Confluence of reading the comments on Sarah Hoyt’s latest blog post, and a couple of observations.
Kids who do math in their heads hate showing their work. And especially since the “gifted” math students can do all the basic arithmetic in their heads, they get to about halfway through algebra and start hitting a brick wall, because they have no idea how they solve math problems and no way of formally applying the mental procedures to more complex equations. Oh, some of them will reason out the answer – in about twice as long as it takes a “less gifted” student to come up with the correct answer through using the formal process and a pencil and paper.
There was a meme about the Common Core math that was floating around on Facebook for a while, and the explanation of it (see link) actually makes sense. Except that the way people do math in their heads is not congruent to the way people communicate through the language of math on paper.
Math is a language. Sure, if you asked me “What’s 82 times 4?” and I didn’t have a calculator or pen and paper in hand, I’d do it mentally like this: “8 times 4 is 32, so 80 times 4 is 320, and 2 times 4 is 8, so the answer is 328.” It’s a neat parlor trick (and my Dad used to do that sort of thing all the time when I was a kid and I found it totally aggravating) but it also requires you to have developed a lot of “working memory” for holding numbers in your head and remembering that the important one is 320 while you’re doing 2×4 and adding the result. I’ll often teach my tutoring students to do this if they’re already quite numerically literate and depending a bit too much on their calculators for simple arithmetic. But if I were going to show you “the math” for doing it, I would write it down and do it the “formal proof” way.
Because math is a language that has certain ways of expressing ideas. Which is exactly what I tell my students about a lot of the formulas they use – they’re just abbreviations for certain ideas. And it just helps you do your math homework/tests faster if you have the basic ones memorized… as long as you’ve internalized what they mean so that it’s not just alphanumeric soup to you. (Of course, there’s a certain amount of “fake it til you make it” that works out okay, as long as you can identify which problem type calls for which formula and where all the numbers go.)
Sure, you can do 30 minus 12 by asking yourself “How far is the distance between 30 and 12 on the number line?” – in which case, the correct answer is “eight to twenty, ten to thirty, therefore 18.” And yes, I think this ought to be taught… along with the idea that there is a “formal proof” method of expressing the operation mathematically. Because not every child is going to be able to do all that in their heads, but asking them to write what amounts to an essay on basic arithmetic is hardly any better than making them use a “traditional” method of showing their work. Unless your goal is to ensure that no parent will be able to help their child with their homework so that the kids with invested parents won’t fare any better than the kids with neglectful parents. (But they’ll be equal!)
And yes, I do believe there are teachers dumb enough to require their students to “show their work” using a method that’s meant to be entirely mental or they’ll downgrade the students. (Probably mostly boys, and the exact same ones who already think showing their work is bogus.) Especially since, as a mental technique, there are multiple ways to approach the “leapfrogging” from one number to the next. The point of having a commonly accepted way of expressing mathematical operations on paper is so that people don’t have to spend a lot of mental energy figuring out exactly what’s going on in between each step of the math problem! Especially when they have sixty papers to grade by tomorrow.
I’m sure if you threw out all the grammar and spelling rules and told kids to just make it up as they go along based on what English sounds like, there would be a lot of people screaming bloody murder, too! Thing is, nobody takes the jokes about dropping “ph” for “f” – and so on – seriously. Although I do have to admit, I have heard that there are some benighted souls who think that having their grammar corrected is racism. Hopefully that is an urban legend. But unfortunately I’m sure there are some radical Marxists who would be perfectly willing to promulgate such a belief…