Technologically Embodied

Geometric Functions

Geometric Functions

About this website

With Technologically Embodied Geometric Functions, students develop conceptual metaphors that directly relate computer-based sensory motor experiences of abstract function concepts. This approach relies on four foundations:

Use the Materials, Support the Work

We’re working hard to create the sketches, worksheets, and support materials for these Web Sketchpad activities. Ours is a volunteer effort on the part of everyone involved, both curriculum developers and field testers, and we make these activities freely available for you to download and use with your own students.

As we develop, revise and expand the activities, we need your help. Email your commentary (What worked well? What didn’t?) and your suggested improvements to the webmaster.

*T _{v}*(

by vector

by scale factor

Try dragging

News Flashes!

**IUSE Project:** From 1 July 2017 through 30 June 2019 we're working on an NSF IUSE project: Forging Dynamic Connections. (IUSE stands for *Innovation for Undergraduate STEM Education.*) Our project focuses on helping undergraduate preservice teachers make two kinds of connections:

- connections of which they're likely unaware between mathematical topics with which they're already familiar, and
- connections between physical manipulatives, a virtual constructive environment (Web Sketchpad), and abstract mathematical concepts.

In the course of the IUSE project we are designing and implementing lessons for a capstone course targeted to preservice secondary math teachers. Each lesson for the teachers is based on a lesson for secondary students, and the undergraduates experience the secondary lesson, observe and analyze videos of secondary students engaging with the lesson, and engage in discussion and reflection about how they would teach the secondary lesson, about what worked well and what didn't as they experienced the lesson, and about how they would modify the lesson to use it themselves.

**ICME Presentation:** During the last week of July 2016, Scott Steketee presented a Geometric Functions paper at the International Congress for Mathematics Education, the premier international conference that meets every four years. Scott used a Bluetooth mouse to engage audience members in doing the mathematics described in the paper. The figures in the paper are live, so if you go to the paper you can interact with the figures yourself.

**NCTM Honor:** The Geometric Functions article "Connecting Functions in Geometry and Algebra" (by Scott Steketee and Daniel Scher) from the February 2016 issue of Mathematics Teacher was recently chosen as the favorite article of the year by the magazine's Editorial Panel. Like the ICME paper, the figures in the NCTM article are live if you read the article online.

Welcome to Geometric Functions

Functions are an essential part of mathematics curricula, yet research shows that students struggle with this topic. Students’ early experiences with geometric transformations (reflections, rotations, translations) are rarely used to develop function concepts, but provide compelling kinesthetic experiences and opportunities for explicit discourse about important function ideas. Dynamic geometry programs such as Web Sketchpad enable students to transform a point, drag it as an independent variable, and explore properties of the resulting function, fulfilling CCSS-M Standard G-CO2: students should “describe transformations as functions that take points in the plane as inputs and give other points as outputs.”

Recognizing the importance of this opportunity to support students in their understanding of functions, and taking into account considerations of cognitive science, mathematics, technology, and pedagogy (see the Foundations described below), we’ve been hard at work developing a series of activities to implement this vision.

Geometric Functions Activities

- The first unit of student activities on this site (Introducing Geometric Transformations as Functions) introduces function concepts by way of geometric transformations, providing students with sensorimotor experiences of variables, functions and their families, domain, range, and so forth. Because students themselves create, manipulate, observe, and analyze mathematical objects, these activities establish a secure groundwork on which students can build a deep understanding of functions.
- The second unit (Connecting Geometry and Algebra Through Functions) connects the world of geometric transformations, where variables are points on a two-dimensional plane, to the world of algebraic functions, where the numeric variables can be thought of as points on a one-dimensional number line.
- The beginnings of a third unit (Variables and Functions on the Complex Plane) is available in draft form.
- Use the Activities menu in the navigation bar above to directly access individual activities.

The activities in these two units, though new, have already been classroom-tested, with very promising results. The most critical student materials (the websketches and worksheets) are very complete and usable, though we remain committed to further testing and refinement (for which reason we welcome your suggestions and other feedback). The how-to videos (short silent videos, less than a minute long, that illustrate particular worksheet steps) are available for some activities, but not yet for all. Performance-based assessment websketches are available for a number of activities, and will be added to others. Teacher-support materials (learning objectives, sample lesson plans, suggested strategies for launching and wrapping up each activity, sample classroom videos, and so forth) will eventually be available for each activity, but for the most part remain to be created.

Web Sketchpad Technology

The activities in both units use Web Sketchpad (WSP), a new technology produced by McGraw-Hill Education and derived from The Geometer’s Sketchpad. WSP supports a simple yet powerful user interface by enabling the activity designer to provide tools that are both easy to use and precisely targeted to a specific activity. WSP activities are compatible with any modern web browser, requiring no other software or licensing.

McGraw-Hill Education makes WSP free for non-commercial use. Though WSP is still undergoing development, and has not been officially released, it has an active beta-test program which allows us to host the geometric functions activities for you and your students. (For more details about hosting websketches, email the webmaster.)

Ongoing Work

As we continue to work on these activities, we recognize that much work remains to be done in developing, testing, refining, and evaluating both student materials and teacher-support materials. We are actively seeking funding to support that work.

We hope to develop additional units that use a geometric approach to further elaborate functions concepts. (For instance, we’d like to develop a unit in which students use geometric functions to plan and create their own interesting graphics special effects. We’d also like to develop a series of extensions to help students leverage the understandings they develop in Units 1 and 2 in their more advanced mathematical work in Algebra 2 through Calculus.)

We also recognize that the “geometric functions” approach to learning function concepts is a significant departure from traditional methods of introducing and developing these concepts, and we very much appreciate the support of teachers who see the promise and use the activities with their students. We hope you will give us feedback and suggestions—both your own and that of students—to help us improve the activities and other materials.

Foundations of Technologically Embodied Geometric Functions

Effective use of a geometric approach to learning functions rests on four foundations:

- Mathematical Foundation: Geometric transformations are mathematical functions.
- Cognitive Science Foundation: Students develop abstract concepts of function through sensorimotor experiences.
- Technology Foundation: Dynamic mathematics software enables students to create and manipulate geometric functions in a particularly direct and vivid way.
- Pedagogical Foundation: Activities are designed to connect students’ concrete experiences to their conceptual understanding.

Earlier Work

Earlier versions of several of the activities were developed through the support of the NSF project on Dynamic Number, and are freely available on the Dynamic Number site in a form that requires The Geometer’s Sketchpad Version 5, available from McGraw-Hill Education).

Update History:

02 January 2014: Created the page.