is that the time needed for total fluency and expertise in nearly anything is about 10,000 hours. Whether he's got it nailed or is off by a few percent, I have to agree with the basic premise.Practice works. I know that I never really got the whole addition and subtraction thing down cold until I had to run a school snack bar. I could do it, just not quickly. When you're selling, speed is critical and there was no cash register to add for me. It was all mental and the kids were adding right along with me to be sure. SOHCAHTOA didn't happen until I had brutally memorized the formulas and then did 30, 40, or a couple hundred of them. It wasn't totally clear until after umpteen million R->P and P->R shifts. (Rectangular to polar coordinates, for any who wandered in here unawares)
Every day in my high school classes, I repeat the mantra. "Estimate mentally. You must know where you're going before you can be sure the calculator got you there." Every time there's an extended fraction or calculation on the board, they try to get a better estimate than I. Even questions as simple as 18*7: "10*7 and 8*7 is what?" I mention the distributive property. It's now been 3 months and some are trying to survive without a calculator as the first response.
Practice, Practice.
In addition, we teachers need to find those problems that are more difficult to do with a calculator than without - things like graphs that are nowhere near the "standard window". I might use y + 15 = ( x - 23 )2 or an exponential function that intersects another at (1000000, 21). Or my favorite old-school SAT problem (I remember looking over at another kid - back in the scratch paper days - who was doing a verrrrry long multiplication and wondering what I had missed. Turns out, nothing. He was doing it in the wrong order)
142802/145726 * 291452/71401 = ?
If you make the numbers at least 6 digits, you are guaranteed they will have transcription errors and the calculator will be more difficult than just working it out. It is one of the nice things about the SAT. The problems are solvable without a calculator. Practice, practice, practice.
In Math Tales from the Spring: Calcaholics, Mrs. H. is having these issues right now with her students - too much reliance on the machine instead of thinking and reasoning. I love the "Calcaholics" term for them, by the way. We're all having it and we all have to help the kids overcome it.
Practice, practice.
This issue comes up around here because the elementary school teachers are, by and large, unable to understand math themselves and so they pass on their overwhelming math phobias and calcuphilia to their students. Along with a healthy dose of "It's Drill and Kill not Teaching", they are ceding control of their classrooms to the whims of the students under the guise of "following their interests." What student wants to play with numbers? Much easier to punch buttons and get it over with.
Takes a while to overcome but it's worth it. First semester of college is over and the kids are coming back to visit. "I am sooo glad you kept making us do ..."
Such a nice feeling.
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