You have 5000 gumballs.  You want to put them in a container for a Christmas present for a young relative.  What size container would you need?
Are any of you still reading? Did you bail at the sight of a dreaded math ‘word problem’?  Or maybe you saw that as a fun brain teaser (which it is!), which may even have intrigued you, but not particularly enough to tackle (especially since you’d have to have more info).
Doesn’t matter.  For my sister-in-law, it was neither. It was a ‘real world’ problem that needed solving, preferably before gift-wrapping time.  My sister-in-law is one sharp cookie, as they say, so I think she already had a plan of attack – maybe even a solution – but she sent the problem to me, hoping for verification.  (Math types are always ‘on the spot’ like this! J)
Now, granted the solution of this problem does not contribute to world peace (except maybe in one small corner of my sister-in-law’s house) or help put a satellite in orbit or reduce the national debt.  But the situation is ‘real world’ and does touch on several points that relate to preparing our students to tackle (and hopefully solve) the variety of problems they will encounter in life.  So, begging your indulgence, let’s take a brief diversion and discuss a few points about solving problems in general.
FIRST:  I suspect you never saw this problem in a text!   I think there must be this general feeling that problems in the real world all fit some nice neat category/method that we were taught in school, but have somehow forgotten.  (Ok, is this a pool-draining problem, or a ‘rate-time-distance ’ problem?)
I’d go so far as to say that this is rarely the way it happens.  Problems come in all shapes and don’t necessarily fit examples in texts.  I’ll repeat an insight that I shared in a previous column: In the real world there is no such thing as a ‘math problem’.  There are only ‘real world’ problems that may need math to help solve them. 
So, the bad news is . . . there is NO set approach.  But the good news is . . . there is no set approach!!  One person’s panic is another person’s freedom.
SECOND:  The solution here does NOT have to be exact. Not at all!!  It only has to be practical.  We don’t care if there’s a little extra space in the final container, we just want to reasonably hold them without renting a dump truck. More freedom.
THIRD:  We need a little more information, eh?  What size are the gumballs? I had a real dilemma when I gave the problem:  do I include that data up front or not?  Some folks almost literally choke on information in a problem, others are frustrated if they don’t immediately have enough to help them.
Either way, I will tell you now: the gumballs are 0.5 inches in diameter (across).  If you get a solution you’d like to share, e-mail me.  There’s a lot to follow up on here.
One of my favorite maxims with teachers is that the way to spot (and help) inexperienced problem solvers is that they ‘don’t know what to do when they don’t know what to do’.  As we prepare our problem-solvers of the future, they need to know that not knowing what to do – or where to start – is normal. Once they have some good math tools in their tool bags, there are always LOTS of ways to start – and to solve – a problem, even if it’s about gumballs.

For my own proposed solution to this problem, see Gumballs.Soln?