Time spent proctoring exam are the best time for brewing mathematical ideas.  In a teaching school like penn State Behrend, I spend more time teaching than doing research.  And if you ask any teacher what the worst part of the job is:  they will invariably say "grading".  Even my brother-in-law who was once a "dataBase Management system" teacher hated the part of grading.

For that reason, I always spend a lot of time making the exam "easier to grade", but not necessarily easier.  A small "mistake" and your exam might demand another 2 to 3 hours grading.  And time spent grading is unlike time spent waiting for the bus or time spend cooking.  A minute grading is ten or twenty times longer than a minute doing the thing you hate most.  Unless one really, really love to grade.  And that number could increase as you spend longer time grading.  So one hour grading is equivalent to ten hours working in the hot sun.  And two hours grading will not be equivalent to twnety hours in the hot sun, it’s more like twenty-five or thirty hours.  If I may bore you with a formula, it’s like T = 3*(1.02)t + t3 + 3t2.  Here T is the time grading and t is the time spend doing the thing you hate most.

So one of the thing I do, is to grade during exam (of another class).  And somethimes I do not have papers to grade in an exam, that will be the best time to brew mathematical ideas.  It’s hard to describe what that means to non-mathematicians (or people who do not do similar things to a mathematician).  It’s not like you stand around and E=mc2 + will just come and occupy your head.   Or we stand around trying to solve a riddle or puzzle.  We have to dream of a problem and then try to solve that problem.  Dreaming and coming up with a good problem perhaps is the most difficult part of the "brewing process".  This is when creativity equivalent to that of a musician or artist is required.

Once you come up with the problem, only the seeds are ground.  The whole "brewing" process has not quite taken place yet.  You twig and change the problem until

1. you can solve the problem
2. yet the problem is interesting enough to be worth solving
3. and not any Tom, Dick or Harry can quickly come up with a solution.

So some persistence and perserverance is good for this brewing process.

So right now, I’m proctoring a Calculus 2 exam.  I actually chose to blog than to do a mathematical brewing.  Let’s hope this writing spark an idea that help me come up with a really good mathematical problem!