Learning to Divide

In November 1972 my family moved from CT to NH. I was 10, in the fifth grade.

I liked reading and writing. Math wasn’t my best subject. So in sixth grade, when we got to fractions, no one was surprised that it was my worst subject. Fractions were hard, I hated doing them, and I always got bad grades in the fractions work.

Seventh grade, and geometry was easier. Not an A-level student, but passable, maybe B- range, and I understood the concepts pretty well. I liked the pool table analogies and I could make pictures in my mind that mostly conveyed to myself how to solve the problem.

In eighth grade, being a good student overall, I was in pre-high school algebra. Within a month it was like fractions on steroids. I was getting A’s in everything else, and D- or F’s in algebra. One day I stayed after class and stood at the chalkboard with the teacher.

“I don’t get what’s going on.”

“Well, let’s take a look at a few things.”

He wrote an equation on the board.

“What would you do first?”

I forget the specifics, but I said, “Well, I’d do this…”

“Right. And then what?”

“I’d move this over here.”

“Right. And then what?”

[This went on for slightly longer than I’m representing here….]

“Okay, here’s where I don’t know what to do.”

“You’re done! That’s it, just divide by X and you’re done.”

“Right, what do you mean?”

“What do you mean, ‘What do I mean?’ That’s it, divide by X.”

“Yeah, I keep hearing this “divide by X” but I don’t know what to do.”

He looked at me and tilted his head. I looked back, shrugged my shoulders, opened my eyes a little, like, Why are you looking at me like that?

Then he thought for a minute and wrote another equation on the board. “How would you solve this?”

It had more steps, but basically I did them all, until the end.

“So now, again, just divide by X and you’re all set. Do you see how these are the same?”

“Well, I keep hearing people talk about dividing, but I don’t know what that is, or what to do. Can you just show me what you mean by ‘divide by X?'”

He counted on his fingers: “Addition, subtraction, multiplication, division. The four operations.”

I counted on my fingers: “Addition, subtraction, multiplication. The three operations. I don’t get what you mean by division. Can you show me how to do it with multiplication instead?”

He sort of shook his head a little, and erased the board. “Okay, how would you solve this one?”

I can’t relay what was different, but essentially I managed to turn it into a multiplication problem, and solved the equation.

He wrote another equation and asked me to solve it. Again I was able to transform it into a problem I could solve with my three-operation repertoire.

He wrote the last equation a different way, and I got to the end and was stuck. “Divide by X?”

“Exactly! See, you get it…”

“Well, except that I still don’t know what you mean by divide.”

“So, it’s like fractions. The X is the denominator, but it’s a symbol.”

“Right, I get that it’s a symbol, but I failed fractions.”

“How did you get into pre-high school algebra if you failed fractions?”

“I dunno, you tell me.”

What followed was a half-hour where he figured out that I didn’t know what division was – I hadn’t really heard the phrase except in passing, and had managed to turn every math problem to date into something I could solve using my three operations.

At the end of this little session I was signed up for some remedial math tutoring, and a whole series of administrative meetings were called to figure out how a smart kid got through 5th, 6th, 7th, and part of 8th grade without knowing there were four math operations. The teacher probably went home and had a drink.

It turns out that just before we left CT the class had finished multiplication, and were about to start division. In NH, they had just finished division and were moving on. I didn’t know it at the time.

It blew their minds that I had succeeded in turning all those problems into a form I could solve, fast enough that no one noticed my methods.

When we had family supper that evening, I said, “Hey did you guys know there are FOUR math operations?? It’s wild: Addition, subtraction, multiplication, AND DIVISION!” Mom and dad looked at each other: “Um, yeah.”

My younger brother was like, “Duh! Don’t you know anything?”

“Well, it’s no wonder I failed fractions, it’s all division! Crazy.”

The next day they got a call from the school to schedule a meeting, where this line of thinking started to make sense to them.

I wonder if this is the root of my communitarian spirit. Being a late bloomer to division, I’d probably be a terrible politician, or general, or Wall-Street quant.

The tactics I invented to get through those four division-free years served me well. In college, as a junior with a psychology major, I wanted to take an acoustics engineering course. The professor looked at my ‘add’ slip: “Have you taken physics?” “No.” “Have you taken calculus?” “No.” “Have you taken the first-year engineering seminar intro?” “No.” He signed my Add slip. “You’re going to have a difficult time. Good luck.”

To no one’s surprise, acoustics engineering has a significant amount of division involved in the maths. My approach was to memorize the equations, and how they worked together. The class had a mid-term and a final. I approached both the same way: For each test question (three, I think, on each) I wrote out what equation I’d use, and what variables would go where, but since I couldn’t solve the equation I’d show how to take the result and use it in the next equation, and so on down the line. Some test questions used five or six or eight equations, and I didn’t solve any of them, I’d just lay them out, take the ‘result’ as a new variable in the next equation, and explain the final result in narrative form.

I got 30% – 40% credit on each question, failing both the mid-term and the final. I didn’t really care though, because I had learned the material to my satisfaction.

When I received my transcript at the end of the term, he had given me a “C” for the course. It was the fairest grade I ever received.