Wasting for Stuporman

I dragged myself to Waiting for Superman last night.

What a confused movie.

Have you ever been on the subway with a crazy person? I am from Boston, where I actually can’t remember this happening to me one time, although we do have crazy people; but it happens a lot in New York. You’ll be going from here to there and somebody on the subway car will just start discoursing, usually to no one in particular but as though they’re having a normal conversation. Sometimes angrily, which is always disconcerting to be around because if you’re simultaneously angry and totally disconnected from reality, who knows what you’ll do next. Often enough, though, it’ll be totally harmless. (Somehow, the time that the man next to me explained “when I cock my fist back, that’s potential energy; and when I throw it toward your face, that’s kinetic energy” managed to fit into the totally-harmless category. He was jovially illustrating a point. You could just tell.)

One of the most striking things about the discourse, though, whether harmless or angry, is that the person is usually speaking with conviction, but not making any sense. This is what it felt like to me watching Waiting for Superman.

Let me try to summarize this movie for you. SPOILER of sorts.

Geoffrey Canada (Harlem Children’s Zone) loved Superman when he was a kid. Davis Guggenheim (the filmmaker) decided to send his kids to private school. They have really cute kids in Boyle Heights, Harlem, the Bronx, and DC. Academic achievement in the US has not improved since 1971. In 2002 there was a moment when it looked like FINALLY THE SCHOOLS WOULD BE FIXED!!! because a republican (Bush) and a democrat (Teddy Kennedy) collaborated on a piece of legislation (NCLB). But it’s 8 years later and we still suck. People used to think that failing schools came from failing neighborhoods but now we realize it’s the OTHER WAY AROUND!! Our schools totally suck and that’s why our neighborhoods have crime and drugs. There are lots and lots of shitty teachers. Randi Weingarten is some kind of mediocrity nazi rallying the national teacher corps into a frenzy of mediocrity. The national Democratic party is basically owned by the teachers’ unions. Teacher tenure used to be useful, back when administrators were arbitrary and exploitative, but now all it does is keep useless, worthless humans in front of children. Even if you have a really kick-*ss teacher, you can’t pay them more money, even if you want to, because they already have a contract that says what you pay them. But then from out of the sky comes MICHELLE RHEE!! The public education bureaucracy prevents teachers from giving students the proper infusion of learning fluid. Michelle had a plan to save us all. Too bad the mediocrity nazis stopped her. US kids suck at math and think they rock, as we learn from Green Day. Tracking is evil because even though it’s supposed to be based on test scores, sometimes kids get tracked based on behavior. But actually, 50 years ago tracking was awesome because it reproduced the class structure, which was awesome. But the world has changed. Suburban schools have all the same problems as urban schools but the kids are higher-skilled so the grades are inflated. Urban schools have problems suburban schools don’t have to deal with. It’s really hard to be a teacher. Davis once made a movie about that. A great teacher is a work of art. Because all great teachers work for charter schools, the cute kids’ parents want them to go to KIPP, SEED and Harlem Success Academy, and basically they’ll DIE if they don’t get in. Bill [Gates] knows education.

There, that about covers it.

Then, as the credits roll, the film acts like this incoherent pastiche has added up to both a clear recipe for action and a movement. We get a summary of what Davis Guggenheim apparently thinks are the self-evident conclusions of the film –

The problem is COMPLEX
But the steps are SIMPLE
It starts with GREAT TEACHERS
More time in school
Getting the bureaucracy out of the way [I’m not remembering this word-for-word but this is the idea]
World-class standards
Real accountability

and a “change starts with you” message. Text “POSSIBLE” to such and such a number, we’re told.

I felt like the crazy person from the subway had just shown up on the corner wearing a PIRG t-shirt and holding a clipboard, trying to get me to sign a petition and donate money. And he was totally sure I was going to sign. It was really weird.

* * * * *

Let me be a little less coy about what I have to say about the content of this film.

Davis – I’m glad you got the draft in on time. You’re showing a lot of passion, but we’ve got to work on the clarity of your thesis and your evidentiary structure. In the meantime, you need to engage with some key sources of information you left out entirely:

1) Good teaching that is happening inside public schools.

You depict failing public schools, portrayed as the norm, and a handful of highly successful charter schools. This narrative makes successful public schools invisible. Have you never encountered one?

I have. Where I went to school, and where I learned how to teach. Unionized workforce and everything. And some of the best teaching you will see anywhere.

2) Teachers getting better.

It’s a shame that you left this image out of your narrative because this is the whole secret to successful education.

Where did you think great teachers come from? That they spring fully formed from the head of Zeus? Just about everybody who’s an accomplished teacher used to be an ineffective teacher, and as the maker of a documentary about first year teachers, I’m totally confused that you don’t seem to understand this. If you want to talk about great teachers, but don’t have anything to say about the conditions under which teachers become great, you are at a different stadium than where the game is happening.

(Hint, by the way: in order to become great, teachers need to make and then learn from their mistakes. What kind of environment fosters making and learning from your mistakes? Fear that you will lose your job over your kids’ test scores? Or maybe transparent, non-defensive collegiality? Okay, good job on that one, now the followup: what kind of education policies are going to create the environment that fosters growth?)

Conversely – where do you think incompetent burnouts come from? The League of Committedly Useless Humans? Do you think anybody gets up at 5:45 every day and gets in front of kids and wants to suck? I know hundreds of teachers, and I don’t know ONE who is honestly okay with doing a bad job. Be that as it may, teaching is actually very hard, a fact to which you pay lip-service, and that means that in a difficult situation and with an absence of support, it can be a pretty crushing experience. (I will go on record with this: teaching is way, way harder than math. Galois theory is a walk in the park next to figuring out how to alter your planning, presence, discussion facilitation, assessment, etc. to get better results for your kids. No contest.) Lots of folks leave the profession; plenty more stay on board and give up. If you want to decrease the amount of incompetence in front of kids, and you don’t have anything to say about how to support teachers in growing, then again, you’re at the wrong stadium.

* * * * *

A lot of the above has already been pointed out by others. Let me direct you to one excellent critique among many –

Ben Allen belongs to a category of person I think I pretty much always get along with: he’s a professional mathematician (a complex systems theorist) who spent time (3 years) teaching math in urban public school. So, when he talks about Waiting for Superman, I’m listening.

Ben calls attention to the absurd scene in which what education is “supposed to be” is depicted in a cartoon as a teacher opening up students’ heads and pouring in a liquid (“knowledge”), before this process gets interrupted by public education’s bureaucratic constraints. I’ll add that I used to use more or less this exact metaphor as a send-up of how people who don’t understand education imagine it works. Learning as some kind of IV drip.

* * * * *

Okay, I thought I was done but I have one more thing to say.

What’s with the creepy appropriation of civil rights language?

Creating Balance III / Miscellany

The Creating Balance in an Unjust World conference is back! I went a year and a half ago and it was awesome. Math education and social justice, what more could you want?

If you’re in NYC and you’re around this weekend, it’s happening right now! I’m going to try to make it to Session 3 this afternoon. It’s at Long Island University, corner of Flatbush and DeKalb in Brooklyn, right off the DeKalb stop on the Q train. I heard from one of the organizers that you can show up and register at the conference. I’m not 100% sure how that works given that it’s already begun, but I am sure you can still go.

* * * * *

I’ve just had a very intense week.

I want to get some thoughts down. I’m going to try very hard to resist my natural inclinations to a) try to work them into an overall narrative, and b) take forever doing it. Let’s see how I do.

(Ed. note: apparently not very well.)

I. Last spring I wrote

20*20 is 400; how does taking away 2 from one of the factors and 3 from the other affect the product? We get kids thinking hard about this and it would support the most contrivance-free explanation for why (neg)(neg)=(pos) that I have ever seen.

Without going into contextual details, let me just say that if you try to use this to actually develop the multiplication rules in a 1-hour lesson, all that will happen is that you will be dragging kids through the biggest, clunkiest, hardest-to-swallow, easiest-to-lose-the-forest-for-the-trees, totally-mathematically-correct-but-come-now model for signed number multiplication that you have ever seen (and this includes the hot and cold cubes). This idea makes sense for building intuition about signed numbers slowly, before they’re an actual object of study. It does not make any sense at all for teaching a one-off lesson explicitly about them. (Yes, the hard way. I totally knew this five months ago – what was I thinking?)

II. I gave a workshop Wednesday night, for about 35 experienced teachers, entitled “Why Linear Algebra Is Awesome.” The idea was to reinterpret the Fibonacci recurrence as a linear transformation and use linear algebra to get a closed form for the Fibonacci numbers. Again, without going into details –

I gave a problem set to make participants notice that the transformation we were working with was linear. I used those PCMI-style tricks like giving two problems in a row that have the same answer for a mathematically significant reason. This worked totally well. Here is the problem set:

View this document on Scribd

Oops I guess I failed to avoid going into details. Anyway, the question was about how to follow this up. I went over 1-4 with everyone (actually, I had individual participants come up to the front for #3 and 4) at which point the only thing I really needed out of this – the linearity of the transformation – had been noticed by pretty much the whole room. One participant had gotten to #9 where you prove it, and I had her go over her proof.

I think this was valueless for the group as a whole. The proof was just a straight computation. You kind of have to do it yourself to feel it at all. It was such a striking difference watching people work on the problem set and have all these lightbulbs go off, vs. listening to somebody prove the thing they’d noticed. It almost seemed like people didn’t see the connection between what they’d noticed and what just got proved. I told them to take 5 minutes and discuss this connection with their table, but I got the feeling that this instruction was actually further disorienting for some participants.

I’m trying to put the experience into language so I get the lesson from it.

It’s like, there was something uninspired and disconnected about watching somebody formally prove the result, and then afterward trying to find the connection between the proof and the observation. Now that I write this down, clearly that was backward. If I wanted the proof (which was really just a boring calculation) to mean anything, especially if I wanted it to be at all engaging to watch somebody else do the proof, we needed to be in suspense about whether the result was true; either because we legitimately weren’t sure, or because we were pretty sure but a lot was riding on it.

This is adding up to: next time I do it, feel no need to prove the linearity. Let them observe it from the problem set and articulate it, but if there is no sense of uncertainty about it, this is enough. Later in the workshop, when we use it to derive a closed form for the Fibonacci numbers, now a lot is riding on it. If it feels right, we could take that moment to make sure it’s true.

III. As I work on my teacher class, something that’s impressing itself upon me for the first time is that definitions are just as important as proofs. What I mean by this is two things:

a) It makes sense to put a real lot of thought into motivating a course’s key definitions,

and maybe even more importantly,

b) Students of math need practice in creating definitions. You know I think that creating proofs is an underdeveloped skill for most students of math; it strikes me that creating definitions might be even more underdeveloped.

Definitions are one of the most overtly creative products of mathematical work, but they also solve problems. Not in quite the same sense that theorems do – they don’t answer precisely stated questions. But they answer an important question nonetheless – what do we really mean? And to really test a definition, you have to try to prove theorems with it. If it helps you prove theorems, and if the picture that emerges when you prove them matches the image you had when you started trying to make the definition, then it is a “good” definition. (This got clear for me by reading Stephen Maurer’s totally entertaining 1980 article The King Chicken Theorems.)

Anyway this adds up to an activity to put students through that I’ve never explicitly thought about before, but now find myself building up to with my teacher class:

a) Pose a definitional problem. Do a lot of work to make the class understand that we have an important idea at hand for which we lack a good definition.

b) Make them try to create a definition.

c) If they come up with something at all workable, have them try to use it to prove something they already believe true. I’ve often talked in the past about how trying to prove something you already believe true is very difficult, and that will be a problem here. However, unlike in the cases I had in mind (e.g. a typical Geometry “proof exercise”), this situation has the necessary element of suspense: does our definition work?

If they don’t come up with something workable, maybe give them a not entirely precise definition to try out.

d) Refine the definition based on the experience trying to use it to prove something.

I’ll let you know how it goes. I’m excited about it because it mirrors the process that advances mathematics as a discipline. But I expect to have a much better sense of its usefulness once I’ve given it an honest whirl.

Honor your Dissatisfaction

Two things I forgot to say last night.

I. The reason I’m excited about the idea of having my class use its own self-made definitions to try to prove things is not just, or even primarily, because it will help them realize the inadequacies in their definitions. Although it will do that for sure. Even more than that, it seems to me the perfect way to support them in coming up with better definitions. This is what happened to Cauchy: he defined the limit verbally and a little vaguely, but then when he actually tried to use his definition to prove things, he started writing down precise inequalities. He didn’t have a teacher around to point out that this meant he should probably revise his definition, but my class does.

II. Yesterday when I asked my class to try to make a precise definition for what it means to converge, or for something to have a limit, some of them who took real analysis long ago began accessing this knowledge in an incomplete way. They started to talk about and , but in vague, uncertain terms. It looked as though others might possibly accept the half-remembered vagueries because they seemed like they might be the “this is supposed to be the answer” answer. I had to prevent this. (The danger would have been even greater if these participants had correctly and confidently remembered the definition.) I stepped in to the conversation to say, yes, that thing you’re half-remembering is my objective, but what’s going to make you understand it so you never forget it again is to fight till you’re satisfied we’ve captured the meaning of convergence. You can either fight with the definition you half-remember or you can fight to build a new definition, but you have to go through your dissatisfaction to get there. You have to air all this dissatisfaction.

Afterward, I thought of a better language. I’ll give this to them next time.

Honor your dissatisfaction.

Dissatisfaction is the engine that created analysis. This content, more than any other content, is both confusing and pointless if you bury your dissatisfaction rather than allowing it to thrive and be answered. The primary virtue of the tools of analysis is that they are satisfying. Only if you bring forth your dissatisfaction will this content have a chance to show you its value. So. Honor your dissatisfaction. It is the engine that will move us forward.

Over the Course of an Instant…

As you may recall, I’m teaching analysis to this class of teachers, developing the – limit. Two weeks ago I bewildered everybody. Last week and this week, I set out to bewilder everyone even further.

Let me say what I’m going for here. The – limit is a notoriously difficult definition.¹ How to scaffold my class to handle this difficulty? I am banking on the following strategy: make them need the definition. Make them unsatisfied with anything less. Continue poking holes in their current understanding, continue showing them inconsistencies between what they believe and the language they have to describe it, till they have no choice but to try to build something new. Then, let them try to build it. If they build the very thing I’m going for, rejoice. If they build something equally precise and powerful, rejoice. If they cannot build either (the most likely outcome, since the “right answer” took the world mathematical community 150 years to come up with), then it will still make powerful sense to them because it satisfactorily answers a question they were already engaged in trying to answer. That’s the plan anyway.

I will leave you with the two problem sets from the last class, and the readings and presentation from this one. I am very proud of the presentation. After that, I’ll write down one new thought for where to take this.

We engaged people’s attempts to define infinite decimals from the previous class, then abruptly shifted topics:

View this document on Scribd

I let them work long enough so everyone got to do the first section of problems. My goals were:

1) Make participants recognize that they believe the speed of a moving object is something that exists in a particular moment of time.
2) Make them recognize that their naive definition of speed (distance / time) doesn’t actually handle this case.
3) Realize that we thus have a similar definitional problem as with repeating decimals.

We got this far. Then, with just 7 or so minutes left, I gave them another problem set:²

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This problem set was designed to get somebody who has never studied calculus basically to take a simple derivative, to bring them into the conversation, and to refresh everyone else’s memory about the basic idea of derivatives. The last problem was on there just so that the calculus folks had a challenge available if they wanted it. Anyway, I had people finish the “Algebra Calisthenics” and “Speed” sections for homework.

This class, we began by engaging this homework, getting a feel for the standard calculus computation in which you identify the speed of an object in a moment as the value toward which average speeds seem to be headed as you look at smaller and smaller intervals. Then we began to press on what this really means.

I handed out a xerox of the scholium from the end of the first section of Book 1 of Newton’s Principia. (The last page of this pdf.) This is where Newton tries to explain what the hell he’s even talking about. I directed their attention to this telling sentence:

An in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities, not before they vanish, nor afterwards, but with which they vanish.

Then, I showed them the following presentation. Wanting to share this with you is the real reason for this blog post. I had a lot of fun making it.

What’sCalculusReallyDoing (as pdf)

What’sCalculusReallyDoing (as powerpoint)

Then I passed out a choice excerpt from the awesome criticism of early calculus by Bishop George Berkeley. (Specif, section XIV.)

I asked for the connection between the definitional problem we have here and the definitional problem we had 2 classes ago regarding infinite decimals. (“They both involve getting closer and closer to something but never getting there.”) Then I asked them to try to come up with definitions to address these problems.

This is such a non-sequitur but here’s my one additional thought. I’ve been thinking about how to push participants to recognize a definition as unsatisfying. Tonight, reading Judith Grabiner’s 1983 essay in the AMM about Cauchy and the origins of the – limit (here it is as a pdf), I had an idea that is totally new to me. Retrospectively I think it’s sort of obvious, but I totally never thought of it before:

To get people to recognize that a definition is mathematically inadequate, have them try to use the definition, for example to prove something! In my case, all of them think that 1/3 = 0.333… Great. So, if we have a candidate definition of the meaning of limits or convergence, can we use it to prove 1/3 = 0.333…? If not, maybe we need a better definition.

(I had this idea when I read Grabiner’s statement that thought Cauchy gave the definition of the limit purely verbally and a bit vaguely, he translated it into the more rigorous language of inequalities when he actually started using it to prove theorems.)

[1] This is for at least 2 distinct (though related reasons): first of all, it’s got three nested quantifiers. “For all , there exists a , such that for all satisfying …” That just makes it inherently confusing. Secondly, it does not in any way psychologically resemble the intuitive image it is intended to capture. This is the definition of the limit. When I think of limits I have these beautiful visual images of little points getting closer to something. When I try to identify a limit, I just imagine the thing that they’re getting closer to. That’s the whole story. When I try to get rigorous, I replace this beautiful and simple image with three nested quantifiers. Yuck.

[2] You will notice some interconnections in the sequence of problems. After a few good experiences with this last year and then hearing how much fun everyone had at PCMI, I am beginning to feel like these sequences of densely but subtly interconnected problems are really, really awesome. Constructing them is a deep art and I am a tiny apprentice. But you can get started humbly and still see payoff: it was certainly a cool moment today in class when we went over these problems and a number of folks who had done out Speed problems #1-3 “the long way” realized that they could have applied their answer to Algebra Calisthenics #2 to do these three problems in moments in their heads.

Vader

I’m working with a new tutoring client, and therefore starting at the beginning in training her to stay engaged with the math rather than getting frustrated or trying to read the answer from my reactions to her guesses. The other night, I came up with a new metaphor to help her with this as she was trying to calculate the area of a circle section. I can’t believe I never thought of it before, it’s so obvious.

[N is visibly struggling to unify the geometric and algebraic information. I love it, I feel like I can literally see her brain growing, but she’s getting frustrated.]

Me: You’re going to grow from this.

N [skeptical]: Really?

Me: Yeah. You’re going to figure this out, and then you’re going to understand that you already had everything you need to figure it out. Have you seen Star Wars?

N: No, but I know it.

Me: You’re Luke, this is Vader. You face Vader and then you become a Jedi.

[Long pause while N thinks about the problem, punctuated by occasional exchanges like, “if I divide the circle area by , does that give me the area of the wedge?” “I don’t know, make up numbers.”]

N: Oh! 360 divided by will give me the fraction of the circle that’s the wedge.

[4 or 5 second pause]

N: Right?

[N has evidently been watching my face intently for the last 4 or 5 seconds, trying to get external confirmation of her insight. When it isn’t forthcoming, she begins to doubt.]

Me: The reason I’ve been looking out the window is so you won’t try to get a cue off of my face. [Stands up] I’m going to do the more radical version of this and get out of your visual field. [Starts to head out of the room.]

N: You’re not going to tell me if I’m right?

Me: You’re trying to get me to fight Vader for you.

0.99999…

So I’m teaching this course this year. It’s for the math faculty of a high school. It’s called:

MA600 Algebra and Analysis with Connections to the K-12 Curriculum

I am unspeakably excited, and want to do the best job possible.

The class: 7 teachers, deeply committed to kids, serious, not real talkative, rightly protective about their time, which is in short supply, but eager to get sh*t done.

The content: Basically, all of mathematics, seen as a unified whole.

It’s met twice. The second class was last Thursday. I need to get my thoughts sorted out here. I’m expecting this to help me visualize the next moves more clearly, just by doing it, but I’d love your thoughts too.

I didn’t really know anything about the mathematical background of the group when I wrote the syllabus, so for the first class I gave them a getting-to-know-you problem set with a wide range of problems and just let them work the whole time. Magically the experience of watching folks work on the problems and then later looking at what they did on paper gave me just enough information to plan the direction of the class’ first unit. We’re beginning with analysis. My first goal: the – definition of the limit. (I.e., the definition of the limit, for the snobby among you.) My second: the completeness axiom.

The plan: generate the need to define the limit by working with 2 everyday concepts that are actually limits. Namely, infinite decimals, and instantaneous speed. My hope is that by pressing on these concepts, we’ll see that in spite of our familiarity with them, we don’t actually understand them unless we have a precise way to talk about limits. Then, develop the definition out of the need to fully understand the familiar. Then, develop the completeness axiom out of the desire to make sure infinite decimals have a limit.

Here’s what we did:

I opened class with a problem set designed to get them thinking about the meaning of decimals in particular, and various other contexts for the idea of limits. I shamelessly bit the format from PCMI. The problems span a wide range of skills and I didn’t leave enough time to do them all, so people could attack problems appropriate to their skill level. This is now my favorite way to differentiate problem sets, a propos of a) using it in some NYMC workshops last year, and b) hearing about how wonderful it was for everyone at PCMI.

View this document on Scribd

Then, since we are all just getting to know each other, I did a short presentation on the mindset I wanted us to be in:

View this document on Scribd

(Scribd did not handle slide 6 very well, which is too bad because I was proud of that slide. This is my first PowerPoint presentation ever. Actually I did it in Keynote.)

Then, we got to business. I put this up:

I asked them to talk about it with their tables. (I had them in 2 pairs and a group of three, in three tables in a horseshoe shape in front of the board. I like this and think I’ll keep it. Easy transitions from pair/group to whole-class; tables feel separate enough so you don’t feel like your conversation with your partner is in front of everybody; but everyone’s close enough so we can all talk. On day 1 I put us all around one table, for a sense of collegiality and common purpose, but it was too close; you couldn’t discreetly check in with your neighbor, for example.)

There was a widespread sense of mathematical discomfort, and rightly so. Infinite decimals enter most people’s math educations with no attention to the fact that they actually violate everything you’ve learned about math up to that point. You don’t get the full story until analysis, but unless you really get intimate with and own that content, you probably don’t connect what you learn there to what your teacher introduced without comment somewhere between 3rd and 7th grade, as though it weren’t a mind-boggling idea. “When you expand 1/3 as a decimal, the 3’s just keep going.” Or, “3.14159… It never ends or repeats.” Um, excuse me? It NEVER ENDS?

So it’s no surprise everybody has an underdeveloped idea of infinite decimals, and therefore that objects like 0.99999… cause some dissonance. This is very productive dissonance. I’m hoping it carries us all the way to the completeness axiom; we’ll see.

One of the three tables produced the standard argument that if x = 0.9999…, then 10x = 9.9999…, so 9x = 10x – x = 9, so x = 1; but even this table found this conclusion unsatisfying. I asked them why. The table that had produced the argument said, “usually this method gives you a fraction.” I asked for an example. They produced one from the problem set:

x = 1.363636…
100x = 136.363636…
99x = 100x – x = 135
x = 135/99 = 15/11

I asked how many folks found this argument convincing. 7 out of 7. (Well, one raised hand was kind of hesitant.)

I asked the same question about the same argument with .9999…. 4 out of 7. Then I dropped this:

View this document on Scribd

How many people found this one convincing? 0 out of 7.

Reasonable.

Right?

Then what’s the difference?

At first, they cast about a bit, but then one of them said, “1.363636… has a finite limit, but …9999.0 doesn’t.” Their ideas began to coalesce around this type of language. Another one said, “we can actually estimate 1.363636…, for example we know it’s between 1 and 2.”

From the point of view I am ultimately heading for, this is the rub. Infinite decimals suggest convergent series, and the standard way to give them meaning as real numbers is that they are equal to the limit of the convergent series they suggest. …9999.0 suggests a wildly divergent series, so it cannot become a real number in the same way. (To bring home that convergence is the heart of the matter: there is an alternative way to define distance between numbers, the 10-adic metric, according to which it is actually …9999.0 that has the convergent series, and in this alternative system the above proof is valid and it actually does equal -1.) What I’d like us to do is a) define limits precisely; b) use this to prove that when a series has a limit, you can do the above type of manipulations to find it; c) try to prove that the series suggested by an infinite decimal always has a limit; d) realize that we can’t prove this without articulating the completeness axiom; e) articulate the axiom; and f) prove from the axiom that any infinite decimal has a real number limit. (Somewhere along the line, produce an – proof that 0.9999… = 1.) Now, how to orchestrate this…

For next time I told them to try to craft a definition of the meaning of an infinite decimal 0.abcd… I gave them a few minutes just before the end to discuss this with their groups. I’m expecting to learn a lot about their thinking from what they come up with, but I’m not counting on anyone to have a mathematically satisfying answer. I’ll be pleased if somebody does though.

As I think about next class, here’s what’s on my mind:

1) When we develop the – limit, what I’m going for is for this definition to feel like a satisfying relief. I know how easy it is for this definition instead to feel like a horrible monstrosity designed to oppress analysis students. I think what I have to do is keep them thinking about the reasons why anything less than this definition is too vague, which means I need to keep coming up with objects and problems that throw monkey wrenches into whatever more naive definitions they go for. (Of course, if they come up with something equally precise as the – limit but different, that would be amazing.) I feel like we’re off to a good start on this, but I want a fuller catalogue of head-scratchers (like …9999.0 = -1) to push the level of precision higher.

2) Relatedly, I sense a danger that the “real answers” will be unsatisfying because it’ll feel like “wait, I already said that.” For example, the participant who said that the difference between 1.3636… = 15/11 and …9999.0 = -1 is that “the first one has a finite limit”… I mean this is basically the answer. But it’s not based on a precise definition of limit yet, so it’s not what I want yet. I’m afraid of a “what was the big deal?” moment when we’ve got the real sh*t up there. I think the way to avoid this lies in that catalogue of head-scratchers I need to develop, so that nothing less than the real thing is satisfying. What do you think?

3) Where to go immediately next. Basically the question is: stick with decimals? Or change gears completely and press on the notion of instantaneous speed? Most (not all, I think) of the teachers have had a calculus course, but think at most 1 or 2 of them have internalized the philosophical lesson that instantaneous speed needs to be defined as a limit in order for us to even access it. I’m attracted to the idea of switching gears because I’m drawn to the connection between the disparate realms: two highly familiar, but totally different, objects – infinite decimals and speed in a moment – both getting pressed on to the point where you realize you never fully understood either one, and then you realize that the missing idea you need is the same thing in the two cases. (A precise way to talk about what number some varying quantity is “heading toward.”)

Actually as I write this out, it seems clear to me that switching gears is the way to go. I think it’ll give us a clearer understanding of what we’re missing with the decimals. Also, it’ll allow us to access all this rich historical stuff around the development of calculus. For example, maybe I’ll share with them some choice quotes from Bishop Berkeley’s The Analyst, to help articulate why the 18th century definition of the derivative was inadequate.

Anyway. Very excited about all this. Will definitely keep you posted.