>The discussion about what is basic is a very important one for educators. >I found your concern that students know that multiplying always makes >things bigger is more basic than knowing 7 x 8 = 56 curious. What about >fractions? Students are frequently confused when they multiply a whole >number by a fraction or a fraction by a fraction and get a smaller number >than they started with. Always and never are very big words. > >Janet Smith >San Jose, CA
When I reread that message I wrote, I realized it wasn't worded as I wanted, and led people to misunderstand the point I was trying to make.
The point was, teachers seem to be *so* concerned with students' learning such basic facts such as 7 x 8 = 56, that they OVERLOOK the fact that many of their students *think* that multiplication must always make things bigger.
Yes, yes, yes, *I* know that multiplication doesn't necessarily make things larger.
To me, the fact that multiplication can make things larger OR smaller (or not change things at all) is a far more *basic* fact than those "times tables" students are forced to memorize.
What, IMHO, are some of the "basic" facts of multiplication?
Multiplication is commutative. Multiplication is associative. 1 is the multiplicative identity. 0 is the multiplicative "annihilator." Multiplying x by -1 yields -x. Given x > 0: if n > 1, then n * x > x if 0 < n < 1, then n * x < x (an example where multiplication "makes things smaller") Additionally, students should be able to represent multiplication through various models (area, arrays, etc.)
It is my feeling that it is far more important for students to explore and understand these concepts than it is for them to know that 7 * 8 = 56, 6 * 9 = 54, etc. I am not saying it is not worthwhile for them to know their "times tables." I am only saying that I think the above list is far more "basic." Yet, it *seems* (I have no evidence other than my observations) that many teachers are far more concerned that students memorize the "times tables" than they are about students understanding the above concepts.
Norm Krumpe Indiana University of Pennsylvania Indiana University at Bloomington