Identify, Manipulatives, and Mistakes: A Better Way to Teach Math
January 10, 2023
January 10, 2023
By Sarah Flippen
The way we teach math is changing, and while those changes are almost all for the good, they do come with some challenges for educators. Years of research are showing us that we need to move math classes away from sedentary environments where students just sit quietly and take in information toward more engaging situations filled with critical thinking, collaboration, and discussion. The challenge for many teachers is that they never experienced this as students.
As a beginning teacher, I launched into math instruction as someone who’d been a strong student and found math work easy to complete. However, I quickly realized that just because I was a strong math student did not mean I was a strong math teacher. I had a difficult time explaining how and why math worked because while my math experiences taught me to follow procedures to get the correct answer, there was no focus on why the answers made sense and rarely anything about how math was applicable to the real world. After hunting for new ideas and ways to improve, I began to unlock my own conceptual understanding and found new instructional tools, including number sense routines, tasks, different types of assessments, quick checks, CRAs, lesson studies, counting collections, and so much more. In fact, there were so many amazing resources and strategies that at times I felt overwhelmed by the things I wanted to fix, change, or add to my teaching. After serious consideration, I’ve pinpointed three areas that I’ve found most transformative in my classroom. They’re easy places to begin, but they’re also sometimes easily overlooked:
Identity and community. As educators and as a society in general, we’ve begun to prioritize strengthening our social and emotional health and developing a positive self-identity. When we think about self-identity, we often think about how people view themselves, their appearance, personality, and place within their community. We don’t often consider one’s identity as a learner or, more specifically, as a learner of math. Math has a tough reputation for being a boring subject, one few people truly enjoy. Imagine being a child and hearing your family discuss how to figure out the tip at a restaurant, having a guardian not understand your math problems when helping with homework, or even hearing your favorite teacher remark on how they didn’t enjoy math when they were in school and you will all just “get through it together.” All these subtle messages work their way into how our math identity is shaped, in addition to what I find challenging or enjoyable, and what I understand as a student.
A simple strategy for any math teacher is spending some time fostering a positive math identity in their students. Talk with your students about what they—and their families and friends—think about math. Check in with them as they work and learn about what they’re understanding and be intentional about showing how what they’re learning will be useful in everyday life. With my primary grade level students, I’ve added “I can” statements into math lessons, which focus on their growth as a learner. Some examples: “If it gets hard, I can persevere,” “I am a critical thinker,” or “I am a mathematician.” We talk about what the words mean as we read them and do this routine daily so the students get the message that they’re capable of doing and thinking about math. I also want them to know that even when they’re capable of it, it doesn’t mean math will always come quickly and easily; they’d have to persevere. I point these traits out throughout lessons, saying things like, “I like how you persevered when working on that problem and even though it was challenging you were able to figure it out.” If you want more information in this area, one of my favorite books is Productive Math Struggle by SanGiovanni, Katt, and Dykema. They include tons of ideas for building math identity for students and class communities.
Using manipulatives. This is another area where most of us can improve. Think about when you learn something new. You may ask things like “Can you just show me?” or “Let me try because I learn better by doing.” If we know that we prefer a hands-on approach to learning, why don’t we use it in math instruction more frequently? There is a learning trajectory with new information or unfolding the ideas behind a procedure. We start in a concrete stage, using manipulatives and models to see and move things around to deepen our understanding. As our brains familiarize themselves with the patterns and what’s happening, we can move to a representational stage, where we can work without physically having something but instead use pictures and drawings to represent what’s happening in the math. Finally, we can reach an abstract level of understanding where we can use numbers and equations to solve the math.
An example of this would be using base-10 blocks for place value, transitioning to drawings, and then only needing to look at the digits to determine what each place’s value means. You could also consider the concept of area. Students begin using tiles to build and determine the area of a space, then transition to drawing it out on grid paper, and finally move to multiplying the numbers to determine the area. There are many ways to do this, but many math classes solely function at the abstract level. In these situations, what tends to happen is only the students who can memorize the procedure succeed. And often, even these students don’t fully understand why the math works, which can make it hard for them to determine the reasonableness of their answers. This can also explain why students return from summer having forgotten some of what they learned, or why you may struggle to help someone with math work that you don’t teach—you may not remember how to solve it because you didn’t understand how or why the math worked.
I was in this same boat and honestly didn’t even realize it. Though math was one of my best subjects as a student, when I began teaching, I realized I had a hard time explaining it to struggling students. It wasn’t until I was working on my master’s that I realized I had a very surface level understanding of math. In that coursework, I had to use manipulatives and take pictures, write out my thought process and how I knew my answers were right, and compare my work with the work of others who solved the problem in a different way to determine why both methods worked and if they’d work in every situation. These experiences opened my eyes to how much we expect students to understand using just a piece of paper, pencil, numbers, and symbols, and I decided to make math a hands-on subject in my classroom.
I began using as many types of tools and manipulatives with my students as I could. This meant that they became familiar with 10 frames, rekeknreks, number bonds, counters, cubes, base-10 blocks, and more. Students could see that there are many different ways to solve a problem, and they learned to use different tools. If they didn’t understand the content by using one tool, there was another I could use to help them make a connection. Experimenting with different tools also allowed them more autonomy. They were able to choose which tool was best for them, giving them the ability to ask for what they needed. It was a beautiful sight to see students move flexibly between tools and pick out the one that worked best for them. I saw them moving away from needing concrete materials to solve problems representationally; some began drawing pictures or using their fingers to solve problems. By the end of the year, I also had students who had moved to an abstract mastery of content using tools only as an afterthought or when they needed to verify their thinking.
Mistakes. Research shows that as we make mistakes our brain grows, yet for some reason many have come to believe that mistakes are a sign of weakness. I’ve worked for years on developing an understanding of math learning and instruction and know that being able to rapidly recall facts or solve mental math problems quickly doesn’t make you more or less smart. Still, when someone asks me a math problem on the spot, I feel my anxiety rise and worry that my inability to correctly answer their question quickly will take away my reputation as a math leader or an intelligent person. Luckily, I know better and after the initial seconds of panic, I can remind myself that I don’t need to know the right answer immediately; if I get it wrong it says nothing more about me than I’m human.
Students don’t automatically know this. When they’re not able to quickly devise an answer or when they make a simple mistake, they feel “less-than” or question their intelligence. It’s our job to show students that mistakes are a normal part of the learning process. However, we have to go beyond even that and teach that making a mistake can help us to persevere and adjust our strategy. Allowing students room to make mistakes and the power to grow from them gives them autonomy not just in the classroom but sets them up to be lifelong learners. Have students reflect and talk about what it feels like to make a mistake. Compare it to other areas that our society is more comfortable with when it comes to making mistakes, like a baby walking for the first time or a child learning to ride a bike. Help students make the connection that growth from mistakes and struggle ultimately leads to learning and mastery. Ask them to reflect on mistakes they’ve made or connections they learned from making the mistake. When you have students learn from a mistake, ask them to show the class where they went wrong and what strategy or steps they took to get to the right answer. Taking the time to teach the value of mistakes might be the most important lesson any human can learn.
As we allow our classrooms to be places where students’ mistakes are welcome, we also need to give ourselves the same room and freedom. As you begin to experiment with different strategies and try to add new things, you’ll make mistakes and some efforts will not work out perfectly. I remember hearing about a new math strategy and being very excited to implement it and share it with co-workers. They all came into my classroom to watch me try it out—and it was an utter failure. Many of the students were unable to finish the activity. After a bit of initial disappointment and embarrassment, and a lot of reflection, I tackled the activity again and again, improving my plans and allowing myself to see the possibilities in the missteps. Flash forward: I’ve now presented on it at a state conference. Give yourself a chance to try something new, allow yourself to take some missteps along the way, and allow students a better understanding of math and of themselves as math students, and let them be learners who can make mistakes!
Flippen, a member of the Caroline Education Association, is a math specialist at Madison Elementary School.