The First Rule of Program Optimization: Don't do it. The Second Rule of Program Optimization (for experts only!): Don't do it yet - Michael A. Jackson
A lot of things taught in math classes are not basic concepts. They are shortcuts - tricks to make problem solving easier. Cross canceling is really just fraction simplification done before the multiplication. Cross multiplication (for which cross canceling is often confused) is simply the combining of two algebraic steps to solve a very particular sort of equation. The rule of "move it to the other side of the equation, and switch the sign" is result of applying the same operation to both sides of the equation. "Just add the opposite" as a replacement for subtraction is similarly the result of an equivalence relation. "Multiply by the exponent, and then reduce it by one" is a common rule for differentiation that completely misses the essence of what differentiation is.
The list could go on. Mr. AB complains of having to teach the ten constituent steps of adding mixed numbers. And he's right, of course. Adding mixed numbers should be three steps: (1) convert to improper fractions (2) add (3) convert back to mixed numbers.
Sure, each of those steps has its own set of steps. And those steps have theirs. (For some of my kids, adding 8+3 has 3 steps.)
Still, if the process for adding fractions has been established, providing a different process for mixed numbers does nothing to reinforce the concept, and only provides distraction.
The quote above refers to writing software. The implication is this: first you should focus on making sure it works. Once it works, if and only if it is too slow, you look for the one place that can make it faster, and fix that. You never introduce speedups unless they are absolutely necessary because, while they make things run faster, they usually also make it more difficult to understand what's going on. If you have to go in and add something else, that optimization is the most likely place for something to break.
Our kids give up because they see math as a bunch of arbitrary, unconnected rules that they have to memorize. This is exacerbated by the fact that more than half of those rules are shortcuts and mnemonics that don't address the basic knowledge.
We need to get rid of those.
We need to teach the basic concepts, and build on those basic concepts to establish new concepts. Yes, some of that is tedious, and yes, those shortcuts can make life easier. But let those shortcuts come on their own. Those shortcuts are for the experts, and they'll come when everything else is well settled. Once the basics are ingrained enough that they are automatic, the kids will recognize the patterns on their own. Not only will they understand the basics, they'll understand why the shortcuts work, and have a knowledge base that can be built upon, and have learned the process of discovering that knowledge for themselves.
Can you add to the list of math stuff we'd be better off not teaching?
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{ 5 } Comments
Couldn't agree more. I hate cross-cancelling and cross-multiplication. They're the source of so. much. error. I don't much like FOIL either, though I will go along with it if that's what the students suggest. Then there's "A negative and a negative gives a positive", which is in a league of its own in terms of generating nonsense.
Bravo! Well said! Cross multiplying is my all-time least favorite, but I also dislike teaching them to add & subtract exponents before they see the pattern because then when we raise a power to a power, they can never tell when to multiply and when to add. The whole problem comes from being taught the shortcut before they understand what's really happening. Another common mistake is trying to find common denominators when multiplying & dividing fractions. If they really understood the purpose of a common denominator, they wouldn't make that mistake.
And the shortcut that they all mess up at some point -- to change a percent to a decimal, move the decimal point two places to the left (or was it to the right. . . . .?).
Another Amen. (Especially with mentioning the confusion between cross-canceling and cross-multiplication. My students don't know what cross-multiplication is, but keep wanting to use it.)
I'll second H and "two negatives makes a positive."
I'm also not a fan of "Please excuse my dear Aunt Sally" for order of operations (even though it's what I personally used) . Nor "invert and multiply" for dividing fractions.
I figure you'll appreciate mnemonic-abbreviations-only-as-necessary. A lot.
On second thought, I'd hate to convert to improper fractions before adding 5 1/6 and 2 1/3.
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