Mathematics Educators Asked on September 5, 2021

For sake of argument, consider that skill of a topic is spectrum from "new and learner" to "experienced and expert." Where should you *relatively* be in order to teach the topic effectively so that your students could do well in the long run?

Omitting the situation where the student is more knowledgeable of the subject that the teacher themself, take the example where the teacher is more or less at the same level of the student. They would seem unprepared when a student asks a question that is beyond the scope of that current lesson or a question that demands more than a "Wikipedia-level" familiarity of the subject.

Then again, take the other extreme where one could be a PhD in number theory but teaching addition to kindergarten. Let’s assume that this person is also good with young kids and is effective with primary school education, I think it’s safe to say that being so advanced is overkill.

So if we say the teacher is more skilled in the topic than the student, the question becomes **how much more skilled?** For example, could a pre-calculus teacher effectively teach the subject thoroughly without not knowing measure theory? It seems very possible (and is probably overwhelmingly common). In fact, they could probably not know analysis and still teach well. But could a pre-calculus teacher not know calculus?

If we consider post-secondary teachers at colleges and universities as standard, they generally have a PhDs and they often teach 1st year courses. In the spectrum of Baccalaureate-Masters-Doctorate, then it seems that it’s 2 "degrees" ahead is what’s necessary?

There is no generally applicable answer to your question.

In my own career I often taught *while* learning the material as my mathematics department began to offer more and more computer science. I learned the material a course at a time by signing up to teach it. I talked a lot with colleague mentors who knew the material.

I always told my students in the first class of the term that we would be learning together. I think some weaker students thought about writing the governor to complain that their professor was unprepared. The better ones knew we would have an adventure. On the first exam in my assembly language programming course one student earned an A and my grade was just A-. He had better answers to my questions than I did. We are still friends.

Although I did not "know the material" my professional experience as a mathematician and a teacher helped me have a good sense of what was important, where the difficult ideas would occur, how to learn them myself, and how to stay far enough ahead of the class as the semester went on.

At the other extreme of "how much more should the teacher know" the answer for elementary school teachers is "a lot more". I have spent time in elementary schools helping the teachers see how the material they are talking about in class fits into a much more sophisticated abstract framework they may never have been exposed to. The good teachers are thrilled to see that.

Answered by Ethan Bolker on September 5, 2021

I teach at community college. That means I have a masters degree. When I first taught statistics, I had never taken a statistics course. Luckily, it was before I adopted my son. I spent 60 hour weeks, studying from many different textbooks, and asking myself questions. (I proved some results that we would never prove in our stats class.) I was an ok teacher that first semester. (I never got great, not because of my limited content knowledge, but because of my limited enthusiasm for stats. You need to keep coming up with interesting data sets. Not fun for me.)

When I teach Calc II, it would help if I were also teaching Calc III, so I could talk more about how the topics continue on. (I have never taught Calc III. I once learned it. But that was long ago.)

To me, it's not so much what degrees you have, as how deep your understanding of the content of a course is, along with what it connects to in following courses. I'm not sure why you're asking this question, so I'm not sure how useful my answer is.

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