In Steve Strogatz’ Infinite Powers, the reader for our calculus course, Strogatz describes Archimedes search for a way to calculate the area of the curved region between a parabola and a line.
Archimedes came up with the strategy of reimagining the parabolic segment as a collection of infinitely many triangular shards glued together. Strogatz reflects that it was through his “noodling around” (2019, p42) that he developed intuition and that it was “an honest account of what it’s like to do creative mathematics” (2019, p42) Mathematicians typically develop intuitive ideas before a formal proof, but we rarely ask students in K-16 mathematics education to use their intuition or to think creatively about mathematics – these important acts are devalued or completely absent. In an important experimental study Schwartz and Bransford (1998) found that students were more interested, engaged and academically successful when asked to use their intuition about a solution before being taught formal methods, this made the methods meaningful and gave students an intellectual purpose for learning them. This activity is an occasion for creative and intuitive thinking about the areas of curves, that could be the opportunity for the learning of Reimann Sums and definite integrals.