Leaving aside specific forms mis-teaching, many of which are avoidable, there's the general issue of how much the process of un-learning may be inherent to the nature of math, the mind, and learning. Could one create a 100% honest pre-college curriculum that wouldn't drive 99% of the students crazy with all its caveats and footnotes and the frustrating asides that end with "... but you won't get to learn about that till you're older"? Better, I think, for curriculum designers to frankly acknowledge that oversimplification and unlearning are part of the way human beings operate in every field of endeavor. Think of Tiger Woods, spending many months unlearning his 1990s-style swing so he could painstakingly develop a new, even better swing. Could he have developed that championship swing in his teen years, by thinking things through more clearly? Probably not; it took a Tiger Woods 1.0 to design a re-training regimen that would create a Tiger Woods 2.0. Similarly, when a young mathematician learns the axiomatic approach to the real number system (possibly via several steps, starting from the natural numbers), it's a process of re-construction. A student comes to this process already "knowing" a lot about the real numbers (otherwise the whole enterprise would be unintelligible and unmotivated), but she has to put aside her prior knowledge and intuitions in evaluating a proof for correctness. Unlearning facts and definitions is easy compared to unlearning the whole uncritical approach that dominates pre-college mathematics. On the other hand, my wife says it drove her nuts that her pre-college math teachers "kept changing the rules"; she would probably have appreciated something more like 100% honesty. And I think we can all agree that some of the lies math teachers tell are actually more cognitively indigestible than the truth! Also: math teacher preparation programs should be designed so that teachers who oversimplify are *aware* that they're oversimplifying. Jim Propp On Monday, May 25, 2015, Bill Gosper <billgosper@gmail.com> wrote:
A lot of stuff we're taught is oversimplified. We have to unlearn it and relearn it later. But this is ridiculous. Is there some way we funsters can gang up against this idiocy? --rwg