While standard mathematics courses can be quite effective, all students (and especially those who intend to become teachers) should experience forms of mathematical learning that are less cut-and-dried than many courses now are. Effective approaches include discovery methods, working in groups, significant applications, and large-scale projects, as well as exposure to historical and cultural aspects of mathematics. The mathematics curriculum also should include more threads than just the path to and through calculus.
Students should also have opportunities to learn with technology, especially for visualization. This does not mean courses about computers or calculators, but courses which use computers or calculators. While applications should play an important role in mathematics education, they should not displace the strengths of traditional mathematics, namely the power of general--yes, even abstract--ideas.
Mathematics departments should do a much better job presenting the history and the present status of their subject. Some courses should connect with contemporary mathematics research, even though it is a difficult challenge to design such courses. All teachers, especially teachers of mathematics teachers, should find out more about how mathematics is used in the world of work.
There is far too much mathematics to learn to complete the job of educating a teacher during an undergraduate or even a graduate program. It is also too late to wait until Eduction School to begin to think about ways of learning mathematics. Mathematics teachers must expect that the job includes life-long learning, so higher education should provide challenging and stimulating opportunities for adult learning.
We in higher education expect a lot of teachers. Therefore, we have a great responsibility to practice what we preach and model what we advocate.
W. James King, Professor of Mathematics at the University of Washington, works with teachers to help expand their knowldge of geometry.