I didn't really "get" algebra until I had taken geometry for some reason. I remember I always had a very difficult time discerning between things that followed from mathematical law and things that were rather arbitrary, or done by convention. Plus, there's a very big difference between understanding how to solve a given problem and understanding the algebraic "big picture." My grades really suffered in 1st-year algebra because I was always thinking in a stepwise fashion---step 1, step 2, step 3--and didn't grasp the concept of an equation as a tool that you can do almost anything to as long as it balances.
In the case of your wife, I think I would try to emphasize the distinction between what's arbitrary and what's not, and I would try to focus on what it means for an equation to balance. E.g.: Saying, "Let N=.454545_" is arbitrary. It lets me create an equation. Once I have an equation I can manipulate it in almost any way I want, as long as both sides *balance.* Balancing both sides just means that whatever I do to one side I have to do to the other. If I multiply both sides by 100, then I get "100N=45.4545_." Note that just because I'm required to multiply both sides by 100 doesn't mean I'm obliged to numerically calculate both sides. For an analogy, imagine an old fashioned scale. If I put a cup of water on one side, I have to put an identical cup of water on the other side (or something that weighs the same) or it won't balance. Now, if I add a cup of milk to one side, I have to add an identical cup of milk to the other, right? So far I've done what's *necessary* to make both sides balance. But I also have some freedom to do certain things arbitrarily. For instance, I can mix some milk with some water on one side of the scale but not the other. The overall weight remains the same, so the scale balances, but I've also manipulated one side in a way that I haven't manipulated the other. That manipulation affects the appearance, but not the overall weight, so it's OK. Back to numbers. If I write "N=.4545_" and multiply both sides by 100, then I can re-write the equation like this: "100 x N = 100 x .4545_" and we know that both sides are still equal. At this point I've done what's mathematically necessary, but now I have some freedom to do what I think is more useful. It's useful to go ahead an multiply 100 x .4545_ and write out 45.4545_. However, since my *goal* is to have an equation in the form "N=a/b" such that a/b equals .4545_, it's of no use to me to actually calculate 100 times the value of N, because then I lose my variable. (Aside: it was also important for me to learn not just what a variable is, but why one can arbitrarily say that N=whatever and why one can define a goal in advance, that N be rendered in a new symbolic form.) In other words, at the end of this process I'm going to want to have N by itself on one side of the equation again, and I can't get there if I actually multiply 100 by the value of N at this time. But I'm not required to *calculate* 100 x N so long as I remember to keep the 100 there to balance out the equation. The numerical calculations here only change the forms of each side of the equation and not the values. As long as the values are equal, the forms can be changed in a large number of ways--the trick is to make sure that whatever symbolic form I use keeps the same value once I calculate it using mathematical rules. So, in this case, on the left side of the equation I'm keeping the water and milk seperate, while on the right side of the equation I've allowed the water and milk to be mixed. Since I have the same amounts of water and milk on both sides, though, everything's OK because the equation or scale still balances. And so on and so on...pick your own analogies. The big trick, I think, it teaching someone to distinguish between what one *must* do and what one *can* do, and why. I was always tripped up because whenever the teacher manipulated an equation arbitrarily I went looking for a mathematical rule to explain *why* she did it--but there was no rule, just the preexisting knowledge that manipulating the form in such-and-such a way would help us get to the ultimate goal. Then, when I would try to do the problems on my own, I would be lost because I would look for something to compel me to take the next step, when in fact I needed to look at the overall goal and infer that performing a given operation would help me recast the equation in a way that would make it possible to move forward. Marvin Long Austin, Texas
