Summer Workshops for Math Teachers

In this three-day workshop, I will present a wealth of visual approaches to the teaching of algebra, including:

- Lab Gear manipulatives for basic symbol manipulation

- geoboard lattices for slope and radicals

- a powerful parallel axes representation for functions

- intelligent use of technology

- three distinct visual paths to the quadratic formula

Participants will learn techniques that will allow them to serve the whole range of students better by offering:

- greater access, because of addressing multiple intelligences

- greater challenge, because of expecting multi-dimensional understanding

- greater variety, because of using manipulative and electronic tools

In addition, participants will work on teacher-level problems rooted in high school subject matter, and strengthen their understanding of the underlying mathematics.

In this two-day workshop, we will rethink every aspect of a high school math program:

- Pedagogy: How can we do better than the I-explain-you-practice model, in order to reach a broader range of students, and to help them develop their ability to make sense of what they are learning? How do we balance discovery and instruction?

- Learning tools: How can we use technology support these changes, and move our program to become more visual, more interactive, and more creative? How can manipulatives simultaneously help us increase both access and depth?

- Curriculum: How can we escape the tyranny of the textbook, and develop our program by moving, removing, and adding topics in order to reflect the needs of our particular students? What electives can we introduce to get off the freeway to calculus and bring to students the variety and richness of mathematics?

- Assessment: How can we complement tests and quizzes with additional approaches that reveal different strengths and weaknesses? How can we give feedback to students based on our learning priorities rather than just on the now-obsolete emphasis on accuracy and speed?

- And finally: How do all these ideas play out if you teach in the long period? How do they affect our ability to work with heterogeneous classes? (All classes are heterogenous!) How do we navigate the pressure towards limitless acceleration? How do we surmount the obstacles in the way of progress?

For these changes to take root, it is essential for the department (or parts of it) to work collaboratively, to share successes and challenges, and to take this on as a step-by-step, gradual, and permanent project. This is what we have been doing for 25 years at the Urban School of San Francisco, and we are happy to share what we have learned. Bring your colleagues, your questions, and your ideas!

In this three-day workshop, I will present kinesthetic activities, as well as many hands-on and electronic tools and activities, and some enrichment lessons. This curriculum is intended to complement related work in paper-pencil and compass-straightedge environments: it serves to preview, review or extend key concepts in geometry.

• Tools include manipulatives (such as pattern blocks and geoboards) and puzzles (such as tangrams, pentominoes and supertangrams).
• Activities include "walking geometry,” "soccer angles," "tile design," and "rep-tiles."
• Participants will use interactive 2D and 3D geometry software to extend these activities, and to work through a challenging and highly motivational construction unit.

I will also present an authentic approach to proof, which tries to navigate a middle course between the too-abstract traditional curriculum and the insufficiently rigorous nature of some reform programs.

These lessons were developed in somewhat heterogeneous classes, and reach a wide range of students. They provide support for the less visual by complementing the drawing and studying of figures, and enrichment for the more talented by offering deep and challenging problems. In addition, participants will get to take some of the problems to a deeper teacher level.

This two-day workshop is designed for high school mathematics teachers who want to make Algebra 2, Trigonometry and Precalculus more accessible, richer and more fun. I will present a number of units to enrich and deepen the corresponding lessons in any textbook, whether traditional or reform.

• Concrete introductions to exponential, logarithmic, inverse variation, and trig functions
• Functions from patterns; functions from geometry
• Function diagrams: upper level applications
• Iterating functions as a gateway to sequences, series, mathematical induction and chaos
• A visual approach to complex numbers and matrices
• Geometry of the parabola (2D, 3D)

A frequent challenge in teaching upper level high school classes is the limited pedagogical range of most textbooks and curricula. This is particularly harmful to the students who find symbol manipulation difficult, but it is also cheating our stronger students of the multi-faceted understanding that would serve them best. To address this, several of the units presented in the workshop involve the intelligent use of electronic tools, hands-on activities with concrete materials, and/or creative alternate representations.