In our discussion the other day of whether the difference between van Hiele levels 0 and 1 is the use of official math vocabulary terms, or whether there is something more there, a student asked this (I am paraphrasing):
Is this like the CGI research, where children know things about addition and subtraction before they are taught in school?
Here’s what I love about this question:
It demonstrates the power of having these students for a second semester;
It demonstrates that students will—over time—make these ideas their own; if we keep at it, and build cases for the importance of the ideas in our courses, students will play with these ideas, and they will look for opportunities to apply and connect them; and
It made me stop and think. I’m still not 100% sure whether it’s like that or not. I have something new to think about.
Today was day 2 with my geometry students/future elementary teachers.
Homework due today consisted of (1) a reading about the van Hiele model, and (2) participation in a discussion on Canvas about geometry terms; they were to choose one term from an extensive list we generated whose meaning they are confident of, and whose meaning they do not know well.
In that discussion, several students chose concave as the term whose meaning they did not know. Another student replied (for they are required to reply substantively to at least two other students’ posts):
Concave is a shape that curves or angles inward and convex is like most regular shapes, and has all its angles pointing outwards.
Now class time is about integrating ideas (does it count as flipping the classroomif they read first, then discuss in class?) So they were given the task of deciding which van Hiele level best describes this claim about the meaning of concavity.
The brave soul who offered her take said, (and I am paraphrasing):
A major difference between level 0 and level 1 in the van Hiele model is whether we are naming the properties; whether we are using geometry vocabulary.
Lovely. This gave us something to work with.
I proposed a game in which we delete all of the official mathematics vocabulary words, replace them with more informal language and ask whether we have changed the van Hiele level. Now we had:
This is a shape that curves or points inward and that is like most normal shapes, and has all its corners pointing outwards.
That first brave volunteer felt that the nature of this claim is now different; that this edited version is a level 0 statement, while the original is level 1. Others were not so ready to commit in either direction. I argued that this second claim has a similar structure to the first, and so really ought to be at the same level.
To illustrate the nature of the claim, I asked them to explain what is meant by pointing inward, or angling inward.
Pointing towards the center of the shape was a definition they could all agree upon. I drew a few examples of concave shapes whose concavities didn’t really seem to point towards the center of the shape, they drew a few examples and they began to formulate other ways of saying what concave means.
In the end, we agreed that it’s awfully difficult to characterize pointing inward in a rigorous way, and that this may not be the best way to state what concave means.
And we agreed to revisit this term later on.
I suppose I left dangling the fact that an important part of the structure of the original claim is that it rests on terms whose meaning is imprecise to the speaker. It’s not about whether I use the word angle instead of corner;it’s about whether I have a precise and defensible meaning for the word I am using.
Do I mean for my word to point to a particular class of objects, while excluding others, and can I tell the difference between these two classes in a principled way? That is the difference we need to unearth.
An essential piece (perhaps the essential piece) of this whole teaching sequence was that it was based on what my students saw in those hexagons. I designed a diverse set that I thought had interesting properties, but this wasn’t about them guessing my properties. Instead…
This was about my helping them to better articulate exactly what they saw in these hexagons. I helped them to develop good language for describing properties (this is not easy to teach, nor to learn!). And I helped them to understand differences between properties and definitions (again, not easy for either party).
Josh wrote in the comments, “I would have worried about leaving so many types of hexagons uncategorized or about not recognizing some of the important types.” Which reminds me to state that this activity is all about process, and not at all about content.
Perhaps that last claim is overstated. I had a few items of content to sneak in there. Vocabulary such as concave. The difference between equilateral and regular. That sort of thing. But there was no one thing about hexagons that I needed to get on the table. Instead, hexagons were our territory for exploring mathematical practices.
This was my second time through the activity. Last semester, my students noticed somewhat different things in the hexagons, and we ended up with candlesticks, chevrons, shields, starfish and rectilinear hexagons. Again, the exact properties and resulting categories aren’t what matters here.
I did a much better job of integrating the hexagon sorting into the proof work. The Venn diagrams and the hierarchy were important to that.
If you read carefully, you’ll see the van Hiele model play out pretty tightly as the underlying model for the sequence of activities. We did not proceed to Level 4.
True confessions: I find a great deal of the school geometry canon tedious.
Does a trapezoid have exactly one or at least one set of opposite parallel sides? Circumcenters and orthocenters. Dull, dull, dull. Boring, boring, boring.
School geometry seems to me one of the most lifeless topics in all of mathematics.
And the worst of all? The hierarchy of quadrilaterals.
This representation of relationships among the special quadrilaterals bored me in fourth grade and I cannot muster energy for it as an adult. But I gotta teach it with my future elementary teachers.
We cut these out. I had students choose one that seemed special to them for some reason, and to identify what property or properties make the hexagon special.
Students identified this one as being special because it has all right angles:
We clarified, defined interior angle and right angle, and agreed that this hexagon is special because it has exactly five right angles. The shape needed a name and we chose Bob. So a Bob is a hexagon with five interior right angles.
We also agreed that we would only specify interior in the future if there was likely to be confusion; we gave ourselves permission to refer to the interior angles of a polygon simply as angles in most cases.
Students identified the next figure as being special because it has three congruent acute angles.
Again, we needed a name and it became a Stacy. So aStacy is a hexagon with three congruent acute angles.
We identified several of our hexagons as being concave, so we defined a concave hexagon as one that has at least one interior angle greater than 180°. (Side note: It turns out that this is the standard definition; I had remembered something about diagonals staying in the interior. In any case, we had to do some work to get from the visual shape with a dent definition to this one.)
I threw a couple of useful terms into the mix: equilateral and equiangular, and pretty soon we had enough to work with.
Having polished off all of the pairwise possibilities, we took to the whiteboard to categorize and to argue.
Concave hexagons, Stacys, equilateral hexagons and equiangular hexagons are all special hexagons that don’t necessarily have anything to do with each other. But you can have an equilateral hexagon that is also equiangular. We named that a “Norm”. And a Stacy can be equilateral. That’s a Mercedes. The Stacy above is a Mercedes. We weren’t sure whether a Mercedes must be concave.
My students proved that no Bob is equilateral.
I would like to repeat that.
My students proved that a Bob cannot be equilateral.
I have never before been able to say that my future elementary teachers proved something. I could say before that they followed a proof I presented. Or that they produced a proof that closely mirrored one they had seen. But never that they proved something. This group did.
Their argument was based on parallel sides in a Bob-how there are two sets of three sides, and that the lengths of two sides in a set add to the length of the third. If a Bob were equilateral, one of these sides would be of length zero, which means it’s not a hexagon and so not a Bob. QED. I have spared the reader some details.
Behold the hierarchy of hexagons:
After this, the hierarchy of quadrilaterals was a mostly trivial exercise. We built it in like 20 minutes and used it as practice for the skills we developed with the hexagons.
We marveled at the bizarre relationship between the definitions of quadrilaterals and their relationships. Why is a rhombus defined in terms of its side lengths, while a parallelogram is not? This makes it hard to see why a rhombus is a special parallelogram.
The question of the concavity of Mercedes was an open one for a couple of weeks. Then yesterday we got out the polystrips. Boom!
Not all Stacys are concave.
If we had more time, we would revise our hierarchy to incorporate this fact. But we have to move on to measurement.
This week, we were discussing definitions. In particular, we were trying to decide whether we needed to include the measures of the congruent angles in our definition of a rectangle, or whether simply stating that they are congruent would be good enough.
That is, can we define a rectangle as an equiangular quadrilateral? Or do we need to specify a quadrilateral with four 90° angles?
This led to discussion of whether it is possible for a quadrilateral to have all four of its angles obtuse.
One student argued “No”. She argued that a rhombus typically has two obtuse angles and two acute angles; if you try to make the acute ones obtuse, the obtuse ones become acute.
The class accepted that argument (much to my chagrin).
So I stepped in, saying that I wasn’t convinced. Specifically, I said:
You are arguing that a quadrilateral can’t have four obtuse angles by showing that a rhombus can have at most two right angles. But I don’t think that’s true of quadrilaterals in general. I think it may be possible for a quadrilateral to have three obtuse angles.
If that’s possible, then your argument doesn’t show what we all seem to think it shows. And maybe four obtuse angles is possible.
A few minutes later, a student produced the diagram above on the board. A quadrilateral with three obtuse angles. Furthermore, we all agreed that we could imagine tweaking things so that all three of these angles would be congruent.
What have we learned? That we still don’t know whether we need to state that the congruent angles in a rectangle are right.
