OK. So we can't agree about long division. Let's try something simpler: single digit addition. 1. Is there any rational basis (along Lou's line of thought) for requiring memorization? 2. Can anyone PROVE that addition even needs to be taught?
Now take the above and successively substitute single digit subtraction single digit multiplication single digit division mlutiple digit, multiple addend single digit division long division with remainders perfect squares square roots trig functions Bessel functions the Turing busy beaver function, W(n)
Now add on to that the question of when each topic should be introduced and when formal instruction should be finished. It would probably be an enormous waste of human potential if everyone was required to include essays on each of the last three in their portfolios as a condition for graduation of high school. Should there be some evidence of ability to do the first 3? Should proficiency and speed on the first 6 be an ABSOLUTE requirement for ALL students? I don't think so.
My personal spin on long division is that it is no big deal. Introduce it it the second grade, formal instruction is over in the 3rd. If a particular student can't do it by the 9th grade and in the absence of any other knowledge, my recommendation would be to give up. If a large identifiable group of students can't do it, there then arise the questions of why?, what can be done?, and should we? Is it worth the additional expenditure of scare resources? By that point, the arguments are so complex, it seems unlikely to me that even (especially?) a proof could be "clear".
---------- > From: firstname.lastname@example.org snip > > > > iii. Is no improvement necessary, much is ill in public education but > > there is nothing we can do about it or > > There is much we can do about it, but we must be clear about our > purposes, and about our reasoning, and about what we know and what we > don't know. I, for one, will be the first to admit that I don't know > very much. And one of the things I still don't know is this: "What is > the place of long division in what Mark Van Doren once called 'the > natural history of a mind'?" An attempt at a rational, instead of a > rationalized, answer to that question might be a good place to start. > K/M is full of sound bites and fury bites, but they are all also > signify-nothing bites, as Andy Isaacs has amply shown. > > And it might prove, in the end, to be an intractible problem. Or it > might prove to be intractible in a culture that, say, does not perceive > teachers to be working if they are not directly engaged with students or > with students' work. Simply changing this single cultural perception > could have a salutory effect. K--12 teachers work in dreadful isolation > from each other, and one of the things that Ma has identified as an > important part of something successful is the collegiality of the > Chinese system; *that* is something that is not predicated upon a single > personality. How does that fit into your suggestion that we emulate > what has been shown successful? (Hint: "Exactly what has been > 'shown'?" is the first question a rational skeptic must ask.) > > --Lou Talman