Dad, How Do You Do Fractions?
I am relaxing, watching something about space on The Learning Channel, when my 10-year-old son strikes.
“How do you do fractions?” he asks.
I hate that question because it means I am about to fly right into the teeth of his beloved classroom teacher who does it a different way. I am about to become really old in the eyes of my kid because I obviously come from the deep dark past where math was done differently. I think he suspects that we counted with dead animal bones or something!
Ten-Year-Old Son does not know this yet, but — even worse — I am about to insist that he actually understand what he is doing instead of just spouting some stupid formula that will give him the answer with no understanding.
I have watched for years as one system of teaching mathematics after another has gained popularity. I have watched rote memorization of tables and endless “kill and drill” of math facts. All that produced was kids like me who could fire off the nine times table at the drop of a hat — but who understood nothing.
I have seen concrete-based programs using “magic manipulatives” become the rage. In these programs kids in kindergarten through Grade 8 play with multicoloured cubes and strips of plastic, in a futile search for the meaning of mathematics.
I have seen the abandonment of all memorization and concept teaching in favour of learning aids such as calculators, number lines, finger counting systems and matrixes. The theory here being that memorization is not necessary if kids just learn the “facts” and “concepts.” Sounds great, but who was going to teach them the facts or the concepts?
Today’s students do not understand the concepts
If you want to confirm that assertion, ask any Grade 7 or Grade 8 student to explain why you invert the divisor and multiply when you are multiplying fractions — not just the memorized rule but a real explanation of the reason so that even a dummy (like a parent) can understand. After all, if a thing cannot be explained by a student in the student’s own words, it was never understood in the first place. It was just memorized!
When Ten-Year-Old Son asked his innocent question, I knew it was because he had forgotten the rule that he was supposed to have memorized. Like many kids, he does not have a great memory for verbal information. What he was really asking was for me to remember the rule for him.
Alas, I can’t do it! I too have forgotten the rule.
Forgotten Rules
But I do understand the concept of fractions and I know that together we can figure the rule out. Of course, I wouldn’t tell him even if I could remember, because it wouldn’t really be helping him. And that’s my job as a parent — to help him if I can.
He sees that gleam in my eye as I reach for some support materials and he begins to panic. “Jeez, Dad, I don’t want you to teach me the thing. Just tell me the answer.”
“Sorry, Buddy,” I say, “but we’re going to have some fun. We’re going to figure this out together! Won’t that be cool?”
His look of panic was eloquent response enough. The concrete-bound education system has trained him to see the purpose of school as memorizing answers so you can get marks on tests, pass to the next grade, and get great presents each summer for doing so well in school. And, here comes old Dad with his goofy belief that Ten-Year-Old Son really wants to know about fractions.
The real problem is that he doesn’t really want to know. He just wants to answer the questions assigned by the teacher for homework and get on to something interesting like playing video games. This is challenge enough for any parent.
I search through a drawer and come up with a fork, a bottle of hot sauce, a salt shaker, a toy and some old poker chips that have somehow survived four kids. There is groaning in the background. He sees that I am excited by the challenge. This probably means that he isn’t going to get into the fourth level of Ultra Doom tonight. Worse than that, he may even have to learn something. This is a gross idea to kids who make a business of surviving in school simply by using their memories.
My excitement does not sweep him in yet. He sits back, feeling helpless, and watches blankly while I go through a few preliminary exercises. He is not getting it. I know that. I expect that. He is not used to starting with an overview of what is coming. He wants facts that can be memorized, just the facts! He is not used to integrating new information into old information in order to build new knowledge.
I continue telling him about the idea of units. I want him to see that numbers by themselves mean nothing — that they are just funny marks representing some real things somewhere else. I line up a salt shaker, the hot sauce, a fork and a small cereal box toy.
He looks at me. I begin to explain about Place and Quantity. We are concentrating on the fork. It is the third object in the line. We begin to make up stuff about the line of objects. Suddenly, he sees something for himself and we write down an equation. (Okay, maybe I did think up the idea of the equation, but he discovered the idea by himself!)
Hot Sauce + Salt Shaker = Fork
We start to talk about the line again. How many of each object is there in the line? Just one. Now we discover that we can add something new to this equation.
1x Hot Sauce + 1x Salt Shaker = 1x Fork
He is getting interested. This is not like math at all. We now discover that in another situation 1x Fork equals only the place on the table where Mr. Fork is standing in line with Mrs. Hot Sauce, Professor Salt Shaker and Yellow Toy. We even discover that Yellow Toy is not even real yet because we didn’t go past Mr. Fork. Yellow Toy does not become real until we pass him or stop on him in line. His whole life depends upon his place in the line.
This is neat stuff! Without realizing it, Ten-Year-Old Son has just discovered two very fundamental principles of mathematics: the principles of Quantity and Place.
We continue. If we know about Place, then we can begin to consider a new idea. Each one of these units represents a single thing, a one. Without intending to, Ten-Year-Old Son now understands this. He even added the 1x to each unit in the second equation by himself. (Okay, maybe a little prompting from the cheap seats)
But, what will happen if sometimes there are fewer than one? Let’s say we are part of an ancient tribe and the hunters return from a successful hunt with one deer, getting ready to skin a deer. How will we divide one into parts smaller than one?
“This is a problem!” I declare.
Ten-Year-Old Son responds, “No way! You just tear the deer into the same number of parts as there are families!”
“Brilliant!” I say. “You have just discovered fractions!”
He frowns. He thought we were just playing! He didn’t know we were doing math! Eventually we arrive at a plan for dividing the meat and, to make things easier, we start to put symbols down instead of a full description of the piece of meat each family gets. We make up weird looking symbols. Using our formula, we realize that each symbol stands for one exact piece of meat in the line (Place), but it can also stand for the sum of the quantity behind it (Quantity). He doesn’t know it yet, but we just invented numbers!
In all, we spent two hours together and, in the end, he invented the way to divide fractions himself! I would not swear to this in court, but I think he actually forgot about Level Four in Mega Doom and had fun doing math. Zounds! What a weird thing to do — have fun learning!
As for me, quality time with Ten-Year-Old Son is hard to find these days, so I will just lie in wait until he forgets and yells out, “Dad, I don’t understand this.”
It’s about understanding
The moral of the story is that if you just memorize facts or formulae or orders of operation or tables, you don’t actually learn anything! Unless you can explain the new concept in your own words, unless you can see the relationship between abstracts (numbers) and real live things, unless you understand why the inversions happen, you just have a bunch of “stuff” floating around in your head that bears no relationship to reality and is completely useless.
Except on a challenge test in school. Does that seem right to you?
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