(At work..)
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I have a good friend who teaches algebra in middle school. We've had many a conversation over how to improve teaching. Until, No Child Left Behind and other more recent changes the effect fed and state support for local schools, he used to constantly play around with different ways to get his students over the various conceptual humps.
But in the last few years he just stopped. The feeder highschool teachers were up in arms over the results of Andy's methods. They want old fashioned flat out competence in manipulation and calculation, period.
He used to call them the math nazis...
But the most important point is that he took the engineering series, which does not require some of the advanced algebra topics (groups, rings, fields, vector/matrix/linear algebra, isomorphisms, etc.) or number theory---and yet it is these subject IMHO where the greatest insights into algebra lay---and in turn which can lead to a much higher standard of understanding, even at an elementary level. For example, I could never get my algebra teacher buddy to get through even my level of understanding. He just didn't have the patience and wasn't exposed to these topics in either his engineering or teaching courses. So he could see where insights might be that could lead to ways of approaching elementary algebra...
Tonight I'll get back to all this, maybe. It's something I am really interested in and make all kinds of conceptual mistakes because I am entire on my own here...
CG