The types of hexagons we are familiar with today include a regular hexagon along with the concavity of the hexagon giving us the two types of convex and concave. When faced with the task to create our own types of hexagons there were many similarities that popped up between our separated groups within the class. Many of our types had to do with interior angles of the polygon and began overlapping with one another. I, first and foremost, think that this is due to a lack of thinking outside of the box. The angle types are the types that really deal with the visual learner. It is quite easy to identify a right or 90 degree angle as opposed to some random angle with the naked eye. This relates back to Van Hiele's ideas about the levels of geometric thinking. We picked out the right angles because of "what they look like" with our shapes such as "Lagon", "Bowtie", and "Right Hexagon". These three shapes really capture the essence of the visual, descriptive learner and exactly why we need to move past those novice ideas and gradually build, as we did after discussion and arguments in our classroom. The next level of geometric thinking, informal deduction, was reached when we had to come up with hexagons that were considered 3 types of hexagons at once. This made us look deeper into the properties than just what "it looked like". We now grouped properties together to discover more and to form our own conjectures from that exploration. For example, one conjecture I formulated when I was working with the hexagons that night after class was that, "If a hexagon is classified as a Lagon then it also always falls under two more types, Right and Concave." I conjectured this after trying to draw different Lagons in a variety of ways and finding the pattern. To somewhat prove that conjecture the definition of a Lagon is necessary, a hexagon with 5 interior right angles. Also, the definition of a Right Hexagon is necessary, a hexagon with at least one right angle. Therefore, by definition a Lagon is also a Right Hexagon. Also, in order to have 5 right angles within a 6 sided closed figure there must be one reflux angle or angle that is larger than 180 degrees. Therefore, by definition of a reflux angle the hexagon is also concave. This type of proof-like thinking is what Van Hiele would classify as clear thinking, being able to go through each step and explain it to a peer sitting next to you. Venn Diagrams also challenged our informal deduction abilities along with the hierarchy trees. These activities allowed us to explore and come up with our own examples of what we believed to be good for our classroom one day in the future. It also encouraged students to persevere in attempting to fill all of the spaces within the diagram. To make the activity more valuable to younger students it may have been beneficial to incorporate more examples within each section in order to let them see the examples along with the non-examples to better understand the properties. Overall, I think the exploration with hexagons took us through the levels of geometric thinking just as we will take our students through those same stages one day soon into the future. I look forward to that day!

'til next time :]

'til next time :]