“School geometry that is presented in a [Euclidean] axiomatic fashion assumes that students think on a formal deductive level. However, that is usually not the case…” (Van Hiele, 1999)
This excerpt from Van Hiele’s article, Developing Geometric Thinking through Activities that Begin with Play, immediately made me think of one topic in particular, the bane of (most) high school geometry students’ existence: proofs. The type of formal reasoning demanded by deductive proofs is seldom easy for students to attain, even in high school. If students have one complaint about geometry, it is usually the proofs.
As such, I am always looking for ways to make proofs easier for students to understand. Before we get into geometric proofs, we always play a “proof game” with simple rules for manipulating strings of letters, to get them used to justifying every statement they make. I also try to emphasize the sequence and structure that is common among many proofs, like using definitions to change words into equations, after which properties of equality can be used to show a range of things. We also try to scaffold heavily as students learn how to write proofs, slowly bringing them along from filling in blanks to eventually writing complete proofs on their own.
But Van Hiele’s article led me to think about students’ struggles with proof in a different light, and question whether my students have been given or are given enough opportunities to progress through Van Hiele’s levels of geometric reasoning. Perhaps some of their difficulty with proof stems from having too little time to engage in informal reasoning.
Van Hiele’s Levels of Geometric Reasoning
1 - Visual
2 - Descriptive
3 - Informal Deductive
4 - Formal Deductive
5 - Axiomatic
I know that my students do not have many opportunities to do this, to take their time and consider what they know about geometric figures, and to discover out of their own knowledge further relationships and geometric truths. On the first day of class, they are introduced to four postulates, and at least one or two postulates or theorems follows in each subsequent lesson. They do come in a logical sequence, but still they are presented for students to take at face value most of the time, unless we take the time to prove them. It would be better if my students could be presented with new ideas along the way, and allowed to explore and struggle to find new relationships on their own, before having the formalized theorems presented to them. And they could practice justifying their discoveries, without immediately being tied to structure of a proof. I think they would much more thoroughly internalize important concepts than they do now.
Formal proof writing will always be an adjustment, though. It is a new style of writing for students. But if they have had more time to explore their way through new concepts, they might be better at making the connections that tie given information to a desired conclusion.
Van Hiele’s mosaic and the activities he describes to go with it are impressive. Its careful design provides a basis for a wide range of discussions and student discoveries. The example of the mosaic makes me want to find more exploratory activities that can apply to the concepts my high school geometry students will work with this year. I will certainly look for these, or try to come up with some, and look for opportune places to tie them into our curriculum.
Time might be the biggest challenge to integrating more exploratory activities. It is at such a premium already, with all that is supposed to be covered in a term. Giving students a day here or there to explore, rather than presenting material directly and moving on is not the most efficient way to go, time-wise. Our math department already has moved to include days dedicated to problem solving group work near the end of each unit. I wonder now if in geometry it might be better to use some of those days for exploratory activities at the start of a unit. That is a suggesting I will make to our geometry team. At the very least, I will be more apt to provide more time within lessons to have students discuss key ideas with each other.
ReferenceVan Hiele, P.M. (1999). Developing geometric thinking through activities that begin with play.
Teaching Children Mathematics, 5(6), 310-316.