For several years now, a number of mathematics educators and educational researchers have advocated using mathematical functions as the unifying principle of secondary students’ study of algebra. But for too many students, our current approach to teaching about functions isn’t working.
The abstract concept of function (including independent and dependent variables, domain, range, covariation, related rates of change, composition, inverse, and so forth) is difficult for students to master in the symbolic realm of algebra. Students’ understanding of functions is insufficiently grounded in concrete experiences varying the independent variable and observing the resulting behavior of the dependent variable. Too many students graduate from secondary schools with a poor understanding of these concepts.
Various studies have examined the effects of studying geometric transformations on students’ development of a sophisticated concept of function, and some secondary-­‐ education curricula have been developed that present geometric transformations as functions. It seems reasonable to think that students may find the concrete, visual nature of geometric transformations (such as translation, reflection, rotation, and dilation) more accessible than abstract symbolic forms (such as f(x) = x2 ‑ 3x + 2).
Modern dynamic mathematics software has shifted the ground by enabling students to directly and continuously drag geometric variables (points) and to observe and analyze dynamic visual depictions of function behavior. Integrating a technology-­‐based geometric approach into students’ study of algebraic functions would seem to have considerable potential benefit to students’ development of a robust and flexible conception of function.
Nevertheless, few of today’s students have the opportunity to integrate transformations into their understanding of functions, and even fewer begin their study of function with geometric transformations.
We start our discussion from this point of view: that understanding functions is hard for many students, that students can benefit from integrating geometric transformations into their study of algebraic functions, that technology facilitates the manipulation and visualization of transformations as functions, and that too few students experience transformations as functions.
One aim of our discussion will be to examine why geometric transformations are not already more widely integrated into the study of function. What are the benefits, and what are the obstacles?
A second aim will be to share our experiences, both successes and failures, in efforts to develop and promote such an approach.
And a third will be to strategize with each other, to consider how best to encourage and facilitate such a change in students’ experience of function.
Thus it is our hope that through this discussion group we can consider some of the issues involved in integrating a geometric approach into the study of function, identify potential advantages and pitfalls, discuss how this approach might relate to—and modify—existing curricula, contemplate the need for the professional development of teachers, and generally help to jump start the related processes of research on and implementation of such a curricular change.

The following questions are suggested as starting points of the discussion:
• What are the potential advantages and disadvantages of integrating a geometric approach into the study of function?
• What role does technology play in integrating a geometric approach into the study of functions?
• Twenty years after the introduction of dynamic mathematics software, why have there not been more research studies on this approach, and why has there been so little adoption in schools?
• What research has actually been done, and what additional research needs to be done, to validate this approach?
• How well are students able to integrate the geometric and algebraic views of various concepts? For instance, how well does a student’s understanding of covariation in the geometric context transfer to the symbolic context?
• What factors bear on this transfer of learning? How can we best encourage students to connect their learning in the two realms?
• What other cognitive obstacles must students overcome?
• Which aspects of the study of function can be developed geometrically, and which cannot?
• What approaches to professional development are likely to be effective in helping teachers understand the approach and develop enthusiasm for it?
• How does this approach relate to ongoing curriculum-­‐reform efforts in various regions or countries?
• How can we in our professional roles best help move this process forward?