When I say math class, what do you think of? One of my favorite math podcasts, Making Math Moments that Matter, asks their guests to share a memorable math moment. Pause for a second, close your eyes, and think of math class. What do you see? What do you remember?
For me, the first thing that comes to mind is third grade. During third grade, we did timed math tests every Friday. The page was full of problems. Once you hit a certain level with addition, you moved to subtraction, then multiplication, then division. I hated timed tests. I got through addition and subtraction all right but remember getting stuck on my multiplication facts. For whatever reason, I could not get over the hump on the multiplication timed tests. Here’s the thing though – I did well in math, I always scored high on tests and quizzes, and was successful with my homework assignments. But because I didn’t do well on my timed tests, I felt like I wasn’t good at math.
The next thing that comes to mind is high school math and my Trigonometry / Pre-Calculus class. We would walk into class, sit down, open a notebook, and then take notes of sample problems for the entire 45-minute class period, then we’d go home and do 15-20 homework problems. The monotony of this process was broken up only by test or quiz day. While I liked my teacher, I did not feel like I learned a lot in this class. Often when doing my homework, I would go back to my notes because there was something I didn’t understand, but my notes would not help me solve the problems. I felt like this was a point where I ran into a wall. I did all right in my math classes up to that point, but for whatever reason, that year, I felt like I was no longer a mathematician.
Even with these two experiences, I generally enjoyed math. I felt like I was pretty good at it, so in college when I had to select an area of endorsement to go with my degree in Elementary Education, I selected math. When I was hired for my first teaching position, my school saw that I had a math endorsement and had me teach the advanced math class for our fifth grade. As I think back on how I taught math those first couple of years, I utilized some of the strategies that I hated about math when I was younger. I fell back to those strategies because that’s what math class was supposed to feel like, right?
Recently, I’ve been reading the book Limitless Mind: Learn, Lead, and Live Without Barriers, and it’s taking me back to my days as the advanced math teacher early in my career and wishing that I would have thought of doing things differently. The premise of the book is that there are six keys of learning. Each chapter of the book presents one of the keys. Chapter 4 is titled “The Connected Brain” and focuses on the following learning key:
“Neural pathways and learning are optimized when considering ideas with a multidimensional approach.”Jo Boaler, Limitless Mind, 2019, p. 101
Reading this chapter has been mind-blowing in helping me understand why I struggled so much with the timed tests in third grade. To me, those papers were just a jumble of numbers on a page. I couldn’t make sense of it all. Even today, I see pictures of worksheets like this, and I feel the anxiety kick in immediately.
So, what is this multidimensional approach that Boaler is talking about? Did you know that even when working on a simple math problem, there are five different areas of the brain that are put to work, and two of them are visual? Our brain wants to take those numbers on a page and create something visual! And when we, as teachers, help our students to access multiple parts of the brain, and communicate with one another, the learning is so much greater!
As I reflect on my learning, I don’t recall exactly when math stopped being visual, but I’m pretty sure that change happened somewhere in my elementary school experience, quite possibly in third grade when I was struggling with those timed tests! So, what are some ways that we can add more visuals to our math practice? Boaler suggests that instead of having our students practice a series of nearly identical questions, have our students practice a small number of questions (like three or four) and think about them in multiple different ways. These are some of the questions that Boaler shares in Limitless Mind (2019, p. 109):
- Can you solve the questions with numbers?
- Can you solve the questions with visuals that connect the numbers through color-coding?
- Can you write a story that captures the question?
- Can you create another representation of the ideas? A sketch, doodle, physical object, or form of movement?
What does this look like? Here’s an example that I created using a simple division problem:
Boaler credits this “Diamond Paper” approach to Cathy Williams, one of Boaler’s colleague’s and fellow director of youcubed. When I saw Diamond Paper, I immediately thought of the Frayer Model. Earlier in my career, I taught science, and one of the ways that we learned some of our difficult vocabulary words was through the Frayer Model graphic organizer. I found that my students had a better conceptual understanding of our vocabulary words and used them appropriately in their lab write-ups. The reality is that something like the diamond paper method or the Frayer Model can be used in lots of different subject areas. The multi-dimensional thinking that is required to complete something like this activates more regions of the brain, creating stronger connections within the brain, which leads to greater learning.
Those timed tests that I struggled with were not helping me learn. While I did eventually pass my multiplication and division timed tests, it was purely memory-based. I did not have a conceptual understanding of what was going on. I still believe that the wall that I ran into in my year of Trigonometry / Pre-Calculus came about because I didn’t have the flexibility of numbers to understand the concepts behind what happened in multiplication and division.
I’ve defined learning on this blog before, and ultimately that’s what the focus is here. How do we learn and grow, and how do we help our students have greater levels of learning? The more we can do to connect different parts of the brain around a topic, the better the conceptual understanding we have in the long run.