Classroom Design
Every Student Can Be a “Math Person” with the Right Curriculum
By Bill McCallum
When I was visiting my daughter the other day, my 6-year-old grandson, Arthur, was counting the slats on his 5-month-old brother's crib. He counted 14 slats on one side and said with joy, "That means there are 28 slats on this crib!"
One possible reaction I could have had is, "My grandson is a math person!"
I interpreted it differently: If kids are allowed to play around with math problems, they will enjoy math more. Enjoying math is just like enjoying a sport; it works better if you are doing it with other people. And learning math is like learning a sport; you learn by doing it, not by watching somebody else do it.
Many who start a sport as a child continue playing into adulthood. That's not the case with mathematics. By the time they reach adulthood, many people would not classify themselves as a “math person”—in part because they experience math in isolation and don't have the opportunity to collaborate with their peers. Somewhere along the way, they lost that natural joy that my grandson showed. Could it be because they spent too much time watching someone else do math (the teacher in front of the classroom, for example) and not enough time trying it for themselves?
Of course, tackling problems of any kind alone can be a challenge. That's why educators should bring collaborative activities and embed collaborative practices into their classroom environments. Students will have a range of problem-solving methods at their disposal, with some strategies proving more efficient than others for certain students. Collaboration with other students to solve a problem helps all students in that group because those who found efficient methods gain a deeper understanding by explaining how they came to an answer, and students with less efficient methods learn by seeing what works for their peers in action.
A combination of solo and collaborative work caters to the interests of all students. Some prefer to have some quiet time to think before talking about a problem, while others are ready to start talking right away. Whatever the mix, students need opportunities to tackle problems for themselves before being given explicit instruction on how to solve them. Explicit instruction is important, but it should come after the student works through a problem first—fnot before. That is when students are primed to receive explicit instruction because of their previous efforts. The teacher also has an important role in making sure students understand the problem, then walking around the classroom observing the work, asking questions or giving information that advances their thinking, selecting student work to present to the class, and making connections between different methods. This approach is called “problem-based instruction.”
Curriculum that supports problem-based instruction has shown results in my teaching experience. Lessons that start with warm-ups that activate prior knowledge, and continue with a sequence of activities that students work on by themselves and in groups, bring out student potential. Activities that are designed with multiple ways for students to approach them, depending on what skills they bring to the problem, give students multiple entry points, whatever their proficiency. Once students have a chance to think and explore problem-solving strategies individually and amongst each other, a teacher-led synthesis ensures students have achieved the learning goals of the activity.
The problem-based instructional model provides a way of avoiding the "I am not a math person" mentality. In a traditional classroom where the teacher shows how to solve a problem first and then gives out a worksheet and walks around to help students, the students quickly become stratified. Students who keep getting things right are math people; students who don't, aren't. In a classroom where students are collaborating to solve a problem, that variation in proficiency becomes a way of forming a team—seeing where student strength lies and working together to succeed. On a soccer team, for example, some kids are less proficient than others; this does not mean they are not "soccer people." Everyone can enjoy the sport, and everyone on the team is needed to maximize the team’s success.
Problem-based instruction can be a shift in practice for a teacher, but that is where instructional routines become key. These work the same way as other routines students might be used to (e.g., bedtime routine) as they allow the teacher and students to know what to expect and make them feel comfortable in the learning environment. For example, in a “think, pair, share” routine, the teacher poses a question to the whole class and asks students to think about it for a few minutes. Then they form pairs and explain their thinking to each other, coming up with the answer as a team. Finally, the teacher selects a few students to share their thoughts with the whole class.
This routine allows for all students to respond and participate, not just the ones that always raise their hand. And the stakes are lower when you are sharing with a partner rather than sharing with the whole class. Those who are asked to share with the whole class have an opportunity to prepare. Structured routines like this allow teachers and students to avoid wasting time and mental energy figuring out who is supposed to do what and when, allowing them to focus on the mathematics and opening conversations that might not happen in a less structured activity.
I have a vision of a classroom community where teachers are supported in implementing a high-quality curriculum with a problem-based instructional model that allows students to collaborate to reach their full potential. Every student can be a “math person” if they know, use, and enjoy mathematics in a way that empowers and engages them.
Bill McCallum is co-founder, CEO, and Chairman of the Board for Illustrative Mathematics. He holds a Ph. D. in Mathematics from Harvard University and is an emeritus Distinguished Professor of Mathematics at The University of Arizona. Bill has worked in number theory research and in mathematics education scholarship, outreach, and policy. He was one of the lead writers of the Common Core State Standards in Mathematics. For this work he received the Distinguished Public Service Award from the American Mathematical Society and the Dolciani Award from the Mathematical Association of America. He cares about mathematical coherence and student learning and has devoted his career to working with teachers, educators, mathematicians, and policy makers towards an education system that shows fidelity to both.