This mathagogy video has generated some behind-the-scenes discussion in recent months.

This week, I got a note from Chris Hill with some questions. I wrote him a long email in response. Here it is:

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First a disclaimer.

I have never taught high school geometry.

I taught middle school, including some geometry there, have written some middle school geometry tasks, and I now work with future elementary teachers. That last is the context in which I developed my hexagon thing. The sequence I describe in the video is the end result of several years worth of trying to work out a way to get my students to geometric proof.

Of course there is a lot more involved in the process than can be conveyed by a short video. Have you seen my blog posts on the sequence?

And now your questions:

**Do students develop the ability to move up the levels through the entire year, or on each unit?**

I think of the van Hiele levels more as an organizing principle for instruction than as a principled diagnostic for students. But the levels do serve both purposes.

Here is what I mean…I do think that most of my students (high school graduates; some young, some older; all students at my community college) come at a pretty easy to observe Level 1. They know names and properties of shapes but cannot talk much about how these properties relate to each other. That, I think, is pretty standard for entering high school geometry students too.

I do not think that they become Level 3 geometers over the course of our work.

Instead, I use the van Hiele levels as a way to organize instruction. So we start where they are with some noticing and naming of properties. Then we move to a tremendous amount of level 2 work. I probably describe some of that in the blog posts, and I should probably write up more of that.

Over the course of several class sessions (we meet for 80 minutes twice a week), I push a little bit harder, and then harder still, on the arguments students are making. Eventually, through this modeling and through my explicit talk with them about it, they begin to be more critical of their own arguments, and eventually we reach a question that sort of requires proof; it seems true, but is non-obvious, and it has arisen from the questions we have been asking (at level 2) about how properties relate to each other. Creating and organizing the hierarchy of hexagons is often a useful part of this sequence.

And then we move on. If we had the whole semester (or whole year!) to spend on geometry, I would imagine that level 2 work would be an important introduction in each unit. This is Serra’s conjecturing phase in the Dan Meyer comment.

*Is it reasonable to expect every high school geometry student to get to the top two levels within a unit (like circles)?*

I don’t think the top level (level 4, rigor) is a reasonable goal for every student. Sure, you can do Taxicab Geometry and notice that “circles” there look like Euclidan squares, but I am not convinced that this has a lasting effect of rigor on students.

I do think having all high school students (for whom future elementary teachers are my proxy) use their ideas about relationships among properties of shapes to build a more formal argument that a proposition must be true is reasonable. Different students will be capable of different levels of sophistication in terms of the propositions they argue, and in terms of the arguments they make, but they can all do some level 3 work for sure.

*If I try to talk about the levels explicitly, do you think high school students will be insulted, not care or would they feel validated when they take a long time to get to a higher level?*

This is a good question. I debate this a lot in my own teaching. How much of the motivation and justification should I provide my students for what we are doing? I tend to talk about things in that spirit—we are doing X today because it will address needs that I know we have; here is how I see this activity getting us to our goals. And I try to provide specific evidence and the source of that evidence. A research basis is a good source of evidence, as is student work on quizzes and things kids say in whole group discussions.

I would not spend time telling students what van Hiele level they are at. And I might not use the actual language of the van Hiele levels with kids (but then again I might).

Rather, I might say something like this:

Our goal is to make formal mathematical arguments (proofs) in this course. Mathematical arguments take a different form, and have a different standard than a scientific argument or an argument you may have with your mom. But that is a goal, not our beginning place.

I know from research I have read, from my experience in previous years, and by observation of your work and the ways you talk about geometry that an important part of getting to that goal will be exploring properties of shapes and how those properties relate to each other. So we will do a lot of that work.

We will argue with each other. And over time, I will help you to make these arguments more mathematical. Eventually, you will be able to make such a great mathematical argument that we can call it a proof. Some of these proofs will be developed and written by the class collectively, and I will ask you to try do write some on your own. But it will be a while before we get there.

Then, when we do get an example of a solid mathematical argument (whether because the class developed it or because I provide it), I make sure to stop, notice it and to have a conversation with the class about how that argument is different from the other work we have been doing.

*Do you know of any standard assessments that I can use to track my students’ progress with these levels?*

No. I ask my students to make short arguments on quizzes, and try to pose questions that admit both informal and more formal arguments. But I don’t know of good vH level instrumentation. It may exist. But I don’t know about it.

The van Hiele research often shows students moving through the levels for each topic. So for each new topic, the need to progress through the levels. I forget the study that shows mathematicians doing this, too, though they progress through the levels remarkably quickly.

So in instruction I look for tasks that are open to students at multiple levels, that offer encouragement and experience to move up a level. I select students to present at multiple levels and then ask about connections. I look for places to argue about differences at the same level, which often require moving up a level to solve. Is it a polygon. Yes. No. Well, what’s the definition of a polygon? (Yes!) Informal arguments get tightened to formal. Etc.

Great post and discussion!

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