Graph Theory With Dolls

Katie really likes playing with dolls.  Actually, she will ‘role play’ with all types of objects (I’ve seen her even do it with pencils!) but barbie sized dolls are by far her favorite.  Recently she has been complaining that she has way more girl dolls than boy dolls and that she absolutely needs another male doll.  For a while I was not giving in to her arguments and pleas because I felt that she already has too many toys in general, and dolls in particular.  However, a few days ago I ordered a new boy doll online for Katie because I realized that I could use it (together with her other dolls) to serve a purpose: introducing math ideas through fun games, stories, role play, etc.  This particular idea was inspired by something I saw on Moebius Noodles where they suggested using dolls in a similar way.

So the doll arrived today, and of course Katie was very excited.  I let her play with it for a while and introduce it to her other dolls.  Then I said that the dolls were going to go to a ball.  The idea intrigued Katie.  We moved to a different part of the living room and put out some small pieces of paper in a row to serve as chairs (initially six in total).  Then the dolls started arriving at the ball in pairs.  The rule was that girls only wanted to sit next to boys and boys only next to girls.  After three pairs arrived Katie had them all sitting on the chairs in an alternating girl-boy pattern.

Everything was going well and the dolls were enjoying talking to each other, when I pointed out to Katie that the two dolls on the ends (one boy and one girl) could only talk to one person each, whereas everyone else could talk to two each.  Naturally, this was making the end dolls a bit sad.  Was there some way to remedy the situation?

Katie almost immediately said ‘yes’.  When I asked her how, she started moving around the dolls without touching the chairs.  The solution wasn’t presenting itself so I left her to it for a bit and wandered off to a different room.  In a few minutes Katie said that she had to go to the bathroom, which she proceeded to do.  Then suddenly I hear a voice calling from the bathroom, ‘I know how to do it!’, she says.  She goes back to the living room and arranges the dolls and chairs.  Then she calls me over and I see that she has arranged them in a circle.  I was very happy and proud.

We then did a few more variations with more dolls, but that was certainly the climax of the ‘lesson’.  We had a fourth girl come to the ball and I asked her whether we could have a similar arrangement to the previous one with every girl talking to two boys.  She immediately said ‘yes, if we also have another boy come.’  I told her that another boy did not come yet, and she spent some time trying to solve the problem with the 7 dolls.  Not surprisingly, she did not succeed :-).  I did not tell her that a solution does not exist (math shouldn’t be about accepting things on faith), but I did tell her that I couldn’t come up with it either.  At the end the dolls got to dance, because it wouldn’t have been much of a ball without that.

A few afterthoughts.  I think that if she had solved the problem right away
I would have still been happy but I probably would have decided that it must have been too easy.  What impressed me the most was that she came up with the solution when she was not actually looking at the physical objects.  Also, if she had not solved the problem I was determined not to tell her the answer but to come back to the problem later, maybe in a different form.  I think that it is very important for kids to solve things on their own because that way they remember the solutions better and feel more confident in their abilities to solve problems on their own.  In the future I hope we come across problems that she finds interesting but does not solve right away.  It would be great to see her have a eureka moment a few days, weeks, months, or even years after the initial introduction to a problem!