Geometry and language

Interesting conversation on Twitter today with Bryan Meyer, Denise Gaskins and Justin Lanier. It began with these tweets on my part, the result of grading some student work.

Oh dear. Class inclusion is giving some of my elementary kiddos fits. Do we have any post-Piagetian research on this?

— Christopher (@Trianglemancsd) February 7, 2014

I suspect some of it is linguistic and probably appears not just in their math work but in their writing too.

— Christopher (@Trianglemancsd) February 7, 2014

Things quickly got too nuanced for Twitter.

An example of something my students struggle with is answering a question such as, Is a square a rectangle?

This type of question asks about class inclusion. Is an element of a subset also an element of the larger set?

Many useful and interesting questions in geometry have to do with whether one class is a subset of another class. Do all isosceles triangles have a pair of congruent angles? Are all quadrilaterals formed by connecting midpoints of other quadrilaterals parallelograms? Are all Stacys concave?

I am trying to sort out the extent to which my students’ struggles with questions of this sort are linguistic, and the extent to which they are about struggles with the idea of class inclusion.

Justin suggested this wording, which I will investigate:

Is a square an example of a rectangle?

Or, more generally:

Is an X an example of a Y?

My suspicion is that this will be helpful for some students when asked in this direction. But I also suspect that asking it in the other direction will be problematic.

Is a rectangle an example of a square?

See, part of what I wonder about is whether class inclusion—and the fact that it doesn’t have to be symmetric—is at the heart of a particular kind of struggle in geometry, and whether this is also related to the ways students think about and use language.

I hope these three (and others) will weigh in here where we have more space to work than we do on Twitter. The ideas are really useful. If you’d like to follow the prior discussion, you can follow this link.

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Reading Children’s Minds

The title does not mean what you likely think it means. I cannot read children’s minds.

But I am reading Children’s Minds.

Michael Doyle recommended it to me.

Well, that’s not quite right. He called me the successor to Margaret Donaldson (author of the book in question). I had never heard of her. I consulted Amazon and now possess a first American printing of the 1978 book.

There is lots of interesting stuff in it. Good, thoughtful critique of Piaget; a lovely read.

I have been enjoying and identifying but not really seeing Michael’s point.

And then this.

To Western adults, and especially to Western adult linguists, languages are formal systems. A formal system can be manipulated in a formal way. It is an easy but dangerous move from this to the conclusion that it is also learned in a formal way.

Replace language with math and you pretty much have everything I’ve been saying on this blog.

I cannot say that I’ll live up to Ms. Donaldson’s legacy, but Michael Doyle is astonishingly good.

Khan’s kindness

Say what you will about Sal Khan (and I have certainly said a lot), but he communicates a tremendous amount of patience with his students.

I watched his video on “Basic Addition” the other day.

He begins with the assumption that the viewer has absolutely no equipment for finding the sum 1+1.

This bears repeating. He assumes absolutely no knowledge of the meaning of the addition symbol in the expression 1+1. None.

As he does so, Khan is patient, supportive and encouraging. He does not condescend and he even apologizes for the word basic in the title of the video-worrying that his viewer may be put off by the term.

When I think of the culture of many math classrooms, in which students don’t ask questions out of fear of looking stupid, or in which instructors use words such as trivial and obvious without apology or concern for the effect these words can have on learners, I get a glimpse of what people find so appealing about Khan’s videos.

Khan gives permission to not know. He reassures the viewer that it’s OK to still be figuring things out. And of course he is happy to repeat what he just said as many times as the viewer likes. Just stop and rewind. The calm, patient demeanor never changes.

The field could learn from Khan’s kindness.

 

“and”

How many dalmatians were in that movie again?

It was one-hundred-and-one, right?

Have I angered you yet? If you teach math, I probably have.

See, we math teachers are precise people. We like things to be just right. Part of being just right is using language correctly. In English number language, and is reserved for separating the whole number part of a number from the fractional part.

One and one-half

One and seven tenths

That sort of thing.

So when we hear one hundred and one, we freak out.

But separating the whole number part from the fractional part? That’s just one perspective. Sure, it’s the grammatically correct one.

But here’s the thing about grammar rules…they are arbitrary. Totally and completely arbitrary. Name a grammar rule and there’s a language somewhere that violates it completely.

Yet we English-speaking math teachers act as though the use of and were a signifier of mathematical understanding or competence. Which it is not.

Here’s another interpretation of and in English number language. Maybe and signifies a change of unit. In one and seven tenths, one is counting the original unit, seven is counting tenths. The and helps the listener to follow along; it signifies this shift.

In that case, one hundred and one is the same way. The first one counts a unit-hundreds. The second one (following the and) counts a different unit (which, awkwardly, we call either ones or units).

If I’m right about this, then you will have heard native speakers of English say something such as,

Three hundred and four thousand and twelve.

Three hundred and four counts thousands. Twelve counts ones. Then within the three hundred and four part, there are two different units also. Hence the and.

If I’m right, then you will probably not have heard native speakers of English say something such as,

Thirty and four.

Both of those words count the same units.

So I say let’s give up on this little obsession we have about and. Let’s not let it get in the way of effective, efficient communication in mathematics classrooms.

Let’s save our wrath for this:

Three point twelve.

Place value and language

Timon Piccini writes about a conversation he recently had with his niece.

My niece is in grade 1, and she is adept at adding single digits.  With little hesitation she can do her basic addition.  She even showed me that she could do things like add 100 + 100.  I thought this was really neat so I asked her some questions.

Me: What’s 1+1?
Niece: That’s easy it’s two.

Me: What’s 100+100?
Niece: It’s 200 duh!

Me: What’s 1000 + 1000?
Niece: 2,000 these are easy!

Me: What’s 10 + 10?
Niece: … I don’t know.

This leads Timon (TIM-in) to wonder about how language is related to early numeracy and later mathematical development. Interesting stuff.

I learned after commenting that his blogging platform doesn’t allow html code. So my comment is hard to read. I have reproduced it below. But go read his full post. And read the comments while you’re there.

My comment

This is where my mind has spent the last few years. So lovely to see that I’m not the only one intrigued by this sort of stuff.

I love that conversation with your niece. Just like we need to read aloud to our children, we also need to talk math with them. I don’t think we do any damage when we move to symbols (as you did in this conversation), but I don’t think we have any evidence that it’s really helpful, either. Like teaching a pig to sing, I suppose (wastes your time and annoys the pig).

What does seem to be helpful is that you’re interested in the child’s ideas. This can take many, many forms. One interesting activity for a curious teacher such as yourself is to take a moment to formulate a hypothesis and then a question to test it. Here you noticed that she could do 1+1 and 100+100 but not 10+10. Your hypothesis (which I also believe to be correct) is that this is language based. So ask her, What is 1 ten plus 1 ten? I’m curious whether she would say “two tens” or “two ten”. I can’t tell from your transcript whether she said “two hundreds” or “two hundred”.

Anyway, I think you and I would both be surprised if she had no answer for 1 ten plus 1 ten. When she offers it, follow up with How much is that? How much is two tens?

Many thanks to all the folks who have contributed references. Those will be helpful as I develop my own understanding of this territory. I’ll add my own (a bit self-serving, admittedly). I wrote a paper on relationships between quantity, numeration and number language (my bit starts a few pages into the file). That paper grew out of work I do with future elementary teachers and research from Karen Fuson. It contains several worthy references.