Experience the joy of discovery.

• the sense of humor to say “I want to whisper the function in your ear because I’m too lazy to do the math.”

• the honesty to say “I want to whisper the function in your ear because I need help with the math,” and quite plainly, “I have no idea.”

• the confidence to correct a teacher’s mistake

• the ability to remember alternative approaches

• the ability to put theory into practice (praxis)

• the persistence to try again and again and again, even to continue the effort at home

• the ability to spot and question assumptions

• speed coupled with accuracy in calculation

• the wisdom to know when to abdicate mental math and instead think with a pencil

• the groundedness to keep things real (“Did Euler lose his sight before he figured that out? What does your function machine look like?”)

• the focused thought required to turn the concrete into the theoretical (“You don’t need to draw a picture of this.”)

• the courage to come into a new and strange situation and contribute to the group effort

No one in our group, or any group for that matter, has all of these skills. But everyone in our group has some of these skills at least some of the time. This means that we have been able to tackle problems that would have been frustrating and probably impossible for any one person in the group to do individually. I want to give the credit for their contribution to our collective success. But, I am purposely not attaching names skills listed above because these skills develop and change over time. By the time you read this, skill sets will be different.

For our last session today, we used these skills to list students’ birth years chronologically so that we could present students’ function machines from youngest to oldest. (My expectation was that they would get harder that way, but, as usual, that isn’t necessarily how it played out.) Getting the age list took some time, as our time line by year had almost everyone on the same point. “You should have the month and year,” suggested G. They ended up with 3 time lines on the board: one for years (2001-2003), one by month just for 2002 (which “is special because when you turn it around it’s the same), and one by date for July of 2002. Then came the machines.

D, the youngest, asked “Am I the only one who brought a function machine?” as he showed his tote bag with “modifications” to be added as needed. Others started suggesting numbers and he realized that he hadn’t fully formed his function, so he decided to produce his later. The functions that people had prepared generally involved either interplay of digits or parity (even and odd). The collaborative approach was definitely needed for N’s machine (If it’s odd, double it, and if it’s even, triple it), G’s (the units digit of the out number is 2 more than the tens digit of the in number and the tens digit of the out number is 3 more than the units digit of the in number), and M’s (a more complicated variation of G’s machine). But we weren’t calling them “machines.” Almost immediately, the kids suggested that we not spend time drawing, and just get to the rule. So what we had been calling “function machines” were now simply “functions.”

We spend some time discussing the difference between 1-step/2-rule (i.e. N’s) and 2-step/1-rule (i.e. G’s) functions. Then K challenged the group with a function that first adds 8 and then takes away 3. It was fun to see explicitly how functions can be simplified. And then D was ready with his function: if you put a number in the bag, the number that came out was 3 more. But, once the bag was modified by inserting a toothbrush, the function appeared to be “if it’s odd add 1 and if it’s even add 2.” D did not accept this as the rule, though, because he remembered from last week that it could be simplified into a 1-step rule” the next larger even integer. (So, our youngest student defied my expectations and presented the most challenging function.) Finally, as we were running out of time, our photographer R (1999) stumped the group with a function that involved both ranges and digits.

For the most part, the kids did not use the mathematical terminology that I did, but I modeled it and eventually they’ll be using those terms too.

With about 15 minutes left, we debated how to use the time. Suggestions included “drawing” (compass art or on the board), polyhedra construction, stories (I had one about Euler, and X repeated her plea from last week: “What about Plato?”), and more function machines. I decided to go with the Euler story to give the group something to ponder on their own. We’ll have to do the rest in a future Math Circle.

I told Ed Sandifer‘s story of how Euler discovered America. * It’s a silly story that demonstrates how proofs depend upon how you define your terms, and encourages independent thought by discouraging blind acceptance of facts and algorithms. This discussion could have taken the whole session instead of the scant 10 minutes we had, but I think it got people thinking. We also voted (and disagreed) on whether math is “invented” or “discovered.” I concluded with Sandifer’s quote that “beautiful mathematics is discovered; ugly mathematics is invented. Euler discovered. “

As the students listened and debated, some were constructing polyhedra. When X asked the group “What’s this called?” about her creation, one person said “hexapolygonamon” and the rest echoed until it was a chorus. (Try to say that out loud!) This naming process is valuable for learning.

There is one skill that everyone in this particular group (but not in all groups) possesses: the restraint and kindness to hold back and say nothing upon recognition of a solution in order to let the others also experience the joy of discovery

What a pleasure it has been to work with people who can do this! (Of course, this skill can be learned, but in this group, everyone had it from day one.) Thank you all for sharing your children with me.

— Rodi

*Sandifer, Ed. ”How Euler Did It.” (pdf) 2006.

** “The constructivist perspective allows for the creation of meaning and the naming of the world. Learners are not restricted or channeled by mythologies and hierarchical prerequisites. Humans are theorizing and shaping their perspective continuously in an interactive process which includes the possibility of critical thought and challenge… If you are constructing your meaning and naming your world, you will create tools as you need them for this construction.”   Read more…

(Photo Credit: Donna Larsen)