# Outlook on teaching math

I maintain a resume. Years ago I added to it an “Outlook on Teaching Mathematics” page. I jammed as much of what I thought was most important as I could into one page (without using small fonts or obnoxiously narrow margins). Some of the items might be too compressed, or too specific, or too personalized, but that’s the danger of trying to put it “all” into one page that you plan to show employers, or programs that you want to accept you. The advantage, of course, is that otherwise I might have run on for twenty times longer.

Here it is:

• * Best mathematics teaching inspires. At all levels and all ages it is possible to communicate some of the elegance, power and beauty of this most abstract subject.
• Mathematics encapsulates abstraction from the real world. A child learns to count spoonfuls, learns to count people, learns to count fingers, learns to just plain count, and in the process acquires the abstract concept of, for example, “two.” The child takes ownership of this concept, and can reapply it freely. As adults we may take “two” for granted, but we have never met it, never touched it, never tasted it. It is one of the first completely abstract concepts that we ever owned.
• Learning mathematics involves skill acquisition, drilling, repetition, and instruction by an authority. It also involves independent construction of knowledge, connection to physical or real world situations, reflection by the learner, and independent reapplication to new situations. Traditional instruction has been overwhelmingly weighted to the former list, standards based instruction to the latter. Neither by itself gives learners adequate opportunity to take ownership of the abstract concepts that make mathematics beautiful and powerful. Best mathematics instruction carefully blends traditional and standards-based techniques.
• I strongly believe that instruction should be adjusted or modified to meet the needs of the current students. This entails a constant process of carefully planned experimentation, reflection, adjustment, and evaluation. Further, I have found it valuable to share with students information about modifications (pacing, depth, styles of instruction, balance of traditional/non-traditional work), and to solicit additional feedback from them.
• Concept ownership takes place more readily when the learner considers him or herself a stakeholder in the process. To this end it is desirable to foster a sense of control or ownership of other aspects of the classroom, including, as appropriate, involving the students in some decision making (see above). It is also possible to make students part of the subject itself, whether through data studies of the class or students’ families, or the creation of geometric figures based on the students’ own birthdays.
• An effective instructor is also a learner. I continue to take courses in mathematics and to study on my own. I am an avid problem-solver. I have never stopped trying new techniques in the classroom, and modifying, or rejecting them based on actual experience. As a role model it is necessary to share this love of learning with students. I freely admit when I do not know, and gladly share with students how I intend to search for “the answer.”
• A teacher of mathematics must be able to distinguish between right and wrong answers. A teacher evaluates alternate approaches, and distinguishes between minor and conceptual errors. The teacher can place a topic into a broader mathematical context, and answer the questions “Where is this topic next applied within mathematics?” and “Where is this topic applied outside of mathematics?” (if there is an answer) Grade level curricula are a subset of mathematics as a whole. It is the teacher’s responsibility to ensure that what is being taught not only leads to a correct answer, but is mathematically valid and will not need to be corrected at subsequent levels. It is not acceptable to know just a bit more than the students. An effective teacher’s knowledge of mathematics must be extensive.*

### Trackbacks

- And now for some different fare… « JD2718
- Math A: What was it? « JD2718
- New York State: looming geometry teacher shortage « JD2718
- Should they know what’s on the test? « JD2718
- My outlook on teaching math « JD2718
- June 2001 letter about the Math Wars « JD2718
- I was once in a math war skirmish… Aftermath… We win… and lose « JD2718
- Math Teachers at Play 30 « JD2718
- Quotations XXVI: On Teaching Math « Let's Play Math!

I love this. I particularly like your closing bit about teaching grade-level math in the context of the larger world of mathematics. I never want to teach my students something that will hurt them in their future mathematics studies no matter how much benefit it gives me at that time.

Thank you. I have been fortunate to work around and study with highly skilled, knowledgeable mathematics teachers. I think this represents some of the best they have taught me.