(3/8/2018) During our course on Invariants, the eight-year-olds spent most of the time exploring the Euler Characteristic (click here for details). This report is essentially a list of other activities we did to start or finish our sessions. An invariant is something that never changes.

Piagetian Conservation Tasks – We did every activity in this article: http://www.cog.brown.edu/courses/cg63/conservation.html. Conservation tasks basically are about invariants. Some cognitive psychologists posit that the ability to do these tasks is not coachable, while others believe it is. Our group had some variety in who could recognize the invariance. All could spot it sometimes. Most had at least one case of not being able to spot it. Interestingly, whether someone had success at these tasks had no bearing on their examination of the Euler Characteristic.

• Line up cubes and count them. If you change the order or the distance between them does it change the count?
• Redistribute blocks into sets – how does that affect the sum? (Note – the students loved using these wood cubes so much that I had to set up before class second time just to give students a chance to play with them
• Pour water in differently-shaped containers. Is there more in a taller thinner container than in a shorter wider one?
• Flatten a play-doh ball. Does this affect how much play doh is there?
• Weigh a play-doh ball in different shapes – does changing the shape affect the weight?

1-2-3 Fingers  – kind of like mathematical Rock Paper Scissors – I say “123” and both you and I hold out however many fingers we want.

• Multiply them – if odd, you win, if even, I win (I win every time – tee hee hee!)
• We did this for three weeks, and by the final week most but not all students had figured out the strategy. Lots of fun! (Thanks to Maria Droujkova for this activity.)
• Play this at home!

Collaboration through NIM

• We played the game NIM, which you can learn about here: https://mathforlove.com/lesson/1-2-nim/. We played several versions of the game.
• I told the students that the goal of this game is collaboration, since real-world mathematicians get help from each other in solving problems
• What is the best way to collaborate if we play a game that’s me against the team of all of you?
• Students played this for 5 weeks, with their collaboration and mathematical strategies evolving over the weeks. While the game was fun and the thinking got deeper and more sophisticated over the weeks, the collaboration that I demanded was stressful. I didn’t tell them how. Each of them had different ideas. Some people cared more that their ideas (for collaboration methods and for NIM game strategies) got tried. Others cared more that conflict be avoided. I talked about the challenges and benefits of collaboration a lot!

Cup game

• You get 7 cups: 5 upside down a 2 rightside up. Your goal is to get them all rightside up by flipping 2 at a time. (Thanks to Maria again.)
• We had very deep math conversations about this game, getting into parity, testing of many cases, changing the rules to see what would happen, and what would proof require if you want to make generalizations.

Cross-country race

• We played the game that is “Example #4” on this handout from the Waterloo Math Circles: http://www.cemc.uwaterloo.ca/events/mathcircles/2010-11/Winter/Senior_Mar23.pdf
• Students changed the names of the cities from unfamiliar Canadian locations to things they made up. This made the game more accessible.
• We played it several times, but not enough to be able to make generalizations. The students who did play it most want to play more to discover what happens when you try other starting points, etc. I promised these pictures so that kids can play at home.

Strings on Cans

I brought in a bunch of cans of many sizes. I had multiple strings cut to the length of the diameter of each can.

• How many strings does it take to wrap around the can with no overlap or gaps?
• Turns out everyone found that it takes a little more than 3 strings to wrap around the can, no matter what can they used.
• Is that an invariant? The students thought no. I asked how many tire-diameter-length strings it would take to wrap around the circumference of a tire, and everything thought a lot more than three, despite our hands-on results here. Piagetian cognitive psychologists posit that there is a fixed developmental stage at which students can transfer mathematical patterns to other examples. (Of course, not everyone agrees, and I think that with the discovery of brain plasticity and the modern research on mindset, more people are seeing things like transference as something coachable.) I promise you that I am not using your children as my mini-cognitive-psychology lab!
• A parent in the background asked “Are you talking about pi?” It turns out that yes, we were! That number a little more than three, that ratio of circumference to diameter, is pi (my favorite invariant!)

Euler History

• I read a little bit about Leonhard Euler in some of the classes so that the students knew that there was a person being the main problem we were exploring.
• I read from Historical Connections in Mathematics – a series that I love.

Function Machines

• One goal was to introduce how to play function machines to students who never did. They are super fun. Ask your children how to play and do it at home! (We used rules like x+1, x+20, 100x+1, x-2, but didn’t discuss them in algebraic terms as I am here.)
• Another goal was to do function machines with invariants and have the students figure out what was invariant, so wanted to use rules like subtracting itself (x-x) or 1 if odd and 0 if even. Ran out of time, though.

Thanks to all of you for sharing your wonderful children in the extra-fun course!
Rodi