My theory on this is quite a bit different. The adoption of these advancements in notation was driven by understanding, not the other way around. In other words, these changes in notation and theory were not adopted for pedogagical purposes, but simply for the fact that they made better sense to the people who already understood arithmetic. Music has went through the same magnitude of advancements in both notation and theory but certainly not for the purpose to make music easier to understand. The new notation and theory just made better sense to the people that already understood. It seems more natural, but again, only to people that already understand. Not to those that don't. I am saying that the natural evolution of notation and theory, in any domain, is simply not a matter of pedagogy.
On Jun 25, 2012, at 4:53 PM, Haim <firstname.lastname@example.org> wrote:
> Robert Hansen Posted: Jun 25, 2012 1:41 PM > >> Yes, but Jonathan's version this time is different. >> Whereas before he was looking for tricks (like some that >> announce a "better" multiplication algorithm, or >> a "better" postulate for area) he is going right to the >> quick. How do you make arithmetic simpler? Or how about, >> how do you make music simpler? How can you make anything >> that is understood by scores of people, simpler, so that >> it can be understood by the scores that don't? I don't >> think you can. I think the question itself is a >> contradiction. > > Bob, > > I am not opposed to the idea, in principle. We have historical examples of just this sort of thing happening. > > The obvious example, appropriate to this forum, is the supplanting of the Roman numerals with the Indian numerals. This was a serious simplification and a great achievement. > > Another great example is the supplanting, in most places, of ideogrammatic writing (like Egyptian hieroglyphs) with alphabetical writing. Although the Chinese have produced a simplified system in the 20th century, theirs is still ideogrammatic and remains a difficult writing system. The Koreans jettisoned Chinese ideograms in favor of their own alphabet. The Japanese did something like that. So, simplifications have happened, to the great benefit of a great many people. > > The flip side is simplification to a fault. I.e., how far can you simplify an idea before it is no longer the idea? Or, in that great aphorism sometimes attributed to Einstein: You should make things as simple as possible, but no simpler. In other words, some things really are complicated, and there is nothing you can do about it except roll up your sleeves, spit on your hands, and get to work. > > So, the question we are being asked to consider is: Where, in the grand scheme of things, are the Arithmetic Simplifiers? Are they going to inaugurate a New World Order of Mathematics (if you really can simplify arithmetic, who knows where the ramifications will end?), or will they take their place with the angle tri-sectors and the perpetual motion machine engineers? > > I am not holding my breath. > > The only thing I would say to you, Bob, is that a guy like you may underestimate the deep hunger for a Great Simplification of arithmetic. Because you "get it" (and you probably got it very early), you think arithmetic already is simple. You think it is so simple that we teach it to small children. But you are mainly wrong. Arithmetic (most especially the arithmetic of fractions) is something we mainly fail to teach to children. (Their own teachers do not really understand it.) > > For the vast majority of people on the planet, the arithmetic of fractions, ratios, and percents is an upper bound on their comfort level with mathematics. They do not really understand it in a clear and compelling way, and they certainly do not like it. I have consulted with lawyers who are not comfortable calculating a percent, for goodness sake! Ask the average man on the street what they think a mathematician does, and you will have people tell you that mathematicians spend their days adding, subtracting, multiplying, and dividing really, really big numbers. > > The only reason I spoke against the Arithmetic Simplifiers, the last time, is because I am gun-shy of Grand Educational Ideas. In the U.S., too many times has someone produced some cockamamie idea (Fuzzy Math, anyone?) and taken it into full scale production (i.e., implemented it in the schools, full bore) contrary to all evidence against it. So long as the Arithmetic Simplifiers play their games on their own dime and on their own time, I have nothing against them. Just keep this stuff out of the schools until it is well and truly proven. (I am not holding by breath.) > > Haim > Shovel ready? What shovel ready?