Structure in math

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I read the news at NCM regularly. I think all math teachers in Sweden should (that and Nämnare). Today I came across this really cool new project and idea:

Structure in problem solving is an endless theme with me in my classes. I have just as much trouble convincing the top performing students to have more structure as I do the struggling students.

On the plus side, I had recent confirmation that I am making progress in this area (which is a nice, though rare, occurrence). During our regional finals in the Pythagoras Quest contest, our team was complemented on their high quality explanations (redovisningar). It was nice for me and my students to hear that from an outside source.

I wonder how many of you out there (assuming anyone is out there) have had this problem with a student: the student is very talented and has a natural instinct for math. They are able to solve all the “standard” problems quickly and easily. Despite efforts to teach them formalized methods, structure and techniques, they like their own personal “winging it” methods. You warn them that these are great for the easier problems, but at some point they will hit a wall and have no idea how to move beyond it. Sure enough, you toss them something more challenging and they are completely stuck with no idea how to proceed (that’s when they hopefully become more receptive to using other mathematical techniques and structure). An even more interesting twist on the story is that on occasion when this has happened a student who is normally considered “weaker” is able to solve the harder problem because she has been practicing the structure and techniques the whole time, and so for her the “harder” problem is actually that different than the “easier” problems, since she just uses the same methods and techniques.

What I would really like, if anyone starts to read this someday, is to have a place for math teachers in Sweden to share good ideas and material. Or at least have meaningful discussions about teaching.


Japanese math

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I recently read about the failure of Japanese students to answer 5 questions posed by the Mathematical Society of Japan. After much searching, I was able to track down one of questions used:

I don’t know if it’s the translation or a fault of the original text, but as it stands the wording of this question has a lot of problems that make it (in my opinion) a poor gauge of a student’s understanding of statistics.

I will focus on just one of these problems: the use of the word “average”. This was the same mistake made by the Swedish national test for 9th graders a few years ago. They had a problem asking for the “genomsnitt” (average). I spend a lot of time explaining to my students that one of the strengths of mathematics is its ability to be precise with definitions. And in particular that the word average has no precise mathematical meaning. If you want the mean, ask for the mean, if you want the median, ask for the median. Yes, in common language “average” usually means “the mean”, but this has no place in a national math test in any language.

There are of course a number of other problems with the text in this question …