Today I gave Katie the following problem:
A mother has five sons. Each son has one sister. How many children does the mother have?
Here is a short version of the conversation that followed:
Katie: That’s easy, I don’t even have to count! She has 10 children.
Me: Why don’t you think about it a bit.
Katie: What’s there to think about? 5+5=10!
Me: Yes, 5+5=10, but you should still think some more about the problem.
Me: Ok, lets suppose that I have a son. How many brothers will you have then?
Katie: You’re not going to have a son!?
Me: We are just supposing here. So how many brothers will you have?
Me: And how many brothers will Zoe have?
Me: And together you will have…
Me: So I will have two sons.
Katie: No! We will have only one brother! We will have the same brother! (Finally some progress :-))
At this point I was willing to leave things the way they were and move on with our evening. However, Katie started whining that she wants to solve the original problem (even though she thought she solved it, she could sense that I didn’t agree with her answer). I ask the same question again and get the same answer (although less confidently this time). Finally, I start giving the sons names and say, “John has a sister Mary, what is the name of Adam’s sister?” Mary! “And Michael’s sister?” Mary! And so on. So how many total sisters? One! And how many kids does the mom have? You can guess the answer :-).
There was actually a lot more back and forth than what I wrote up here. At times Katie would say that she was confused, but she refused to give up. I got this problem (as well as a number of my other ideas) from the wonderful book Math from Three to Seven by Zvonkin. There he discusses giving this problem to his son and friends, with similar results.