University of Massachusetts, Amherst                                                                                                                                                                                     Fall 2018

## MATH 491: Putnam competition preparation seminar

(schedule number ...)
• ##### Some ideas we have covered A. Sets and numbers (This translation is the basis of combinatorics.). Standard operations on numbers come from operations on sets: zero $\emp$, sum, product, exponentiation of numbers (measures the size of sets of functions), binomial coefficients measure operation n choose k'', factorials come from permutations. B. Power series: the coefficients in Taylor formula. .

• HOW TO LEARN abstract MATHEMATICS.
The following is what I see as the {\em basic} approach towards learning mathematics at the conceptual level. The procedure is
• (0) You start by hearing (or reading) of a new idea, new procedure, new trick.
• (1) To make sense of it you check what it means in sufficiently many examples. You discuss it with teachers and friends.
• (2) After you see enough examples you get to the point where you think that you more or less get it. Now you attempt the last (and critical) step:
• (3) Re-tell this idea or procedure, theorem, proof or whatever it is, to yourself in YOUR OWN words.
• More on step (3).
• Trying to memorize someone else's formulation, is a beginning but it is far from what you really need.
• You should get to the stage where you can tell it as a story, as if you are teaching someone else.
• When you can do this, and your story makes sense to you, you are done. You own it now.
• However, if at some point you find a piece that does not make sense, then you have to return to one of the earlier steps (1--3) above. Repeat this process as many times as necessary.