I try to learn from watching them struggle

I’ve written several times that my biggest struggle teaching the boys is that I’ve never been through any of this material before. One of the ways this particular struggle manifests itself is that I have a really hard time knowing when problems will be easy or difficult for them. I guess wrong (both ways) all the time.

Today my older son really struggled with this problem:

Problem 11 from the 2006 AMC 10b

The problem is:

Find the tens digit of the sum 7! + 8! + 9! + \ldots + 2006!

I certainly wouldn’t have considered this an easy problem, but since we’ve spent quite a bit of time on similar number theory problems – including last digit problems – I wouldn’t have thought this one would have completely stumped him.

Having recently finished up a chapter on “last digits” with my younger son I thought he’d be interested in helping us work through this problem, too. So we all went through it tonight for a fun Family Math project.

We started talking through it and I asked the kids what they noticed. My younger son thought that the solution probably would not require us to calculate all of the factorials. My older son thought maybe factoring out a 7! from all of the terms might be a way to start.

The next thought that my older son had was that finding the units digit of this sum would be fairly easy. This idea is, at least in my mind, an important first step to solving the problem because the reason that finding the units digit is fairly easy is essentially the same reason that finding the tens digit isn’t too difficult.

Making the transition to the next step was a little difficult, though

 

I broke the conversation at about 5 min just to start a new video. When we restarted the conversation I reminded them of one of James Tanton’s strategies – do **something**.

My older son suggested that something we could do would be to make a simpler problem. His suggestion for a simpler problem was to find the tens digit of the sum 7! + 8!. Making a simpler problem is often a great way to see what’s going on with the more complicated problem, and the simpler problem chosen by my son got us going down the right path.

We stayed on the path and calculated the units digit of the sums 7! + 8!, 7! + 8! + 9!, and 7! + 8! + 9! + 10!. Once the kids saw what happens with 10!, they were able to see how to the end on this problem.

 

The lesson I hope the boys take away from this talk is the James Tanton strategy – try **something**. Even, and maybe especially, when it isn’t obvious where to start with a problem, trying something is a great way to learn what works and what doesn’t.

What I hope to take away from this conversation (and really all of our conversations, of course) is a better understanding of the problems that the boys will find easy and the ones they will find to be difficult. Unfortunately, I think that I’m not very far down that path, yet.

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Comments

2 Comments so far. Leave a comment below.
  1. Two things occurred to me as I read this note:
    (1) I recently saw an observation from another teacher that kids often won’t get started if they don’t see a clear path to the answer. This seems to be a common issue.

    (2) What fraction of your activities and assignments are investigations, explorations, solving problems, or directed proofs (given X, prove Y)? I wonder if there is a bias toward the latter 2 (hinted in the name “Art of Problem Solving”)?

    Solving problems and directed proofs share the similarity that you know where you are supposed to be going, while the other two are much more open. I feel that using the open language/problem phrasing could short-circuit the problem of getting stuck for not knowing the full solution.

    FWIW, my definitions:
    Investigation: usually directed by teacher/professor, the student knows/are told there is something interesting to find, but not what it is. For example, investigate continued fractions where the coefficients repeat.

    Exploration: playing and examining with the hope, but not guarantee, there is something exciting to discover. I guess this is most of mathematical research.

    • Textbooks don’t help with this, either. They present material as a perfect progression of thought without all of the (often frustrating) exploration that students need to experience in math class. It would be better to allow students to discover for themselves since mere textbook knowledge will not help them in life and math.

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