Thank you for your clarifications. You have clearly thought out your position and it is based on much experience.
I, on the other hand, am a student teacher in my third year of a three-year program doing my B.Ed. concurrently with my B.Fine Arts, Music. The fact that I enjoy and have proven good at teaching math has been a welcome surprise.
As you may imagine, and your comments indicated, teacher's colleges these days are promoting alternative assessment, NCTM standard-based active learning, etc., etc..
I had some problems this year teaching integers using these new methods (alge-tiles and other, number-line - based manipulatives), to the point that the kids themselves said "Can't you just teach us the rules and give us the homework? - this is too confusing!". I ended up using both and, as an experiment (loose use of the term) gave them the same test their teacher had used last year. The marks were an average of 18% higher, and the lowest score was a C+. I think this actually bolsters what you were saying - that new methods often overlook focus and practice. In this I am in total agreement with you. Yet in spite of their professed confusion, the kids actually discovered the rules for themselves - by the time I told them they all either knew them or had some intuitive idea of them.
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