Tuesday, May 12, 2009

Fractions

Rational Numbers

I taught a course in arithmetic for elementary school teachers and I did a terrible job. Among the many reasons that it was not a good course was how I dealt with fractions.
After all, I thought, the fractions are just the field of rational numbers and there aren’t a lot of field axioms to memorize. You don’t care where the axioms come from; they’re axioms for God’s sake. You don’t have to know where they come from, you have to accept them.
Mea culpa, mea culpa, mea culpa.
At a later time I thought about how I would teach fractions if I were given a second chance...which I was, probably wisely, not given.
I started with integers as counting numbers. I realized that some memorization was needed. Not being familiar with Arabic, I get no clue from the symbol 2 that it stands for ** that many stars. I must have memorized a meaning in terms of how many things I was looking at and associated one of the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 or 0 with that meaning.
I picture the first attempt at transferring numerical information from one person to another as holding up fingers and some agreed upon hand signal to indicate repetition. Next I see a need to give name to the number of fingers I hold up.
I then go into a story that sees the need for arithmetic and a reason for developing addition etc. I make no claim to historical accuracy which is irrelevant. What I want is a history that makes some sense and gives a cohesion, a framework for further ideas.
I also realized, as I thought about it, that small integers were different than big integers. Homer talks of as many Greeks before Troy as there were leaves on the trees of Agamemnon’s olive orchard. Something like that. We hear on the TV that 102,000 people attended the USC-Notre Dame game. How much more does 102,000 convey than ‘the number of leaves on the trees’?
If I had a class of 20 students, I would have to count them to see if one was missing. If you were a crow and watched n hunters go into a barn and (n-1) hunters come out, how big would n have to be before you couldn’t tell the difference between n and (n-1)?
I define multiplication as fast addition. n x m means ‘add m to itself n times’. An obvious question: Is ‘adding m to itself n times’ the same as ‘adding n to itself m times’?
I am driving down a road and on the right side are 734 piles of bricks and in each pile are 2147 bricks. On the left side of the road are 2147 piles and 734 bricks in each pile. Which side of the road has the most bricks? And why do you believe your answer is correct?
“Well”, you might say, “The number of bricks on the right side is 734•2147 and the number of bricks on the left is 2147•734. Since 734•2147=2147•734, there is the same number of bricks on both sides.”
That last equality isn’t obvious to me. I can’t realistically count the bricks to verify it. I don’t have a have an intuitive sense of 734 piles or 2147 bricks much less of adding 2147 to itself 734 times. Why should I believe your answer?
But I do have an intuitive sense of adding 2 to itself 3 times and adding 3 to itself 2 times. I get the same number of fingers either way I do it. For integers that are small enough for computation on my fingers, I can see that m•n=n•m. Not only that, if I look at it geometrically and compute the area covered by square unit tiles, I can see why it works and see no reason why it wouldn’t work for any integers, no matter how big they were. I believe that multiplication commutes for large numbers because I believe it for small numbers. Thus I believe your answer.

I should explain again that I am after belief. I have gone through proofs that I could see were logically correct but I couldn’t see why they worked so I couldn’t really believe their conclusions.
While pictures are not acceptable in a formal proof they do wonders for belief and understanding. There are no rules in a knife fight.
Principle of Induction (modified): If something is true for a few, consecutive small integers starting at one, then it’s true for all integers.
I seldom use more than the first ten.

At last I come to finding a numerical way to describe parts of things. I have an estate valued at $21 and I want to divide it equally between my three children. I want to divide 21 into three equal parts. I’m going to define 1/n•A to mean breaking A up into n equal parts and 1/n•A stands for the size of any one of them.
I’m going to break my $21 into three equal parts and give each of the three children one of the three equal parts. I will give each child (1/3)•21=7 dollars and end up with three piles of dollars with 7 dollars in each pile. 3•7=21
New problem: I’m going to give a kid $3 to help sweep out my bowling alley. If I have $21, how many kids can I hire? The question here is how many times can I subtract 3 from 21? But this concept already has a name, division, and a symbol 21÷3. I can subtract 3 from 21 seven times and 21÷3=7. Now I have seven piles of dollars with three dollars in each pile. 7•3=21.
New problem: I cut a whole bunch of pizzas into thirds. (I am in the Ideal World. You can’t cut a real world pizza exactly into thirds. Actually, cutting something in half needs an “I cut, you choose”procedure. I passed the pieces out to the multitudes and after everyone had eaten their fill there were 21 pieces left. How many whole pizzas would that make?
This is the problem of adding up 21 pieces of pizza, that is, adding 21 thirds.
I can write this as 21•(1/3)=7.

The last example takes some short cuts but the idea is that there is a separate problem with a separate intuition for each of the three, numerically equal, expressions.

7=21•1/3 = 21÷3=(1/3)•21

They correspond to the word problem types: 7•?=21,?•3=21 and 7•3=?

The rules tell you that (1/3)•21=21÷3=21•(1/3) but the rules don’t help in deciding which form of the rule to use in a particular problem.

I am not necessarily saying that my ideas should be used verbatim if at all, but I don’t see how a person, say, a student, can deal with these kinds of word problems and not at least be aware that there are three kinds of problems. How can a person teach these kinds of word problems and not understand the basic ideas?

I don’t know how much stuff should be put into the introduction of a kid to fractions. I have not had a whole lot of success in teaching fractions to kids. One problem was getting the kid to realize that fractions have a relevance to their lives.

How do you design a course for people who are sure that they will never think of the subject material again? How do you teach ‘the null course’? What about teaching to students who will, indeed, never think about it again?

I think that one of the basic questions is what if anything should be taught. When I think about how I’m going to teach something, I assume at least some minimal interest in learning what I trying to teach. I make this assumption because otherwise anything I might say about fractions has the same teaching value...none. So, if knowledge of fractions is judged to be an important part of knowledge, you have to start somewhere other than fractions.

Any damn fool can teach students with a hunger for fractions. Hungry students force the person in front of the class to at least do no harm.

But back to fractions:

I thought about how to bring intuition to the rules used in computing with fractions and started running into trouble with division by a fraction. Trying to give a useful intuition to (1/2+1/3)÷(1/5+1/7) was beyond me and I haven’t yet come to using large integers in the fractions. And the “simplify the following fractions” problems seemed to have no other way to solve them than mindlessly following rules. The application of the rules is not entirely straight forward and there are often many ways to simplify a fraction. Why is one form simpler than another?

It seemed to me that a ‘rule intuition’ had to be developed. Simplifying fractions is more akin to chess than mathematics. In both endeavors, if you learn a few basic rules you can play the game but it takes practice to gain intuition for the rules and be good at it.

There is a segment of technology that is devoted to making sure that simplifying fractions is an obsolete skill. I think that part of fractions could be done away with, or at least that possibility considered. Decimals are the big rage now, decimals.

It would seem that educators would spend time discussing what to teach and then discuss how to teach it. And there are good cogent reasons to teach fractions...but maybe not for everybody. For my language in Jr. Hi, I chose Latin. And I probably would have chosen fractions as well. But I am not the only person in the world.

My point is that the understanding of fractions is non-trivial and it isn’t clear that it is worth the effort to force it down the student’s throat. I think some more thought could be put into arithmetic courses, in particular what should a course in arithmetic accomplish?

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