Two of my students stared at me, and then at each other. Each held up a sheet of graph paper, but their graphs were different. And then, one of them asked the question I knew was coming.

“But, who is right?”

I looked at them and said, “You tell me.”

The activity I had planned for that day involved showing a video of a man running around a baseball diamond in typical order: home, first, second, third, back to home. I gave the class some guidelines: independent variable was time, they chose the dependent variable, and they had to be able to explain their reasoning. There were things in that lesson that I expected to, and did, see that day. This disagreement between the two students wasn’t one of them. They had chosen the same dependent variable—why were their graphs different, they asked?

In the world of personalized learning, it’s no longer good enough to preach to students about how to solve problems, nor should teachers rely on complicated tech tools for instruction. Rather, it should be about allowing students to get there on their own—which requires a couple of key mindset changes, my fellow teachers.

## Using Language Students Already Know to Teach Math

Traditionally, math education has been based on teaching students algorithms to explain how things work. Two plus two equals four because someone along the way showed us that this is true.

But a few years ago, I realized that as a math teacher, if my job is supposed to be to help students learn and develop the math skills they need after they leave my classroom, and ostensibly after they walk out of the doors of the school they attend, then my goal shouldn’t be algorithms. Rather, my goal should relate to relationships that already exist, and the language that describes those relationships.

My classroom today is different from what it used to be even just three or four years ago. Trying to change the way students not just think about math—but also how they communicate about it—has challenged me in a way that I hadn’t expected.

When I started teaching at Research Triangle High School, a STEM charter school, I found an opportunity for personalized learning that set me up to encourage more students to lean towards using language to describe mathematical relationships. The problem I’ve always had was that I can’t expect all my students to come into my classroom knowing all the same algorithms and processes; traditional teaching doesn’t always allow for me to address that effectively.

On the other hand, I do know that they’ll come in being able to use some kind of language, and so we start there. When it comes to personalized learning, I spend less time talking at my students and more time talking to them, using the language they already have to help them develop the language of mathematics they need to know.

Take, for example, drawing lines. Almost all students know what a line is and how to draw one, but the struggle comes with how to describe a line mathematically and with precision. Thus, students come in knowing the words “slant” and “straight,” and through conversation, eventually evolve to using words like “slope” and “linear function.” By creating direct connections between what they already know (slant) and what they need to know (slope), they find that the latter is a much more precise and specific way to get at the concept they’re trying to describe.

## Letting Students Discuss and Determine Their Own Conclusions

Personalized learning in my classroom doesn’t always take on a specific form, but it always takes on a specific purpose: to meet my students exactly where they are and to give them what they need. Even though they don’t all start at the same place, as students develop their math knowledge and start to apply it in the classroom, we eventually—and organically—develop a common language with each other.

Then, we really start to talk. We talk about why something works or why it doesn’t, we talk about why they think they’re right or wrong, we talk about how to understand and describe relationships that already exist using math. But we talk. The students talk to each other and to me. We talk a lot. And when we do, my classroom is loud and boisterous and full of students pushing each other to understand better.

As I watch my students work through their own fear of an algorithm, they start to increase their own descriptive vocabulary. They gesture. Sometimes, they draw or stand up to try and describe what it is they’re thinking. They’re explaining it to each other, and in doing so, are developing both skills and mathematical knowledge.

While my class was once built around me telling students what algorithms they needed and why, it’s now built around the students wanting to know what the algorithms are because they already know why as the result of our classroom conversations. In most cases they have some idea of what the algorithms are and sometimes the connection there is faster and much deeper. In some cases a recognition of the algorithm is all they have and the discovery of why happens mid-conversation. The students eventually learn that these algorithms give them another way to talk about something, a way that’s more efficient, more elegant, and more purposeful than what it was they were using before and so they’re linking those algorithms to the language they’re already using. They all reach this point at different times. Some of them might not reach it this year and others will reach it more quickly.

But that’s the purpose of personalized learning: for students to get there on their own, using what they already have and know.

And what was the end to that debate between those two students, you ask?

My students ended up getting into a pretty heated debate about who was right. But over time, their conversation slowly turned from general descriptive terms to math-specific descriptive terms. I listened to their exchange and at first heard words like, “Going this way,” and “Turning over here … ,” and then eventually, “Distance directly from…” and “… comparing distance to time.”

Both students were right about their graphs. One student had graphed distance as the base runner ran from base to base; the other had graphed distance from home plate directly to the base runner. In the end, though, I didn’t have to tell those students who was right.

They figured it out together.