Commentary: Reasoning in Number Theory (Plus Definitions)
Number
theory is a wonderful vehicle for helping teach students about counterexamples,
converses, and other elements of mathematical logic. I have students look for patterns in the
multiplication table and then make conjectures.
Last year, I compiled student conjectures, but
the year before, I rewrote them in mathematically correct language, grouped
them, and added some of my own. Both
approaches have merit. I used them both
in Math 130 last year the class was for prospective preK-6
teachers, the year before it was for students who did poorly on our then
placement test. In all cases, the
logic/reasoning is difficult for students, but I think one of the most critical
things for them to struggle with in a math course
especially for the future teachers. Reform curricula give students open-ended
problems, and teachers need to be able to evaluate student responses. Of course, in this world where we are all
lied to so often, reasoning is important for everyone, not just future
teachers.
The sheet includes places for students to write definitions at the beginning. It can be confusing to sort out the types of numbers etc. 8/05