A teacher educator friend of mine put it this way:
As an example, 7 x 0 = 0, and 7 x 1 = 7 ... that is, in the first case, the product is equal to the smaller of the two factors. In the second, the product is equal to the larger of the two. Therefore, if we multiply 7 x any proper fraction F, since 0 < F < 1, we should not be surprised that the product P would fall between 0 and 7!
Investigations by students along these lines allows students to recognize multiplication by fractions as part of the general concept of multiplication, not as special cases of it.
On Thu, 29 Jun 1995, Janet V Smith wrote:
> 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 > >