I came to realize that my job was not to tell students what to do. My job was also not to show them how to do a problem. My job was to create interesting situations where they could think about mathematics and learn from the work and discussions that followed. My job was to assist students in anyway I could to develop their own understandings of important mathematics. I have documented these thoughts and processes many times in earlier posts. Here is a new thought for me.
This same process applies to teacher improvement. Those in charge of teachers at the local, state and national level are ill served by piling on more and more rules and mandates about how teachers should teach. If they really want to improve instruction, they must follow the same model that I followed in my classroom. The supervisors need to see that their job is to do whatever they can to facilitate teacher learning (therefor student learning), as opposed to requiring teachers to do certain things in a certain way. Teachers, with proper resources, will find ways to reach students. As things stand now, those in charge are working very hard to make a teacher's job as hard as possible.
Administrators need to observe teachers teaching. Administrators need to listen to what teachers have to say about the difficulties they have, and administrators need to work hard to help the teachers solve their problems. I have had to good fortune of working for a few such supervisors, and it makes more of a difference than I would have ever imagined. And the very best supervisors worked hard to help the top-level administrators understand that learning to teach well is a very difficult process, that it takes time, and that it requires support and nurturing. Learning to teach well does not require demands and punishment on the part of the Administration.
It is part of the job of experienced classroom teachers to facilitate this process with newcomers and to help administrators understand what they need to do to be effective.
If I can sound corny in the comments, I'd say that your second paragraph points to the most important shift going on in math education today: away from teaching kids specific facts and algorithms, and towards engaging them as mathematical thinkers. As always, you're ahead of your time!
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