## Monday, April 23, 2007

### What's so hard about piecewise functions?

I'm teaching about absolute value functions these days, and about rewriting absolute value functions as piecewise functions. The students have a lot of trouble with this. In the hope that I'll remember to check out my notes on what they're struggling with before next time I teach this stuff, I'm storing them here.

The big problem seems to be to understand that the restricted domain is just that, a specification of what interval of the independent variable we are concerning ourselves with, and that the domain is not somehow "a solution" or part of the actual function or something. I am clearly unclear about just what is the issue, and how to go about filling in the missing parts. Apparently Dan Greene has figured out ways of approaching students' tendency to "flip back and forth between x and f(x) in their minds". I guess I should backtrack and spend a whole lesson or two on picking out the segment of a graph corresponding to a particular x-interval. Not that there's really time for that, but... it is needed.

## Thursday, April 12, 2007

### A meeting of Bay Area math teachers

Dan Greene organized a meeting of Bay Area math teachers today. We were not many, but it was good. We met in a cafe in San Francisco and chatted about teaching basic skills and building classroom culture, about school administrations and useful software for math teachers. Hopefully this will develop into a bigger thing, a forum where math teachers can share ideas and materials, dissect concepts and debate how to teach them, learn more about technology from each other, and maybe find encouragement from knowing that others are struggling with similar challenges. It would make sense that there would be some kind of society of Bay Area math teachers. (Might there be one out there already? If so, it certainly is not very visible! Hello..?! Has anyone heard of such a thing?)

We also talked about elementary math instruction, and about finding ideas and resources by seeking out materials designed for K-4 teachers. Both math and English teachers in high-poverty high schools can benefit a lot from materials and staff development designed for elementary school educators. We have to teach place value and multiplication, reading and capitalization - and the pedagogical approaches designed for teaching these things are hardly dealt with in Single Subject credentialing programs. Not that these programs are generally all that helpful with grade-appropriate pedagogy either, but the point here is about where to look for what we need, and that may well be in the Multiple Subjects teachers' bags of tricks.

Given the level of skill that our students have when they enter 9th grade, one can't help wondering about what goes on in middle and elementary school classrooms. When the students enter without mastering 4th grade standards, is that due to major classroom chaos in earlier years? Long-term teacher vacancies? Lack of subject matter competency among teachers? Anything else? Maybe some time in the future a local math teacher society could have monthly meetings where elementary school teachers could get help to deepen subject matter knowledge from high school teachers, while high school teachers could get input from elementary school teachers on making foundational concepts accessible. Just a thought.

While we may be inclined to think of teaching basic skills as a bit of a hassle, Dan enthusiastically insisted that teaching his Numeracy course is a lot of fun for the instructor as well as for the students. Making remedial instruction as enjoyable as possible is certainly the way to go: Since we'll need to do a lot of it, we might as well do it cheerfully.