Simple but serious problems for the edification of non-mathematicians

manpreet Best Answer 3 years ago

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:

• 3×5=5+5+53×5=5+5+5 and 5×3=3+3+3+3+35×3=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