In these lessons, we explored the affordances of drawing functions in the Cartesian plane with 3D pens in a high school calculus classroom. During the lesson, students were to draw the graphs of basic functions, such as y=x^2, as well as a “line” in 3D. Although the task is 2D in nature, in that students drew parabolas and lines, the third dimension came into play when they began to manipulate with the line so that it would “just touch” the curve at one point.

Having already learned the meaning of tangent, the students used their fingers to manipulate the “tangent line” by moving it along the curve and observed the change of its slope at different points of tangency. We found that this task offered a physical substantiation of tangent to a curve, in that student could feel the idea of local linearity by the sense of touch. The students also easily grasped the idea that they could pick up the parabola and translate or reflect it when working with functions such as y=x^2+1 or y= -x^2, which allowed them to explore the relationships between the graphs of derivative functions.

In another pilot calculus lesson, we invited students to draw the “solids by revolution” before they learned to solve for their volume using definite integrals. The students employed various 3D drawing strategies which are worth describing because of the interplay between the drawing process and the solid formed—the gesture-diagram interaction—that was facilitated in the process. For example, when asked by the classroom teacher to visualize the solid formed by revolving a curve about the x-axis, they invented a strategy that made use of the “x-axis” as a manipulative and the action of spinning the axis. Having drawn a curve and the coordinate axes with a 3D pen, they picked up the drawing from paper, hold the two ends of the x-axis and began rotating it physically and rapidly.

These examples not only highlight mathematical thinking as an embodied activity but also the intricate interactions among gestures, diagrams and mathematical thinking that are mobilised in a 3D drawing environment.

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