Peter Gray writes about a study that elementary school children do better at arithmetic if you teach them less arithmetic. This is a bit of a mischaracterization, since he describes a school where they simply delayed teaching arithmetic until the sixth grade. The study was this: there was an experimental group of students who receive no arithmetic training until the sixth grade, and a control group that had learned arithmetic in the usual sequence.
The results: A pre-test was given at the beginning of the sixth grade. The control group did better on your standard arithmetic problems (of course), but the experimental group actually did better on common-sense story problems, measurement, and number sense. At the end of sixth grade (after the experimental group had been taught arithmetic), the experimental group did at least as well as the control group on everything.
It appears that children may not be developmentally ready for the formalism of high school arithmetic until later (like sixth grade). When they are ready, they can very efficiently learn arithmetic.
This is one reason why I think that we are doing a disservice by pushing algebra and geometry down to middle school, and calculus down to high school. I am generally a fan of slowing down, rather than accelerating, the curriculum.
The study also reminded me of Kamii’s work. She also had a control group of students who were taught in the usual way, and an experimental group that were not taught standard arithmetic. This time, I believe they were first-graders. The experimental group of first-graders spent the time they would normally spend on “arithmetic” by playing carefully-chosen games. These games would require them to roll two dice and add the results, or add the face-values of two cards in a game of “Double War.”
The results: once again, the experimental group did significantly better than the control except for on the most algorithmic of problems (maybe 2-3 problems out of 30).