http://resources.lowellschool.org/blog/learning-math-then-and-now
I keep a strict different in my high school science class between math and arithmetic.
Use an equation to find relationships between quantities, to understand the pattern, it's math. It says the two things are equal.
Use the same equation as a formula to find a numeric answer just by plugging in numbers and cranking out the answer on a calculator, I tell them it's arithmetic. You're assigning a value, what's calculated on the right, to the symbol that's on the left. They're "equal" in a very trivial sense.
They don't get math. Even most of the smarter or harder working ones. They've been so taught to get to the right answer they don't stop to look at what the infinite set of solutions has to look like, what non-solutions you can get out of an equation. Even more mathy kinds of solutions like "what does this equation look like?" they just need to identify the right answer on a multiple choice test, and for that there are graphing calculators.
In other words, after two years of algebra, a year of geometry, and a few months of pre-cal they look at p = mv and don't see a straight line. It's meaningless without numbers. And the idea of F being the slope of p to a p-t graph makes them scratch their heads in bewilderment. Because slope isn't a relationship, it's what you get when you plug specific numbers into a formula.
Level science classes are bad, but it's even worse to think about what it means when you see kids in AP Physics and AP Chemistry doing the same thing for all the same reasons. And the good math teachers wailing and gnashing their teeth because they know they're committing educational malpractice but have administrators looking at the latest "data" from a campus or district common assessment. Even the new form of tracking we have now, with level, pre-AP, and GT classes isn't cut any slack as more and more schools and district, in the search for Big Data, require them all to be in lockstep and teach "the basics."
"Getting to the right answer" is a winning strategy for those who have gaps in background and who are at not just the trailing edge of the achievement curve but the trailing edge of the ability curve. To limit the teaching to just that for all kids, though, to miss the hard basics for easily improved test scores, is criminal, because then the GT kids are punished and we fail the high achievers. When I see a smart, hard-working kid get to college and get ground up because colleges haven't fallen for test-score mania, I think, "We built that." It's a warped system which has as its goal making sure the worst are at least mediocre instead of making sure that the best and brightest thrive. Leave No Child Behind is not the same as Help Each Child Be a Success.
And, sadly, the "accountability" drive for colleges and universities can't help but lead to the same test-score mania. It doesn't matter if you can do research, build a good bridge, or derive a novel synthesis. What matters is that the college has trained you to bubble in the correct answers on a limited, circumscribed body of information that can be objectively evaluated by a computer-based modality.
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