Parts:

Do you ever use words without quite knowing what they mean? And you sort of assume that others will natrilently understand what you’re talking about even though the word isn’t exactly what you want? If you just keep using it, it’ll start to take on the meaning you need it to, and you don’t know the precise word you need, so what the hell, you’ll just keep calling your favorite professors “erudite” until you eventually look the word up in a dictionary?
It sort of works. But until you look the word up, you feel awkward using it. You float it out in conversations and watch for reactions. Most of the time no one else can define it exactly either, and they get the gist of your message anyway, so conversation can proceed. The conversational ground is tenuous, but sometimes that doesn’t really matter – it’s easier to just continue.
I care about the phenomenon of building on ideas we’re not perfectly sure about. Can you guess where this happens four times a week in fiftyminute chunks? I’ll give you a hint: it starts with “R” and ends with “iley’s Algebra class.”
No one can build understanding on a foundation of vaguely expressed ideas. In our classes, we must not allow vaguely expressed ideas to be final products, because until those ideas are sufficiently specific our students will not be able to build on them. If a student offers an answer that is correct but too vague for our needs, we should avoid saying, “right, and…” and then supplying the rest of the answer ourselves or from another source. Doing that would give our student the impression that he was right when he wasn’t, and that vague answers are good enough. Instead, we should say “your answer is incomplete,” or, “we need something more specific,” or, “what do you mean?” In the end, before you think he knows what he’s talking about, he should give a complete answer.
Such a demand for specific clarity can seem overly picky, or even mean, but it’s vital because of a human ability to get the gist of things and to use that gist to reach conclusions that sound reasonable but are in fact bullshit.
Hilariously, I want to leave off here and try a different tack. Please keep the vagaries above in mind as you consider the following. Take two:
We are so smart that we can parse badly formed language (I can remv vwls frm mst of my wrds and you hv no prblm undrstndng me, I can skewer the definitions of words, and I don’t have to even use grammar that good). This is an incredible feat, but there are two major problems it causes:
 When we use imprecise expressions, we lose the benefits of generalization and specification, because both acts require very precise definitions, and
 When we use imprecise expressions, we lose the benefits of generalization and specification, but we don’t realize it, and we generalize and specify anyway. Incorrectly.
Now, it is hard to recognize a badlyformed sentence in a language when we are not fluent in that language. When we aren’t fluent, we don’t have that immediate reaction of something being out of place, like we do when I say, “what time are it right now?” And since our students do not have the fluency to automatically detect discrepancies in their math – to notice immediately that their units do not match or that their answer is an order of magnitude bigger than what they would expect – it is hard for them to recognize these badlyformed, imprecise statements in math. We cannot expect them to build any significant understanding until they are fluent enough to catch this sort of thing. They learn the formula for the area of a circle, and then apply it to an ellipse, because they haven’t internalized the precise definition of “circle.” My students say “x squared to the third” to mean and then think I’m just giving them a hard time when I insist they say “x cubed” or “x to the third power” – they really don’t see the big deal. So they fall especially in to problem 2 above – thinking they can build on concepts that they think they understand, and then being confused when their calculator spits out .
On the other hand, I got around alright in Italy having taken two years of high school Spanish. Sometimes all we need are nouns and hand gestures. In part two of this series, I’ll write about one key difference between math and spoken languages: that mathematical notation, not just the meaning behind the notation, can be manipulated to find more information. Stay tuned! But for now, let’s remember that we don’t need to start with number theory for everything; we can memorize some multiplication tables and an algorithm and then get a very thorough understanding of geometry and calculus without ever understanding (or even thinking about) the underlying mechanics of how that multiplication works. I conclude this post with a question: how do you decide where to draw the line? When do you say “this is fundamental and we need to understand it before we move on,” and when do you say, “you can sort of see how this works from this picture; now let’s move on?”
This looks like Lemov’s “Right is right”.
If we want to encourage student precision, we need to have authentic situations where it matters. We can’t just demand it because some day it might make a difference. That’s just not relevant.
What is an example relevant to your students where imprecision in language caused difficulty? That would be easier than finding a place in school mathematics that matters to them.
I find your arguments to imply that the gist is in fact the most important thing. Most of our students are foreign language speakers on a brief sojourn through mathland. Unless we find a way to make it more relevant to their lives, two years of mathish might be all they need.
That’s an interesting point. Precision for its own sake is trite and pedantic, and I can agree with you on that. Why would anyone demand precision for its own sake?
But there are some benefits of deep precision that you’re overlooking, John. When we understand the reasons for rules and algorithms, we can use them confidently and identify their misuse. We can extend them to more general cases and creatively adapt them to different situations. Fluency can bring a confidence that makes the math fun and intrinsically rewarding.
If our goal is just to prepare our students for a “brief sojourn through mathland,” to give them just enough command of the language that they can talk to their mortgage broker with head nods and hand gestures, then we needn’t bother with calculus, trigonometry, imaginary numbers, or even functions and graphing.
In part three of this series I hope to share my definition of “relevant” math, which I’ve hinted to already in some other posts (search for “reality”). I basically disagree with you that math needs to be relevant to something already in students’ lives – I think that math can be a part of students’ lives if we give them the time they need to build up their language skills.
I can’t wait for the rest of this series.