Changes

"Problems solved only with proof-by-contradiction".  As above, I consider the method of contradiction to be so fundamental that I'd be reluctant ever to say that somethng can be proved '''only''' that way.  Better:  In the introductory explanation of proof by contradiction, using the "always a prime between N and N+1" theorem, or the "infinite number of primes" theorem, as an example to motivate it.

"Problems solved only with proof-by-contradiction".  As above, I consider the method of contradiction to be so fundamental that I'd be reluctant ever to say that somethng can be proved '''only''' that way.  Better:  In the introductory explanation of proof by contradiction, using the "always a prime between N and N+1" theorem, or the "infinite number of primes" theorem, as an example to motivate it.

Definition of the integers.  Do you really want to do a Peano-postulate program here?  As in Landau's "Foundations of Analysis"?  I don't think people in this age group will actually appreciate that the tediousness of this is worth it.  (Of course, '''we''' know that it's worth it, but '''they''' donn't.)  Also, a Dedekind-cut construction of the reals would actually contain material that the students don't already "know", in that it shows, for example, that the square root of 2 really exists.  But of course, skipping over the integers and going straight to the reals is somewhat unesthetic!  The students already "know" the result of the Peano program, they just don't realize that they don't really know it.

Definition of the integers.  Do you really want to do a Peano-postulate program here?  As in Landau's "Foundations of Analysis"?  I don't think people in this age group will actually appreciate that the tediousness of this is worth it.  (Of course, '''we''' know that it's worth it, but '''they''' don't.)  Also, a Dedekind-cut construction of the reals would actually contain material that the students don't already "know", in that it shows, for example, that the square root of 2 really exists.  But of course, skipping over the integers and going straight to the reals is somewhat unesthetic!  The students already "know" the result of the Peano program, they just don't realize that they don't really know it.

"Interesting problems in number theory and Euclidean geometry".  Yes!  There's a lot of really cool, engaging, and surprising material in geometry, above and beyond its use to show the axiomatic method.  Geometrical inversion, for example.

"Interesting problems in number theory and Euclidean geometry".  Yes!  There's a lot of really cool, engaging, and surprising material in geometry, above and beyond its use to show the axiomatic method.  Geometrical inversion, for example.