Thursday, April 29, 2021

Use Your Imagination!

Albert Einstein sticks out his tongue.
“Imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.” ~ Albert Einstein

I recently had a conversation with colleagues about the need for some families to have their kids accelerated in math.  The idea that a district would de-track their math classes was met online with cries of "socialism" and "liberal equity agenda!"  The less extreme messages were about "providing the best possible experience to allow my child to excel" or "making sure my child gets into a good college."

Whether these are knee-jerk political reactions, NIMBYism, or honest feelings of wanting the best for our students, I think they all display a fundamental misunderstanding about the nature of mathematics.  Unfortunately, the topics and ways we teach and the ways we assess math contribute to this misunderstanding.  Too often, math is seen as a set of skills to be checked-off.  The more you can check off the list, the better college you can get into.  Or, if you're feeling math-phobic, what's the minimum you have to check off to graduate?

But mathematics is far richer than a set of skills or a method for tracking success.  There is lots to be noticed, and lots to be wondered about.  Whether you're in grade 5 math, 9th grade Algebra 1, or 12th grade Calculus, there are so many interesting, useful, beautiful, amazing, and curious paths to explore that everyone can find something to enjoy, and no one should be bored.  True mathematics takes imagination, and every answer can prompt a new question.  Many of the skills are important, but mathematical thinking goes far beyond being able to memorize multiplication tables or solve equations.  And imagination is not built or nurtured by remaining in homogeneous groups with the same set of classmates on the same track, whether it's the "higher" or the "lower" track.  To be good mathematicians, to be good thinkers, to be good people, we need to surround ourselves with thoughts and ideas that are not always like our own.  Detracking is not an attempt to "dumb it down", but a way of expanding everyone's understanding and imagination.  And people with imagination are never bored.

Here's a math problem to get your imagination going: Roll six standard dice.  Find a way to separate them into two groups so each group has the same sum, if possible.*  Can you predict when it's possible and when it's not?  Can you create an algorithm to solve this problem?  What if you used only four dice, or used seven dice, or eight dice, or N dice?  What if you roll seven dice and have to discard one; can you predict which die to discard?  Back to the original problem, what's the probability that the six dice can be separated into equal groups?  Can a set of dice be separated into three equal groups and how many dice would you need to guarantee this is possible?  Can you create a game out of this, and would different colored dice or differently sized dice add some new twistsand questions?  There are a lot of skills built into this problem, from addition and division to conditional probability, to computer science.  With some imagination, math can be a far richer experience for everyone, and having students who think about it differently all in the same class will only enrich the experience for everyone. 

If you want to see more examples of imaginative math check out James Tanton (@jamestanton) or Numberphile (@numberphile) or Sunil Singh (@mathgarden).  And there are many others!

*I first saw the dice problem here: