When people graduate with honors from prestigious universities thinking everything in math is already known and the field consists of memorizing algorithms, then the educational system has failed in one of its major endeavors.
If members of the next freshman class will take just one one-semester math course before becoming the aforementioned graduates, here's what I think I might do (and this posting is indeed a question, as you will see). I would not have a fixed syllabus of topics that the course must cover by the end of the semester. I would assign very simple but serious problems that I would not tell the students how to do. A few simple examples:
- $3 \times 5 = 5 + 5 + 5$ and $5 \times 3 = 3 + 3 + 3 + 3 + 3$. Why must this operation thus defined be commutative?
- A water lily has a single leaf floating on the surface of a pond. The leaf doubles in size every day. After 16 days it covers the whole pond. How long will it take two such leaves to cover the whole pond. (Here lots of students say "8 days". I might warn them against that. This is the very hardest problem assigned in an algebra course that I taught, according to most of the students.)
- Here is a square circumscribing a circle. [Illustration here.] Here is how you use this to see that $\pi<4$. [Explanation here.] Now figure out how to prove that $\pi > 3$ by a similarly simple argument.
- Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, ...... Multiples of 18 are 18, 36, 54, 72, 90, ..... The smallest one that they have in common is 36. Multiples of 63 are 63, 126, 189, 252, 315, 378,..... Multiples of 77 are 77, 154, 231, 308, 385,.... Could this sequence go on forever without any number appearing in both lists? (Usual answer: Yes. It will. Because 63 and 77 have nothing in common.) Is it the case that no matter which pair of numbers you start with, eventually some number will appear in both lists?
I said simple but serious, the latter meaning they will actually learn something worth learning about mathematics or about how to think about mathematics. Not all need be as elementary as these. With some of the less elementary problems I might sketch a solution or write out a solution in detail and then ask questions about the solution.
I would not fix in advance the date at which problems were to be turned in, but would set deadlines after discussion reveals that serious difficulties are overcome. I might also do some "teasing" concerning various math topics not covered.
HERE'S THE QUESTION: Which published books of problems can participants in this forum recommend for this purpose? Why those ones?