Challenge 07 from Collaborative Mathematics is here! Watch the video, try out the probability experiment, and send in your data. Or, tackle it theoretically!

Plus: we've received a great response to Challenge 06 from Dani, and a fantastic new approach to Challenge 03 from Paul.

There are several very cool blogs out there -- among them Math Mistakes and Bridging the Gap -- that highlight the value of thinking deeply about student responses, the mistakes they make, and the explanations they give for doing things the way they do.

One of the ideas behind Collaborative Mathematics has always been that participants could learn something from exploring other people's solutions, in addition to exploring the challenges. Some of the response videos in the archive contain mistakes, and I think that's fantastic. Rather than being an answer key of tidy solutions, they represent a library of arguments and ideas that require additional critical thinking.

One aspect of this is that I avoid posting my own answers to the challenges. I don't want my way of solving a problem to be interpreted as "the right way" just because I'm the one who posed the challenge in the first place. Over the next few posts, though, I would like to share my thoughts on a particular solution as a way to talk about my own mathematical curiosity, and what I've learned from a really creative approach to one of the video challenges.

But before I start going on and on about what I think, I want to give you a chance to explore for yourselves. Here's what I suggest:

• Take a few minutes to visit the archive page for Challenge 03: Finger Counting.
• Solve the challenge on your own, if you haven't already.
• Watch a few of the response videos.
  
• In particular, watch Maile's solution and see what makes you curious.

Next time I'll share how I've come to think about the mathematics of Maile's approach.

This is my response to the first mission from Exploring the MathTwitterBlogosphere. My whole thing lately has been about trying to bring more open-ended problem solving into my teaching, so I'll write a bit about a trio of connected problems that I really like, and which I hope others might find interesting.

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A note from the future: I've got to say that as I'm writing these problems, I'm both tempted and reluctant to talk about the solutions. I'm tempted to talk solutions because I find myself wanting to be specific about the kinds of conversations I like to have with students about these problems. I'm reluctant because I, personally, would (almost) always rather explore a problem on my own, at least for a few minutes, before reading someone else's ideas about it... and maybe some of you are like me in that way.

I've chosen to avoid solutions, though I think it might make this post sound a bit vague and hand-wavy. I welcome your thoughts in the comments.
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You may already know the first one. It's a classic:

I like this problem because it lends itself to a variety of approaches. Students might try acting it out or drawing a picture. They might jump right in and try to answer the question for a large number of participants. Or, they might try solving a simpler problem: starting with smaller numbers of participants, organizing their data in some way, and looking for a pattern.

Then, once we find the pattern (and perhaps even an equation for the number of handshakes given the number of people), I like that we can explain that pattern in a number of different ways. We could explain it purely using the context, or using arithmetic in a clever way, or using algebra, or by making a geometric argument... And, even better, we can connect all of these different explanations.

I like to follow-up the handshake problem with a seemingly unrelated problem:

I like this problem because it forces students to employ an often helpful (and often necessary) problem solving strategy right from the start: solve a simpler, related problem.

Students realize immediately that 170 diagonals is ungrokable; we have to start smaller. But then students usually start by drawing polygons and counting their diagonals, without realizing that they have inverted the problem. That is to say, the problem gives you the number of diagonals and asks you for the number of sides. Students usually start by fixing the number of sides (for example, by drawing a pentagon) and counting up the diagonals. I like that the strategy of solving a simpler, related problem emerges so naturally.

Plus, the polygon diagonals problem has all the great features of the handshake problem when it comes to explaining the pattern and the equation that are eventually revealed.

Then, I like to conclude with a problem that seems both (a) unrelated and (b) totally tedious:

After their initial groan, I'm quick to remind students that we've learned a lot that might help us to solve this problem without having to slog through adding up all of the numbers. This becomes an exercise in creative application, as well as a chance to explore other interesting patterns that emerge.

In fact, if you haven't ever explored the multiplication table problem, I encourage you to kick it around for a few minutes. I was surprised at the number of cool connections hiding in there.

I look forward to hearing from people in the comments!

As I get ready to participate in the missions from Exploring the MTBoS, I thought I'd do a quick test of MathJax on my Weebly website.

Formatting seems to work for inline equations such as \(x - h\) and \(c = \sqrt{ a^2 + b^2 }\), and for equations that will appear \[P(E) = {n \choose k} p^k (1-p)^{n-k} \] centered on their own line. But, equation rendering does not seem to be working for the mobile version of the site.

I recently received response videos to a number of earlier challenges, so I'll summarize the updates here. Click the links below to visit the archive page for each challenge, where the new videos are highlighted. Plus, you can watch the original challenge video and the other responses, and take on the follow-up challenges.

• A new solution to Challenge 01, in which our problem solvers go back to the beginning and hope to count the even "Ermer Numbers".
  
• A new solution to Challenge 03, in which our problem solvers take on the most popular challenge from the spring, "Finger Counting".
  
• A new solution to Challenge 04, in which our problem solver sets about to explain how we know we have found the minimum number of button presses on the less-than-useful "Two-key Calculator".
  
• A new solution to Challenge 05, in which our problem solvers bust out the rulers for a practical approach to "International Paper".