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I'm not completely sure how well this addresses the question, but here is my best response. A few years ago I was teaching a Methods course for preservice secondary math teachers. Over the course of the semester there was a fair amount of griping, not about my course but about the other courses they were required to take, and in particular about their ...

Here is some text from the paper "Teacher Characteristics and Student Achievement Gains: A Review" (Wayne and Youngs, 2003): Three analyses take advantage of the detailed teacher data in NELS:88 on degrees. The analysis by Goldhaber and Brewer (1997a) illustrates the key finding most clearly. No differences were evident when the authors examined ...

I have faced exactly the same issue. I often teach masters level math courses to students getting a masters in education with a focus in mathematics; so these are future high-school math teachers. The way I look at it, my real task is to provide them with mathematical enrichment and sophistication, to show them a larger context and provide a deeper ...

The question is this: "why do you want to teach math ?" I'll try to stay positive here, I think anyone who has had experience with potential teachers is well aware that bad teachers are probably first bad students. Moreover, as one of my colleagues said, difference between doctors and us: they bury theirs we just graduate them. That aside, I would like for ...

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....

Of course, as already mentioned, it is important for a teacher to know more than her or his students. But one thing where this principle is especially important is exercise (and test) designing. Let me give a few simple examples. How do teachers manage to chose the second degree polynoms that have not-too-complicated roots? They start from the factorization,...

While teaching polynomials to future high-school teachers, I realized one example of pretty advanced mathematics that is needed to understand why a certain important (but often kept implicit) result in high school mathematics is a result, and not something so obvious that it need not be even stated: Two real polynomials taking the same value at each ...

To teach mathematics, you need to know, first, what mathematics is, no exceptions. The problem of school-math being a whole different thing having little in common with mathematics, is caused, among others, by people who teach school-math, but have a bad idea about what mathematics is. However, it's a hard chance that someone will have contact with ...

This heavily-cited paper examines data from the National Educational Longitudinal Study of 1988 and concludes that "teachers who are certified in mathematics, and those with Bachelors or Masters degrees in math, are identified with higher test scores." More specifically, the study finds that "the results reported... show that a teacher with a BA in math or ...

When my high school offered the AP course in Calculus, the teacher chosen to teach the class was the one who had studied it most recently. She was rusty and it showed. She was continually making mistakes in her demonstrations and the students wound up frequently correcting her. When I tutored college algebra, I frequently encountered prospective elementary ...

Examples from Abstract Algebra The reason why the product of two negative numbers is positive boils down to a proof from abstract algebra. $(-3)(4)$ should be the opposite of $(3)(4)$ because you add the two to get zero. Then $(-3)(-4)$ should in turn be the opposite of $(-3)(4)$. Why negative numbers exist, and why subtraction is just adding a negative. ...

This is but a part of the bigger picture and certainly not the most important one: Speeding up the legwork. Obviously you have to be better than your students in what you teach to, e.g., design and correct tests in a reasonable time. There are two ways of getting better at such things: Getting more routine with the techniques you are teaching or having a ...

First off, I would like to commend your desire to teach at a high school level. It is a very challenging and tiring profession but it is so rewarding and IMHO is one of the most useful settings that one can teach in (aside from maybe elementary school but that is a whole different ball game :) It sounds like you already understand one of the most basic ...

I would start with a discussion with the Engineering department about what it is they want the students to learn. Teaching different skills will need different methods, so you might as well start with learning what will help you most in the short term. Personally, my experience suggests proofs are likely to be very low priority, with a focus on calculation ...

In terms of researchers who I am personally familiar with: I would recommend work by Orit Zaslavsky. From her NYU-Steinhardt bio: Orit Zaslavsky's research focuses on mathematics teacher education. She has been among the first scholars to study the development of teacher-educators, from both theoretical and practical perspectives. Within this broad ...

For math pedagogy, I recommend exploring some of the blogs out there, in particular Dan Meyer's blog. He also has a great TEDx talk. The whole online community of math educators, sometimes called the mathtwitterblogosphere, is worth exploring. What's out there ranges from virtual filing cabinets of creative lessons, to folks venting about common frustrations,...

In my particular case, I was lucky to have some very inspiring teachers, and to be teaching assistant in large, well-organized courses. Yes, other teachers were awful, but I learned how to teach from the ones I admire (and I hope I'm doing them justice in my work). I learned a lot from colleagues, even (or perhaps particularly) from ones in other fields. ...

While my answer is not a book, author, idea, or article, it is a source of ideas: conferences of people who are interested in such things are very good for inspiring you to learn. The American Mathematical Association of Two-Year Colleges (AMATYC) has affiliates in each region and many of them host annual conferences. My first experience at the Illinois ...

I think you need to build a support network locally. Even if it means going outside the university to a nearby university, you will need mentors, fellow TA's, administrators, and others to help you not just understand and solve the problems you currently see, but to anticipate problems and improve your efforts as a teaching assistant. Even going outside ...

"What is my mission when I teach math for engineering students?" I think it is possible to zig-zag between applications and appreciation of the history and the beauty of the mathematics underlying those applications. An example is the "simple suspension bridge," which follows a catenary. There is interesting history here, in that Galileo realized the curve ...

If you are still doing research on PD, here are references specifically related to implementation of reform recommendations. These articles are based on the fact that successful implementation of reform requires most teachers to make fundamental and perhaps even radical changes in their teaching practices. Policies That Support Professional Development in ...

The best way is to surround yourself with people who are creative, smart, and actively thinking about and using effective pedagogy in the classroom. Twitter can be great for this. I'm fortunate to be in a department full of such people, such as David Coffey, John Golden, and others. I can just walk down the hall and embroil myself in a fascinating ...

A perhaps unusual recommendation: Lakoff and Núñez's book Where Mathematics Comes From, despite suffering from being mildly repetitive, has made me think very differently about mathematics education, as long as this is defined as the activity of instilling mathematical understanding in students (versus "mere" mathematical knowledge). Lakoff and Núñez, more ...

To the question about whether or not there is existant literature around some of these ideas, I believe there is a great deal. It seems you are getting at conversations around proof and how this plays out in teachers beliefs about mathematics. Several researchers have engaged in such pursuits. Immediately two things come to mind. The ICMI study on proof ...

If I were to take my master's degree again, I would take: 1-2 courses in abstract algebra, 1-2 courses in analysis, 1-2 courses in topology, 1 class in statistics, and 1 class in combinatorics for a total of 5-7 classes to get a good solid background in mathematics. I would then set up a year long apprenticeship with a master teacher. I guess this would ...

I don’t think that you can explain this to the students. Perhaps instead, you should better familiarize yourself with the high school standards and draw the connections. A high school mathematics teacher will teach material that heavily overlaps with all undergraduate coursework, and is only a slight extension of most ideas. While a high school ...