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This page describes and illustrates eight elements of the mathematical foundation underlying the Geometric Functions approach—eight ways in which students’ sensory-motor experiences provide a route to mathematical understanding.

1. Functions need not be algebraic formulas with numeric inputs and outputs.

We aim for students to move beyond the limited notion that a function must take a number as input and use a formula to produce a second number as output. The definition of function is very general, and allows the variables to be of any type whatever.

2. Variables really vary.

By dragging the input point of a transformation, students experience variation kinesthetically. The input is truly a variable—and particularly compatible with human-computer interactions. By using the two-dimensional computer interface, students find point variables and geometric functions to be uniquely suited for manipulating by finger or mouse and for observing on a two-dimensional screen.

3. Variables vary continuously.

By dragging points, students experience continuous variation. This experience of dragging can then be extended to the numeric realm, helping students to more easily imagine numeric quantities varying continuously, not just serving as placeholders for discrete values.

4. Function notation is meaningful in the context of geometric transformations.

The notation r_j(x) can be read as “the reflection across j of independent variable x.” Using function notation in the context of geometric transformations helps students make sense of it:
r_j (“reflect across j”) describes the function and r_j(x) indicates the result of applying the function to independent variable x.

5. Functions map pre-image to image, domain to range.

Students see a collection of points as a coherent set that they can use as input to a function, resulting in a similarly coherent set of output points. Working with these collections of points helps them understand the concepts of domain and range and see how a function maps domain to range.

6. Function behavior is revealed by the variation of the dependent variable.

In the examples above, the student drags the independent variable. This is really easy; the independent variable can go wherever it wants in its domain. But the dependent variable has it a lot harder: its location is determined by the independent variable. By assuming the dependent variable’s role, by trying to follow the function rule as accurately as possible, the student develops a feel for the required speed and direction of the dependent variable (that is, for the relative rate of change of the two variables). Try out several function dances here: Function Dances.

7. Functions can be viewed atomically or collectively.

In the previous examples students first transform a single input point to a single output point, and then vary the input point along a restricted domain to transform an entire set of points, one after the other. Students can generalize this process by using an input point and its corresponding output point to define a Sketchpad custom transformation that operates all at once on sets of points, even the pixels of a photograph. For instance, a student could apply a sine function to a picture of the Golden Gate Bridge. By animating the horizontal shift of the sine function, the student creates an oscillating bridge—a function-based special effect that is novel and appealing.

8. Function behavior is embedded in static Cartesian graphs.

Students typically experience Cartesian graphs as fully-formed static objects whose variables are invisible and whose behavior requires interpretation through a numeric lens, disconnected from concrete sensory-motor experiences. To restore these elements to the Cartesian graph, the student can assume the dependent variable's role, dragging a point to try to match the behavior dictated by the function. By doing so the student matches not only the shape of the graph, but also the dynamic behavior of the function, thus experiencing the function’s rate of change and the behavior associated with extrema in a concrete, physical way. Through this and similar activities, students can internalize the covariation inherent to a variety of functions such as linear, quadratic, exponential, trigonometric, and rational functions—concrete experiences of instantaneous rate of change that serve as a precursor to differential calculus.

Update History:
31 December 2013: Refined the page design.
09 December 2013: Added the first video, and text under the numbered elements.
02 December 2013: Created this page.