Technologically Embodied
Geometric Functions

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

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Technology Foundation of Geometric Functions

Existing Technology

Several forms of technology have been used, to greater or lesser extents, in students’ study of function concepts:

• graphing calculators,
• computer algebra systems (CAS systems),
• computer-based rangers (CBRs) (Bazzini, 2001; Borba, 2004), and
• dynagraphs (Goldenberg, Lewis & O’Keefe, 1992).

These are all useful supports as students investigate various kinds of functions. However, it’s worth noting that these technologies are based for the most part on functions defined by algebraic equations, and that the two most common (graphing calculators and CAS systems) provide little opportunity for students to directly vary an independent variable and observe the effect on the dependent variable.

Geometric Functions Technology

The Geometric Functions activities are based on a geometric approach to function concepts, and provide a particularly close connection between students’ sensory-motor activities and the corresponding abstract function concepts. For instance, students drag a physical finger or mouse in two dimensions (on a touch screen or mouse pad) to vary a geometric point (the independent variable) in two dimensions (on the Euclidean plane). As they drag, they observe (and can trace) the motion of a transformed point (the dependent variable) on the two-dimensional screen. This experience provides an extraordinarily close connection between the concrete physical reality and the corresponding abstract concepts: The two-dimensional physical motion corresponds to 2D mathematical variation, the two points (the dragged point and its observed image) correspond to the independent and dependent variables.

In cognitive science terms, we can use the term conceptual metaphor to refer to the connection between the manipulated and observed concrete objects on one hand (the dragged point and its observed transformed image), and the abstract mathematical concepts on the other hand (an independent and a dependent variable related by a function). This connection seems so close and so direct that we might venture to say that it shows a remarkably short metaphorical distance—that is, the concrete reality is extraordinarily similar to the abstract function concepts to which the metaphor refers.

The activities in their current form use The Geometer’s Sketchpad (Jackiw, 2009), due to its user friendliness and its powerful function capabilities. The affordances of Sketchpad allow the activities to address aspects of function (such as function notation, restricted domain, and composition), to incorporate real-time experience of covariation and rate of change, and to exploit powerful function-embodying technology such as locus constructions and custom transformations of arbitrary objects, including pictures. (See the November 2012 Mathematics Teacher article on “Multiple Representations of Composition of Functions.”)

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