This is sort of an accidental “my favorite” but I spent 30 minutes with my older son today and found myself thinking about how much I love watching problem solving ideas develop in kids.
“Problem Solving” is notoriously hard to define – and since I’m in a happy mood, I’m not going to try to define it 🙂 It is, at least in my mind, though, a skill that you can watch develop over time.
Today my son worked through four old AMC 10 problems that had given him difficulty the first time through them. We had not looked at or reviewed these problems since he worked through these tests, so, although he had seen the problems before, he’s not previously been able to solve them. This afternoon with just a few nudges here and there he was able to work through all of them. Along the way are some pretty nice examples of what a kid looks like doing math.
All of the problems can be found on Art of Problem Solving’s site here:
The 2009 AMC 10 A hosted on Art of Problem Solving’s website
and here:
The 2008 AMC 10 b hosted on Art of Problem Solving’s website
The first problem was #14 from the 2009 AMC 10 A – it is a problem about absolute values:
There are a lot of plus and minus signs to keep track of in this problem, and he does a nice job of approaching the problem in a pretty systematic way to help keep track of all of those signs.
The second problem is #17 from the 2009 AMC 10 a – it is a problem about 3-4-5 right triangles
There are a lot triangles similar to a 3-4-5 triangle in this problem and keeping track of the sides is made a little extra difficult by how the various triangles scale. In one of the triangles the longer leg has length 3 and in another triangle the shorter leg has length 4. That trickiness does trip him up, but luckily he does catch the mistake (because his original answer wasn’t one of the 5 choices).
The next problem was #14 from the 2008 AMC 10 b – it is a problem about rotations:
This problem gave him a lot of difficulty. There’s a little bit of geometry to keep track of and also you have to keep track of a few plus and minus signs at the end. His solution here is a good example of working through a few initial misconceptions to arrive at the correct solution:
The last problem was #19 from the 2008 AMC 10b.
This problem looks like a super challenging 3d geometry problem – it is really easy to have a reaction similar to – “well, I really don’t know anything about water in cylinders.”
What I loved about his work in this problem is that he figures out what he can do and then figures out how to use that information to solve the problem. At one point he says something like “wouldn’t it be great if these were 30-60-90 triangles?” Loved that!
So, the work today really did make me happy. I love having the opportunity to work on math with my kids, and I love watching their problem solving skills slowly develop over time.
Mike's Math Page 
I love that you video-taped the problem solving process!
The first, #16 from 2009, relates to Ducci sequences. Would be a fun investigation, if you haven’t already done it.
This link Math4Love Diffy Sq III is the last post in a little series, but it has links at the top to the earlier 2 posts and other references, esp the NYT wordplay column.
Tried to comment already, so apologies if this is a duplicate.
The first problem, #16 from 2009, relates to Ducci sequences. It could be a fun investigation for your kids.
Best link is Math4Love Diffy Squares which starts toward the end of their discussion, but has a nice set of links at the top. In particular, make sure to read the NYT wordplay column that also has a nice story about someone’s formative math experiences.