For many years the way we approached mathematics was very resistant to new ideas, methods, and strategies. Imagine being in any place in the world where all the people had to think in the same way, following the same procedures to solve problems, and being told that you are not a good person if you do not memorize the rules. How would you feel? Can you see how someone might get anxiety or dislike being in that place? BOOM! That is exactly what some students — and even adults — feel in a math environment, and that has to stop.
A long time ago, speaking different languages was not allowed in some countries, but nowadays that culture has changed because we understand the benefits of bilingualism. The same thing is happening with mathematics. It’s not so much that we have “new math.” Most of these “new math” strategies have been around for years. The only difference is that now the world is opening up to the idea of using these mathematical ideas to explore, visualize, and connect them to support conceptual understanding and create more ideas.
Jo Boaler is conducting research on mathematics, mistakes, and growth mindset with Stanford University professors Carol Dweck and Greg Walton. Reading their research has been eye opening for me in understanding the importance of growth mindset in mathematics, giving me the awareness that everybody can learn math at high levels.
Thanks to this research and my experience working with students and teachers at every level, from kindergarten through twelfth grade, I’ve developed a cycle of three mindsets I encourage teachers to embrace in the math classroom to create a culture of creativity and connection instead of rote memorization or rule-following. When all these mindsets are active in a math classroom, each feeds and supports the next to help students get excited about math concepts and take ownership of their learning.
1. Math is Open
When I was a kid learning math, the world was like I described it above: there was just one way to solve a problem, and math was about memorizing and applying formulas. As you can imagine — and maybe you’ve experienced — that process led to a lot of students deciding they just aren’t “math people” and giving up altogether.
But if we can develop an open mindset in the math classroom, then everybody can be a math person. A “math is open” mindset means there’s more than just one path to the correct answer, and students are encouraged to think critically and creatively, and share their ideas about how to solve problems. One student’s work may look different from another’s, but both can still get to the right answer. When we bring an open mindset to math, we encourage students to wonder and experiment, taking ownership over their learning and showing them that they are “math people” after all.
2. Math is Visual
Just like students need visual comprehensible input in other content areas, they need to be able to visualize math concepts in order to truly understand them. I’ve always loved math, but for a long time, I was only focused on finding the answers. Once I learned to visualize it, that’s when I started to have those “a-ha” moments and really understand the concepts I was studying.
Math visualization tools turn abstract concepts into concrete ideas that are much easier for students to grasp than formulas and theories alone. And there are countless ways to visualize math concepts: teachers and students can draw out their ideas or create graphic organizers like number lines, Venn diagrams, or number webs. Students can use any number of virtual or physical manipulatives, from geometric blocks to fraction tiles. Creativity is encouraged here, too! Anything that creates a concrete representation to help students grasp the otherwise abstract math concepts is a powerful visualization tool.
For more ideas on how to make math visual, especially during remote learning, check out my webinar on using virtual manipulatives.
3. Math is Connections and Conversations
Once students know that they’re free to think creatively about math concepts, and once they can visualize abstract ideas clearly, then it’s time to amp up the use of academic language to help them make connections through conversation.
After all, as John Seidlitz says, “If students do not verbalize, students do not internalize.”
In this phase, students will share the visuals they’ve created to illustrate math concepts, and because each student’s work may look different, sharing will lead to conversation about different approaches and strategies.
It’s important to recognize and reinforce academic language during these conversations, using sentence stems and structured conversation strategies like QSSSA to help students share their work and connect different ideas or apply their ideas to real-world problems.
As students engage in lively (structured, academic) conversation, they’ll be reinforcing that first mindset, that math is open, and the cycle will begin again.
In today’s math classroom, students’ engagement isn’t measured by how quiet they are as they puzzle through formulas. It’s measured by how willing and excited they are to think creatively, collaborate, and share ideas about how to visualize and solve problems. When we implement this cycle of mathematical mindsets, then not only do we give students the opportunity to really internalize the content, but we also show them that everybody can be a math person.