I wanted to answer Dave in this thread earlier, because it strikes me that the concerns being posed as important ones. My current remarks are addressed to him, but let's just say that what I'm attempting to address is difficult for me to express.


> > For mental math, it often helps to change the problem to a bunch of easy steps. To do 18x31, which is ugly if not hard, the first step would be 20x31=620, then 2x31=62, then 620-62=600-42=560-2=558. Lots of steps to it, but all easy ones, so it ends up taking very little time.
> >
> None of these are easy steps to me. In fact, the very fact that I have to hold so many *more* numbers in my head before getting to the number I want probably makes this method *more* difficult for me.
>
> I wish I could think of a good example of what I'm really talking about. Basically, when confronted with a problem and a "short-cut" method I think will work but don't really understand *why* it might work, I usually have to convince myself it'll work by working out a much simpler, more basic problem in this method to see if it works for that. Then I usually try it again with another simple problem before I'm convinced it's not just a fluke and will work for all sets of numbers. But the real kicker is the next time I got to use this trick, if it's been sufficiently long since the last time I used it, I have to convince myself all over again that it really works before I use it again.
>

Again it's primarily the question of being a concrete thinker vs. an abstract thinker. As koalamom says, concrete thinkers like you (Dave), koalamom, and me need to visualize THINGS; abstract thinkers like Wes, gabby, and -- I suspect Arthur and Sam -- don't need to do so. But I think there's more to it than just mental imagery involved. Abstract thinkers can "separate" themselves from the object under question, and consider its qualities apart from the association between an object and its known characteristics. But concrete thinkers like to keep things real, solid, and VISCERAL: you like to "keep your eye" on an object and possess it when you're considering its qualities. And therein lies the difficulty.

I suspect this is why the series of steps as outlined by gabby in solving 18x31 (20x31=620, then 2x31=62, then 620-62=600-42=560-2=558) create frustration because the deep effort it takes to image something -- to make it real for you -- also makes it harder to "let go" of the numbers generated in each previous step. Keeping tabs on too many mental objects becomes confusing and the system breaks down for a concrete thinker. Whereas an abstractor has the advantage of not requiring "real" quantities to actively manipulate. An abstract thinker can separate himself from the object, and therefore is comfortable with the notion of approximating a guesstimated answer.

In other words, the key to doing mental math lies not in duplicating the pencil-and-paper method in your head. Let's face it, doing the latter is HARD when you don't have placeholders to help keep track. The rules are different if you want to do mental math effectively. For me to do mental math, I had to convince myself that (1) it's okay to "cheat" by guessing an *approximate* answer, then correct it by adding or substracting a fudge factor to fix it. AND I had to get comfortable with this idea. (2) Instead of manipulating the numbers directly, I "cheat" again by doing Pattern Recognition: if I could reduce a given problem into a simpler subset of steps which were already familiar (like using multiples of 2, 5, 10, and 25), I could work *those* numbers instead, rather than with problem numbers like 37.

People who are really good with pattern recognition can solve math problems rapidly only because existing numbers remind them of other numeric relationships with which they're already familiar. Example: a quantity like 1731 is probably meaningless to you and me, and if you asked me to take the cubed root of 1731 mentally, my first notmal reaction would be to go into a stutter. But to someone who's familiar with playing with numbers and real-world quantities, they might know already that a cubic foot contains 1728 cubic inches; and so the answer to "what is the cube root of 1731?" would be "a few decimal places over 12." Anyway, that's how Feynman did it.


> Also, I seem to have no faith in my understanding of basic arithmetic rules. For instance, several people have said they would accomplish my sample problem, 24x7, by doing the problem (20x7)+(4x7) instead. I wouldn't be able to do this, because I wouldn't *believe* that it would work. To me, those are completely separate problems that coincidentally have the same answer.
>

But Dave, I don't think "faith" or "believing" has anything to do with solving math problems. You don't need "faith" to do math; you need understanding on how to reframe the questions into something more tangible and concrete *for you*. Knowledge comes in the form of tools which help us to understand how we think, and how to deal with and compensate for our inherent limitations.

I mean, for me the suggested solution to 24x7 reframed as (20x7)+(4x7) is just as bad, if not worse than the original math question. The way I look at the answer is: 24x7 is like 25x7=175 minus 7. I visualize 4 quarters to a dollar, and 7 quarters as "two bucks less a quarter." Then I correct for the fudging by subtracting the excess 7 cents.


>
> As an aside, does anyone else visualize numbers the way I do? Anytime I see numbers in my head, unless I'm arranging them to do a problem, I see them on a clock face. So the numbers 1-12 are arranged in order like they are on a clock, and starting with thirteen, they snake up into oblivion in my mind's eye. [...] Then they slope up much more slowly until 35 or 40 or so, then shoot up again. Somewhere around 70 or so they start bending back towards the left, because I get the distinct impression of 90-100 ascending right-to-left instead of left-to-right.
>
> Ok, I guess I'm just crazy. :-(

It's not that crazy to visualize numbers in units of twelve. You *are* correlating the numbers to a real-world model, an analog clock face. The only misfortune here is that this mental model is not particularly useful, so it creates a stumbling block in how you might approach integer numbers. The way I might reframe this mental image is to think of a corkscrew spring that spirals towards me: positive integers spiral clockwise nearest to me, while negative integers recede away in counterclockwise fashion. So if I drew a straight line connecting all the numbers in the "twelve o'clock" position on the corkscrew, then the numbers -24, -12, 0, 12, 24, 36, etc. would be connected together. If I drew a line through the "nine o'clock" position, the integers -3, 9, 21, 33, etc. would be interconnected. And so on and so forth. The only real advantage of this would be in visualizing the 12x table starting from any digit, which might come in useful in playing some African bean-counting game like Mancala, I suppose.

Wolfspirit