn’t believe it was important. I just didn’t know how to teach this way.

## How Hard Is Teaching Conceptual Math?

One reason I struggled to embrace conceptual math was that I wasn’t completely sure about the difference between conceptual and non-conceptual math. Were word problems conceptual? How about number lines?

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Another issue was that I didn’t have the resources to teach conceptual math. While the internet was around, we didn’t have sites with free 3-Act Math Tasks, or online stores where you could find interactive activities for algebraic reasoning or creating visual models.

There’s also the “easier said than done” argument. It’s one thing to get up and tell a roomful of teachers to teach a certain way. It’s another to actually coach them by collaborating on lesson plans or teaching a demo lesson.

## What Counts as Conceptual Math?

While there are many definitions of conceptual learning, not all of them agree on exactly what qualifies as conceptual learning. Others can be a bit vague, mentioning the importance of understanding. Deep understanding. Many articles address the importance of conceptual understanding for real-world success.

And while real-world relevance is certainly important, it doesn’t necessarily help us to tell conceptual from non-conceptual math.

To that end, I think about indicators of conceptual understanding. In other words, how do we know when a student has truly grasped a concept, and when they’ve memorized a procedure?

I believe the best indicator of conceptual understanding is the ability to make connections.

If you understand that subtraction ‘undoes’ addition or how multiplication is a repetition of addition, you understand the connection between operations.

Conceptual understanding also allows us to make connections outside of math class. When students in music class recognize the math in a rhythmic structure, the are demonstrating conceptual understanding. Same goes for applying their understanding of statistics to a science project or a social studies lesson.

Finally, the gold standard of conceptual understanding is when students apply mathematical understanding outside of school. Maybe they use math to start a business, write software, or figure out how much soda to buy for a party.

## Isn’t That Why We Have Common Core?

In theory, Common Core was supposed to ensure that schools find a healthy balance of procedural and conceptual math. In fact, that’s the definition of rigor provided on the Common Core website.

But over a decade in, and many educators are questioning whether Common Core accomplished its goals. Even supporters of the Common Core philosophy, myself included, acknowledge that changing the culture of math education has proven more challenging than anticipated.

And there have been a number of “unforced errors” in how the CCSS were rolled out. We began testing students on the new standards before anyone really knew how to measure things like how well students “construct viable and critique the reasoning of others“ (math practice standard 3).

While the test makers were still figuring it out, they rolled out common core aligned high stakes tests, with real consequences for students and teachers. Not only that, they began testing all students across grade levels on the new standards, rather than rolling them out one year at a time. So students who had taken traditional tests throughout their schools, suddenly switched over to common core testing in middle school or high school.

And let’s not forget the textbooks. I’m not sure what was required for textbooks to label themselves “Common Core Aligned.” But the process can’t have been too rigorous.

Once these new textbooks were rolled out, the internet was littered with pictures of incomprehensible word problems and angry rants from frustrated parents and teachers.

Despite these challenges, the Common Core standards represent the first serious attempt to implement conceptual math on a national scale. So while the roll-out has had its challenges, without them, we might not even be having the conversation.

## The Three Vehicles of Conceptual Math

So the question remains, what will it take for conceptual learning to become more “common” in math classes around the world?

While better math standards are a good first step, standards alone won’t change how most students learn math.

One way to ensure students understand math concepts is to focus on the five representations of mathematical ideas

Translating mathematical ideas from one representation to another helps students develop an understanding of the concepts behind the representations. The five representations also offer an effective way to assess a students’ depth of understanding.

The three vehicles are a set of classroom practices based on the five representations.

But why call them vehicles? A vehicle describes anything that can take you from one place to another. When using the vehicles in class, first we need to “build each vehicle.” Students must learn fundamentals of scale models, number sentences, and story problems.

Once they’ve learned the basics, the vehicles become powerful tools for introducing new math concepts. Once educators adopt these interactive lesson models, they can be adapted for active learning across domains and grade levels.

### Scale Models

Educators looking to emphasize conceptual math would do well to start with scale models.

You’ve heard that a picture is worth a thousand words. And that seeing is believing. When it comes to mathematics, seeing is also understanding.

Our brains are designed to process objects that we can see and we can touch. Too often, the math children learn in school is disconnected from their experiences in the outside world.

Scale models reestablish that connection. By working with physical and visual representations (two of the five), they begin to connect “school math” with their lived experiences.

There are two reasons why I like the term “scale models.” One is to highlight the similarity between a manipulative model and a pictorial representation. In fact, a great way to help students deepen their understanding is to have students draw a pictorial representation of whatever they’ve demonstrated with their manipulatives.

The other is to stress the importance of scale. There are visual representations that are not scale models, such as number bonds and tables.

Almost any math concept can be represented in physical or visual form. In order to be effective as a classroom practice, students should learn both to interpret scale models and to create their own.

### Number Sentences

Number sentences fall under the “symbolic” category of the five representations of mathematical ideas.

Technically, all of the numbers (3, 6, 28) and operation symbols (+, -, x) we use in math count as symbolic representations.

But unlike traditional algorithms (‘stacking’ and long division ‘houses’), number sentences are incredibly flexible.

A number sentence is a  mathematical statement. It consist of two expressions and a relational symbol (equal sign or inequality sign). Typically, the term ‘number sentence’ is used in the lower grades. But it’s a useful term because it includes both equations and inequalities.

To introduce number sentences, it helps to familiarize students with expressions. Then, add the idea of equality as balance.

With number sentence proofs, students are given a number sentence that may be true or false. They then prove it by calculation (simplifying both sides, symbolic).

Next, they prove through reasoning – this is where the conceptual understanding comes in. The proof via reasoning asks them to either explain why the number sentence is true (verbal representation) or draw a picture (visual) to justify their finding.

Once students complete number sentences with whole numbers, the activity can be expanded to include fractions, decimals, negatives, and so on.

There are even “open number sentence proofs” with variables. Instead of proving them true or false, a number sentence can be always true, sometimes true, or never true.

### Story Problems

Once students have developed their reasoning though visual models and number sentence proofs, they can apply their understanding to solve challenging word problems.

Story problems address the contextual and verbal representations of math. It’s helpful to introduce them last, because there are more steps to a word problem.

First, students must formulate the problem. This is where they “translate” from a verbal/contextual representation to another form (typically physical, visual, or symbolic) in order to solve it. The mathematician Georg Polya developed a four-step process that can be used to solve any word problem.

His process has been adapted to create a number of “word problem strategies” such as CUBES. Unfortunately, most of these modified versions try to avoid the crucial step of formulation, instead trying to employ tricks like key words to allow students to solve problems without understanding them. Spoiler alert: it doesn’t work.

Instead, teach students the original four step process, including strategies they can use for specific problem types (such as draw a picture, guess and check, etc). For younger students, this story problem helper can be a lifesaver. For students in upper elementary and middle school, the Polya organizer serves a similar purpose. Middle and upper school students can use the Multiple Representations of Functions organizer for problems involving more than one variable.

## Bringing Conceptual Math to Your Classroom

Students who regularly learn math through conceptual approaches enjoy a number of benefits. They enjoy their math classes. Their deeper understanding allows them to use math flexibly, and they better retain what they’ve learned throughout the year, and even across grade levels.

And the best way to bring conceptual math to your classroom is one step at a time. You don’t need to throw your textbook out the window. (Though eventually you might not be able to help yourself).

If you’re curious about how to take the next step, consider one of our online workshops for math teachers. If you’d prefer individualized, one-on-one support, schedule a free consultation to learn how instructional coaching can help you connect theory to practice.

Either way, make sure to sign up for our weekly Educator’s Newsletter. That way, you’ll be the first to know when we publish a new blog post or announce an upcoming workshop.

Jeff Lisciandrello is the founder of Room to Discover and an education consultant specializing in student-centered learning. His 3-Bridges Design for Learning helps schools explore innovative practices within traditional settings. He enjoys helping educators embrace inquiry-based and personalized approaches to instruction. You can connect with him via Twitter @EdTechJeff

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