Presentations at
Professional Conferences

Level: Grades 9-12

Description:
Most high school curricula seem to forget that the conic sections are geometric objects! I will explain in several ways that contrary to popular belief, all parabolas have exactly the same shape. I will use interactive software (both 2D and 3D) to construct the conics, prove their reflection properties, and show that they are indeed the result of slicing a cone. Finally, I will explore a question about soccer that unexpectedly leads to a hyperbola.

On this site:

Geometry of the Parabola

Geometry of the Conic Sections

Soccer Angles

Level: Grades 11-12

Description:
An advanced geometry elective I have taught biennially since 1992. Three components: symmetry (introduction to abstract algebra, recognizing symmetry groups around a point, along a line, and in the plane, art projects, tiling); transformations (complex numbers review, matrices, isometries); dimension (polyhedra, Platonic and Archimedean solids, duality, Euler's and Descartes' theorems, the fourth dimension.) Using Cabri 2 and 3D software, building with the Zome system, reading Abbott's Flatland.

On this site: Space

Level: Grades 9-12

Description:
High school math classes look very much the same from year to year and from school to school. Yet, other models are possible! We need not cede every aspect of pedagogy and curriculum to tradition, to textbook companies, or to the College Board.

On this site:

Presentation slides

Nothing Works

Urban School Math Department

Audience: This session will be of particular interest to department chairs and anyone involved in school change.

Description: Teachers value autonomy and specialization, yet the advantages of collaboration and flexibility are many. So are the complications. Hear the rationale for one department's move to intensive mentoring and the development of a collaborative ethic. I will assess decades of experience in this practice, and reflect upon its impact on teachers, curriculum, pedagogy, and learning.

On this site:

Presentation slides.

Teacher Collaboration (an article from Independent School, co-authored with Jonathan Howland).

Escape from the Textbook! sharing and collaboration network.

Level: Grades 9-12

Description:
Teaching high school math is a complex endeavor, where apparently contradictory approaches can complement each other: there is no one way that works with all teachers and all students. I will present my mix of techniques for organizing curriculum, sequencing concepts, designing rich activities, working with (somewhat) heterogeneous classes, leading effective class discussions, using cooperative learning groups, assigning homework, assessing student understanding, and other day-to-day concerns.

On this site: About Teaching

Level: Grades 11-12

Description:
Syllabus and highlights of an alternate math elective after Algebra 2, which I have been teaching biennially since 1991: paradoxes involving infinity, proof by contradiction, Cantor's discoveries, mathematical induction, chaos, fractals; connections to literature, philosophy, science, and computer programming. Readily available materials on these subjects tend to be written for either the general public or college students. My presentation will focus on how to make this content accessible in high school.

On this site: Infinity

Level: Grades 8-11

Description:
A sequence of lessons on parabolas, quadratic functions, and quadratic equations. The unit works well with Algebra 2 students, and includes activities with manipulatives, graphing, and symbol manipulation. These approaches lead to three distinct proofs of the quadratic formula, including a new one.

Bibliography: For the hands-on approach to quadratics and completing the square, see Lab Gear Activities for Algebra 1, by Henri Picciotto, Creative Publications (how to get it)

On this site:

An introduction to the Lab Gear.

Two graphical approaches

Parabolas and Quadratics

Constant Sums, Constant Products

Level: High School Minicourse

Description:
Units from a course I developed with my colleagues in the Math Department at the Urban School of San Francisco. Our approach is to cover fewer topics in greater depth and to use a variety of learning tools, both manipulative and electronic.

This presentation was initially created by Naoko Akiyama and Scott Nelson as a one-hour presentation. I joined them to expand it to a three-hour minicourse. Get the slides.

Bibliography: Much of the material is unpublished, but see

Hands-on approach to quadratics and completing the square: Lab Gear Activities for Algebra 1, by Henri Picciotto, Creative Publications (how to get it)

Geometric approach to complex numbers: Algebra II/Trigonometry: A Guided Inquiry, by Stein, Crabill, and Chakerian (out of print)

On this site: Seeking Depth in Algebra 2

Using Manipulatives to Teach about Angles and to Introduce Trigonometry

Level: Grades 8-11 Workshop

Description:
What is an angle? Interior and exterior angles in a polygon. Inscribed and central angles. Soccer angles. Trig ratios.

Puzzles and problems using pattern blocks and circular geoboards, plus the students' own bodies.

These new approaches to old topics provide both access and challenge and work well with heterogeneous classes. The labs enhance discourse and deepen understanding.

(I also offer an all-day version of this, covering more topics.)

Bibliography: See my book Geometry Labs.

On this site: See Angles and The Ten-Centimeter Circle

Minicourse

Description: Make regular polygons, pyramids, prisms, and antiprisms. Explore the relationships between the dodecahedron, the icosahedron, the rhombic triacontahedron, and more... Identify the components of icosahedral symmetry.

A hands-on lab with an amazing manipulative, making connections with many traditional geometry and trigonometry topics.

Bibliography:
The handouts are excerpted from Zome Geometry (Key Curriculum Press) a book I co-authored with George W. Hart, using the Zome System

On the Web:
For a taste of what is possible with the Zome System, check out George Hart's page about Zome Polyhedra, and his Advanced Zome Constructions.