Math is about more than just finding the correct answer. It’s about applying the concepts of math to problems and thinking them through to come to those correct answers. Educators acknowledge this every time we ask students to show their work and award them partial credit for wrong answers that demonstrate correct problem solving.
But learning to work through problems can be its own challenge, and students need practice thinking through the process. They need to develop the persistence to keep trying when they don’t get it right away and the confidence that they can solve a problem. For students to become persistent, confident, world-class problem solvers, educators need to create math-rich environments that cultivate those skills.
The power of failure
Failure is a powerful tool for learning that takes advantage of the perception-action cycle—a process that takes place in all humans’ brains. Our brains create a model of the world based on past experiences and then make predictions based on that model. In math, our prediction would be the answer. If that answer is correct, our model is reinforced and the neural pathways underpinning it are reinforced.
But if our answer is incorrect, the perception-action cycle says, “Aha! It’s time to learn something.” The neural pathways of our existing model become a little weaker, and we’re able to adjust that model and begin carving new pathways in our brains.
The perception-action cycle works best when the feedback is immediate and as unfiltered as possible. When we explain math concepts to students, that feedback can take a back seat because language works as a filter between explanation and understanding. When students are able to receive feedback directly through visual or tactile stimuli, on the other hand, the cycle becomes much more powerful.
In practice, this means that although it is important to give students academic language they can use to discuss mathematics with teachers and peers, it is also important to give them opportunities to explore math more directly and develop a deep conceptual understanding of math.
Creating a safe environment for exploration — and failure
First and foremost, students need to feel comfortable making mistakes in a math classroom. A math-rich classroom should include plenty of discussion for students to work out their understanding, along with plenty of room for working together, in small groups, with the teacher, and on their own. Students in a math-rich environment should have ample opportunity to work and struggle with math concepts in a real-world situation and successfully solve problems.
At Lordship Elementary School, we like our teachers to begin with some kind of number talk to let students know it’s math time. It gives students a chance to look at concepts as a whole class and hear what different people have to say about how they reach an answer. We practice using phrases like “I respectfully disagree” as students talk about their strategies to solving problems and then move into a think, share, compare routine where we again look at different strategies and assess them together.
The point of all this discussion is not for the teacher to act as the “sage on the stage” disseminating math knowledge but rather to question students and prod at their strategies—and the thinking behind them—to help them develop new ways of thinking about, and eventually solving, problems.
That all speaks to the emotional or intellectual environment, but the physical environment needs to support that approach as well. Students should have access to manipulatives at all times and know how to use those tools so that they can puzzle out new math concepts. As they do, they receive the visual and tactile stimuli that work directly through the perception-action cycle.
We also encourage teachers to use math picture books to help students see through the filter of academic language and engage with concepts visually. We use anchor charts to offer students visual cues for problem solving relevant to the concepts they’re working on, as well.
And, of course, the desks in our math classrooms are not arranged in rows but can be easily moved and are usually arranged in clusters to enable the small-group work we emphasize.
Don’t be afraid of unconventional tools
This year, we’ve had the opportunity to beta test the newest version of ST Math, a visual math game that deemphasizes language in favor of conceptual learning through manipulating objects in space and time (ST stands for spatial temporal). Students see a math problem on the screen and have to figure out how to solve it to move a penguin named JiJi across the screen.
My 3rd-grade teachers were nervous because the ST Math journey for their grade level begins with multiplication and division—we had decided earlier that, for Tier I students, we were going to begin the year with adding and subtracting within 1,000.
“Our students have no idea what multiplication means,” the teachers told me. “What are they going to do? They’re going to need a lot of help.”
And I just said, “Let’s just watch them.”
On the very first day, a young boy was struggling and I said to his teacher, “Can we try giving him a manipulative?”
It was a fun repeated addition game with an animal, and he had to figure out how many shoes go on the animal. I gave him some chips and as soon as they were in his hands, he just started making groups with them. And then he said, “Oh, they just want it grouped. So, I have four groups.” Then he hit the four on his screen, and it was right. He looked at me and beamed, and then he went on to the next one and the next one and the next one. Nobody said the word “groups” to him. That just came to him from watching the math, interacting with the action of the game, and working with some chips in his hands.
It’s a powerful moment to see a child create his own language to describe a concept they’ve made sense of on their own. It’s deeper than learning a new math concept. It goes right to the root of struggling and persisting and succeeding without having anything handed to them except the tools to learn.
Robyn Tedesco is the math coach at Lordship Elementary School. Her district, Stratford Public Schools, began using ST Math as part of its math-rich environments this school year. She can be reached at firstname.lastname@example.org.
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