Math did not come easily to me in school. I remember being bored with long division. I found it difficult to concentrate on the problem-solving processes that we were taught. So many steps and so much room for error. I found myself getting frustrated. I could barely engage long enough to finish my worksheets.

It did not take long for my nine-year-old brain to check out of math altogether. My report card in fourth grade almost reflected an ‘F’ in math. I say almost because I remember passing due to an excessive amount of extra-credit activities. My standardized test scores in math reflected the same – a student who could not grasp math. In fact my scores were low enough to afford me the pleasure of being in “remedial” math for all of middle school.

Then something strange happened to me in high school geometry: I got it. Really got it. Suddenly, I set the highest score on each exam. I went on to do the same in Algebra II and Pre-Calculus. When I went on to undergrad, the university hired me to tutor calculus and physics. This was not quite the trajectory my fourth grade teacher had predicted for me. 

So what changed? The answer, I think, is simple. The answer for me boiled down to the difference between the abstract and concrete sides of math. We learn everything we know first from our own concrete experience before we can think abstractly. We learn to count, not by memorizing the numbers in order but by counting objects. We soon learn that, no matter what objects we are counting, the order of the numbers never change. This simple fact is our first experience in abstraction. 

For me, geometry made sense. There were shapes. The properties of shapes such as triangles and squares were very intuitive. However, in my fourth grade long division experience, all I knew was a process. The goal: master the process. I didn’t understand what exactly it was that I was doing and couldn’t enjoy it. But in geometry, looking at shapes and understanding how these objects were related ignited my curiosity. I recognized the problem and could enjoy working hard to solve it. 

Math is a series of abstractions. Each level of abstraction is simplifying and generalizing the concrete world. It is easy to lose sight of the concrete world in math and rely on memorization, math tricks, processes and shortcuts. Math can become an exercise in mastering processes rather than truly understanding the problem. So before the mind is ready to abstract effectively a foundation must be built.

Take division for instance. When I ask my five-year-old what twelve divided by two is, he has no idea. However, if I give him twelve jellybeans and ask him to share with his brother so that they each get the same number, he divides by two with no problem. He can understand one question but not the other. One question is concrete; one question is abstract. He can consistently do this with any even number. He also realized that when he has to divide an odd number by two there is always one left over. With my seven-year-old, I play with the number twelve a little bit more. I ask him how many jellybeans three friends would get. I ask about four friends, then six friends. He learns that three groups of four make twelve. He learns two groups of six make twelve. In this game he is touching on several mathematical concepts: division, multiplication, factoring and expression. This is now a game where there is a puzzle to figure out.

The more a student interacts with the concrete world in this way, the stronger the concrete intuition is built. Once students have a strong foundation in the basics truths of math, or really any subject, they are then able to build upon them, to think abstractly.  It was this very moving away from processes and tricks and moving toward thinking abstractly while standing on a firm foundation that drew me toward becoming a math teacher.