I Don’t Get It vs. I Don’t Buy It

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 -1 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 i whose square is -1…,” 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 \times. And it was clear to him that in this model, there is a point, in fact two points, whose squares correspond to the point -1 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 \times, 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 -1…”

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)

Shawn Cornally doing his thing.

My new favorite blog, chronicling an adventure in striving to keep math class true to your deepest commitments. (Thanks to Work in Pencil for the recommendation.)

Paul Salomon’s “imbalance problems”. You know how I love a thought-provoking picture.

Math Munch. If you haven’t yet checked out this joint project of Paul, Anna and Justin yet, you should get on that.

Notes from the Learning Lab: How to Dull My Curiosity

I know I say this kind of thing a lot but I’m sitting here studying for a final, and this truth is just glaring and throbbing at me:

If you want to dull my curiosity, tell me what the answer is supposed to be.

If you want to make my curiosity vanish completely, do that and then add in a little time pressure.

There is nothing as lethal to my sense of wonder as that alchemical combination of already knowing how things are going to turn out (without knowing why), and feeling the clock tick.

Wherein This Blog Serves Its Original Function

The original inspiration for starting this blog was the following:

I read research articles and other writing on math education (and education more generally) when I can. I had been fantasizing (back in fall 2009) about keeping an annotated bibliography of articles I read, to defeat the feeling that I couldn’t remember what was in them a few months later. However, this is one of those virtuous side projects that I never seemed to get to. I had also met Kate Nowak and Jesse Johnson at a conference that summer, and due to Kate’s inspiration, Jesse had started blogging. The two ideas came together and clicked: I could keep my annotated bibliography as a blog, and then it would be more exciting and motivating.

That’s how I started, but while I’ve occasionally engaged in lengthy explication and analysis of a single piece of writing, this blog has never really been an annotated bibliography. EXCEPT FOR RIGHT THIS VERY SECOND. HA! Take THAT, Mr. Things-Never-Go-According-To-Plan Monster!

“Opportunities to Learn Reasoning and Proof in High School Mathematics Textbooks”, by Denisse R. Thompson, Sharon L. Senk, and Gwendolyn J. Johnson, published in the Journal for Research in Mathematics Education, Vol. 43 No. 3, May 2012, pp. 253-295

The authors looked at HS level textbooks from six series (Key Curriculum Press; Core Plus; UCSMP; and divisions of the major publishers Holt, Glencoe, and Prentice-Hall) and analyzed the lessons and problem sets from the point of view of “what are the opportunities to learn about proof?” To keep the project manageable they just looked at Alg. 1, Alg. 2 and Precalc books and focused on the lessons on exponents, logarithms and polynomials.

They cast the net wide, looking for any “proof-related reasoning,” not just actual proofs. For lessons, they were looking for any justification of stated results: either an actual proof, or a specific example that illustrated the method of the general argument, or an opportunity for students to fill in the argument. For exercise sets, they looked at problems that asked students to make or investigate a conjecture or evaluate an argument or find a mistake in an argument in addition to asking students to actually develop an argument.

In spite of this wide net, they found that:

* In the exposition, proof-related reasoning is common but lack of justification is equally common: across the textbook series, 40% of the mathematical assertions about the chosen topics were made without any form of justification;

* In the exercises, proof-related reasoning was exceedingly rare: across the textbook series, less than 6% of exercises involved any proof-related reasoning. Only 3% involved actually making or evaluating an argument.

* Core Plus had the greatest percentage of exercises with opportunities for students to develop an argument (7.5%), and also to engage in proof-related reasoning more generally (14.7%). Glencoe had the least (1.7% and 3.5% respectively). Key Curriculum Press had the greatest percentage of exercises with opportunities for students to make a conjecture (6.0%). Holt had the least (1.2%).

The authors conclude that mainstream curricular materials do not reflect the pride of place given to reasoning and proof in the education research literature and in curricular mandates.

“Expert and Novice Approaches to Reading Mathematical Proofs”, by Matthew Inglis and Lara Alcock, published in the Journal for Research in Mathematics Education, Vol. 43 No. 4, July 2012, pp. 358-390

The authors had groups of undergraduates and research mathematicians read several short, student-work-typed proofs of elementary theorems, and decide if the proofs were valid. They taped the participants’ eye movements to see where their attention was directed.

They found:

* The mathematicians did not have uniform agreement on the validity of the proofs. Some of the proofs had a clear mistake and then the mathematicians did agree, but others were more ambiguous. (The proofs that were used are in an appendix in the article so you can have a look for yourself if you have JSTOR or whatever.) The authors are interested in using this result to challenge the conventional wisdom that mathematicians have a strong shared standard for judging proofs. I am sympathetic to the project of recognizing the way that proof reading depends on context, but found this argument a little irritating. The proofs used by the authors look like student work: the sequence of ideas isn’t being communicated clearly. So it wasn’t the validity of a sequence of ideas that the participants evaluated, it was also the success of an imperfect attempt to communicate that sequence. Maybe this distinction is ultimately unsupportable, but I think it has to be acknowledged in order to give the idea that mathematicians have high levels of agreement about proofs its due. Nobody who espouses this really thinks that mathematicians are likely to agree on what counts as clear communication. Somehow the sequence of ideas has to be separated from the attempt to communicate it if this idea is to be legitimately tested.

* The undergraduates spent a higher percentage of the time looking at the formulas in the proofs and a lower percentage of time looking at the text, as compared with the mathematicians. The authors argue that this is not fully explained by the hypothesis that the students had more trouble processing the formulas, since the undergrads spent only slightly more time total on them. The mathematicians spent substantially more time on the text. The authors speculate that the students were not paying as much attention to the logic of the arguments, and that this pattern accounts for some of the notorious difficulty that students have in determining the validity of proofs.

* The mathematicians moved their focus back and forth between consecutive lines of the proofs more frequently than the undergrads did. The authors suggest that the mathematicians were doing this to try to infer the “implicit warrant” that justified the 2nd line from the 1st.

The authors are also interested in arguing that mathematicians’ introspective descriptions of their proof-validation behavior are not reliable. Their evidence is that previous research (Weber, 2008: “How mathematicians determine if an argument is a valid proof”, JRME 39, pp. 431-459) based on introspective descriptions of mathematicians found that mathematicians begin by reading quickly through a proof to get the overall structure, before going into the details; however, none of the mathematicians in the present study did this according to their eye data. One of them stated that she does this in her informal debrief after the study, but her eye data didn’t indicate that she did it here. Again I’m sympathetic to the project of shaking up conventional wisdom, and there is lots of research in other fields to suggest that experts are not generally expert at describing their expert behavior, and I think it’s great when we (mathematicians or anyone else) have it pointed out to us that we aren’t right about everything. But I don’t feel the authors have quite got the smoking gun they claim to have. As they acknowledge in the study, the proofs they used are all really short. These aren’t the proofs to test the quick-read-thru hypothesis on.

The authors conclude by suggesting that when attempting to teach students how to read proofs, it might be useful to explicitly teach them to mimic the major difference found between novices and experts in the study: in particular, the idea is to teach them to ask themselves if a “warrant” is required to get from one line to the next, to try to come up with one if it is, and then to evaluate it. This idea seems interesting to me, especially in any class where students are expected to read a text containing proofs. (The authors are also calling for research that tests the efficacy of this idea.)

The authors also suggest ways that proof-writing could be changed to make it easier for non-experts to determine validity. They suggest (a) reducing the amount of symbolism to prevent students being distracted by it, and (b) making the between-line warrants more explicit. These ideas strike me as ridiculous. Texts already differ dramatically with respect to (a) and (b), there is no systemic platform from which to influence proof-writing anyway, and in any case as the authors rightly note, there are also costs to both, so the sweet spot in terms of text / symbolism balance isn’t at all clear and neither is the implicit / explicit balance. Maybe I’m being mean.


I got home last night from a week and a half of traveling to find a newspaper clipping my mother sent me: the obituary for Bill Thurston in the New York Times. I hadn’t know he was sick.

Thurston was a giant of twentieth century geometry, but more important to me is the sense I always get from his writing – a sense of warmth, the intention to share, an abiding interest in math as a human practice. A complete lack of interest in the privilege of being seen as brilliant. A desire to demystify the process of mathematical discovery.

I spent some time today looking for online tributes. Justin Lanier wrote a beautiful one. My desire to refer you to this was the impulse that prompted this post.

Here’s something else beautiful: Thurston’s profile on MathOverflow, linking to the questions he asked and answered on that site.

And for math and art enthusiasts, here’s some high fashion inspired by Thurston’s work.

Rest in peace Bill Thurston.

Some Miscellaneous Awesomeness

Just some awesome stuff I feel like pointing out:

Vi Hart does it again. That young woman has created a new art form.

Terry Tao’s airport puzzle. If you have to get from one end of the airport to the other to catch a plane, but you really need to stop for a minute to tie your shoe, is it best to do it while you’re on the moving walkway or not? (I learned this problem from Tim Gowers’ blog.)

Paul Salomon quotes Vi Hart quoting Edmund Snow Carpenter, and the quote is absolutely worth me quoting yet again:

The trouble with knowing what to say and saying it clearly and fully, is that clear speaking is generally obsolete thinking. Clear statement is like an art object: it is the afterlife of the process which called it into being.

Dan Goldner is doing my job for me. The original purpose of this blog was to read writing about math education, and to summarize and discuss it. I don’t do this very much any more (although expect summaries of a couple articles from the current JRME in the next few weeks months), but I do have a long list of things I wanted to read and discuss here but figured I’d probably never get to. On this list was the 1938 NCTM Yearbook, The Nature of Proof, by Harold Fawcett. But I’m taking it off; Dan’s got it covered.

Another One to Keep Your Eye On: Anna Weltman

Here’s another blog to keep an eye on:

Recipes for Pi, by Anna Weltman.

I know Anna IRL. In fact, both of us have seen the other one teach. Thus prior to discovering her blog I already knew her as mathematically thought-provoking, endlessly creative, and deeply tuned in to student experience, not to mention a total sweetheart.

So I was excited to learn that she had started blogging in February, and her writing hasn’t disappointed. It’s sporadic, but who am I to complain about that, and more importantly it’s characterized by that same deep thinking about math and student experience that marks her teaching. Check it out.

Aside: Anna teaches at St. Ann’s School, along with Justin Lanier, Paul Salomon, and Paul Lockhart.

What She Said

Three weeks ago Sue VanHattum and Kate Nowak recommended Bob and Ellen Kaplan’s Math Circle Training Institute. If you are looking for a PD opportunity this summer and you are interested in cultivating students thinking for themselves, I strongly second their recommendation.

This is a weeklong training on the campus of Notre Dame in South Bend, Indiana where you learn how to run a math circle in the spirit of the Kaplans. What that means is that you ask thought-provoking questions and you facilitate students discussing them. Heaven, right? The setup is that in the morning, the Kaplans run a math circle on you, and in the afternoon they bus in local kiddies for you to try out your thought-provoking questions on, and watch others do it, and give and receive feedback. At lunch and at night you hang out with like minded educators talking about math and education. The $850 includes room and board for the whole week.

I did this training in the summer of 2009 and it was a key step on my path to being the educator I am now. In 2007-8 I had come to the realization that my most central, pressing goal as an educator was to empower students to find their own mathematical curiosity, and I started stretching my pedagogical boundaries to find out what it would look and feel like to teach with this as the only goal. But I felt like I was reinventing the wheel. Reading the Kaplans’ book Out of the Labyrinth, I felt like I had found my comrades. Going to the Summer Institute, I felt like I had met them.

For example, Sue and Alex, and my fairy blogfamily Kate and Jesse Johnson. See what I mean?

Tangential to the math PD but also a wonderful benefit was the opportunity to spend a week on the Notre Dame campus. As a Jew I did not go into the experience expecting to be so moved by the shrines and sanctuaries of this Catholic institution, but I was. After my first experience with a labyrinth (the meditative kind), Alex McFerron said to me, “the Catholics really ace those sacred spaces.” True that.

Elementary Mathematics from an Advanced Standpoint

Another of the many reasons I’m in grad school. I benefit as a teacher from understanding the content I teach in way more depth than I teach it. (I think everybody does, but it’s easiest to talk about myself.)

This does a number of things for me. The simplest is that it makes the content more exciting to me. Something that previously seemed routine can become pregnant with significance if I know where it’s going, and there’s a corresponding twinkle that shows up in my eye the whole time my students are dealing with it. A second benefit is that it gives me both tools and inspiration to find more different ways of explaining things. A third is that it helps me see (and therefore develop lessons aimed at) connections between different ideas.

So, this post is a catalogue of some insights that I’ve had about K-12 math that I’ve been led to by PhD study. The title of the post is a reference to Felix Klein’s classic books of the same name. The catalogue is mostly for my own benefit, and I don’t have all that much time, so I’m going to try to suppress the impulse to fully explain some of the more esoteric vocabulary, but I never want to write something here that requires expert knowledge to avoid being useless, so I’ll try to be both clear and pithy. (Wish me luck.)

Elementary level: Multiplication is function composition.

I’m developing the opinion that it’s important for especially middle and high school teachers to have this language. The upshot is that in addition to the usual models of multiplication as (a) repeated addition and (b) arrays and area of rectangles (and, if you’re lucky, (c) double number lines), multiplication is also the net effect of doing two things in a row, such as stretching (and possibly reversing) a number line.

The big thing I want to say here is that understanding this is key to understanding multiplication of signed numbers. I would go so far as to wager that anybody who feels they know intuitively why -\cdot -=+ understands it on some level, consciously or not.

When somebody asks me why a negative times a negative is a positive, I have often had the inclination to answer with, “well, what’s the opposite of the opposite of something?” (I have seen many teachers use metaphors with the same upshot.) The problem is that if you understand multiplication only as repeated addition and as the the area of rectangles, I’ve changed the subject with this answer. It is a complete nonsequitur. It’s probably clear why it has to do with negatives but why does it have to do with multiplication?

On the other hand, if on any level you realize that one meaning of 2\times 3 is “double then triple”, then it’s natural for (-2)\times(-3) to mean “double and oppositize, then triple and oppositize.” But for this you had to be able to see multiplication as “do something then do something else.”

Algebra I and Algebra II: Substitution is calculation inside a coordinate ring.

I just realized this today, and that’s what inspired this blog post. So far, I’m not sure the benefit of this one to my teaching beyond the twinkle it will bring to my eye, though perhaps that will become clear later. It’s certainly helping me understand something about algebraic geometry. The basic idea is this: say you’re finding the intersections of some graphs like y=3x+5 and 2x+y=30. You’re like, “alright, substitute using the fact that y=3x+5. 2x+(3x+5)=30, so 5x+5=30…” and you solve that to find x=5, for an intersection point of (5,20). A way to look at what you’re doing when you make the substitution y=3x+5 is that you’re working in a special algebraic system determined by the line y=3x+5, in particular the (tautological) fact that on this line, y is exactly three x plus five. In this system, polynomials in x or y alone work the usual way, but polynomials in x and y both can often be simplified using the relation y=3x+5 connecting x and y. This algebraic system is called “the coordinate ring of the line y=3x+5.”

I can’t tell if it will even seem that I’ve said anything at all here. The point, for me, is just a sublte shift in perspective. I imagine myself sitting on the line y=3x+5; then this line determines an algebraic system (the coordinate ring) which, as long as I’m on the line, is the right system; and when I substitute 3x+5 for y, what I’m doing is using the rules of that system.

Calculus: The chain rule is the functoriality of the derivative.

“Functoriality” is a word from category theory which I will avoid defining. The point is really about the chain rule. The main ways the derivative is presented in a first-year calculus class are as speed, or rate of change, on the one hand (like, you’re always thinking of the independent variable as time, whatever it really is), and the slope of the tangent line of a graph, on the other. There is a third way to look at it, which I learned from differential geometry. If you look at a function as a mapping from the real line to itself, then the derivative describes the factor by which it stretches small intervals. For example, f(x)=x^2 has a derivative of 6 at x=3. What this is saying is that very small intervals around x=3 get mapped to intervals that are about 6 times as long. (To illustrate: the interval [3,3.01] gets mapped to [9,9.0601], about 6 times as long.)

Seen in this way, the strange formula [f(g(x))]'=f'(g(x))\cdot g'(x) for the chain rule becomes the only sensible way it could be. The function f(g(x)) is the net effect of doing g to x and then doing f to the answer g(x). If I want to know how much this function stretches intervals, well, when g is applied to x they are stretched by a factor of g'(x). Then when f is applied to g(x) they are stretched by a factor of f'(g(x)). (Note it is clear why you evaluate f' at g(x): that is the number to which f got applied.) So you stretched first by a factor of g'(x) and then by a factor of f'(g(x)); net effect, f'(g(x))\cdot g'(x), just like the formula says.

(As an aside, for the sake of being thematic, note the role here of the fact that the multiplication comes from the composition of the two stretches – multiplication is function composition. When I say “the derivative is functorial” what I really mean is that it turns composition of functions into composition of stretches.)

Calculus: The intermediate value theorem is f(\text{connected})=\text{connected}. The extreme value theorem is f(\text{compact})=\text{compact}.

This is a good example of what I was talking about at the beginning about the twinkle in my eye, and connections between ideas. When I used to teach AP calculus, the extreme value theorem and the intermediate value theorem were things I had trouble connecting to the rest of the curriculum. They were these miscellaneous, intuitively obvious factoids about continuous functions that were stuck into the course in awkward places. They both had the same clunky hypothesis, “if f is a function that is continuous on a closed interval [a,b]…” I didn’t do much with them, because I didn’t care about them.

I started to see a bigger picture about three years ago, in a course for calculus teachers taught by the irrepressible Larry Zimmerman. He referred to that clunky hypothesis as something to the effect of “a lilting refrain.” I was also left with the image of a golden thread weaving through the fabric of calculus but I’m not sure if he said that. The point is, he made a big deal about that hypothesis, making me notice how thematic it is.

Last year when I taught a course on algebra and analysis, having benefited from this education, I made these theorems important goals of the course. But something further clicked into place this fall, when I started to need to draw on point-set topology knowledge as I studied differential geometry. Two fundamental concepts in topology are compactness and connectedness. They have technical definitions for which you can follow the links. Intuitively, connectedness is what it sounds like (all one piece), and compactness means (very loosely) that a set “ends, and reaches everywhere it heads toward.” (A closed interval is compact. The whole real line is not compact because it doesn’t end. An open interval is not compact because it wants to include its endpoints but it doesn’t. A professor of mine described compactness as, “everything that should happen [in the set] does happen.”)

Two basic theorems of point-set topology are that under a continuous mapping, the image of any connected set is connected and the image of any compact set is compact. These theorems are very general: they are true in the setting of any map between any two topological spaces. (They could be multidimensional, curved or twisted, or even more exotic…) What I realized is that the intermediate value theorem is just the theorem about connectedness specialized to the real line, and the extreme value theorem is just the theorem about compactness. What is a compact, connected subset of \mathbb{R}? It is precisely a closed interval. Under a continuous function, the image must therefore be compact and connected. Therefore, it must attain a maximum and minimum, because if not, the image either “doesn’t end” or “doesn’t reach its ends,” either of which would make it noncompact. And, for any two values hit by the image, it must hit every value between them; any missing value would disconnect it. So, “if f is a function that is continuous on a closed interval [a,b]…”

Never Be Wobbly

I spent at least 9 hours today thinking about squishing baloon-shaped surfaces into other shapes. This is what a PhD program in math is doing to me.[1]

Having learning as a full-time job is really, really delicious. But tonight when I stopped mathing and engaged the edublogosphere it felt like a relief to read about classrooms, populated by humans. (To my fellow humans: I love the differentiable manifolds but I love you more.) Thanks Jesse Johnson, Dan Goldner, and Kate No Wackness[2] for your continuing dedication to learning (your kids’, yours and ours).

Holy Crap, a Three-Legged Table Can Never Be Wobbly.

This is my favorite thing ever.

[1]The particular thought that was driving me crazy at least from 7pm to 10pm, not that you care, was: if f:X \rightarrow Y is any surjective continuous function between topological spaces that maps open sets to open sets, then I can prove that the inverse image of a compact set is compact. I studied a converse if X and Y happen to be smooth manifolds and f happens to be an injective immersion. But these are very very strong assumptions. How much can they be weakened?

[2]Kate, this is A’s name for you. (An homage to your no bullsh*t approach.)