Launchings from the CUPM Curriculum Guide:
Preparing Secondary Teachers

David M. Bressoud December, 2006

This last set of recommendations are aimed at specific groups of students within the mathematics major. This month’s recommendation deals with preparing prospective secondary teachers. In writing this recommendation, the CUPM was very aware that it was moving outside of its area of expertise and into the purview of other committees. In particular, in 2001 the Conference Board of the Mathematical Sciences, an umbrella organization of the mathematical societies including AMS and MAA, issued its recommendations on The Mathematical Education of Teachers [1]. The CUPM recommendation is not intended to serve as a set of guidelines for pre-service teacher education in mathematics, but rather as a distillation of four of the most important and commonsense recommendations that have emerged in recent years.

One of the most important resources that any teacher can bring to the classroom is depth of understanding of the subject at hand. This includes knowing where it came from and why it arose, knowing the conceptual difficulties that people encountered during its development and the difficulties that students are likely to have as they master and learn to apply it, knowing how it relates to other parts of mathematics and to problems in other disciplines, and knowing when, why, and how prospective teachers will need to draw on this knowledge in the future. This depth of understanding includes a rich reserve of examples that illustrate different aspects of the topic, together with good questions that probe and test student knowledge at all levels from basic recall through sophisticated analysis, synthesis, and evaluation. It includes knowing when and how technology can be useful in helping students through a difficult conceptual point. It means knowing where this topic lies in the great web of mathematical ideas and how it relates to the ideas around it, those that should come before, those that will come after, and those that students might never see.

Four years of undergraduate mathematics is not sufficient to create this depth of understanding. Not even the additional years of graduate work will complete it. This is a bed of mastery that takes a lifetime to create, but it must be begun, and begun well, by the time the future teacher graduates with the bachelor’s degree. This includes beginning to string together the connections. This is why it is so critically important that future teachers receive a mathematics education that emphasizes connections. They need to learn to look for them. Whether learning analysis, algebra, or geometry, prospective teachers need to understand how the mathematics they see as undergraduates is connected to the mathematics they will be teaching.

There are now many excellent resources to assist in putting together such a program. The Illustrative Resources lists many of them. An article that I consider particularly insightful is H. Wu’s “On the education of mathematics teachers” [2].See also "What (Future) High School Math Teachers Need to Know about Trigonometry" by Richard Hill[3]. For those who are interested in working with pre-service teachers, the PMET program (Preparing Mathematicians to Educate Teachers) [4] runs workshops, publishes information, helps to create and maintain networks, and provides mini-grants to support work in this area.

[1] The Mathematical Education of Teachers, volume 11 in the CBMS series "Issues in Mathematics Education," 2001, AMS and MAA, Providence RI and Washington, DC,

[2] H. Wu, On the education of mathematics teachers,

[3] Richard Hill, What(Future)High School Math Teachers Need to Know about Trigonometry,

[4]Preparing Mathematicians to Educate Teachers, homepage at

Do you know of programs, projects, or ideas that should be included in the CUPM Illustrative Resources?

Submit resources at

We would appreciate more examples that document experiences with the use of technology as well as examples of interdisciplinary cooperation.

David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, he was one of the writers for the Curriculum Guide, and he currently serves as Chair of the CUPM. He wrote this column with help from his colleagues in CUPM, but it does not reflect an official position of the committee. You can reach him at