On November 6 I had the honor of being one of the panelists in a Symposium Honoring Zalman Usiskin, held to honor Zal’s many years of contributions to mathematics education, from his groundbreaking 1971 textbook Geometry: A Transformation Approach (GATA) to his continuing activities today.

My panel was supposed to discuss his work on the UCSMP curriculum. My two co-panelists had been deeply involved in that work, but I had not, so I took the opportunity to address one aspect of the mathematics that Zal pioneered in GATA and that remains prominent in UCSMP Geometry today: the treatment of geometric transformations as functions (a treatment that I describe here as “geometric functions”).

My hope was to show how activities based on The Geometer’s Sketchpad not only support Zal’s insight from over 40 years ago, but validate it in ways that weren’t even well-understood at that time. I wanted to make two main points, one about cognitive science and one about mathematics:

• Treating transformations as functions is supported by the cognitive science findings regarding embodied cognition and conceptual metaphor, as described (for instance) in Where Mathematics Comes From by Lakoff and Nunez. When students drag a point as the independent variable, they are experiencing variables in a physical way, and the act of varying point x and observing the resulting motion of point r_j(x) (the reflection across mirror j of x) becomes for the student a conceptual metaphor for the function that relates the two points.

• Treating transformations as functions enables students to connect geometry and algebra in a very direct and elegant way. Students can restrict such transformations to a number line, thereby turning the two-dimensional point variables of geometry into one-dimensional real numbers of algebra while simultaneously turning the transformation itself into a linear function. Having done so, they can apply a translation to the dependent variable to produce the Dynagraph representation invented by Goldenberg, Lewis, and O’Keefe, or they can apply a rotation to produce the Cartesian graph of y = mx + b (where m is the scale factor for dilation and b is the vector length for translation).

Here’s a movie I made of my presentation:

And here’s a Web Sketchpad version that shows the restriction of the geometric function to a number line to turn it into a linear function, and the subsequent transformation that represents the function as a Cartesian graph.

It was an honor to participate in this symposium, and I hope I did justice to Zal’s insight from so many years ago by showing its deep connections both to cognitive science and to the unity of geometry and algebra.

Note: In advance of the symposium and dinner, Lisa Carmona (Vice President at McGraw-Hill Education, preK-5 Portfolio) put up an eloquent post on the McGraw-Hill Education blog attesting not only to Zal’s accomplishments, but to the way he inspires so many of us to advocacy as well as a commitment to students’ deep understanding of mathematics.