Blog #1 Geometry

Blog #1 Geometry

NYS Standards

G.CO.C.10 Prove and apply theorems about the properties of triangles.

G.CO.D.12 Make, justify and apply formal geometric constructions.

Goals

            I wanted to put myself in the shoes of a typical geometry student. Because I already had the background knowledge to solve the geometric proofs for the Geometry regents I tried to focus on the complaints the typical geometry student had, which was the level of abstraction and thinking that needed to solve these proofs. Therefore I tried to challenge myself with the same type of abstraction that would take me a while to solve, and which gave little to now clue on how to solve it. I focused on hard geometric constructions because it helped me review geometric concepts and also was abstract enough. The question that I had required creating a perpendicular to a point on a line using only three uses of either creating a circle/arc or using a straightedge to create a line.

1.                                                                                           2.

3                                                                                         4

Reflection

What this exercise made me reconsider is how unintuitive/painfully abstract applying the geometric properties were. The answer to this was to visualize that a triangle whose longest side is the diameter of a circle which creates two perpendicular lines, so the legs of the triangle inscribed in the circle are 1. the lines given and 2. the perpendicular line you need to draw (yellow line). If the goal for proofs and constructions is to provide students the ability to acquire problem solving and abstract thinking skills, students can be better suited to focusing on problem solving using techniques found in math-competitions but don't require this level of frustration and time. Furthermore, if students aren't able to get how to prove a specific property, it only leads students to feel as if they were "unintelligent" when the truth of the matter is that the educational system has never built up this level of abstraction within their education.

Furthermore, looking at the proofs that are part of the curriculum, either the proofs are two column proofs where all of the conclusions are made, and the student needs to simply memorize the property like so:

or they are way to abstract, similar to what I had to face:

In the first case, the proofs don't test abstract thinking, rather it just tests whether the student has memorized all the necessary theorems. In the second case, it is just too abstract.

My conclusion is that proofs should be removed from the geometry curriculum.