## Thursday, March 24, 2011

### Ours Not to Reason Why

Theirs not to reason why,
Theirs but to do and die:
Into the valley of Death
Rode the six hundred.

(from The Charge of the Light Brigade, by Alfred, Lord Tennyson.)

"Ours not to reason why, just invert and multiply."

(origin unknown; describes the rule for dividing one fraction by another.)

The "just invert and multiply" rhyme is often quoted as a parody of bad traditionalist teaching, where kids just memorized algorithms, without understanding why they work or how to use them appropriately.

But I will confess that I've been meaning to teach "ours not to reason why, just invert and multiply" to the Trailblazers-befuddled 6th grader that I'm tutoring.  At least it would help her remember the rule, and if she sees it enough times, we can approach real understanding.

While it may be true that just memorizing algorithms isn't enough, and that kids should ALSO understand how and why they work, you can't claim victory by just avoiding the standard algorithms.

Here's an interesting non-standard approach that I just learned about (from Those Frustrating Fractions):

To divide fractions, can you divide the numerators and divide the denominators?

$\frac{3}{4} \div \frac{1}{4} = \frac{3 \div 1}{4 \div 4} = \frac{3}{1} = 3\;? \;$

Yes…
…but it works only if you are careful to keep all the numbers in the right order.

1. Interesting -- but it also only works if the answers to the mini-division problems are whole numbers. Try 3/4 divided by 1/5 that way.

2. Right, if you go to the page you'll see she talks about how if you don't get whole numbers you'll have to use the standard procedure.

3. There's too much hang up these days about if you teach students the rules without the reasons (i.e the conceptual mathematical underpinning) the student will somehow be injured for life. Obviously giving a context and an explanation is important. Most math books (traditional ones I mean) do this, some better than others. But none I have seen just supply an algorithm with no explanation whatsoever as detractors of the traditional method seem to mischaracterize things. As far as invert and multiply, even Singapore Math doesn't provide an outright explanation. They do, however, provide the pattern of invert and multiply starting with multiplication of fractions (1/2 x 4 is the same as 4/2, and extending that to other situations (1/4 divided by 2 is 1/4 x 1/2), leading up to whole numbers divided by fractions, and then a leap to fractions divided by fractions. The formal explanation comes in 7th grade after students have mastered enough of algebraic manipulation to be able to understand a symbolic explanation.

See http://www.educationnews.org/commentaries/opinions_on_education/93277.html

4. Barry, I think you're right. I tried teaching my student why "invert and multiply" works by walking her through the proof, and I think it just confused her. Of course, that's partly because she's never seen a proof, or really any kind of mathematical reasoning, before (after six years of Trailblazers!)

Now I'm focusing on just getting her capable with the basic fraction operations. I'll try to get back to the reasons later.

I'm also starting to think that it might be easier for a kid to get fluent with the procedures first, and *then* look at the theory behind them.

5. FedUpMom: Can you be my tutor?:) Actually, my (mathy) husband totally agrees with you that kids can and should learn the theory via standard procedures and algorithms. His point is that these algorithms contain within them the concept, which can become clear with use--and proper explanation at the right time.

6. I'm with teach the standard algorithm (with an explanation) -- and with enough practice, things become clearer. Maybe explanations my math-happy mother gave me during school didn't make sense to me until years later, even though I could do the algorithms well.

For 6th/7th graders, I had them draw out pictures to at least have an idea going into the problem what their answer might look like. That is, for 3/4 divided by 1/4, we'd look at a 3/4 "pie" and measure out how many 1/4 "pie" pieces you would get out of it.

That sort of thinking first at least helped them understand how two fractions, each less than 1, turned into a whole number larger than two! It gave them, if not a conceptual understanding of the invert and multiply algorithm, at least a beginning concept of what they were doing overall.