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 classroom if 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.
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.