My wife's sister Susan taught fourth grade for a year at a charter school in Fremont. On the one hand: she worked very hard, and spent some of her own salary o

My wife's sister Susan taught fourth grade for a year at a charter school in Fremont. On the one hand: she worked very hard, and spent some of her own salary on teaching materials, as, is evidently standard practice in our Prop. 13-devastated California schools. Also, in fairness, all her past experience had been teaching first grade. On the other hand: she was completely incompetent to teach fourth grade math. The prescribed subject matter was adding and subtracting fractions, and she had to ask me to remind her how to do it. But it quickly became clear that she had no idea how to communicate the knowledge to her students. All she knew how to do was to assign them dozens of homework problems and spend many hours laboriously grading them -- all completely wasted effort because, as I saw, the students were getting many problems wrong, and they weren't improving over time. This went on for weeks. It was worse than wasted effort, because all she could possibly have succeeded in doing was convincing the kids that they hated math and sucked at it. Thus one generation fails the next; and on it goes. I tried to tell her that she needed to assign many fewer problems. I think that if a student can successfully work five of these problems, showing their work, that's enough to demonstrate mastery. And if they get any of them wrong, they don't get it and they need to be shown how to do it, not just mechanically but with visualization aids so they understand the concept, not just the mechanics. The whole thing should take about half an hour, once they understand multiplication and division. Not weeks!! From the article: One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W’s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it. Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “¼,” larger than the “3” in “⅓,” led them astray. Clearly, people making this error don't know what a fraction means at all. I was stunned when I first read this, but given that my sister-in-law's teaching methods are probably not that unusual, maybe I shouldn't have been so surprised.