At the start of class for the last two weeks, my algebra team teacher and I have been using a discussion on “area under a curve” to extend our students’ thinking. Of course, our eighth-grade algebra I students do not have the calculus tools they need to solve this type of problem, and we are not trying to teach them the necessary algorithms. We’re asking them to envision different ways that the problem might be solved. One of our primary goals in this algebra class is to create divergent thinkers who realize that the process of solving a problem is often more important than the solution. By offering them problems that are “unsolvable” given the math that they know, they are forced to think of the problems as abstractions, to ponder the same ideas that intrigued the likes of Archimedes, Descartes, Leibniz, and Newton.
Our students’ first attempts to best measure the area under a curve mirrored the earliest methods of the Greeks -- the exhaustion method. The students quickly decided that they could fill the area using different size polygons and circles, and they would ultimately get pretty close to a solution. While this was a good first step, and started them down the conceptual road to limits, we ultimately decided that it would be problematic to obtain the measurements of different sized shapes. Some students wondered if using a single shape of the same area would simplify the process. Then one of our students, Rion, said that every gamer knows that shapes on the computer screen are just made up of the same size pixel, and that the problem could be best addressed by filling the area under a curve with pixels that could be counted. Extending this line of thought, Rion thought that he could use the Minecraft creating website, Plotz, to illustrate the idea that reducing the size of the squares, thus increasing the number of squares under the curve, would result in a more accurate answer. Rion shared these images with the class:
After this, we discussed the idea of being able to reduce the error of the area measurement to a point where for all practical purposes it was accurate. While the students certainly did not reinvent the concept of a limit, they did discover, in a general sense, the way a limit could be used to help solve this type of problem. Our hope is that they will be able to apply this concept and type of thinking to other problems that we challenge them with in the future.
My team teacher and I believe that it is vital to continue to push the students’ ability to think of math in abstract terms. This forces the students to see math as something different from memorizing steps in order to follow the predictable, worn-out path of textbook examples and problems. It forces them to engage with math -- to wrestle with it in a way that builds engagement and deep conceptual understanding.
