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.
The major advantage that a teacher has when compared to their students is that they have probably solved many more problems similar to this one before. They are therefore confident that they can find a solution, and have one or two alternative methods in mind they can resort to if their first choice of method turns out not to work so well. While a teacher may not have a clue as to exactly how many steps lie ahead, they probably do have a sense of whether this is likely to be a long, medium, or short problem.
The other “advantage” a teacher has is adrenaline (or is that a disadvantage?). They usually do not wish to make mistakes in front of an audience. While students may not perceive it, every time a teacher’s marker touches the board, the teacher’s mind is probably double and triple-checking the correctness of what they are planning to write, watching out for all the “usual” errors that they make: sign errors, distribution errors, six times seven is not 48, etc.
When I do make an error in working a problem at the board, the faster someone catches it, the better. If I make it all the way to a solution before someone points out an error in my first line, I have wasted people’s time. I would much rather have my mistakes caught quickly, which will help build class trust in both the solution and the process that leads to it.
So, for all Math students sitting in class watching someone solve a problem at the board: please follow the work of the person at the board closely, making sure you understand every step completely, and double-check their work. If you have questions, or think you have spotted an error, please speak up immediately (in a helpful way, of course). And lastly, don’t assume the person at the board knows the entire solution intuitively – they are probably working it out as they go while anxiously trying to avoid errors.
Problem solving is often non-linear, messy, and/or accompanied by false starts – even for teachers. Therefore, we should model it that way for our students! It is important for students to realize that solving problems can involve false starts, dead ends, and/or intuitive approaches.