Week 2 Group 5: Infinity

I wrote my "anticipated responses" down on paper before the circle, so I will include those now, and follow with "what was different than your prediction?"

"What is Infinity?" - Kids won't really know how to respond; they might think there is a right answer. A few may interject ideas. Different: while some students did struggle (thinking there was a right or a wrong answer) most contributed with one of three main answers: it's a really big number, it never ends, or it includes everything

"Drawing Infinity" - A few may draw the infinity symbol, but likely will be stumped. I anticipate statements like "I don't know what do draw" or " Is this good/right?" Different: Actually, this is exactly what happened. Most drawings were the infinity symbol or kids writing a really large number. One attempted a formula, and a couple drew pictures. They liked the examples (an oroboros and a recycle sign). Some kids insisted if they could just draw enough zeros/periods on their number they would have infinity.

"Recursive Poems" - I am sure they will like this, but I am afraid they will just be singing it the rest of circle time, and silliness may ensue. Different: I skipped this activity in the interest of time

"Mirror Image" - I think a few will count and be limited by their mirror angles to a countable number. Others will fabricate a count. Others will recognize it as infinite. Different: the kids all knew that there would be infinite reflections, and were surpassingly uninterested. This would have been a great place for an extension, because I was at a loss for what to say more.

"Fractals" - I think the initial fractal drawing on the board will be difficult for kids (the concept is hard if you've never seen a fractal. Maybe something like Sierpinski's triangle would be easier to give instructions for but less-open). After an example on the board I think seat work will be easier, but anticipate lots of "I messed it up" and "I didn't do it right." I think they'll enjoy the beauty and wonder of the images and videos. Different: sadly we had to skip this in the interest of time, but we did look at a fractal picture and watch two fractal videos at the end. The kids liked the images. They were in stitches over fractal cows, so I am glad it was at the end.

"Squares and Rectangles" I think kids will ask "more squares or rectangle IN WHAT?" There may be some who argue more rectangles because squares are a subset of rectangles. Different: I got an immediate "more squares or rectangles IN WHAT?" and so I just suggested "in existence or possible." I had one student who argued very strongly that rectangle infinity was larger than square infinity because squares are a subset, and I wasn't really clear on how to explain the one-to-one correspondence idea (maybe expound this a little in the book?). I did have children develop their own parameters to the question. For example " there are more squares than rectangles on my page because I drew more of them" or "every rectangle is equal to two squares so there are more squares" - I tried to emphasize that they put their own boundaries on the problem and that under those parameters they are right, but then asked "were those parameters part of my question?"

"Infinite and Unlimited (torn paper version)" - Some may argue that the example is actually finite because of the physical limitations in tearing smaller and smaller pieces. Different: they almost universally agreed that something infinite should be unlimited before the activity, but the kids agreed quickly that I could just keep tearing the paper an infinite number of times. I think the grid/halving the distance version would have been better but I couldn't really set this up.

"Hotel Infinity" - I think this may be easier to understand as a child, and therefore the kids won't be as surprised by the solution. Different: I may have not explained this right, because no one wrestled with the paradox. They all followed that room one would be open and there would always be room for more guests. Either they were agreeing without understanding or I didn't explain it well.

"Dice Roll" - I think they kids will like the physical movement in the game. I anticipate them persisting in this task longer than most adults would. Different: The kids did really enjoy the physical part of this, and they all wanted jobs to do (I only had four jobs and 16 kids). They liked testing of their hypotheses. What I was not prepared for was the answer to "so, is it possible to reach 20?" I assumed yes, because if we continue to roll larger numbers in the forward direction we would slowly make our way forward, and the probability might be incredibly low. But here is a place I would like a solution(s) with explanation so I am prepared to help them.

So I didn't see this post until after I did my group today, but I'm not sure I could have posted much in the way of predicting how the kids would react. I had no clue.

So I did the group today with 8 kids, from age 7 to 11, but mostly 8 & 9 years old. One friend of my twins came who dislikes math so much her mother didn't tell her it was about math until the girl saw the direction signs I had posted that said Math Circle. LOL. On her way out she asked her mom when the next meeting was going to be. I call that a success. All the kids wanted to meet again. So I'm going to put together one session a month starting in late January and see how it goes.

I didn't get a picture of the drawings, everyone wanted to take them home...but we had minecraft related drawings, representations of air, infinity symbols, an equation, numbers followed by ellipses, numbers followed by many many zeros.....and that's all I can remember. Most of the pictures and the explanations were quite concrete (except my little equation guy....Wow, he already understood a lot!). I did expect these to be quite concrete just based on my training in child cognitive development.

The introduction went pretty well, almost everyone offered up a definition of infinity (and that's where I figured out that we had one young kiddo with very advanced theoretical math knowledge). On the drawings about half did actual drawings and the rest were more number or symbol or equation type representations. We skipped the recursive poems (we only had an hour and they seemed to need to move on to something more active at that point). They liked pairing up and doing the mirror images. They tried several different ways of aligning the mirrors and toys, then someone figured out how to see their face infinitely and they all tried that (and they were small candle holder mirrors from the dollar store).

I think the part they liked best was fractals. We did one as a group on the white board. I started it, then each student took a turn adding to it at least once and a few of them finished it out to cover the board after they had done their own drawings. Some colored their drawings, we talked about how that added a different dimension. We looked at a ViHart drawing. We looked at the 3D fractals on google images. They REALLY liked this activity and I could probably facilitate another circle JUST on fractals (and we may!).

We skipped Squares and Rectangles (again time and the need to move around). We did the INfinite and Unlimited activity on a big sheet of paper. We skipped Hotel Infinity, somewhat b/c of time, but more b/c I didn't understand it very well (I've found some resources to remedy that). They really enjoyed the dice roll activity. We did this as 2 groups to give more folks a chance to participate. They rolled and moved and ticked off the times it took. One group actually managed to get to the other end! Unfortunately, they lost count of how many times it took (and I'm not sure they did it exactly right)....we'll probably do this activity again.

The topic of infinity really could fill more than one session (or at least more than the hour and fifteen minutes that we had). I want to get a copy of "The Cat in Numberland" and read more about paradoxes and do another group mostly on that aspect.

As for what to put in the book...

*better indication of how long each activity should take (I realize this depends on the age and size of the group, but estimates for say 10 kids (small group/homeschoolers, etc) or 25 kids (classroom) would be helpful.

*More explanation about Hotel Infinity....I'm mathy, but that confused me just enough to make me unsure of facilitating a student activity/discussion about it.

*Hints on how to deal with a multi-age group, group with different skill/knowledge levels.

*Take home handouts outlining what was covered and websites/activities for follow up at home.

Anything else you want to know? Basically this was pretty low prep time for me and pretty easy to execute. I like that. I think in our situation (homeschoolers that sort of know each other), we may need an initial session of getting to know you activities before jumping into the math just so the kids are comfortable talking more. But otherwise, it went pretty well. Like I said, the kids were all interested in doing it again b/c it didn't FEEL like math. I'd call that a success.