


Keynote Part IProblem #1: Analyzing Errors Let me give the first example of analyzing errors. For many of you, this will be relatively second nature and therefore will not constitute a difficult problem for you but if you've not taught at the grade levels where this tends to come up this may be a more difficult mathematical problem for you. Here's the problem 
Teachers see a wide variety of answers to problems they assign. And so, they are often involved in analyzing errors. A student produces this answer to the problem, 132  57. What method is producing this answer? Take a moment to think about this. See if you can describe to someone else what method produced that answer. Is there more than one possible method? If you've taught second or third grade you've probably seen this a million times. High school teachers and mathematicians in the audience may not have seen what was going on quite so quickly. It's not a trivial problem, even though for some of you this is obvious. If it was obvious, it is because you've actually learned a kind of mathematical reasoning from your practice that makes this question easy to answer. But I guarantee that not every one finds this easy. That's something to notice about the sort of mathematical reasoning involved in analyzing errors. One of things you were doing was you looked at a result and you tried to think about what operations could produce that result and what, therefore, might have been the algorithm for it. That is a significant and productive form of mathematical reasoning. And that's what you were drawing on. Let's try one that is, perhaps, a little more challenging. 

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