The people who learn about FOIL either haven't been doing this multiplication for the last 8 to 10 years, or maybe aren't able to generalize from, say, 23 * 15 to (a + 3)(b + 5).
Yes, it requires much more thought to be able to extend multiplication from the arithmetic of multiplying integers to products of binomials. And the multiplication of binomials is by no means something that I would consider a special case. I would consider multiplying, say, a 7-term polynomial by a 4-term polynomial a special case.Student100 said:Does it require more thought to extend the multiplication in this way, than it does learning an mnemonic built around a very special case?
And so what? Virtually any endeavor, if you want to get good at it, requires a lot of time on the basics. If you want to learn how to play the piano, you have to learn the names of the keys so that you can read music sheets. After that, there's a lot of time spent on practice, building "muscle memory" so that you can play a tune without having to think about each and every note. The same is true for sports of all kinds, with a lot of time spent hitting a ball, or throwing a ball into a basket, or whatever. The more you perform these actions, the more fluid and automatic they become. The same is true for arithmetic and mathematics at a higher level, I believe. If you aren't sure whether it's 6 x 9 = 63 or 54, it becomes much more difficult to do quick and dirty approximations as sanity checks on more difficult problems.
How do you "work out the arithmetic" if you don't have a firm grasp on multiplying two single-digit numbers? Earlier you saidStudent100 said:I don't consider memorizing tables without motivation a viable part of the basics. The basics would be working out the arithmetic , then as you become experienced you'll naturally develop a working memory for these calculations.
First off, 5*6 + 4 is not an equation -- it's an expression, and there's really nothing to solve. If you haven't spent some time memorizing things such as 5 times 6, you'll be stuck right away.solve simple equations like 5*6 + 4 = 34
This is like saying a football team would be more successful just playing football, and not spending valuable practice time doing running drills, blocking practice, running though tires, and all that other boring stuff. I don't think any successful coach would agree with you.Student100 said:And the "and so what" part I guess would be that the class time spent memorizing tables could be better spent working problems or motivating mathematics in general.?