Collaborative Mathematics

Learning from student "mistakes", part 2

4/11/2013

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This is part 2 of a discussion about Maile's solution to Challenge 03: Finger Counting. If you haven't explored that challenge or thought about her solution yet, please start here.

In her solution, Maile observes that the numeral 1000 can represent different numbers. It might represent one thousand. In fact, I think it's fair to say it "usually" represents one thousand because we usually interpret numerals in base 10. But, the symbol 1000 could just as easily represent eight if we interpret it in base 2. (In fact, it could just as easily represent a lot of numbers: sixty-four, four thousand ninety-six, negative twenty-seven... but let's not get ahead of ourselves.)

Maile answers the challenge by claiming that we can stop finger counting at eight rather than counting all the way to one thousand, since 1000 represents the number eight in base 2. She's right in that we do in fact land on the same finger. "But," I wondered, "Does that always work?"

(As a quick aside, I have to appreciate the lucky randomness of the fact that I chose 1000 as the target number. That is to say, if I had asked what finger we'd be on when counting to 500 or to 2000, the connection to binary numbers would never, I suspect, have come up.)

To answer the question "does it always work?", it's helpful to watch Paul's solution to the challenge, in which he makes the connection between this kind of finger counting and modular arithmetic. If a number is congruent to either 2 or 0 modulo 8 it will end up on your index finger. Since \(1000 \equiv 0 \mod{8}\), we end up on the index finger when we count to 1000.

Said the other way around: if two numbers are congruent modulo 8, then they will end up on the same finger when finger counting in this way. So here was the question I had:
Given a numeral made up only of the digits 1 and 0, are the base 2 value of this numeral and the base 10 value of this numeral always congruent modulo 8?
This is a cool problem to solve, and you might want to kick this problem around before reading on. My gut feeling was that the answer was yes, but I needed to crack open a number theory book to prove it.

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Learning from student "mistakes", part 1

11/10/2013

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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.

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New responses to earlier challenges!

3/10/2013

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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".
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