My teaching mission is to

Guide students
as they become
mathematical thinkers
and communicators.

My teaching approach is designed according to that statement of purpose. I've written it like that in order to highlight each part separately.

First and most importantly, my job is to act as a guide.

The best practices recommended by educational research, as well as my experience as a teacher and learner, show that learning occurs best when students are placed in charge of their educational mission. Educational research has consistently shown that knowledge acquired though "inquiry, investigation, and discovery is different, more in-depth, more enduring, and much more powerful than what occurs in a traditional classroom". I see my role in the classroom as facilitator more than as lecturer, as resource more than as transmitter.

I am a guide for students on a purposeful journey, and so I pay careful attention to their learning process. Their path should:

• be towards a worthwhile and appropriate destination,
• to which they make steady progress,
• following a clear roadmap
• that still leaves students free to explore
• and trains them in setting a path themselves.

I help my students set challenging and appropriate goals, and entrust them with the responsibility of meeting those goals. I design instruction and evaluation around those goals. I help students incorporate the results of evaluation into their work (so that they make better progress) and keep summative evaluation focused on final goals (so that they make steady progress). Students (independently and in groups) investigate a variety of interesting, important and useful topics, and have significant latitude in determining those topics and their approach. Finally, I try to make students aware of the higher-order process they're undertaking -- I engage students in experiences that might engender contradictions to their initial hypotheses and then encourage discussion, and I have students reflect on and improve their learning process.

These peregrinations are, however, always focused on a core set of developmentally appropriate mathematical concepts. Mathematics we operationally define as

The use of numbers, symbols, and patterns
to understand and explain our universe

My goal is to introduce students to all the major tools in the mathematical workshop, and give them the time, guidance and freedom required to see that they are capable of constructing beautiful work -- work that is interesting, useful and important.

In doing so, I'd like them to become so fluent in deploying these tools that they reach for them instinctively and apply them skillfully throughout their lives. Rather than regarding the mathematical workshop as a place only visited during school hours at a teacher's behest, they visit often to putter around and make use of the many doors that link the mathematics workshop to their work, school and play settings.

Our operational definition of mathematics carries the following corollary:

If you haven't explained it, you haven't understood it.

I believe that most real-world applications of mathematical reasoning take place in a didactic context: "I feel this is the best strategy regarding alternative-fuels energy policy, and here are the tradeoffs involved"; "I believe this is the best design for our car's suspension, and here is why;" "Here is a notable fact about the sides of a right triangle, and here is why you should believe it to always hold."

In my classroom, learning is a social process. Students frequently present the results of their investigations to each other, and consult their classmates for ideas and instruction. They become critical and responsive consumers of information, and answer to each other as much as to me on standards of proof and discourse.

In a larger sense, I entrust the classroom community (students, teacher, parents and administrators) to appropriately set guidelines for classroom conduct and course direction. Since students are in charge of their educational mission, they are expected to set high standards, work hard, think deep, have fun, and meet goals. Since students are part of a community, they must not only govern their conduct to allow other students to learn, but they must also demand and help produce responsible conduct from their classmates.

Finally, as students take more charge of their education I increasingly allow their interests and responses to drive lessons, shift instructional strategies, and alter content (hewing, of course, to the core mathematical ideas of our course goals).