Over the last week I posted five logic puzzles that push the solver to adopt multiple points of view:  HatsGreen-eyed guruPerfectly logical piratesPrisoners with HatsLeprechauns.  I also posted a solutions page.

So what do you do with these, how, why?

Of course if you are here, you will probably try your hand at solving them on your own. You are mathematically curious.

But with a class?

I am not a fan of tossing the kids the puzzle and letting them try it on their own if they interested. I am a fan of occasionally making time in class to do something completely different…

Format:  I like grouping kids (or adults) up. That way even those who don’t hit the big insights can feel like they are part of what’s happening. I also like the forced communication. Can you explain why you claim X?

#### Understand the problem:

I handle this whole-class. Read it. Ask questions. Then when they all claim to understand, force them to ask more. And if they don’t, I will. Can they mime the words green or blue? Can they peek? Can the gold pieces be divided? What happens if there is a tie? Anticipate the clarifying questions that should arise, and ask them if the kids don’t.

#### Planning:

As the groups begin to work, come around with lots of questions along the lines of “Why…” “Is that enough…? “How do you know…” and the ever so leading “and what is everyone else thinking?”

Let them struggle. Do not rush to give them an answer, or even to point them on the right track. You get a chance to struggle on problems, right? You gain insight through bad attempts, blind paths. They should have that experience.

At some point (but let time pass first!), get groups to report on progress. They can share insights, beginnings of approaches. I usually ask for things that did not work first (What did you try? Why did you think it might get you somewhere? How did you discover you were in a dead end?) So no one gets shut out because they don’t have a final answer. And then the partial approaches let them listen to each other, and change angle of attack mid-problem. Changing the approach is a big deal, kids are not used to that, and it is good that they learn to let go. And then those who are lost get a chance to work productively again.

If a group is truly dead in the water, handing them a key insight may be necessary. You also might ask them to assign roles, and act the problem (or a small version) out. This really works, but needs some supervision (in general).

#### Carry out the plan

When a group grabs the key insight for a problem, I have them share it out, without a complete solution, but just a start. Tell us about pirates C, D, and E. Or about what happens with 4, or 5, or 6, or 7 or 8 leperchauns. Or if there are only 3 people on the island. Then all groups can work on the mechanics.

#### Looking Back:

Finally, as they arrive at the answer, let them celebrate their cleverness for a moment. And then make them consider the process they went through. What made this hard? What were the key insights? What were they tempted to do first, and why was that not productive? I try to get a kid to say that it was important to look at the problem from everyone’s point of view at once… but it can be hard to come up with. When they don’t, I suggest it.

— — —

These are fun. They are surprising. And with some theatrics, they can become incredibly engaging. The idea of adopting multiple points of view is valuable. Letting kids bang their heads for a bit against something that seems intractable also has value. And they are forced to communicate unusual ideas clearly, with you, with their group, and then with their class. I love being able to give up a day or half a day once a month or so to do this kind of thing. Winners all around.

Credit: I don’t know where the problems came from originally (except the Leprechauns I stole from Blinkdagger, and the Pirates are in a SciAm article), but the idea of grouping some of these together came from Jim Matthews, a math ed professor at Siena.

1. August 2, 2009 pm31 10:51 pm 10:51 pm

How do you set up your groups? Like ability or more mixed? It seems that whenever I try to do groups, it seems like it just makes an excuse for the weaker kids to do nothing while the stronger kids work on the puzzle. I have not had success in setting up groups where everyone works and participates. Do you have some tricks for that?