In my view, mathematics education is not about the authority of the teacher, it's about developing the students as their own mathematical authorities. Ultimately, the authority in mathematics is that any one of us can use what we have learned, even via different methods, to verify what other people have done.
In mathematics classrooms, we do not learn only mathematical rules and structure. We learn to think mathematically. And this involves learning something about our identity relative to "mathematics." Valuable things for your students to learn:
- Every one of them can wield mathematical authority; that's the power of learning mathematics.
- In mathematics (and in science, and in life) we have some responsibility to trustworthiness, which means an attitude of eagerness to be challenged.
- Justification is not an inconvenient response to a challenge, it is a vital mathematical practice
- We should avoid anything that teaches the student that making a mistake is something to be ashamed of; when students learn that, they begin to attribute their own mistakes to "not being good at math." They start to believe that someone good at math would not have made that mistake.
A teacher who is relying on cultivating the air of an infallible mathematical authority will have a hard time conveying important aspects of mathematics to the students. But worse, that concern seems to put one's own personal emotional well-being ahead of the students' learning. If this façade helps you avoid challenges from the students, is it for their good, or is it ego-driven? It's a question educators should at least consider. Who benefits?
If students learn from you that even teachers make mistakes, but yet they persevere in correcting them, they may see themselves as possibly reaching your level, with some work. Isn't that, ultimately, our goal as mathematics educators?
I thought I would offer one peek into the implications of the development identity in mathematics classrooms, since I am interested broadly in STEM education and how people come to see themselves as entering STEM fields. Jo Boaler has done research on the implications of different models of teaching in math classrooms, and one perspective she has used to compare traditional and more heavily discussion-based classrooms is to look at the developing identities of the students. One brief excerpt from one of Prof. Boaler's papers illustrates some of what students say when they begin to think mathematics is not for them:
The disaffected students we interviewed were being turned away from
mathematics because of pedagogical practices that are unrelated to the
nature of mathematics (Burton, 1999a, b). Most of the students who
told us about their rejection of mathematics in the 4 didactic
classrooms – 9 girls and 5 boys, all successful mathematics students –
had decided to leave the discipline because they wanted to pursue
subjects that offered opportunities for expression, interpretation and
human agency. In contrast, those students who remained motivated and
interested in the traditional classes were those who seemed happy to
‘receive’ knowledge and to be relinquished of the requirement to think
J: I always like subjects where there is a definite right or wrong
answer. That’s why I’m not a very inclined or good English student.
Because I don’t really think about how or why something is the way it
is. I just like math because it is or it isn’t. (Jerry, Lemon school)
In other words, there was a sort of filtering process going on in heavily didactic classrooms, teaching students to select themselves out of mathematics if they were interested in having control of their own choices (human agency), self-expression, and interpretation. The students who stayed interested were ones who felt relieved at the lack of complexity of "one right answer."
By contrast, the students in a more discussion-oriented calculus classroom did not see mathematics learning as conflicting with their developing identities as expressive, interpretive people with personal agency. Part of the difference between these two types of classroom is in how authority is seen.
It's more than just how mistakes are handled, of course. I offered this example to show that there is much more going on in math classrooms, and part of it is very important to how students determine what they are going to put their effort into.
Something to consider!
Boaler, J. (2002). The development of disciplinary relationships: knowledge, practice and identity in mathematics classrooms. For the Learning of Mathematics, 22(1), 42–47.