I saw an interesting question from Tracy Johnston Zager on twitter tonight. Unfortunately twitter isn’t cooperating with me right this second and I’m having trouble linking to all of the tweets (ARRGH!!!!), so I’ll have to summarize because I’m trying to get to be early tonight.
The conversation went along these lines:
Tracy was in a classroom working on a game similar to Nim. On each turn of the game a player can add either 1, 2, or 3 blocks to a pile. The player who adds the 10th block loses (or maybe wins, I forgot this detail, but luckily it doesn’t matter for purposes of this post).
The one tweet I can get to link are some of the “notice and wonder” questions that the students had:
The question that caught my eye was about the students’ questions about the actual pieces used to play the game. I’m paraphrasing, but the question from Tracy as I understood it was essentially – there’s not really that much math behind the questions about the game pieces, so is it productive to talk about those questions in a math class?
I thought that addressing those questions might be interesting, but I’m also a terrible judge as to what will be interesting, so I decided to talk about those ideas with my younger son (a 4th grader).
Here’s how the conversation went.
First I introduced the game and we played a few rounds (just 2 min here – the rounds go quick!):
Second – I asked him if he thought the game would change if we played with yellow blocks instead of orange blocks. He thinks that the game will not really change, but importantly for this blog post he does not appear to think this is a silly question:
Finally, I asked him what would change if we played with Lego mini-figures rather than blocks. Again, he thinks the game will not change.
Then I asked him what *would* change the game and something really cool happened!
It was an incredibly lucky break that the conversation about what would change the game led to my son figuring out how to solve a new game. But, even without that fun ending, I think talking about what would change the game and what wouldn’t change it was productive.
I can definitely believe that a kid would have questions about the game pieces and wonder if changing the pieces would change the game. In my mind it is similar to a kid learning algebra wondering if the way you solve an equation like 
Anyway, I’m glad I tried out the question with my son – it was interesting to hear his thoughts 🙂
Mike's Math Page 
Sweet reasoning by your son!
I sometimes use this game to get at backwards reasoning with PSTs, but when I play with kids they get to variations so quickly.
NIM, I mean.
That’s terrific, Mike. And it really is interesting that that’s where he took the conversation. I mean I don’t doubt that part of it’s you — you’ve been teaching both of them to look for underlying structures, patterns, assumptions, and vantage points (like ‘first move’) for years — but he went there very naturally, and the first rule he thought to change was to cut out the ability to add one.
I think it really does say a lot about training and the mental overlays we put on things…I’ll have to try these on my daughter. My own first thought, once the Lego figures were in there, was that the groupings you added would have to make sense storywise — allies with allies, etc. Which just gets you to Diplomacy. 🙂 Or Risk. Are you guys playing anything like that?