Date: Oct 25, 2012 1:44 PM
Author: Paul A. Tanner III
Subject: Easier and quicker methods for algebra - rote tricks? (was Re:<br> Why?)

In reply to my 

"Re: Why?"
http://mathforum.org/kb/message.jspa?messageID=7911987

in which I wrote

"There is a way of doing things with respect to isolating target variables that does not going through all that multiplying and/or dividing both sides of the equation to isolate the target variable. I shared some of these ideas at math-learn almost ten years ago in the original post of a thread - see all my posts in that thread in 2003:

"[math-learn] Algebra as a spatial motion game of logic, like chess or checkers"
http://mathforum.org/kb/message.jspa?messageID=1364908

On Thu, Oct 25, 2012 at 5:02 AM, Robert Hansen <bob@rsccore.com> wrote:
>
> On Oct 24, 2012, at 11:47 AM, Paul Tanner <upprho@gmail.com> wrote:
>
> I suspect that you will show the same type of hostility to such ideas
> as these that make things so much easier and quicker for so many
> people, the same hostility I have received from so many in terms of
> making it easier and quicker to perform various skills.
>
>
> This is because your critics (including myself) don't view math as a
> collection of rote tricks to be taught "easier and quicker". Their view of
> teaching arithmetic involves a true sense of arithmetic being transferred
> from teacher to student. They view "pretending to teach algebra" as
> superficial and damaging to that sense. They know that without the sense,
> the student will be unable to apply any of it.


I don't view math as such. It's telling that you and all else like you who think that those who want to *include* (*not* "replace") easier and quicker ways of doing things view math as being such.

And according to your own statement "without the sense, the student will be unable to apply any of it" you have destroyed all criticism of the methods you criticize. How? Because, since *including* these methods you criticize actually can and in my experience do yield better results on tests, they are able to apply it and therefore are getting the sense.

That is, since to you the result on tests is everything, your position that they are not getting "sense" from *including* these methods is destroyed.

Generally, if *including* (I did *not* say "replace") in the totality of what is taught are methods that actually result in the students getting better measurements on tests - that which you put so much stock in, then what is the problem? Either you have to change your position that these measurements - these tests - are everything or you'll have to stop criticizing what actually works for so many because part of what is taught does not meet your personal standards on some philosophy of education.

But 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 inverse-function 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.

>
> You did mention struggling students and grades. I have tutored students in
> algebra before and had to pretend, but these were not students working on
> some math intensive degree. Algebra was a requirement and the last math
> class they would ever take in their life (maybe with a pretend statistics
> class as well). That does change things. By hook or by crook you try to get
> them through the course, so that they can go on with what they are really
> interested in.
>
> Is this what you are talking about?


No. Because 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[h-i])

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 (h-i) 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 well-known saying is not necessarily meant to cover knowledge that can cause one emotional harm.)