# Defining concavity

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.

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.

### 4 responses to “Defining concavity”

1. Barry

“Does it count as flipping the classroom if they read first, then discuss in class”. That’s the original flip! Old-school flipping! I’m always shocked when people refer to the “watch videos and reserve class time for problems or discussion” type of flipping as an “innovative” or “new” way of organizing classes. I have anecdotal evidence that asking students to read ahead in math class was once not unusual.

I am happy to now be in a department where we all share the philosophy that students should come to class prepared, and we all use some method to give students incentive to do this reading and assess how much they are getting out of it (I can’t claim we are great at this assessment — I’d love to hear good ideas for assessing reading facility).

The quality of many textbooks is certainly such that a teacher creating an instructive video is almost guaranteed to be a better resource. But if a textbook is eminently readable and well-designed, I find it hard to believe most teachers would find the time and editing capability to create videos of the same quality. Video has some advantages, like adding a tiny amount of extra overhead before a student can jump ahead. Certainly, it has advantages when it is used in the style of Dan Meyers. But give me a well-written text, and I’ll be happy to skip the videos and have my students read ahead instead.

Also, call me a dinosaur, but I just cannot see video replacing text anytime soon as the primary source for learning difficult technical content. But maybe I’m wrong. Maybe ten years from now, classic MOOC videos will be people’s first choice when needing to learn new stuff.

2. Don

Chris, once more I am envious of your conversation with your students. I will be teaching Geometry for the first time in a block (90 min) format starting Tuesday. I am so torn between asking good questions and prompting discovery or just getting through the curriculum. I know I should do the former, but I will not hit all the CCSS standards. Of course, at 90 minutes for 4.5 months, I also do not think I will hit all the standards with the latter…. Our school just improved on its jr. high math scores, so there is now a lot of pressure put on the high school to do the same…(luckily, I teach students math in both…so I may have a little room to experiment…)
Barry, all my administrators ever talk about is the flipped classroom, and they are hilariously unaware. The other day one commented, “Well, if everyone assigned a 15 minute video, our students would have to watch up to 90 minutes of videos at home.” Now, it could be me, but I almost always had more than 90 minutes of homework in high school…I do, however, make videos of many of my assignments so that absent students or those who need more help than I can provide have a resource. I struggled with this, but my students are doing better on tests and we are having good mathematical conversations in my algebra classes as a result. I have not used a textbook since arriving at my school 3 years ago. The students refused to read it even when quizzes were given…and the text basically said “plug ‘n’ chug” to everything so even if they had read they would not have learned anything. (Gah!)

3. It’s precisely your “dangling” final thoughts that make all the difference, yes?
There is a tendency, when discussing student vocabulary in particular, that once rigorous terms are introduced, the speaker’s sophistication has somehow leveled up. But if you start with a Level n statement, no amount of signifier exchanging can fundamentally alter n, even though it might superficially appear that way. It’s the relationship between the signifier and signified that matters, holistically (i.e., Is it “precise” and “defensible?” Is it self-consistent? Is it part of the public discourse?) The relationship between angled and what(ever) it signifies is equally arbitrary as the relationship between pointing and what(ever) it signifies. If there’s no meaningful, discernible signified object, then the attached word is literally meaningless. It’s a tempting trap to see vertex as more sophisticated than corner, and that’s a tricky thing for people who are learning to parse the thoughts of other, littler people, exclusively through the filter of language. It’s exceedingly awesome that your students are being confronted with these things. Love it. Tell Griffy that I said, “Owhay ouyay oingday?”