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.