Lately, a significant number of thoughts about the classification of necessary mathematical and necessary methodological knowledge of future and current teachers of mathematics is present. Do you think that these (pre-service and in-service) teachers should be informed to these classifications? If yes (or not), why? in which way would such knowledge be able to raise the quality of mathematics teaching?
yes,
mathematics teachers should be informed about the classification of necessary knowledge for teaching.
Yes of course. The more the math teacher is familiar with these classifications of mathematical knowledge components, the better it will be to choose the appropriate method of teaching each of these components, so that the appropriate way to raise the level of learning of students is to raise and improve the level of learning mathematics.
Well certainly yes if you mean algebra, arithmetic, geometry.
Perhaps I would not try to define homological algebra or algebraic geometry.
Yes , of course...
Math is a tool to understand the other sciences...
I would suggest that at least two classifications be made - procedural knowledge (know WHAT to do) and conceptual knowledge (knowing WHY one does it). These two categories reflect not only the characteristics of mathematics, but also map nicely into the psycholinguistics underpinning problem solving and concept generation.
They also create a quick evaluation potential quite useful in classroom instruction. (I know the following article is an oldie, but it outlines what I am referring to...)
Peck, D. M., Jencks, S. M., & Connell, M. L. (1989). Improving instruction via brief interviews. Arithmetic Teacher. 37(3), 15-17.
I would definitely agree with this. There is an art of teaching that is is entirely different than knowledge or even mastery of a field. For someone who had a natural aptitude or prior exposure to a class it is hard to know where some struggle or to remember how someone used to struggle on a concept or idea. Personally, proofs were a very weird transition in mathematics for me. I knew what a proof was and I was motivated to learn more about them, but I couldn't conceive the art behind it. Something just didn't initially "click" for me. I began reading G. Polya's book, "How To Solve it" and it has given me an idea of what a proof should look and feel like. Although, I am certainly no master at proofs now it has really helped me understand them in a much deeper way. This might be a bit of a stretch, but suggesting book like these prior to classes would certainly help build a solid basis before a student begins a course. I'd say the most vital thing is knowing where to send a student who is stuck on a concept or problem to find their answer.
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