For the last two semesters I have spent my lunch hour learning to count, add, subtract, multiply, and measure with 30 of my fellow buckeyes. Math for Teachers has exposed me to the mathematical concepts from my childhood that I thought I already knew well. It has challenged me to become a more critical thinker and has taught me that you do not truly understand something unless you can help someone else to understand it. I have learned to look to my peers to deepen my understanding of content. Working together we can learn more about the patterns in our world and how we can use them.

The focus of this semester has been on geometry. In thinking back to my experience with geometry in my own K12 education I have little memory of geometry lessons in elementary and middle school. I do remember learning that the most efficient way to build a fence was in the shape of a square because it had the smallest area to perimeter ratio – a fact that I have kept with me since third grade in the corner of my brain where I keep my dream plans for a chicken coup.

It wasn’t until high school that I explored geometry in depth and really enjoyed my math coursework. My favorite topic in my sophomore geometry class were proofs, specifically those that worked with arbelos. On the last day of our unit my teacher, Mr. Wilcoxon, pulled up an animation of an arbelos with a circle on the SMART Board. He dragged point D from one side of the largest semi circle to the other. The more even the smaller semi circles were the larger the circle was and vice versa. He told this that we weren’t going to have to prove this one, but that he wanted to show it to us. No matter where the semi circles were positioned the area of the arbelo (green region) was equal to the area of the circle.

Up until this point in my math education everything that I had been taught felt man-made. There was something rigid and lifeless to it. It felt like a means to an often dull end. The arbelo and circle challenged that. It didn’t feel like the circle needed to be even the awkward shape, but it was so perfect that it was. Working with shapes off of a coordinate plane introduced me to the captivating patterns that existed in the universe long before people had even begun to count. Geometry taught me that, at least to me, math was an articulation of those patterns.

### Thinking from a Teacher’s Point of View

My favorite geometry topic this semester was constructing shapes by folding and using a straightedge and/or compass. In my K12 education I always drew shapes (other than circles) with just a ruler. Working without unit measurement forced me to think more creatively and gave me a deeper understanding of the properties of shapes. The relationships between opposite angles, diagonals, etc. became clearer when having to problem solve to build my shapes.

Here are two examples of my work this semester constructing shapes:

### Learning from Others’ Points of View

This semester we were asked to craft a lesson plan for teaching area to a group of second through fifth graders who had never been exposed to the concept before. We shared out lessons with each other through a virtual discussion board. At first I was a little dumbfounded as to what to write. We have spent a lot of time working on correcting students work and explaining specific problems, but rarely do we address how to introduce a topic. Much of what we learn is that elementary math is more complicated than it seems. The task of of introducing abstract concepts feels daunting when you know so many ways that it can be misunderstood. After much tinkering I felt content with plan. However, reading my fellow classmates lessons made me aware of the areas of weakness in my own and gave me ideas of different explorations into the topic. By seeing how we each approached the same task in unique ways showed my the diversity of thought we shared as a class and the diversity of thought I could expect form my own future students. Below is my revised lesson plan with notes of the changes I made.