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.