Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength. – Jordan Ellenberg, How Not to Be Wrong
Math is like… Saturday, Oct 25 2014
Mathematics 8:14 pm
Hard Problems and Hints Friday, Jul 11 2014
I have a friend O with a very mathematically engaged son J, who semi-often corresponds with me about his and J’s mathematical experiences together. We had a recent exchange and what I was saying to him I found myself wanting to say to everybody. So, without further ado, here is his email and my reply (my take on Aunt Pythia) -
J’s class is learning about volume in math. They’ll be working with cubes, rectangular prisms and possibly cylinders, but that’s all. He asked his teacher if he could work on a “challenge” that has been on his mind, which is to find a formula for the volume of one of his favorite shapes, the dodecahedron. He build a few of these out of paper earlier in the year and really was/is fascinated with them. I think he began this quest to find the volume thinking that it would be pretty much impossible, but he has stuck with it for almost a week now. I am pleased to see that he’s not only sticking with it, but also that he has made a few pretty interesting observations along the way, including coming up with an approach to solving it that involves, as he put it, “breaking it up into equal pieces of some simpler shape and then putting them together.” After trying a few ways to break/slice up the dodecahedron and finding that none of them seemed to make matters simpler, he had an “ah ha” moment in the car and decided that the way to do would be to break it up into 12 “pentagonal pyramids” (that’s what he calls them) that fit together, meeting at the center of rotation of the whole shape. If we can find the volume of one of those things, we’re all set. A few days later, he told me that he realized that “not every pentagonal pyramid could combine to make a dodecahedron” so maybe there was something special about the ones that do, i.e., maybe there is a special relationship between the length of the side of the pentagon and the length of the edge of the pyramid that could be used to form a dodecahedron.
He is still sticking with it, and seems to be having a grand time, so I am definitely going to encourage him and puzzle through it with him if he wants.
But here’s my question for you…
I sneaked a peak on google to see what the formula actually is, and found (as you might know) that it’s pretty complicated. The formula for the volume of the pentagonal pyramid involves (or something horrible like that) and the formula for the volume of a dodecahedron involves or something evil like that. In short, I am doubtful that he will actually be able to solve this problem he’s puzzling through. What does a good teacher do in such a situation? You have a student who is really interested in this problem, but you know that it’s far more likely that he will hit a wall (or many walls) that he really doesn’t have the tools to work through. On the other hand, you really want him to find satisfaction in the process and not measure the joy or the value of the process by whether he ultimately solves it.
I certainly don’t care whether he solves it or not. But I want to help him get value out of hitting the wall. How do you strike a balance so that the challenge is the right level of frustrating? When is it good to “give a hint” (you’ve done that for me a few times in what felt like a good way… not too much, but just enough so that the task was possible).
In this case, he’s at least trying to answer a question that has an answer. I suppose you could find a student working on a problem that you know has NO known answer, or that has been proven to be unsolvable. Although there, at least, after the student throws up his hands after giving it a good go, you can comfort her by saying, “guess what… you’re in good company!” But here, I’d like to help give him some of the tools he might use to actually make some headway, without giving away the store.
I think he’s off to a really good start — learning a lot along the way – getting a lot of out the process, the approach. I can already tell that many of the “ah ha” moments have applicability in all sorts of problems, so that’s wonderful.
Wow, okay first of all, I love that you asked me this and it makes me really appreciate your role in this journey J is on, in other words I wish every child had an adult present in their mathematical journey who recognizes the value in their self-driven exploration and is interested in being the guardian of the child’s understanding of that value.
Second: no matter what happens, you have access to the “guess what… you’re in good company” response, because the experience of hitting walls as you try to find your way through the maze of the truth is literally the experience of all research mathematicians, nearly all of the time. If by any chance J ends up being a research mathematician, he will spend literally 99% or more of his working life in this state.
In fact, I would want to tweak the message a bit; I find the “guess what… you’re in good company” a tad consolation-prize-y (as also expressed by the fact that you described it as a “comfort”). It implies that there was an underlying defeat whose pain this message is designed to ameliorate. I want to encourage you and J both to see this situation as one in which a defeat is not even possible, because the goal is to deepen understanding, and that is definitely happening, regardless of the outcome. The specific question (“what’s the volume of a dodecahedron?”) is a tool that’s being used to give the mind focus and drive in exploring the jungle of mathematical reality, but the real value is the journey, not the answer to the question. The question is just a tool to help the mind focus.
In fairness, questing for a goal such as finding the answer to a question and then not meeting the goal is always a little disappointing, and I’m not trying to act like that disappointment can be escaped through some sort of mental jiu-jitsu. What I am trying to say is that it is possible to experience this disappointment as superficial, because the goal-quest is an exciting and focusing activity that expresses your curiosity, but the goal is not the container of the quest’s value.
So, that’s what you tell the kid. Way before they hit any walls. More than that, that’s how you should see it, and encourage them to see it that way by modeling.
Third. A hard thing about being in J’s position in life (speaking from experience) is that the excitement generated in adults by his mathematical interests and corresponding “advancement” is exciting and heady, but can have the negative impact of encouraging him to see the value of what he’s doing in terms of it making him awesome rather than the exploration itself being the awesome thing, and this puts him in the position where it is possible for an unsuccessful mathematical expedition to be very ego-challenging. This is something that’s been behind a lot of the conversations we’ve had, but I want to highlight it here, to connect the dots in the concrete situation we’re discussing. To the extent that there are adults invested in J’s mathematical precociousness per se, and to the extent that J may experience an unsuccessful quest as a major defeat, these two things are connected.
Fourth, to respond to your request for concrete advice regarding when it is a good idea to give a hint. Well, there is an art to this, but here are some basic principles:
* Hints that are minimally obtrusive allow the learner to preserve their sense of ownership over the final result. The big dangers with a hint are (a) that you steal the opportunity to learn by removing a part of the task that would have been important to the learning experience, and (b) that you steal the experience of success because the learner doesn’t feel like they really did it. These dangers are related but distinct.
* How do you give a minimally obtrusive hint?
(a) Hints that direct the learner’s attention to a potentially fruitful avenue of thought are superior to hints that are designed to give the learner a new tool.
(b) Hints that are designed to facilitate movement in the direction of thought the learner already has going on are generally better than hints that attempt to steer the learner in a completely new direction.
* If the learner does need a new tool, this should be addressed explicitly. It’s kind of disingenuous to think of it as a “hint” – looking up “hint” in the dictionary just now, I’m seeing words like “indirect / suggestion / covert indication”. If the learner is missing a key tool, they need something direct. The best scenario is if they can actually ask for what they need:
Learner: If I only had a way to find the length of this side using this angle…
Teacher: oh yes, there’s a whole body of techniques for that, it’s called trigonometry.
This is rare but that’s okay because it’s not necessary. If the teacher sees that the learner is up against the lack of a certain tool, they can also elicit the need for it from the learner:
Teacher: It seems like you’re stuck because you know this angle but you don’t know this side.
Teacher: What if I told you there was a whole body of techniques for that?
Okay, those are my four cents. Keep me posted on this journey, it sounds like a really rich learning experience for J.
All the best, Ben
Sue’s Book Is Ready for Press and Needs Crowdfunding! Friday, Jun 20 2014
Hey y’all, I am incredibly excited about Sue’s book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. If you have been around the math education blogosphere for more than a short time, you probably are too.
It needs crowdfunding to cover publication costs. I am about to help out and I invite you to do so too!
Steven Strogatz on what math education often gets wrong Thursday, Apr 10 2014
If you’re telling a student the answer to a question it would never occur to them to ask, I can’t see how that’s a productive use of anyone’s time.
(This is not verbatim, but it’s the idea.)
New Math Learning Site on StackExchange.com Needs You Sunday, Mar 2 2014
There’s a new proposal at stackexchange.com for a Q&A site on Mathematics Learning, Studying, and Education.
Of course the entire mathtwitterblogosphere is a massive Q&A site on Mathematics Learning, Studying, and Education. But based on my experience of the incredible usefulness of the StackExchange sites Math StackExchange and MathOverflow, I think this site could become a great resource.
Possibly also a great forum for some much-needed productive dialogue between the K-12 and collegiate levels. For that to happen, though, it needs you. The bulk of the folks currently signed up for the beta of the new site are active on Math StackExchange and MathOverflow, which are dominated by college-and-up level math. The conversation is going to be so much richer with serious K-12 representation! Go sign up!
If you haven’t heard of the StackExchange sites before, they are a very thoughtfully constructed Q&A structure. It all started with StackOverflow, which was for working programmers to ask and answer practical coding questions. MathOverflow copied this idea for working research mathematicians. Math StackExchange is in principle for Q&A about math at any level, although as I mentioned, in practice it’s usually (though not always) about college and graduate level. Now there are also StackExchange sites on cooking, gaming, English language learning, and a million other things. The design of the software, and the culture of the sites, do an impressive job keeping the Q&A productive and on topic.
In the case of the math sites, the culture can also feel a little normatively intense (as in, there’s a “way we do things” that can be pretty strongly policed) and not always welcoming. Denizens of the sites will tell you that this is how they keep the conversation so productive and on-topic. But imho, it also stems from the deep ambivalence that the academic math world has about whether it wants to
(A) Share all its goodies and invite everyone into its kingdom, or
(B) Bull-guard the considerable stash of privilege that accrues from its high intellectual status.
(More on this in future posts.) The incredible usefulness of the sites makes it worth it; but also, this is part of why I want you guys to go populate the new Math Education site. You are clear in your hearts that math is for everybody. This is our chance to go talk with some folks who represent a culture that is working through that for itself. Meanwhile, we get to benefit from their perspectives, which have seen very different parts of the mathematical kingdom in their travels.
Disclaimer: I think lots and lots of individuals on Math.SE and MO think math is for everybody. I am not trying to stereotype the sites or mathematicians more generally. And I think it’s likely that the people from Math.SE and MO who gravitate to the new Math Learning/Education site are going to be skewed toward the folks who think that math is for everybody. What I am trying to do is to name some notes that I hear in the cultural soundscape of academic math as a whole, and Math StackExchange and MathOverflow in particular; but I’m not trying to identify those notes with any individuals.
A Critical Language for Problem Design Saturday, Jan 18 2014
I am at the Joint Mathematics Meetings this week. I had a conversation yesterday, with Cody L. Patterson, Yvonne Lai, and Aaron Hill, that was very exciting to me. Cody was proposing the development of what he called a “critical language of task design.”
This is an awesome idea.
But first, what does he mean?
He means giving (frankly, catchy) names to important attributes, types, and design principles, of mathematical tasks. I can best elucidate by example. Here are two words that Cody has coined in this connection, along with his definitions and illustrative examples.
Jamming – transitive verb. Posing a mathematical task in which the underlying concepts are essential, but the procedure cannot be used (e.g., due to insufficient information).
Example: you are teaching calculus. Your students have gotten good at differentiating polynomials using the power rule, but you have a sinking suspicion they have forgotten what the derivative is even really about. You give them a table like this
and then ask for a reasonable estimate of f'(4). You are jamming the power rule because you’re giving them a problem that aims at the concept underlying the derivative and that cannot be solved with the power rule.
Thwarting – transitive verb. Posing a mathematical task in which mindless execution of the procedure is possible but likely to lead to a wrong answer.
Example: you are teaching area of simple plane figures. Your students have gotten good at area of parallelogram = base * height but you feel like they’re just going through the motions. You give them this parallelogram:
Of course they all try to find the area by . You are thwarting the thoughtless use of base * height because it gets the wrong answer in this case.
Why am I so into this? These are just two words, naming things that all teachers have probably done in some form or another without their ever having been named. They describe only a very tiny fraction of good tasks. What’s the big deal?
It’s that these words are a tiny beginning. We’re talking about a whole language of task design. I’m imagining having a conversation with a fellow educator, and having access to hundreds of different pedagogically powerful ideas like these, neatly packaged in catchy usable words. “I see you’re thwarting the quadratic formula pretty hard here, so I’m wondering if you want to balance it out with some splitting / smooshing / etc.” (I have no idea what those would mean but you get the idea.)
I have no doubt that a thoughtful, extensive and shared vocabulary of this kind would elevate our profession. It would be a concrete vehicle for the transmission and development of our shared expertise in designing mathematical experiences.
This notion has some antecedents. First, there are the passes at articulating what makes a problem pedagogically valuable. On the math blogosphere, see discussions by Avery Pickford, Breedeen Murray, and Michael Pershan. (Edit 1/21: I knew Dan had one of these too.) I also would like to believe that there is a well-developed discussion on this topic in academic print journals, although I am unaware of it. (A google search turned up this methodologically odd but interesting-seeming article about biomed students. Is it the tip of the iceberg? Is anyone reading this acquainted with the relevant literature?)
Also, I know a few other actual words that fit into the category “specialized vocabulary to discuss math tasks and problems.” I forget where I first ran into the word problematic in this context – possibly in the work of Cathy Twomey-Fosnot and Math in the City – but that’s a great word. It means that the problem feels authentic and vital; the opposite of contrived. I also forget where I first heard the word grabby (synonymous with Pershan’s hooky, and not far from how Dan uses perplexing) to describe a math problem – maybe from the lips of Justin Lanier? But, once you know it it’s pretty indispensible. Jo Boaler, by way of Dan Meyer, has given us the equally indispensable pseudocontext. So, the ball is already rolling.
When Cody shared his ideas, Yvonne and I speculated that the folks responsible for the PCMI problem sets – Bowen Kerins and Darryl Yong, and their friends at the EDC – have some sort of internal shared vocabulary of problem design, since they are masters. They were giving a talk today, so I went, and asked this question. It wasn’t really the setting to get into it, but superficially it sounded like yes. For starters, the PCMI’s problem sets (if you are not familiar with them, click through the link above – you will not be sorry) all contain problems labeled important, neat and tough. “Important” means accessible, and also at the center of connections to many other problems. Darryl talked about the importance of making sure the “important” problems have a “low threshold, high ceiling” (a phrase I know I’ve heard before – anyone know where that comes from?). He said that Bowen talks about “arcs,” roughly meaning, mathematical themes that run through the problem sets, but I wanted to hear much more about that. Bowen, are you reading this? What else can you tell us?
Most of these words share with Cody’s coinages the quality of being catchy / natural-language-feeling. They are not jargony. In other words, they are inclusive rather than exclusive. It is possible for me to imagine that they could become a shared vocabulary of our whole profession.
So now what I really want to ultimately happen is for a whole bunch of people (Cody, Yvonne, Bowen, you, me…) to put in some serious work and to write a book called A Critical Language for Mathematical Problem Design, that catalogues, organizes and elucidates a large and supple vocabulary to describe the design of mathematical problems and tasks. To get this out of the completely-idle-fantasy stage, can we do a little brainstorming in the comments? Let’s get a proof of concept going. What other concepts for thinking about task design can you describe and (jargonlessly) name?
I’m casting the net wide here. Cody’s “jamming” and “thwarting” are verbs describing ways that problems can interrupt the rote application of methods. “Problematic” and “grabby” are ways of describing desirable features of problems, while “pseudocontext” is a way to describe negative features. Bowen and Darryl’s “important/neat/tough” are ways to conceptualize a problem’s role in a whole problem set / course of instruction. I’m looking for any word that you could use, in any way, when discussing the design of math tasks. Got anything for me?
In fairness, for all I know, somebody has written a book entitled A Critical Language for Mathematical Task Design. I doubt it, but just in case, feel free to get me a copy for my birthday.
I am taking a perhaps-undeserved dig here at a number of in-many-ways-wonderful curriculum and instructional design initiatives that have a lot of rich and deep thought about pedagogy behind them but have really jargony names, such as Understanding by Design and Cognitively Guided Instruction. (To prove that an instructional design paradigm does not have to be jargony, consider Three-Acts.) I feel a bit ungenerous with this criticism, but I can’t completely shake the feeling that jargony names are a kind of exclusion: if you really wanted everybody to use your ideas, you would have given them a name you could imagine everybody saying.
Kids Summarizing Sunday, Sep 8 2013
Back in the spring, I resolved to make a practice of having students summarize each others’ thoughts whenever I have classroom opportunities. This summer, I got the opportunity to give this technique a sustained go, when I taught at SPMPS (which was completely awesome btw). And:
It is an effing game-changer.
This summer, when I or a student put forth an idea, I regularly followed it with, “who can summarize what so-and-so said?” Or (even better), “so-and-so, can you summarize what so-and-so just said?” Following the models of Lucy West and Deborah Ball, I carefully distinguished summary from evaluation. “Not whether you buy it, just the idea itself.” When dipsticking the room on an idea, I would also make this distinction. “Raise your hand if you feel that you understand what was just said; not that you buy it, just that you understand what they’re trying to say.” Then, “leave your hand up if you also buy it.”
These moves completely transformed the way whole-class conversation felt to me:
* Students were perceptibly more engaged with each others’ ideas.
* The ideas felt more like community products.
* Students who were shy to venture an idea in the first place nonetheless played key roles as translators of others’ ideas.
Furthermore, for the first time I felt I had a reliable way past the impasse that happens when somebody is saying something rich and other people are not fully engaged. More generally, past the impasse that happens when somebody says something awesome and there are others for whom it doesn’t quite land. (Whether they were engaged or not.)
A snippet of remembered classroom dialogue to illustrate:
Me: The question before us is, do the primes end, or do they go on forever? At this point, does anybody think they know?
(Aside: This was after a day of work on the subject. Most kids didn’t see the whole picture at this point, but one did:)
[J raises his hand.]
J: They don’t end. If they ended, you’d have a list. You could multiply everything on the list and add 1 and you would get a big number N. Either N is prime or it’s composite. If it is prime, you can add it to the list. If it is composite, it has at least one prime factor. Its factor can’t be on the list because all the numbers on the list when divided [into] N have a remainder of 1. So you can add its factor to the list. You can keep doing this forever so they don’t end.
Me: Raise your hand to summarize J’s thought.
(Aside: although J has just basically given a complete version of Euclid’s proof of the infinitude of the primes, and although I am ecstatic about this, I can’t admit any of this because the burden of thought needs to stay with the kids. J is just about done with the question, but this is just the right thing, said once: the class as a whole is nowhere near done. This is one of the situations in which asking for summaries is so perfect.)
[Several kids raise their hands. I call on T.]
T: J is saying that the primes don’t end. He says this because if you have a list of all the primes, you can multiply them and add one, giving you a big number N. If N is prime, you can add it to the list. If N is not prime, and its prime factors are not on the list, you can add them.
Me: J, is that what you were trying to say?
(Notice that a key point in J’s argument, that the factors of N cannot already be on the list, was not dealt with by T, and J did not catch this when asked if T had summarized his point. This is totally typical. Most kids in the room have not seen why this point is important. Some kids have probably not seen why J’s argument even relates to the question of whether the primes end. All this has to be given more engaged airtime.)
Me: raise your hand if you feel that you understand the idea that J put forth that T is summarizing.
[About 2/3 of the room raises hands. I raise mine too.]
Me: Leave your hand up if you also find the idea convincing and you now believe the primes don’t end.
[A few kids put their hands down. I put mine down too.]
N [to me]: Why did you put your hand down?
Me [to class]: Who else wants to know?
[At least half the class raises hands.]
Me [to T]: Here’s what’s bugging me. You said that if N is not prime and its prime factors are not on the list, I can add them. But what if N is not prime and its prime factors are already on the list?
T [thinks for a minute]: I don’t know, I’ll have to think more about that.
[J's hand shoots up]
Me [to T]: Do you want to see what J has to say about that or do you want to think more about it first?
[T calls on J to speak]
J: Can’t happen. All the numbers on the list were multiplied together and added 1 to get N. So when N is divided by 2, 3, 5, and so on, it has a remainder of 1. So N’s factor can’t be 2, 3, 5, and so on.
T: Oh, yeah, he’s right.
Me: Can you summarize his whole thought?
[T explains the whole thing start to finish.]
Me: Do you buy it?
Me: Who else wants to summarize the idea that J put forth and T summarized?
Unexpectedly, this technique speaks to a question I was mulling over a year and a half ago, about how to encourage question-asking. How can the design of the classroom experience structurally (as opposed to culturally) encourage people to ask questions and seek clarification when they need it? The answer I half-proposed back then was to choose certain moments in the lesson and make student questions the desired product in those moments. (“Okay everyone, pair up and generate a question about the definition we just put up” or whatever.) At the time I didn’t feel like this really addressed the need I was articulating because it had to be planned. Kate rightly pressed me on this because actually it’s awesome to do that. But I was hungering for something more ongoingly part of the texture of class, not something to build into a lesson at specific points. And as it turns out, student summaries are just what I was looking for! The questions and requests for clarification are forced out by putting students on the spot to summarize.
A last thought. Learning this new trick has been for me a testament to teaching’s infinitude as a craft. Facilitating rich and thought-provoking classroom discussions was already something I’d given a lot of thought and conscious work to; perhaps more than to any other part of teaching, at least in recent years. I.e. this is an area where I already saw myself as pretty accomplished (and, hopefully with due modesty, I still stand by that). And yet I could still learn something so basic as “so-and-so, can you summarize what so-and-so said?” and have it make a huge difference. What an amazing enterprise to always be able to grow so much.
I Don’t Get It vs. I Don’t Buy It Wednesday, Jul 24 2013
I was having a conversation a few weeks ago with a computer programmer and math enthusiast whom I’ll call Dorian. He was arguing very passionately that talking about a square root of was the wrong way to introduce complex numbers. He recounted this moment in his own schooling: 16 year old Dorian, told by his teacher “we introduce a new number whose square is …,” asking, “but I can prove that the square of any number is positive, what about that?!” His teacher wasn’t able to satisfy his objection and made him feel that it wasn’t valid. He left the experience feeling angry and frustrated and that his question had been treated as a failure to understand.
Dorian later learned that complex numbers can be visualized as a plane containing the real line; that addition of points in this plane is just vector addition; and that multiplication is done by multiplying the distances from the origin and adding the angles from the positive real axis (see here for a brief explanation if desired). Here was a concrete model for the complex numbers, with concrete geometrical interpretations of the operations and . And it was clear to him that in this model, there is a point, in fact two points, whose squares correspond to the point on the real axis. But philosophically, this fact is a consequence of the concrete geometrical description of the operations in the plane, rather than an ontologically dubious starting point for the whole project.
Dorian concluded that actually this model, via the geometry of addition and multiplication in the complex plane, is a pedagogically superior introduction to the complex numbers. His argument is that it presents no ontological quandary. Nobody will object to a plane. Nobody will object, at least on philosophical grounds, to these new definitions of and , as long as you can prove they have nice properties and coincide with the old definitions on the real line. You’re not saying anything so wildly speculative as “postulate a square root of …”
I am not writing this post to get into the question of whether Dorian is right about this. I see lots to say on both sides. What I am writing this to say is that there is a lesson in Dorian’s story much deeper than the question of how to introduce the complex numbers. That is not the real question here as far as I am concerned.
The real question is this: when you’ve picked your approach and gone with it, how will you deal with the students it doesn’t work for?
Now you can always obsess about how to introduce a topic, and I believe there is basically always value in thinking and talking about the pedagogical consequences of different ways of looking at things. And I think some models for ideas are legitimately better than others. But no model will speak to every student. This point is so important, and was so lost on me as a young teacher, and is lost on so many (especially young) teachers that I have spoken with, so excited that they are about the way they have thought of to present negative numbers or whatever, as though miraculously everyone in the room will get it this time, that I need to repeat it:
There is no model that is the right model for each and every student, each and every time.
No matter how awesome your idea for how to think about XYZ concept is, there will be somebody in your class who will have no idea what you are talking about. To me, the big question here is, what are you going to do about it?
More specifically, how are you going to treat their thinking?
Now, I like to think that nobody reading this blog would be so callous as to intentionally make a student feel stupid for asking an honest question. But there are far subtler ways to do it. The one I most want to warn you against is the sin I know I’m guilty of: being so wrapped up in the awesomeness of your presentation that the kid who doesn’t get it does not compute to you. You say whatever you say out loud but in your mind you’re like, “wait – you don’t understand? Huh?” Or, you’re like, “oh my goodness can’t you just see it as I do?”
Regardless of what you say out loud, having such a response in the back of your mind invalidates whatever obstacle the student is facing. I want to suggest an alternative:
Take the case that any earnest failure of a student to see your point of view is actually coming from a legitimate mathematical objection.
This is how you treat dissatisfaction with honor.
I don’t care what the kid’s IEP says. Mathematical convention does not require us to check somebody’s Wechsler results before they are allowed to raise an objection. If they don’t buy it, they don’t buy it. Now it’s your turn to understand their objection and answer it.
“I don’t get it.” “I don’t buy it.”
A student I’ll call Manny, whom I had in my 2003-4 AP Calculus class, came to me around March and said something like, “this entire class is based on a paradox.” He objected to my (retrospectively totally hand-wavy) discussion of limits. It never gets there, so how can you talk about what happens if it were to get there?
I tried to answer Manny’s objections; I spent some time with him on it; but he left the conversation unsatisfied. Retrospectively it is clear to me that this is because (a) I didn’t get what the problem was, and (b) to my shame I didn’t consider the possibility that there was really much to it. Then, less than a year later, I read The Calculus Gallery, whereupon I learned that actually Manny’s objection was more or less exactly Bishop Berkeley’s famous objection that in due time forced mathematicians to invent real analysis. For a sense of the importance of this development, let me mention that I have read, though I don’t recall where right now, that the development of real analysis was really the event that led to the birth of modern mathematical rigor.
So, yes, I am on record as having treated as essentially invalid an objection that actually led to the creation of modern rigor. Don’t let that be you.
If they don’t get it, take the case that there’s a legitimate mathematical objection behind that. Treat their “I don’t get it” as “I don’t buy it.” Now getting them to buy it is your job.
Sh*t I F*cking Love (Wherein I Am Moved to Profanity by Enthusiasm) Friday, Apr 5 2013
Paul Salomon’s “imbalance problems”. You know how I love a thought-provoking picture.