Module 4: Assignment #1
How would you define abstracting? This complicated topic can be boiled down into one sentence for me. Abstracting is not looking at something in its entirety, but focusing on one feature or viewpoint of it. An abstraction may represent specific properties that are not obvious when viewing a whole object. The goal of abstracting is to find “minimum visual stimulus that can be put on paper or canvas and still evoke recognition without spelling everything out” (Root-Bernstein, 1999).
Abstracting plays a very large role in the field of mathematics. Although we always say that math is “real”, much of the work we do is not obvious. We use abstract symbols, numbers, and equations to represent concepts. Before practicing this cognitive tool, I chose to keep working with the Pythagorean Theorem as the formula is very confusing and foreign to most students.
I focused on the hypotenuse in a right triangle and how it is formed. I considered the “real life” problems where students solve for the hypotenuse of a tree’s shadow or the distance between two points. As I searched through my phone, I looked for something I could connect those ideas to. I stopped when I got to a picture of my MAET group picture from this summer. Whenever I look at this statue I am drawn to his eyes. If that were one of the endpoints of the hypotenuse you could put a point anywhere around him to create a hypotenuse. You could put one on the ground, a person, a piece of grass, etc. and calculate the distance using the Pythagorean Theorem. You would just need to know the height of the statue (from ground to his eyes) and then measure from the endpoint to the base.
My second abstraction took a long time to come up with. I know that the idea of the Pythagorean Theorem came to Pythagoras as he was looking at a tile pattern. I chose to create some graphic art expanding on the basic ideas that Pythagoras used from the tiles. I started with my graph paper and only used 45-45-90 triangles to keep it consistent as each hypotenuse is double the two legs. I placed my focus on that and traced only those lines, not the triangles themselves. As I worked I thought of Tetris and wanted to flip my figures around to piece them together. I traced my final image and highlighted the triangles to show others where the figures came from. I am curious to see what images I could produce using 30-60-90 triangles.
By narrowing my focus to the hypotenuse of right triangles I could think of the Pythagorean Theorem in a new way. I am proud of my work and realize how this may help my students. Abstracting allows students to deepen their understanding by viewing specific parts of something instead of the whole thing. They may see new relationships, create images, develop meaning, or relate to a content area in a way that is not possible otherwise.
Root-Bernstein, R. S., & Root-Berstein, M. (1999). Sparks of Genius: The 13 Thinking Tools of the World’s Most Creative People. Boston, MA: Houghton Mifflin.