Many students perceive their teachers to know more than they really do (Math teachers in particular). If a Math student who just observed a teacher solve a problem at the board is asked “What do you think was going through the teacher’s mind as they solved that problem?”, I suspect the average answer would be something very specific, like “This is a second degree polynomial in standard form, so solving this problem will require precisely six steps, the first of which is…”

In reality, the jumble of thoughts in a math teacher’s mind probably go something more like: “What looks easiest to simplify first? Oh… I see two places, no make that three, that I could start… but which should I choose? Does this look like it will take a lot of room to solve? If so, I had better organize my work a bit more… OK – now that I have simplified things a bit, what options do I see from here?”

Teachers and mathematicians do not “see” the entire series of steps needed to solve a problem before they start work on it. Instead, they usually seek to take whatever step looks like it will simplify the problem the most, and have no clue (yet) what will follow that. Once that first step has been completed, the appearance of the result influences what is tried as a second step. It is an iterative process, one step at a time, with a re-evaluation of the situation being done after each step. There is much uncertainty in the process, even for a math teacher.

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