Aliens, Number Systems, and Contemplative Mathematics

“Some alien spies have been helping our captain.  It turns out that you were right: there are enemies hiding on the field behind the house.  The captain wants to know how many.  The alien photographer comes down and takes a picture of them, gives it to her captain, who writes it down and gives it to our captain.  The alien captain wrote either 7, 13, or 23.  Which is correct?”¹

I drew a diagram of the photo – a square containing 13 dots, each representing an enemy, in the shape of the number seven.  The kids debated whether 7 or 13 was correct until V questioned, “Why would 23 even be on this list?”  This is the real meat of the problem.  V’s question produced many very insightful conjectures about groupings of numbers.  I then drew a diagram of an alien (2 fingers on one hand, 3 on the other), leading to further conjectures, but none were fruitful in the long run.

I posed the alien question after I had begun class with a few rounds of child-directed Exploding Dots, first in decimal, then in octal.  We discussed different base systems, and King Charles XII of Sweden’s contribution to them.²  We even attempted to convert from octal to decimal.  A few students suggested a novel way to do this, but most didn’t understood this conversion.  That’s fine.  Math Circle is not a race to “get it.”  What’s important is that kids are developing mathematical thinking in a collegial environment.  Mathematical thinking is a contemplative endeavor.  Stuart C. Lord, President of Naropa University, states that “the greatest disservice we can do to students is to imprint them with how we assume the world to be, without giving them the necessary tools to investigate and explore the world as it is, with an open heart and discerning mind…  Those whose higher education includes a contemplative education component are imbued with the much-needed capacity for authentic insight and revolutionary thinking; they are poised to meet the world as it is, and have the passion and capacity to improve it.  When these students graduate, they have spent years turning problems upside down, looking at them from the inside out, holding them quietly within, and working collaboratively with others to find novel approaches to solving them – providing not just a quick fix, but a true change that benefits everyone.”³  We have been turning this Signaling Problem upside down for nearly 6 weeks.  While Dr. Lord is discussing college students and beyond, a contemplative approach to mathematics can begin in early childhood.  According to the Berkeley Math Circle, “It often requires several years to become an experienced problem solver and to be comfortable with everything taught by the Circle. Thus, expect the first year to understand on average between 30% and 50% of everything said at the Circle, taking into account that the material covered is very advanced and non-traditional. 30% to 50% is actually quite a lot!”⁴   I must admit that in my Math Circle at Talking Stick, I want kids to understand more than 50% of the material.  I want them to enjoy mathematics and to want to come back for more.  The yin that goes along with this yang, though, and what Berkeley is referring to, is that students need to experience frustration and confusion to have breakthroughs and really own their mathematics.

The mathematics that I’d like kids to own from this course is that counting strategies exist and that our number system depends upon the idea of grouping.  We call this grouping place value.  Once kids grasp the utility of grouping, arithmetic makes intuitive sense and the generalizations that define algebraic thinking can emerge naturally.  Robert Berkman, who created this alien question (thank you Robert!), extends the problem into algebra beautifully in his article “Exploring Interplanetary Algebra to Understand Earthly Mathematics.”¹

In our exploration of the alien-question, I revisited the term “decimal” when the conjectures came to a standstill:

• What does decimal mean?
• What are digits?
• What does the word digit mean in Latin?
• Why do we count the way we do?
• Why is it easy for us to use base 10 (a.k.a. decimal)?

When the group discussed these questions, and members practiced counting on their fingers, the initial question started to make sense.  I needed to add more choices to the initial numerical-only choices, and the kids arrived at consensus just as we ran out of time.  The answer was “(f) 1 or 2 of the above.”

They left asking to act out the problem next week, our final week.  I promised that after we finish Level 3 of the Signalling Problem (goal: devising a code that can send 30 messages) we will.  See you then!

Rodi Steinig, December 4, 2012

¹ Teaching Children Mathematics, v5 n2 p78-83 Oct 1998

² http://en.wikipedia.org/wiki/Charles_XII_of_Sweden (see “Scientific Contributions”)

³ Shambala Sun, v19 n6 p59 July 2011

⁴  http://mathcircle.berkeley.edu/index.php?options=bmc|bmc_elementary|BMC%20Elementary