Participants in the debates on math education argue about many things, but they seem to agree on at least one idea: math teachers should know math, the more the better. Unfortunately, many math teachers in the United States have not studied math much beyond the classes they teach. Others have, but found that collegelevel mathematics courses are not all that related to the material they teach.
One solution lies in developing preservice and inservice education that helps teachers develop more depth of understanding of precollege math. Zalman Usiskin considers this field a part of applied math, and calls it "Teachers' Mathematics". Ideally, each math teacher and each math department would be engaged in continuous studying of teachers' mathematics, in a way that is intimately connected with classroom teaching. Unfortunately, given how little preparation time teachers have, much of this has to happen at conferences and during summer workshops.
On this page, I link to lessons in teachers' mathematics which I have used with middle school and high school teachers. Most of the lessons are based on ideas that are accessible to students, though not at the same depth.
The lessons fall more or less in three categories:
Concept Analysis
 Look at familiar material from unfamiliar angles, in order to increase depth of understanding.
 Function diagrams
 Parabolas and Quadratics
 Iterating linear functions
 Exponential Functions
 Abstract Algebra
Problem Analysis
 Start with a problem that can be posed to students, and end with an analysis at a deeper level, generally by seeking generalization and / or proof.
 The "Mc Nuggets Problem"
 Staircases
 Pattern Block Trains
 Soccer Angles
Formal Development
 Put precollege mathematics in a more formal framework.
 Proof of Pick's Formula
 Geometry of y=mx+b
 Geometry of the Parabola: 2D  3D
 Geometry of the Conic Sections: 2D  3D

(background: Constant Sums, Constant Products)