<div dir="ltr">On Thu, Jul 17, 2008 at 7:16 PM, Albert Cahalan <<a href="mailto:acahalan@gmail.com">acahalan@gmail.com</a>> wrote: <div class="gmail_quote"><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"> <div class="Ih2E3d">Alan Kay writes: > Similarly, from the POV of a former guitar teacher and player, > "playing guitar" has a threshold that excludes "air guitar", > "Guitar Hero", certain kinds of noise, and "not enough fluency > to make music yet". We can use "air guitar" as a metaphor for > (a) taking such a small subset of an activity that only some > form and essentially no important content is being done, and > (b) for using form over content to fool ourselves that we are > "players" and "part of the club". </div>That works. I also like the term "math appreciation". Like a music appreciation course, it doesn't get you to be competent. <div class="Ih2E3d"> > Two of Seymour Papert's most important insights about above > threshold math-with-computers for children were to (a) find and > use the real mathematical thinking that children could do at each > stage of development, and (b) to both pick from the large body of > existing mathematics and to invent new mathematics that embody the > most "powerful ideas" that humans have come up with. One of many > such examples is how to use the children's ability to add (and to > think additively) and to physically move their bodies to make for > them a powerful and valid version of Gauss' Differential Geometry > which covers some of the most important parts of vector calculus > in a way easily learnable by children. </div>I do believe that many children can learn vector calculus, and that this might have some value. However... When you put the cart before the horse, trying to skip all of the arithmetic and such, you're teaching math appreciation. It's air math. An actual bake-off has been conducted. It was the largest educational study ever done, covering 79000 children in 180 communities. I'm sure you've heard of it: Project Follow Through. In that study, Direct Instruction (sage on the stage) trounced all other programs in multiple ways. More here: <a href="http://www.jefflindsay.com/EducData.shtml" target="_blank">http://www.jefflindsay.com/EducData.shtml</a> <a href="http://www.illinoisloop.org/oswegomath.html" target="_blank">http://www.illinoisloop.org/oswegomath.html</a> <a href="http://www.heartland.org/Article.cfm?artId=19790" target="_blank">http://www.heartland.org/Article.cfm?artId=19790</a> That last article has a lovely graph of the results. One may question the ethics of such an experiment, since it does in fact involve experimenting on children in a life-effecting way. I see no alternative though, so I'm glad it was done. Now it is time to accept the results, eat some humble pie as required, and teach children math. It is unethical to endlessly repeat the experiment, hoping that you will somehow get results that support a well-loved hypothesis. Vector calculus is a fine subject for children. Set high expectations for daily progress, eliminate distractions, and soon enough most kids will reach vector calculus. (for real, not vector calculus appreciation) High expectations go something like this: multi-digit add/subtract with traditional procedure: 1st year multiplication table memorized: 1st year long division: 2nd year 2-step word problems: 2nd year 4 basic operations on fractions: 3rd year 4-step word problems: 4th year order of operations: 4th year algebra (with proofs): 5th year geometry (with proofs) and trig: 6th year regular calculus: 7th year vector calculus: 8th year It's doable, but you won't get there if you waste time or if you use educational methods that are proven to be horrible. In case anybody wants to look at current curriculum that work: The two best math programs, unfortunately subject to copyright, are Saxon Math and Singapore Math. Saxon Math is better for the slower students, particularly if students are missing school or transferring in from places that use a different math program. Singapore Math is better for the faster students. Both are available for purchase.</blockquote></div> hi albert, thanks for raising an issue sharply that does need to be discussed misunderstandings and misrepresentations aside, I'd raise this point in response - your assumption is that a large scale study is more important than the individual research and findings of one person I don't see why this assumption should necessarily be true - ie. historically it has been shown many times that lone individuals or small groups have turned out to be correct and the predominant or mainstream way of doing things has eventually been displaced - that is the nature of scientific revolutions it could be that whole systems have been built and maintained for generations on principles of direct instruction - that various challenges to this have arisen and been trialled, some good, some  not so good - but throughout this process the predominant form of teaching has remained direct instruction it seems to me that in a system that has evolved in that way, that due to forces of inertia and group think mainstream studies would tend to show that mainstream ways of doing things are the "best way" Piaget did many studies and wrote many books and papers based on the study of 3 children - that does not in itself make him wrong. He might be wrong but I can see many advantages of doing in depth studies based on a small group. I would like to discuss this issue more, just raising it here in simple form --> minority views are not wrong because they are minority views Of course your challenge still applies as a practical issue for those who want to go beyond direct instruction at least in some respects cheers, - Bill </div>