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Re: Easier and quicker methods for algebra  rote tricks? (was Re: Why?)
Posted:
Oct 27, 2012 4:38 PM


...
> ...to address this "rote trick" comment: > > They are actually not rote tricks, but are certain > theorems applied as methods, where their applications > as methods simply take note of the visual attributes > of their written form. > > These certain theorems are based on this well known > theorem of groups: For all a,b in group G, the > equation a = xb has a unique solution in x, ab^{1} = > x. > > This implies for all a,b,x the theorem equivalence > > ab^{1} = x if and only if a = xb. > > Written in terms of addition/subtraction and > multiplication/division we have the theorem > equivalences of > > a  b = x if and only if a = x + b > > and > > a/b = x if and only if a = xb. > > Note that in the context of semigroups  where > minimally defined there are not inverse elements  > like the natural numbers these equivalences can be > taken as respectively the inversefunction based > definitions of subtraction and division. > > (Side note: I kept the b on the right side of the > equation on the right side of the expression to cover > those contexts where the commutative property does > not apply. So when written that way, these > equivalences already apply without having to be > modified in more advanced math contexts where the > commutative property does not apply.) > > The "rote tricks" are these theorems applied as > methods, where their applications as methods simply > take note of the visual attributes of their written > form, of pushing the symbol b back and forth either > horizontally with a sign change > (addition/subtraction) or diagonally > (multiplication/division). > > What we teach the students in algebra classes is > actually not these theorems applied as methods, but > their proofs applied as methods, their proofs being > this "doing the same thing to both sides of the > equation" stuff. > > Applying a proof of a theorem as a method many times > means a method that is clunky and takes more time and > effort in comparison to applying the theorem itself > as a method. >
...
> ...again, these methods almost all are what > so many smarter kids already notice on their own and > use as they see fit. "Pushing symbols around > according to certain rules" is how I (in general many > people who "went on to higher math") think when my > eye scans a mathematical expression for the purpose > of rewriting it as an equivalent expression. I (and > many others) most certainly do not think in terms of > clunky methods that take much longer and tax the > working memory more and so on. > > Case in point: One of my favorite evaluations is to > ask a person to solve (you know what I mean  isolate > the variable) for h in > > (ab)/(cd) = (ef)/(gh) > > and > > (ab)/(cd) = (ef)/(g[hi]) > > and get to the point of writing the final expression > as quickly as possible. > > For students who are not particularly good at > algebra, just sit back and watch the disaster unfold > before your very eyes. If they already don't know > quicker and easier ways of doing things, of pushing > symbols around according to certain rules, I quickly > step in and say something like "We can solve for h in > just one written step in the first case and in just > two written steps in the second case." After I see > the jaws drop (for those who do not already know > quicker and easier ways of doing things), I show > them. We then practice for a while choosing any of > the variables at random to solve for using the > quicker and easier way. > > If you don't see it: > > For the first, using the multiplication/division rule > of moving factors diagonally, we do the following: > Simultaneously in the mind's eye move the h from the > lower right position to the upper left position, move > ab from the upper left to the lower right, and move > cd from the lower left to the upper right. (Note that > the first two means that ab and h simply exchange > positions.) This yields > > h/1 = (cdef)/(abg) > > which we write as the final solution > > h = (cdef)/(abg). > > (One of the multiplication/division rules is that if > we take all the variables out of a position, then we > leave behind 1. And included in all this is the fact > that for any expression x we have x/1 = x.) > > For the second, we do the same while understanding > that (hi) is a factor, where we write > > h  i = (cdef)/(abg) > > and then we use the addition/subtraction rule of > moving terms (addends) horizontally with a sign > change to obtain the final written solution of > > h = (cdef)/(abg) + i. > > A good manipulative to show this at least for the > multiplication/division context and practice visually > solving for any randomly chosen of the 8 variables  > especially with a group of people  is to use those > children's blocks with letters of the alphabet > written on them and a device that has the four > positions of upper left, lower left, upper right, and > lower right to put the blocks on. > > So again, I ask, in a different way, "Why should it > be that those who need more help and especially the > most help and who could and even would (since "would" > has already been demonstrated to hold for so many in > my experience) benefit from obtaining certain > information be disallowed from obtaining that > information?" > > To anticipate an answer that would try to justify > disallowing them such information: I utterly reject > the idea that there is such a thing as harmful > knowledge. Only ignorance can be harmful. Knowledge > is power, and ignorance is weakness. (I'm of course > excluding knowledge that can cause one emotional harm >  and this wellknown saying is not necessarily meant > to cover knowledge that can cause one emotional > harm.)
Here are some extra thoughts on making things easier and quicker and getting students to measure better:
I cited in
"Learning math through nonnumeric symbols, even right from the first grade " http://mathforum.org/kb/message.jspa?messageID=6769784
and
"Re: Learning math through nonnumeric symbols, even right from the first grade" http://mathforum.org/kb/message.jspa?messageID=6772078
some research showing that general examples can work better than concrete examples, for not just college students but for 6th graders.
Here is one take on this:
Algebra Without Numbers.
One good exercise in this context for algebra or even prealgebra students would be to turn the complex fraction with numerator and denominator each being a sum/difference of fractions into a single noncomplex fraction, using the relevant formulas for fraction multiplication/division and addition/subtraction, in just two written steps like so  and although it looks bad because of having to type here in ascii text, it looks actually OK using regular handwritten form without most of the parentheses and brackets:
(a/b + c/d)/(e/f + g/h) = [(ad + bc)/(bd)]/[(eh + fg)/(fh)] = [(ad + bc)fh]/[bd( eh + fg)].
One good side aspect of this exercise is that it teaches the necessary skill of equivocating as needed on what the letters of the alphabet represent in formulas. That is, for instance, the first fraction addition/subtraction formula below applies to the dominator of the first expression on the left above even though different letters of the alphabet are used, and the product "ad" in the last expression above on the right is not the same "ad" in the formula as written below for fraction division.
For fraction multiplication and division,
(a/b)(c/d) = (ac)/(bd)
and
(a/b)/(c/d) = (ad)/(bc)
and for fraction addition and subtraction,
a/b + c/d = (ad + bc)/(bd)
and, using m as any common multiple of the denominators like the least common denominator, a formula for adding/subtracting any number fractions
t_1/b_1 + ... + t_n/b_n = [(m/b_1)t_1 + ... + (m/b_n)t_n]/m
such that we simply perform "m over the bottom times the top" n times to quickly and easily directly obtain the numerator of the single fraction to be obtained from the adding/subtracting of any number of fractions.
(I think that the vast majority of people have never been taught the first formula for fraction addition/subtraction  and I know no one has been taught the second one except via my presenting it here at Math Forum unless someone else came up with it somewhere else, in which case I would sure like to know about it. I think that it is insane to be against people ever knowing these formulas, since knowledge of them could prove so useful in both arithmetic and algebra.)
This is about the patterns, the patterns, the patterns, even in arithmetic (see again the title of Keith Devlin's book published in a Scientific American series of science books, "Mathematics: The Science of Patterns"), not just about the numeric in arithmetic.
Wayne Bishop very recently said somewhere that it's been found that fluency on such as turning a complex fraction with numerator and denominator each being a sumdifference of fractions into a single noncomplex fraction is predictive of success in future mathematics based academics. But I addressed this idea a couple of years ago in the original post of this thread:
"Mathematical literacy and the form a/b" http://mathforum.org/kb/message.jspa?messageID=7211837



