Cathy L. Seeley - Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver (ASCD Arias)

When I learned how to teach mathematics many years ago, it seemed like a fairly straightforward taskprepare well and explain clearly. Whenever I could, I tried to elaborate why a particular procedure worked or a particular kind of problem might be solved a certain way. If I wanted the students to really stay with me, I learned to focus on asking good questions and challenging students to go beyond their comfort zone, always with enthusiasm and a smile on my face. I worked hard to find interesting puzzles and games that might have some slim connection to what I was teaching. And I made myself available for students before and after school. I like to think that I got better every year, and I also like to think that most of my students thought I was a pretty good teacher overall. Many of them came to not hate math, and perhaps they even learned most of what they needed to know in order to move on. I tried not to focus on the fact that the math I was teaching might not have been very relevant to their lives or might not make sense to some students, even with my "clear" explanations.

I've learned a lot since those early days in my teaching career. Today we know much more about what it takes to equip students to become mathematical thinkers who can take on any problem they encounter. We also know that studentsall studentshave more ability and even more intelligence than we might have imagined. As we think about how to nurture and help students develop their abilities and intelligence, I'm convinced that their success in the future depends at least as much on how they think as it does on what they know. Likewise, I'm convinced that if we're going to help them succeed, we need to pay at least as much attention to how we teach as to what we teach. We need to challenge some of our old ideas about struggling and frustration and consider structuring our classrooms differently from how classrooms might have been structured when we were students. We may even need to turn those structures upside down. And we need to recognize that professional learning communitiesif used appropriatelycan offer a powerful vehicle for teachers to learn how to more effectively help students gain the mathematical knowledge, problem-solving skills, and habits of mind they need for living and working in the 21st century.

In this brief look at mathematics teaching, let's think together about what it takes for every teacher to help every student become a mathematical thinker. To look at broader issues related to prioritizing math at the school level and creating and supporting strong math programs, see my companion volume for leaders, Building a Math-Positive Culture (Seeley, 2016).

Students have difficulties in mathematics for many reasons. They may have learning problems. They may not speak English as their primary language. They may be too shy to ask questions or have behavior or attendance problems. They may have missed or misunderstood an important concept in the past. They may not see the relevance of what they're learning. But the biggest barrier for many students is their belief that some people are naturally good at math and some people simply aren't. Students come to believe this myth largely because the adults around them, including their parents and teachers, may also believe it. Unfortunately, this idea has been reinforced by some long-standing practices and underlying beliefs, particularly structuring classrooms around teacher-centered lectures, using timed tests to assess mathematical fluency, and believing that good students shouldn't make mistakes.

Traditionally, we may have thought of a "good" or "smart" math student as one who is quick to respond to a teacher's question, accurate at computing the answer to a computation problem, and able to apply a newly learned procedure to solve a word problem. But the truth is that there are many ways to be smart in math. Some students may be creative problem solvers. Others may be visual thinkers who can see and analyze quantitative or spatial relationships. Still others may be thoughtful, slow processors of information and generators of multifaceted solutions to complex problems. When we expand our ideas about what it means to be smart in math, and when we help students develop their mathematical talents in a variety of ways, we're likely to see many more smart students. Of greater importance, more students will see themselves as smart . And when they see themselves that way, they're far more likely to be willing to tackle the next mathematical idea or challenging problem they encounter.

We now know that nearly any student can learn mathematics and succeed if we shift our practice a bit. We'll take a look at some of those shifts shortly. For now, let's consider what it means to be smartsmart in general, and specifically, smart in mathematics.

Thanks to advances in cognitive psychology and technological breakthroughs in studying neural connections in the brain, we now know more about the nature of intelligence than ever before. In the groundbreaking book Mindset , Carol Dweck (2006) brought these notions to a broad audience as she discussed the differences between a fixed mindset about intelligence and a growth mindset . In the past, many peopleincluding some expertsbelieved that intelligence was all about the genes a person is born with. This fixed notion of intelligence would mean that a person is as smart now as he or she ever was and is ever going to bethat even if someone learns new things, his or her basic intelligence would remain the same.

However, researchers are accumulating a growing body of evidence in support of a growth mindsetthe notion that intelligence is far more malleable than some may have thought. We now know that genes are only a starting point in determining a person's intelligence. Researchers studying intelligence and analyzing brain scans of individuals working on different types of problems have found that as people work through an activity that is difficult, they can actually grow new neural connectionsin essence, their intelligence increases, and they become smarter. The experiences in a person's life and, more significantly, how that person processes those experiences, can influence that person's intelligence. If someone understands that certain kinds of experiences can increase intelligence, it can influence not only how the person approaches school (especially hard subjects), but even how that person relates to others and functions in everyday life.

A student's mindset about his or her intelligence plays a huge role in the student's willingness to tackle and persevere through a challenging problem. Imagine how a student with a fixed mindsetwho believes he's only as smart as he's ever going to bereacts when encountering a hard problem that he doesn't know how to solve. He's very likely to believe that the problem is beyond his intelligence and capability. His response may be something like, "You never taught me that. I don't know how to do this. I can't do it."