That's the best way I have to sum up what I've learned about teaching over the past few months. It's a simple statement with deep consequences.
Consequence #1: Every piece of learning worth remembering needs a name.
It's not important that the name be particularly descriptive. Names aren't usually descriptive. "Michael" doesn't describe my features in any way, but it is a hell of a lot easier to drop "Michael" in conversation than "that red-haired math teacher from NYC."
I almost always ask my students to come up with the names for these things, because it's fun, and that's how we end up with "The Feces Theorem" or "The Friday the 13th Theorem."
Consequence #2: Thinking is having a way of talking. Giving kids new ways to talk helps give them new ways to think.
By the end of our polygons unit we were able to go far beyond talking about this shape as "looking like a rocket". We were talking about pointiness, and the number of points, and which sort of points counted as sharp points, and we were able to talk about acute angles and obtuse angles and all sorts of other things. (Documented here.)
I put in all this work into helping my Trig kids see that the sin(45) is not 0.5 on the Unit Circle. We investigated the Unit Square, and it's "sine" didn't have this property either. This tends to be tough and mind-bending for my students, but the learning is slippery. You know how I can make it more effective, though? By introducing the language of "linear" and "non-linear", and giving them a way to talk about these functions and their properties.
Having a concept is deeply connecting to having a way to talk about a thing. Not having a way to talk about a thing makes it harder to have a concept.
Consequence #3: You can't learn math from Khan Academy.
Learning from Khan Academy means learning alone, and that means never developing a language to talk about concepts. Which means that you probably aren't really learning them.
(Or do you believe in the possibility of a private language?)
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Credits:
Trying to implement Christopher Danielson's Orpda and Hexagons lessons sensitized me to the connection between language and concept development.
Naming Infinity is a really interesting book about intersections between mysticism and math that influenced some of my thinking here. Ditto on my philosophy courses in Philosophy of Language, in particular "Two Dogmas of Empiricism," which is arcane but I take the paper to be an attack on the hard distinction between knowledge and language.
The title of this post has been lifted from Bret Anthony Johnston's book.
A couple of points.
ReplyDeleteFirst two quick videos of Richard Feynman being interviewed:
(1) (~2 min) The difference between knowing the name of something and knowing something:
http://www.youtube.com/watch?v=05WS0WN7zMQ
(2) A longer version (~5 min) cut from the same interview. Same topic, with a few other interesting thoughts:
http://www.youtube.com/watch?v=chKyBUOtfws&feature=youtu.be
I think Feynman's idea about the difference between knowing the name of something and knowing something doesn't quite align with your first point.
Second, and I'm not 100% sure on this one, but quickly flipping through "Naming Infinity" made me think you might like the book by Amir Aczel - The Mystery of Aleph. I read it a year or two ago and really enjoyed it.
Third. It may indeed be the case that you can't learn math from Khan Academy, but the reason given that "learning from Khan Academy means learning alone" doesn't seem convincing to me.
Plenty of people have learned quite a bit of math alone. Heck, plenty of extremely respected people in the math world have done lots of amazing work alone. Since I mentioned him above, a good example is Feynman who developed his own language for quantum mechanics that took the physics community quite a long time to understand and then adopt.
Some people may work and learn better alone, others will work and learn better in a group. For me anyway, differences in learning or working style would not diminish (or enhance) my view of the learning / work.
An interesting example from this year gets to the point. When Tom Zhang from UNH proved his result about prime numbers, he did that largely working alone (at least as I understand it). Subsequently, there was quite an amazing collaborative effort on the internet involving many mathematicians to understand and then improve the result. Both the individual work and the collaborative work led to fascinating results.
These are some good, thoughtful criticisms Mike. Thanks for them.
DeleteWhile I might concede some of my ground -- of course you can learn math alone -- I'd like to stand by the basic notion that having a way to talk about something is intimately related to knowing something.
My third point needs some more subtlety, so here it is: of course you can develop a language for communicating some concept on your own. Feynman did it, Zhang did it, I've done it, you've done it. But working alone is hardly the most natural context for creating language. The most natural context for creating language is the need to communicate, and you can't do that unless there are other people around.
Of course, of course: there needs to be time to be alone and time to be with a group of people. Of course, of course: you can learn stuff all on your own. But this aspect of learning is hard to do all by yourself, but creating an teaching model for all kids out of the learning habits of great mathematicians and physicists is probably not the best of pedagogical ideas out there.
Basic point: the social context helps kids develop the language of math, with is often crucial for developing, or even synonymous with learning math itself. The social context of school is a feature, not a concession to budget or convenience.
Oh, and I loved Amir Aczel in high school. I read Mystery of the Aleph a bunch of times, it was my first introduction to diagonalization and all that beautiful stuff.
DeleteMike Lawler beat me to the punch. As I was reading this post I was also thinking of the great Feynman video. Michael, I assume that your students move past the phase where they refer to the Friday the 13th theorem. Having a name that makes personal sense is obviously great while developing understanding, but recognizing that common names for ideas is a key to communication with the greater world is essential. I think that this focus on language is so powerful. When students tell me that they don't understand something I often urge them to work on identifying exactly where they lost the thread and to work on specific language identifying their misunderstandings.
ReplyDeleteMichael (and Mike Lawler and mrdardy): Thanks for the post, for the references to the books, and also to the Feynman clips. It is a valuable discussion with very important implications for math education in elementary school (especially primary grades K-2) when much of the vocabulary and terminology is first introduced.
ReplyDeleteI have posted some of my thoughts on my brand new blog:
exit10a.blogspot.com
which I was inspired to start by the work done here at rational expressions, as well as on blogs by folks like Dan Meyer, Christopher Danielson, and Andrew Stadel, to name a few. Hope I can add my voice to those trying to reconsider what it means to be a math teacher.