My further below is not what Haim was talking about - he was dumping on the idea of trying to make things more rigorous for no other reason than to have something to point to while trying to claim that things are more rigorous.
My message further below is about trying to make it so that people have less problems with algebra when they finally get into a formal algebra course by using algebra "power tools" like letters of the alphabet representing numbers to make it more likely to acquire higher levels of arithmetic knowledge and skill.
Take fraction division for instance. Lots of young people don't have the knowledge and skill that we would want them to have on this topic, since either they don't memorize verbal instructions like "invert and multiply" or "multiply by the reciprocal of the denominator fraction" or whatever, or of they do memorize them, they don't do so knowing or understand what they mean well enough to apply them without getting things all tangled up somehow.
I'm willing to bet that if they successfully memorize the formula I wrote below, by using such memorization aids as recognizing that the two outer numbers go on the top and the two inner numbers go below, they would be more successful with fraction arithmetic involving complex fractions.
Same with fraction addition, especially with using that more simple formula that does not involve the least common multiple of the denominators.
And all the talk about proofs was simply that maybe these young people might be smarter than you think they are and of you gave them the chance, they might surprise in terms of what they can handle.
If you don't think so, then consider that East Asian curricula seems to be actually a lot more meaty in terms of proof-based theory than the typical US textbooks show, and I mean in the lower grades. Take that Liping Ma book, and read what the Chinese teachers say they do in terms of explaining and justifying things to their students. It''s clearly more theoretically meaty than what you see from the US teachers when they explain or justify things, even if the theory is very basic.
On Tue, Oct 23, 2012 at 12:53 PM, Robert Hansen <firstname.lastname@example.org> wrote: > Did you send this post to Haim before he replied to my post? Or did Haim just guess that you would post this before you posted this? > > Bob Hansen > > On Oct 23, 2012, at 12:39 PM, "Paul A. Tanner III" <email@example.com> wrote: > >>> Why is the attached page in a 4th grade math text? >>> >>> Teaching algebra in 4th grade IS NOT the path to >>> algebra. >>> >>> Teaching arithmetic in 4th grade IS the path to >>> algebra. >>> >>> Bob Hansen >> >> They are not mutually exclusive. >> >> I would think that the quicker we can get young people comfortable with the idea of using letters of the alphabet to represent numbers, the better. >> >> There are all kinds of reasons for this. >> >> One is that it makes it easier to memorize algorithms - like the algorithms of arithmetic, especially the ones dealing with fractions, when they can be put into the form of a compact formula using letters of the alphabet representing numbers, in comparison to their being *only* in the form of a long and verbose set of instructions. Note that my use of "only" means that I am not advocating eliminating the traditional use of verbose instructions.) >> >> For the arithmetic of fractions, we already know about fraction multiplication, division, addition, and subtraction being put into the form of single-equation formulas, after finally getting into a formal algebra course in late middle school or early high school, but why is waiting so long good? I would think that more young people would show more arithmetic skill on tests - especially with respect to fractions - if they sooner rather than later had the "power tools" of abstract representation (of using letters of the alphabet to represent numbers. >> >> (These formulas for fractions would of course be something like the following where here I have to use more parentheses because of writing in ascii text - they are of course easier on the eyes and so probably much easier to memorize in one's mind's eye using the standard written form: >> >> Fraction multiplication and division: >> (a/b)(c/d) = (ac)/(bd) >> (a/b)/(c/d) = (ad)/(bc) >> >> Fraction addition/subtraction >> a/b +- c/d = (ad +- bc)/(bd) >> >> Fraction addition/subtraction using the least common multiple of the denominators, with m being this least common multiple (or more generally, it can be any common multiple of the denominators including bd): >> >> a/b +- c/d = [(m/b)a +- (m/d)c]/m >> >> Here is my most recent post on this last formula: >> >> "Re: To meet a challenge" >> http://mathforum.org/kb/message.jspa?messageID=7901059 >> >> And not only that, it gives young people the tools that they need to understand simple proofs sooner, and I would think that the quicker we can get young people to the point of being able to grasp even just the simplest proofs, the better. >> >> If you wish, I can provide a proof of each of these above even without using an inverse element, each as so-called inline proof which is a sequence of expressions connected by equality such that the first and last expression is the equality to be proved (I've already done so for each formula above in the past here at Math Forum), which means that these formulas hold true even in the natural numbers and integers when x/y means "x divided by y" and when "x divided by y" is defined as a natural number or integer. >