Let me save you a lot of investigative work and typing, Robert:
here's a problem: you tell me what "kind" it is:
There are 877 coins (all of the same denomination, say, US quarters) lying face up ("heads") flat on a table and some other coins of the same denomination lying face down ("tails") also flat on the table, no coins overlapping, not in any particular arrangement.
Your job is to arrange them into exactly two piles so that the number of heads in each pile is the same.
You have to do this blindfolded.
Is this a "Mensa" problem? A puzzle? Or what?
While you're thinking:
Do you understand that the people with whom I'm most generally in agreement understand that teaching involves a knowledgeable teacher leading the class? That no one suggests that the teacher sits around twiddling her thumbs while the kids discover the Pythagorean Theorem or prove that the primes are infinite or anything of the sort? That what the Kaplans actually do, what Deborah Ball does, what Maggie Lampert does, what many gifted, thoughtful progressive teachers do is difficult and requires not only mathematical knowledge, about which you claim to know, but a very different kind of knowledge, generally called 'pedagogical content knowledge' about which I have to say you appear to know nothing at all?
If you did, of course, you'd have a hard time claiming that what the Kaplans are doing represents "traditional" mathematics instruction. But then again, as GS Chandy recently suggested, nearly everything in their teaching fails to live down (or up) to your principals. No tests, not grading, no homework. Heavens! And no reference anywhere on their site or in their writing or work to standardized test scores. Somebody call a Math Cop!
Quoting Robert Hansen <email@example.com>:
> Another thing to discuss is the types of problems used in these > circles. I have noticed that some choose problems that while > interesting and deep are also well represented in mathematics that > is within reach of the class and lead to the teacher led format > (actually a lecture in many cases). These seem to represent the > problems chosen by the Kaplans (and RSM). And some choose the more > "puzzle like" questions which are easy to state but are more easily > explained (solved) with intuition than "math". They actually often > have a mathematical basis but the foundation required to even tackle > it in that manner would be far beyond the class's ability. Mensa > problems are often like this. > > The advantage of the first type is that they are both interesting > and lead to rigor and formal concepts. The second category are > enjoyable but do not lead very far. I'll post examples later to more > clearly show what I mean. > > >
- -- ************************** Michael Paul Goldenberg 6655 Jackson Rd Lot #136 Ann Arbor, MI 48103 734 644-0975 (c) 734 786-8425 (h) firstname.lastname@example.org rationalmathed.blogspot.com It was when I found out I could make mistakes that I knew I was on to something. - Ornette Coleman