Prof Klespis remarks that secondary math methods courses have many students who have had calculus but cannot give you a cogent argument or example of why the product of two negative numbers is positive. I am curious about what he would hope his students would say in defense of this; something that would be convincing to the person-in-the-street I hope. I think it seems quite arbitrary to many ordinary people that a product of two negative numbers should be positive -- they don't see the sense in it. Indeed, I have seen volumes written by people we would likely call cranks who argue that we mathematicians got it wrong when we defined multiplication that way, and that it has caused untold trouble.
The usual purely algebraic argument for "neg*neg is pos" depends on assuming that the distributive law holds for ALL triples of numbers. But of course that just begs the question of why we decided to define multiplication for negative numbers so that the distributive law would continue to work (having earlier checked that it works for positive numbers). What would you like your students/teachers to argue about this? Should their argument start with "Yes it is arbitrary, but not unthinking. Here is why the conventional choice is a good one. ..."?
On the other hand, suppose we take a geometric approach to numbers as recently advocated by Hy Bass at the Wash. Conf. sponsored by MSEB and NCTM on the Nature and Role of Algebra in the K-14 Curriculum. We take any Euclidean line, mark point 0 on it, choose a unit length and mark point 1, and then the non-negative "numbers" are associated (identified) with all the points on the ray from 0 through 1in the usual way. The negative numbers are then identified with the remaining points on the opposite ray in the usual way so that we have now a Cartesian coordinate number line L. Now choose a second line M through 0 and mark a 1 on it at the same unit distance as we used on L, and complete this so M is also a Cartesian coordinate line. Now for any number b (point) on L, and any nonzero number c on M, the product point b*c can be found: let J be the line from 1 on L to c on M, and K the line parallel to J through b on L, then b*c is the intersection of K and M. Now just check what this gives for all cases of pos and neg numbers b, c. [Of course if M is parallel to L and their positive directions are the same, and you do the same construction but using 0 on L instead of 1, then the intersection point is b+c.] Do secondary math methods books point these things out and discuss them as alternative ways to think about the questions? Is this any more convincing than an argument in favor of preserving the distributive law?
Ladnor Geissinger Math Prof at UNC Chapel Hill & Math Chair at IAT email: firstname.lastname@example.org or email@example.com phone: 919-405-1925 address: Institute for Academic Technology 2525 Meridian Parkway, Suite 400 Durham NC 27713 USA IAT phone: 919-560-5031 IAT fax: 919-560-5047 IAT web home page: www.iat.unc.edu LEARN NC home page: www.learnnc.org Mathwright Library: ike.engr.washington.edu/mathwright/