It has been so long since you referred to my stance on coin and digit problems that I thought you had forgotten me. I am deligted that you still remember what I said.
I do wish, though, that when you extrapolate from my statements, you carefully indicate that it is your extrapoation or your conclusion based on your reading of what I said. In your latest restatement you said:
>Remember how Prof. Jacobs argued against coin and digit problems: >"This has nothing to do with solving a problem. This is understanding >numeration. Why make-up a situation that is never faced in life to >demonstrate to students that mathematics is useful. If solving problems >is a reason to know mathematics and you present me with problems in which >1) you know the answer if you know the data or 2) relate to mathematics >as an abstract science, and neither of these concern me, then why would >I want to learn mathematics."
That's fine up to here. You then go on to say: > >Pay attention that Prof. Jacobs is against relating to >mathematics as an abstract science! This does not concerns her!
This is not what I said; it is not what I believe. I do not have the same view of mathematics that most k-12 students have.
I do want students to relate to mathematics as an abstract system. I want them to appreciate the fact that when it is impossible to do an operation on two number, for there is no number in a set that represents the answer or (forgive me Andre for this example) there is no number that describes the solution to a real world problem, like owing money or how three people can share two cookies), we simply invent new numbers. And, when we combine the new numbers with the old numbers (until we get to the complex numbers), we do not lose one single property that held for the original set and gain some new ones in the superset. I also want them to realize that these structures in mathematics do not apply just to numbers. Looking at what happens in geometry, we find that we get a commutative group with rotations (turns) but when we include reflections (flips) we lose commutativity.
I want students to explain why the divisisbilty rules that they intuitvely figure out for 2, 5 and 10 work. Then I want them to examine the way we write numbers and how our numeration/place value system works so that they can figure out what other divisibilty rules can be developed. I am not interested in the ones for 7 and 11, but the one for 12 is powerful. This requires them to look at mathematics as an abstract sytem.
Andre, I am an algebraist and have done some work in number theory. Once I get started on these topics I could go on forever. I do recognize that those in applied mathematics or analysis see this stuff as mind games and of little real value. Similarly, many students do not find the above intrinsically interesting. When I am able to relate these excursions to something that interests them or might make doing mathematics easier, I stand a better chance of keeping them engaged in looking at mathematics as an abstract system. No Andre, coin and digit problems just do not do it for me. And, you can quote me on that.
======================= CEEMaST ================== Judith E. Jacobs, Director Center for Education and Equity in Mathematics, Science, and Technology California State Polytechnic University, Pomona 3801 West Temple Ave. Pomona, CA 91768 phone: 909-869-3473 fax: 909-869-4616 email: firstname.lastname@example.org