Pedagogy: Outlook on teaching math
Dave at Mathnotations wrote this big ideas about teaching math post. It got me thinking.
I maintain a resume. Years ago I added to it an “Outlook on Teaching Mathematics” page. I jammed as much of what I thought was most important as I could into one page (without using small fonts or obnoxiously narrow margins). I’ll post it here, in three pieces, over the course of the next few days. Read on for the first piece.
• Best mathematics teaching inspires. At all levels and all ages it is possible to communicate some of the elegance, power and beauty of this most abstract subject.
As adults we may take “two” for granted, but we have never met it, never touched it, never tasted it.
(more below the fold) —————->>>
• Mathematics encapsulates abstraction from the real world. A child learns to count spoonfuls, learns to count people, learns to count fingers, learns to just plain count, and in the process acquires the abstract concept of, for example, “two.” The child takes ownership of this concept, and can reapply it freely. As adults we may take “two” for granted, but we have never met it, never touched it, never tasted it. It is one of the first completely abstract concepts that we ever owned.
Best mathematics instruction carefully blends traditional and standards-based techniques.
• Learning mathematics involves skill acquisition, drilling, repetition, and instruction by an authority. It also involves independent construction of knowledge, connection to physical or real world situations, reflection by the learner, and independent reapplication to new situations. Traditional instruction has been overwhelmingly weighted to the former list, standards based instruction to the latter. Neither by itself gives learners adequate opportunity to take ownership of the abstract concepts that make mathematics beautiful and powerful. Best mathematics instruction carefully blends traditional and standards-based techniques.
At all levels and all ages it is possible to communicate some of the elegance, power and beauty of this most abstract subject.
[The whole piece is available here.]
JD2718 
I’m not sure what “traditional” means either. I imagine that when most of us uninitiated use the term that we mean something like the way things were before 1970…and in the United States at that.
But some of these “traditional” approaches resulted in only a fraction of the class making it through to the end.
It’s hard for me to comment in debates over this issue without seeing exactly what is being criticized. Given a specific topic,e.g. long division, and a well-defined approach it’s easier for me to see what’s going on. I have certainly seen silly kindergarten lesson plans involving calculators. On the other hand, I’ve rolled my eyes at seeing worksheets in a “traditional” program where the student has to be prompted on where to carry the one when doing addition.
Since I can’t cherry-pick teachers for my children the best solution that I’ve come up with is to learn math myself.
…
“As adults we may take “two” for granted”
Ugh, not any more. I’ve got to make it past “one”.
…
“Think independently. No, not thatway. Thisway.”
beautiful thoughts…
now i will feel guilty if my lessons for tomorrow are less than inspirational! now what was that message i read on the back of some athletes’ shirts at my school: “When you chase perfection, you catch excellence!”
Since math may be pretty close to the embodiment of perfection, I guess we are trying to chase that perfect math lesson…
BTW, I think I’ve grasped the essence of ‘ONE’ – it’s the loneliest number! Of course if we could make it prime it would have some friends…
Dave
My sentiments echo Dave’s. Two math teachers that made a difference for me were Mr. Frezza and Ms. Fogel. I struggled with Algebra until summer school, after my 9th grade year. Mr. Frezza was my summer school math teacher and during breaks and lunch, he would sit with me, explaining poblems in various ways and encouraged me until the light bulb went off in my head and I got it. Ms. Fogel was a semi-retired teacher who taught part time, but gave it her all. One day when a kid smarted off to her and brought up her pay, she told that kid that the school didn’t and couldn’t pay her enough. She was there because she cared and wanted to be there and then told the kid to get his ass out of her classroom. Just like that! A little unorthodox? Yeah, but it was just what was needed. She sent the message, at least to me, loud and clear that she was more concerned about us than her check. I’ll never forget either of them.