How can you hope to clearly explain something you yourself do not understand well?

I have found the following line of reasoning is quite convincing for many skeptics:

How hard did precalculus seem when you took it? After taking a semester of calculus, how did your perception change? How about after taking 3 semesters of calculus?

Mathematics is cumulative. The more you learn, the more you reinforce the basics.

By the time you finish the standard sequence of several semesters of calculus, differential equations and linear algebra, you start to find that algebra and the basic precalculus material is equally difficult with basic arithmetic. When first confronted with derivatives, maybe they seemed weird/difficult/tricky, but after a couple of semesters of material building on the basics, taking a derivative is like adding 2 and 2 or breathing.

The point is, the more you know, the more you will consider basic/easy/trivial. When teaching a subject, you want to be able to focus on how you are presenting the material - how to best communicate this to you students. You *don't* want to be fighting with understanding it yourself.

Along the same lines, the more you know, the better perspective you have as to why different topics are important. The more math you learn, the better you will be armed when confronted with a skeptical student of your own who asks, "Why do I need to learn this?"

Admittedly modern algebra as taught at most universities is a hard sell for you typical unmotivated future highschool teacher (why you would enter such a profession half-hearted, I do not know, but it seems common). If you focus on ring theory, you have a chance at selling the connections to factorization and the like. Groups are trickier because, honestly, that specific material may very well be completely unapplicable. However, students who spend a semester really engaging their modern algebra (or any other proof oriented) class should leave with at the very least bullet proof logic skills. This of course is of immeasurable value when teaching mathematics.

Anyway, that's how I've approached this topic with many students (and been successful as far as I can tell).

2I'm confused by the first sentence. The high school teachers don't need a bachelor/masters degree? – JDH – 2014-03-27T22:31:05.017

3@JDH No, they don't need it, but (at least) in German universities, most of the courses in mathematics are for both, bachelor candidates and high school teacher candidates. – Markus Klein – 2014-03-27T22:40:10.930

4If I recall correctly, it's not. For example, teachers with Masters degrees don't perform better than those with just Bachelors degrees. Knowing the abstract stuff doesn't really help them teach the high school stuff, and in any case you can easily get a degree in math education without taking an upper level proof-based math class. – Potato – 2014-03-14T17:57:30.020

1Are you looking for personal stories? – Thomas – 2014-03-14T18:39:02.157

2@Thomas: Sorry, no. I am looking for issues like "Without learning about <university subject>, it is difficult to understand the background of <school subject>". Or some arguments like "Only if you saw (at some point) the edge of reaearch based questions you really know that mathematics is and can teach" – Markus Klein – 2014-03-14T18:44:23.690

3I know this question is old, but wanted to note (for future readers, even if the OP knows this well) that this can be country-dependent. The life of a "high-school teacher" can be rather different in Western Europe (non-UK), Eastern Europe, the UK, Canada, the USA... – Yemon Choi – 2016-05-26T02:25:26.037

@Potato Discrete Math and Abstract Algebra are both required at my university for a Math Ed degree. – David G – 2014-03-15T07:02:00.843

1How are these educators going to be able to teach things to students that don't want to learn the material when they themselves are students that don't want to learn other material? – Robert – 2018-01-02T23:06:07.040