Friday, August 10, 2012

Mathematical thinking

When I have time, I read other math blogs and a couple of discussion groups on LinkedIn. One of those discussions pointed me in the direction of an article entitled The Calculus Trap. In the article, Richard Rusczyk says:
Developing a broader understanding of mathematics and problem solving forms a foundation upon which knowledge of advanced mathematical and scientific concepts can be built. Curricular classes do not prepare students for the leap from the usual ‘one step and done’ problems to multi-step, multi-discipline problems they will face later on.
The curricular classes he's talking about are pretty much everything we teach in K-12 and lower division college mathematics, up to and including calculus. From my own experience, I treated math as a tool to force to my will and Get The Answer. And I was pretty good at it. Until I reached the upper division college math classes required for my major. Abstract Algebra and Analysis of the Real Variable completely derailed me. Because math is NOT just about Getting The Answer. It's about logic and philosophy and a way of thinking. But we don't teach that. We teach students to Get The Answer. And it's possible to do well in math all the way through calculus with that mindset. But doing this stunts the advanced students and does a disservice to the average students. How can we get around this? When teaching the cornerstone math courses, especially Algebra, how can we get students to think more holistically? The suggestion in the article is to create clubs that encourage intellectual curiosity, and that's fine, as far as it goes. It seems to go against the trend of including all abilities in one class. As Rusczyk says, this is self-selecting for people who like to think about things. This seems a little like what would happen when students were "tracked". The smart kids are together in a group and push each other to perform better. While we now have classes that have mixed ability levels, extra curricular activities could help with this. But, how about bringing the average students up to a more abstract level of thinking? Can we employ techniques in the classroom that do this?

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