By Timon Piccini, posted November 6, 2017 —

“It’s super simple; just do the opposite operation to find x.” We’ve all been there. The way that we were taught seemed so simple. We have these sound bites of sound advice that we have picked up from learning math: Find the common denominators, and add the numerators. Cross multiply. To divide, just invert and multiply. Yet we have also seen the dumbfounded looks on our students’ faces: This is one more reason why math is an impenetrable fortress that is more mystical than scientific in the eyes of many beholders. 

My Moment of Reckoning 

In my first year of teaching, I spent 90 percent of my time reteaching a concept after I had introduced it because the students I was teaching had zero context for why my “tricks” could even work. My math class may as well have been in Hogwarts because what I was teaching seemed to them to be more sorcery than any other subject they had ever learned. It was my introduction to linear equations that solidified this realization.

I remember thinking to myself, “Self? Do you think I really need to prepare some big lesson to teach kids to divide when they see multiplication?” 

“Surely not, self, that seems ridiculous.”

I patted myself on the back and went to bed early. When morning came, I prepared to head to work. Once I was in front of the class, I was ready to cast my spell. I worked through addition, subtraction, multiplication, and division.

“What’s the opposite of multiplication?” 

“Addition?” 

“What? No! Why would you think that? It’s obviously division because it undoes multiplication!”

That’s right, I taught that division was the opposite of multiplication because it was the opposite of multiplication. I think a few students went cross-eyed. They definitely did not understand why this method of solving equations had any logical basis because they were never given the opportunity to think about it. 

We Don’t Understand What They Don’t Understand 

In this moment, I flashed back to my high school days. In grade 11 math, I coasted along with nary a problem, but I noticed my fellow classmates did not comprehend so easily. I remember a student getting frustrated because my teacher kept explaining some form of math, but this student simply did not get it. The teacher got frustrated because the student could not get it, and the student got frustrated because the teacher was getting frustrated. Tears happened, and I realized that my teacher just didn’t understand what this student didn’t understand. Math facts can become self-evident principles when we finally learn them; we cannot even fathom that they were at one point in time incomprehensible. When we find someone who doesn’t seem to get it, we think, “But that is so easy!” Thing is, it is not. If we were able to learn from the pedagogy of lecture and quick tricks, we were the lucky few.

Inside the Magician’s Hat 

These concepts, however, are not incomprehensible, but we need to equip students. We need to find ways that students can discover and reason. I am not against mnemonic devices, but I am against removing mathematical discovery from context and inquiry. The fact is that every math student reaches a point when something feels like wizardry or a secret code to a hidden club. The first time this happened to me was in my third year of college, when I asked my teacher how to form some proof in group theory. He explained the proof, and I said something along the lines of, “How would I ever come up with that?” 

His response was this: “Mathematics is just more creative at this point!” At first I thought that this wasn’t fair because math should be clear, and in all honesty, I meant algorithmic, but now I realize it is not fair because math should always be creative. I had been robbed during my whole education of the joy of discovery. Now it only remains to begin the process of giving students a strong entry point into conceptual discovery. 

Timon Piccini is an elementary school teacher who has a strong love of getting students to see that mathematics is more than just numbers. His favorite sound is when an entire grade 7 class cheers because they are starting to understand a base five number system (a true story). Piccini considers himself a jack of all trades, and when he is not teaching, he can be seen running, hiking, playing guitar, playing video games, and attending concerts, and pursuing just about anything to do with good food. He especially loves doing all this alongside his better half, Kelli.