For the past two weeks we have been doing an interesting activity in my math class that focuses on teaching geometry for elementary students. We have been creating hexagons and classifying them. At first I was uncertain about what I would learn about teaching geometry to elementary students from creating and coming up with names for these hexagons that we had created however I found the process to help my understanding and move me forward in my Van Hiele thinking. I am certain that if adapted to their level, this could be a very beneficial activity for elementary students to help them better understand the concepts associated with classifying quadrilaterals rather than memorizing the properties that are commonly used in classifying them.
The first thing that we did as a part of this activity was we created as may different hexagons as we could. Our only limitation was the dot paper we were using, however even in this there were many, many hexagons that we could create that all looked very different. Of these shapes there were some that were concave, convex, had reflex angles or more than one reflex angle, there were some with right angles, and of course the regular hexagon was one of the first for almost everyone to draw. These varied shapes allowed for use to be as creative as we could and discover as many “odd” hexagons that we could until we had exhausted all possibilities.
If elementary students were able to do this activity using quadrilaterals instead of hexagons I believe that this could be an incredible activity that could help move them up in their Van Hiele level of thinking as well and allow them to discover and learn through inquiry. This of course is one of the best ways that students can learn is though self-exploration. In coming up with all the possible quadrilaterals they can I think that they could become more comfortable with some of the shapes that they do not see as frequently such as trapezoids. By giving them this time to play and explore these non-standard shapes they will become more comfortable simply by familiarity.
The next thing that we did in our class exploration with hexagons was that we took all of the hexagons that we created and defined a “type.” We then proposed these types for the class which as a group we either decided was a good type or we vetoed it and decided not to make it a shape. As we were doing this I found myself asking the question “What makes a good type?” After some deliberation in my group we thought that the answer to this would be that a good group has very specific characteristics that would include a population of hexagons but would also eliminate a significant portion as well. As we thought about our definition of what a type should be, I believe this influenced how we thought about the propositions of what types we thought that we should keep and which one we thought were not necessary.
I love this aspect of the activity for elementary age students. By creating their own categories and deciding what they think would be a good type could help them in as they learn what the mathematically accepted types of quadrilaterals are and why we have chosen these to be types. I also like the critical thinking that is involved as students propose types they must think about the specific examples of what rules will fit the shapes they have created as well as the additional rules they must come up with to include or exclude other specific shapes that can be made.
Now that we had our categories the next thing that we did was we were asked to create shapes that would fit in exactly 3 categories, 2 or more categories, and exactly one category. I found this to be one of the most challenging activities that we did with this activity. I think that this is because we had created all of rules, it was difficult for me to find one shape that fit exactly in one of my categories because I was unsure as to what I was trying to not create as well. After recognizing that we had both concave and convex as a category, I realized that in order to be just one category, the hexagon would have to fall in either of these categories yet none of the others. After discovering this it became easier for me to come up with an example that would fit just one category. I also tried to create a venn diagram for the shapes that we had created and I found this incredibly challenging as there was a difficult aspect in knowing what to do with categories that overlapped in some categories but not in others. After modeling my venn diagram after that of quadrilaterals it became somewhat easier to create yet still proved to be a mathematically challenging activity. I think that this could be an extension for some students who show that they are proficient in their math abilities in the previous activities with quadrilaterals. As students categorize and organize these categories they will begin to question and further examine their definitions and ask themselves if their definitions are inclusive or exclusive. This would lead to a good discussion to the mathematically accepted definitions of quadrilaterals that are inclusive and why they think this could be. Students could also use this time to see if they would like to eliminate certain categories that they had created and come up with reasoning why they should eliminate it. Some categories could be redundant or only cater to a small population of hexagons that may seem irrelevant or grouped better by another definition.
In participating through these activities, I would think that these critically thinking questions can help students to move up in their thinking. Hopefully by carefully analyzing and looking at all properties of quadrilaterals, students could possibly begin to think formally and continue to realize that very frequently it is the shape properties are what we analyze in mathematics.