Prior to introducing the concept, a teacher asked students to try to determine a way to calculate the area under a curve on their own. They were asked to write down their ideas including what worked and what did not work.
This student looked at the problem and decided on an initial approach. When the process was identified as “not good enough”, the process was changed and re-evaluated based on the same initial criteria.
This student started by drawing the problem to get a better understanding. She noticed the general area under the curve was roughly triangular and thought of determining the area of the triangle. She also tried to think of it as a quarter of a circle. She graphed the quadratic equation and tried to find rectangles and triangles. But she realized that the Pythagorean theorem could not be applied to find the area of the “triangles” as the hypotenuses were curves rather than straight lines.
Finally, she tried to find the area by thinking about the mass of the graph in relation to the mass of the paper and the area as a proportion of the total. This is an unorthodox approach to area and not the method she will be taught, but it represents a serious attempt to generate and analyze approaches based on known mathematics concepts.