Learning Math

Math did not come easy in school. I remember being bored with long division. I found it difficult to concentrate on the processes taught to me to solve the problem. 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 9 year old brain to check out of math altogether. My report card in 4th grade almost reflected an ‘F’ in math. I say almost because I performed many extra-credit activities. I boosted my ‘F’ to a tolerable ‘B-.’ 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 Geometry. I set the highest score on each exam. I went on to do the same in Algebra II and Pre-Calculus. When I went to college, the university hired me to tutor Calculus and Physics. This was quite a switch and not the trajectory that my 4th grade teacher predicted for me.

So what changed? Why all of a sudden is a switch thrown in the brain that allows a student to go from almost no comprehension to mastery in mathematics. The answer I think is simple. It boils 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 abstract. 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. We abstract the idea of the number 2 that exists independently of the things we are counting. For me geometry class made sense. There were shapes. The properties of shapes such as triangles and squares were very intuitive. However, In my 4th grade long division experience, all I knew was a process. The goal: master the processes. I didn’t really understand what it was I doing. The process of long division is uninteresting to me. Looking at shapes and understanding how these objects are related ignited my curiosity. I could now work hard in my math because I am solving problems.

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 and not understanding. Before the mind is ready to abstract effectively a concrete intuition must be built. Take division for instance. When I ask my 5 year old what 12 divided by 2 is, he has no idea. However, if I give him 12 jellybeans and ask him to share equally with his brother so that they each get the same number, he divides by 2 with no problem. He understood one question but not the other. One question is abstract, one question is concrete. He can consistently do this with any even number. He also realized that when he has to divide an odd number by 2 there is always one left over. (Dividing and having a remainder is pretty advanced for a 5 year old). With my 7 year old, I play with the number 12 a little bit more. I ask him how many jellybeans do 3 friends get if you have 12 jellybeans to share. I ask about 4 friends, then 6 friends. He learns that three groups of 4 make 12. He learns 2 groups of 6 make 12. 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.