Author: John Novick, Jr., NPES Head of School 

Contemporary research into the functioning of the human brain continues to change how we understand teaching and learning. Intellectual capacity, once thought to be fixed, is more malleable than previously understood. We rewire our own brains as we live our lives, with our mindset, choices about how we spend our time, and our priorities impacting what we’re capable of. It turns out that the conventional wisdom about people having or not having “math brains” (or being right- or left-brained) is void of wisdom. There is no such thing as a “math brain”; not being able to “do” math is a conditioned behavior. Approximately 97% of the population’s brains have the potential to learn all levels of mathematics. Success and interest in math is due to other variables, not innate ability. Our brains prioritize what we ask them to, to the point where even when we’re sleeping our brains are working on problems of all sorts that we’ve identified as important (the idea of “sleeping on it,” in fact, derives from centuries-old instinctual wisdom that is now being proven to be a solid strategy for deep
thinking).

At NPES we keep this brain research at the forefront of our minds when shaping our math classrooms and making pedagogical choices. As a result, if you walk into an NPES math classroom you will most likely see students with heads together grappling with a few select and complex real-world problems or holding manipulatives and visual representations of shapes and quantities. Students turn to their peers or come to the front of the room to explain strategies and thinking, or share their wonderings and questions. Students continually revisit previously-learned content to build upon their knowledge and deepen understanding, with a playful attitude with math games and flexible thinking around problem solving. At NPES, math is alive and accessible to everyone; we are all mathematicians.

In Math education, our notions about using tools have also been changed by brain research. While it has been considered essential to use materials (counters, blocks, drawings, fingers, etc.) to help students develop conceptual knowledge, we have encouraged students to get to the numbers and leave the “crutches” behind as they develop. But contemporary research shows us how our brains can connect numbers with visual representations we’ve created, supporting more robust understanding (beyond memorization). So instead of, “Eliana, I know you know 3 x 7, you don’t need to draw that picture anymore,” we are likely to say, “I see you know that 3 x 7 equals 21, Eliane, but can you represent that with a visual model and justify your answer to your group?”

Note the word group in the example above. We know today that individually we can memorize and rehearse a series of steps–like following a recipe to bake a cake–to solve mathematical problems that have been prepared for us, and that we can do so (given time for focused practice) fairly quickly and accurately. But in order to develop a conceptual understanding of mathematics that will enable us to apply what we have learned to situations that are not scripted, it is imperative that students collaborate with others, learning how to make their mathematical reasoning explicit while integrating approaches from around the table they may not have seen (think of baking a cake but having to substitute for a number of missing ingredients, being open to making a delicious dessert that may not exactly resemble a cake but is more like a cousin to it). This is how students engage in authentic and purposeful mathematical inquiry, and how they prepare for problem-solving in high school, college, and beyond.

Our attitude toward mistakes has also been challenged by brain research. Students were once told to use a pencil for math so they could erase their mistakes. Now we know that synapses fire in our brains when we make mistakes and that the brain lights up with activity when we realize and analyze mistakes. This is what we want, so mistakes in our classrooms are expected, inspected, and respected. They are like finding gold when it comes to developing deeper understandings. Students are asked to give their mistakes to the class for the greater good…and doing so, rather than erasing them or hiding them, takes practice.

Math in the 21st Century is more than arithmetic. It is not enough to compute accurately or quickly (how fast one processes mathematically is not really of value at all; learning math is more about depth and complexity than speed). Students need intentional practice striving to make connections, reason with the numbers, develop mathematical representations of the problems, and use mathematical vocabulary to communicate processes to others and justify their solutions. Often students today mistake fast processing of computation as having “mastered” a concept in mathematics, developing a misinformed pride in being advanced in their problem-solving skills…even as they appear unable to describe what they’re doing, other ways they might do it, and why it matters in the first place. Despite the Hollywood-reinforced mythology, mathematical inquiry is not about intense narrow focus, rigid thinking, or innate giftedness in working with numbers that always leads to the correct answer. It’s about flexibility, openness, questioning, creativity, exploring, expression, collaboration, analysis, synthesis, documentation…and pushing yourself to make mistakes.

There are many ways to solve nearly all problems, but in the U.S. we have been taught that the algorithms our educational system arbitrarily picked to teach a century ago are not only the best, but also the only algorithms. This, more than almost any other factor (setting aside mindset, perhaps), has limited the success of our nation’s mathematics programs over the last one-hundred years. There are numerous efficient and useful algorithms used around the world for doing arithmetic and mathematicians often use number sense and reasoning to solve problems in other ways. So in classrooms today, you see students solving the same problem in a variety of ways. This does not mean that learning and being able to recall math facts–and developing a degree of automaticity in working with numbers–are unimportant things. They’re necessary tools for engaging in mathematical problem solving and inquiry. But they’re not learning mathematics for understanding or application. They’re baking a cake.

The National Council of Teachers of Mathematics developed habits of effective math learners for the 21st Century, which you can see alive in our own NPES Philosophy of Mathematics. I hope you find them helpful as a parent/caregiver. For those of us who learned math in a very different way–and who did not have the benefit of today’s brain research in our schools–it is incumbent upon us to not reinforce the myths about mathematics with children, but instead to teach them what we know about the human brain. Their mindset truly matters.

• I make sense of problems and persevere in solving them.
• I can solve problems in more than one way.
• I can explain my math thinking and talk about it with others.I see math in everyday life and use math to solve everyday problems.
• I check for details and accuracy when doing math problems.
• I know how to choose and use the right tools to solve a math problem.
• I can see and understand patterns in math.
• I notice when calculations are repeated while solving a problem.

Note: Researchers and practitioners whose work is reflected in this piece include Carol Dweck (Stanford), Jo Boaler (Stanford), Anne Anzalone (Wright State), and the leadership team at the National Council of Teachers of Mathematics.