# Everybody is very nerdy about geometry

This is just a short post cataloging some occasions when people geeked out on me about geometry in non-mathematical contexts. I feel like everybody loves geometry.

• When a family member explained to me how to fold fitted sheets. They were so, so excited to give me the big-picture concept of how, while the elastic fundamentally compromises the process of creating a perfect rectangle, the proper technique coordinates and isolates the elastic so as to minimize its impact on the overall effect.
• When an old friend explained to me why you need to be more careful to cook thoroughly, from a food safety standpoint, when cooking hamburger than when cooking steak. They pointed out to me that with steak, the exterior of the cut of meat is at the most risk for developing bacteria before cooking, but it’s also the most exposed to heat in the cooking process. In contrast, with ground meat, exterior parts (bacteria risk) end up on the interior (protected from heat during cooking), hence higher total food safety risk. I don’t think I can do justice to how excited they were to tell me this.
• When the person who sold me my bike lock pointed out that, if you put the lock through the part of the rear wheel that’s in the interior of the rear triangle of the bike’s frame, then the lock will not be hooked around the frame at all but it’s still impossible to steal the bike without going through the frame. They literally said, “it’s the geometry!!”

# My work on the AMS Teaching & Learning Blog

I don’t know why I didn’t think to tell you this earlier, but: in 2019 I joined the editorial board of the American Mathematical Society’s Teaching & Learning Blog, and I’ve written several pieces for it. I’m extremely proud of each of these, and would like to share them with you.

• Some thoughts about epsilon and delta (August 19, 2019) is a deep dive on student difficulties with a notoriously challenging definition from calculus. I got pretty scholarly and read a bunch of research for it, but the core of the post is a discussion of challenges faced by specific learners I’ve known, one of whom is my own self. I also include a brief history of this definition.
• The things in proofs are weird: a note on student difficulties (May 20, 2020) is a meditation on the nature of the objects we use in proofs, and the difficulties students have in getting used to working with objects with this strange nature. I again got pretty scholarly and read a bunch of research. Nonetheless, it includes an extended riff on Abbott and Costello’s Who’s on First?
• A K-pop dance routine and the false dilemma of concept vs. procedure (August 18, 2020) is a… ok let me back up. People used to fight about whether conceptual or procedural knowledge was more important. I think we’ve more or less reached a place in the public conversation about math teaching where there’s an official public consensus that conceptual and procedural knowledge are both important and are mutually supportive. But just because we all can say these words doesn’t mean we’ve necessarily fully reconciled the impulses behind that older fight. For example, in spite of firm intellectual conviction that this view is correct, I have a bias toward the conceptual in my teaching, in the sense that I have a strong tendency to assume any student difficulty is rooted in a conceptual difficulty. This bias is really useful a lot of the time, but sometimes it can lead me to misdiagnose what a student needs to move forward. Anyway, so one day I was learning a BLACKPINK dance and the learning experience just really eloquently illustrated both the advantages and disadvantages of that exact bias. Hopefully you’re intrigued!
• The rapid expansion of online instruction, occasioned by the pandemic, has forced academia to contend with the limits of the control that its usual physical setup allows it to exercise over students’ movements and choices. One place this manifests very clearly is in the setting of timed tests, which are historically proctored in person. Remote proctoring: a failed experiment in control (January 19, 2021) is my heartfelt contribution to the pushback against the Orwellian trend of turning to “remote proctoring” (where the student is surveilled in their home during tests) to try to claw back the lost control, rather than accepting that the game has changed and rethinking assessment from the ground up, as the situation demands.
• Three foundational theorems of elementary school math (November 22, 2021) could have been titled, “The logical structure of elementary school math is actually extremely beautiful and intricate, and I want everyone to pay more attention to this.” It’s a love letter to three closely related facts from elementary school math that I think often don’t get their due, making the case that they deserve to be thought of as theorems. I discuss proofs (including some relevant student work) and connections. (If any long-time readers of this blog are still here in 2021, this post is a distant but direct descendant of this post I wrote nearly 12 years ago, when I was a baby blogger.)

I also solicited a piece from Michael Pershan, which I am also extremely proud of:

• What math professors and k-12 teachers think of each other (November 18, 2019) is Michael’s synthesis of and meditation on an informal survey he ran, canvassing math educators teaching in schools and universities about what they think about the differences in the shape of math education at these different levels. Michael’s characteristic thoughtfulness is on full display here, and it’s all with an eye toward how we can collaborate effectively. I love it.

# Math, Democracy, Equality, and Classroom Culture

This is a contribution to Sam Shah‘s Virtual Conference on Humanizing Mathematics.

As a secondary matter, it fits into my series of posts exploring the relationship of math to democracy.

One aspect of this exploration has been experimenting with explicitly framing mathematical knowledge building with students as a democratic process, analogous to being part of a democratic polity. In a democracy, at least according to the ideal, the direction of the polity is determined by its members, all having an equal say. In the same way, I’ve been striving to build a way of working with students in which they see the knowledge as determined by themselves engaged in a collective process in which they are all equal participants, substantially inspired by Jason Cushner and Sarah Bertucci’s Consensus Is the Answer Key.[1]

My interest is in having students walk away from mathematical experiences knowing that math is nothing more mysterious than communities of humans trying to figure things out together; that the process that led to all mathematical knowledge is something they, and anyone, can participate in; that they can be the authors of such knowledge. That they are entitled to a say in what the community they are part of believes about math, and that their own sense of what to believe benefits by being part of a community thoughtfully working together to try to figure things out.

I hope these overall goals give you a sense of why I wanted to write about this as part of the Virtual Conference on Humanizing Mathematics: while math is often seen as some kind of disembodied and strangely history-less ancient wisdom handed down by specially-anointed priests (“math teachers”), themselves entrusted with it by an even higher priesthood (“mathematicians”), the truth is that it is nothing but the product of humans trying to figure things out together, and I want this to be what students experience it as.

While I do plan in the future on writing about the specific instructional protocols I’ve been exploring to accomplish this, I’m going to keep the scope limited here, and tell you just one story, about a time when the frame of “democracy” unexpectedly gave me a new resource in handling a situation to do with classroom culture that I priorly would have found challenging.

In 2017 I tried my first experiment building a whole learning experience around the math-knowledge-building-as-a-democratic-process metaphor. (It was a course at BEAM.) On the first day I explained that mathematical knowledge is democratic in character[2] and that they would be working democratically as a community to decide what’s true. They bought in.

They were working through something, I no longer remember what. C asked a question or made an argument and A replied. A’s reply was mathematical and on topic, but his tone was a little condescending. Just a little, but it was there.

This is a type of situation in which I’ve historically found it a little hard to exercise my authority to move the classroom culture in a positive direction. I don’t want a room in which it’s okay for people to be condescending to each other. That’s a recipe for the class to start to feel emotionally unsafe. On the other hand, I’ve often had trouble finding a way of intervening in this type of situation that would have felt fair to A. If I said, “that was disrespectful,” well, perhaps it was, a little, but it was also on topic and advanced the conversation, and, well, “disrespectful” is a powerful word. Furthermore, this intervention would not have been very actionable for A: the thing I didn’t like was not located in his choice of words, but in a subtle tone thing. If he felt defensive at all (and who wouldn’t?), it would be difficult for him even to perceive what he had done that was being criticized; how would he correct it?

I think some teachers deftly handle this type of situation using light-touch humor, but that has always been a difficult tool for me to wield when being corrective. I’m too earnest; it’s hard for me to get that dial just right.

I’ve found myself in situations like this countless times, but this was the first time I had encountered it while teaching a class that had explicitly bought into the idea that they were a democratic community. I found myself, quite to my surprise, with a confident new move:

“A, you’re saying something very interesting, but your tone of voice is a little like you’re the teacher and C is the student. In a democracy, you’re equals. So can you try making that exact same interesting point except from one equal to another?”

And what was beautiful was that he completely, happily, undefensively took it on. In fact, he seemed excited to try. And, he did it! He said the exact same thing, except from one equal to another. The conversation proceeded with a new foundation of safety and mutual respect established.

I knew I wanted to teach them that math was something humans make by coming together as equals and trying to figure stuff out together. I didn’t know this would also give me new moves to support the development of a healthy mathematical culture. Retrospectively, maybe I should have.

#### Notes

[1] Sarah wrote an essay on this pedagogical principle which unfortunately has never been published, but the link above is a nice description of a session she and Jason and their students led at the Creating Balance conference in 2008.

I’ve been working to develop a community of educators interested in this “math-as-democracy” pedagogy. I facilitated a minicourse and a professional learning team at Math for America this past year on the subject, and James Cleveland, who was part of both, led a session at TMCNYC19. This fall I am co-facilitating another professional learning team on instructional routines, one of which is democracy-focused. If you’re interested in thinking about this circle of ideas with me, get in touch!

[2] I explained the underlying philosophy here, and also see the first minute or so of my TED talk.

# I Just Started a Math Blog!

Hey y’all, I just started a new blog for completely random thoughts about math. I just figured something out yesterday morning that would probably never go in a paper or anything, but I wanted to record it somehow, and why not publicly? So, new blog for that kind of thing:

Every Single Problem

It’s gonna be pretty math-ola. Enjoy!

p.s. No promises it will ever contain more than this single post! But, it might!

# Math is Democracy II: Math is Democracy!

I announced a series on math and democracy back in October.

It will deal with a lot of concrete areas. Last time I talked about a case that is before the Supreme Court and will influence voting law throughout the land. In the future I’ll be talking about voting, political participation, technology and who has a say over its development, and of course the classroom.

But I want to properly kick things off with a post that is essentially philosophical. I am here to assert the following proposition:

Math is democracy!

What do I mean?

Democracy — from Greek — literally, “rule by the people.” I am referring to the ideal itself, not any particular system of government. Throughout the world we have various systems attempting to implement this ideal. One can ask questions about the degree of success of these attempts, but that’s not what this post is about. I’m just isolating the ideal — democracy — rule by the people.

Mathematics — from Greek — literally, “learning.” Of all the domains of human inquiry, math occupies a privileged place in terms of our confidence in its conclusions. It is the only field where practitioners regularly express unqualified certainty about its results. We sometimes discuss the wisdom it gives us as some sort of celestial gift (as in Wigner’s classic essay on its applicability to the sciences).

I am about to draw a connection. I expect it is still opaque at this point, but hang on.

If math is a miracle, then there is a second miracle: the divine gift was implanted in each of us, since it springs solely from the universal human capacity for rational thought. The wisdom of mathematics was not given us by way of Mt. Sinai, handed down from on high by somebody with privileged access to The Boss. Although many people think back to childhood and recall inscrutable formulas dispensed by a teacher who mysteriously knew the answer (how did they know??), this memory conceals the real truth, which is that the only place mathematical knowledge comes from is a community of peers reaching some kind of consensus after a period of engaged discussion. Furthermore, at least in principle (if not always in practice), anybody in this community has the right at any time to raise good-faith questions about the logic underlying any of our mathematical knowledge, and the matter is not really settled unless these questions have a good answer.

Thus, the only true source of mathematical authority is the consensus of a community of equals.

The principle of democracy is that this is also the only true source of legitimate political authority.

Broadening further, I offer that the principle of democracy holds that the only source of authority (of any kind) over a community is consensus of that community. So math is literally democracy.

This is edited from the version I posted yesterday, where I used the phrase “functional consensus” instead of “consensus.” This was to acknowledge that in a large-scale community such as a nation, or the international community of mathematics researchers, true consensus is not a viable goal. That said, the “functional” didn’t sit well with me overnight, because I thought it could be taken to suggest some sort of majoritarian principle. To me, majoritarianism is a fatal compromise of the principle of democracy articulated here, and it defeats the purpose of the analogy with math.

The thing about math is that, in principle, if an objection is raised to what is regarded as established fact, then that objection needs to be dealt with. Maybe something was overlooked! In actual practice, it may or may not be, because the question of whether you can get people to pay attention to your objection depends on things like if you’re famous, if you’re well-connected, how much work other people have to do to understand it, etc. But mathematicians’ collective understanding of what we’re doing holds that if somebody raises a new objection to something thought to be well-established, we have to answer it, not ignore it, in order to hold onto the established knowledge. This ideal isn’t attained, but it is still how we think about it.

By the same token, it seems to me that the democratic ideal insists that a minority view has the right to be processed rigorously by the community. I am making a high-level analogy so I’m not getting into what that processing might look like. But the failure of a community to take into account minority constituencies in some way is a failure of democracy.

I want to acknowledge some intellectual debt!

In 2008, I went to the Creating Balance in an Unjust World conference and saw a presentation by Sarah Bertucci, Jason Cushner, and several of their current and former students, entitled Consensus is the Answer Key: Empowerment in the Math Classroom. The presentation was on using consensus as the source of mathematical knowledge in the classroom. Later (in 2009?), I visited the school in Vermont where Jason and Sarah were then teaching, and saw Jason’s class. (Random aside: I also met Jasmine Walker!) The ideas have shaped how I saw both mathematics and the classroom ever since. You can see their clear imprint above (and in many of the things I’ve written on this blog over the years).

In about 2010, I was having a conversation with Jay Gillen of the Baltimore Algebra Project. At the time, I was preparing to apply to graduate school in math. Jay asked me many questions about how I thought about the math classroom and the subject itself. At some point he paused and said, “Everything you love about math is what free people love about democracy.” This comment has been continuously blowing my mind for 8 years, and again you can see its clear imprint in the above.

# Math is Democracy I: The Citizen and the Gerrymander

I am intending a series of my typically long, elaborate blog posts entitled Math is Democracy. The ideas have been brewing for years although they have been rapidly expanding and taking on new urgency since January. I alluded to this intention previously.

I wasn’t ready to start it yet, but I feel I must. I was reading the oral arguments in Gill v Whitford, the Wisconsin partisan gerrymandering case currently before the Supreme Court. I had to stop and have a moment when I read this:

CHIEF JUSTICE ROBERTS: Mr. Smith, I’m going to follow an example of one of my colleagues and lay out for you as concisely as I can what — what is the main problem for me and give you an opportunity to address it.

I would think if these — if the claim is allowed to proceed, there will naturally be a lot of these claims raised around the country. Politics is a very important driving force and those claims will be raised.

And every one of them will come here for a decision on the merits. These cases are not within our discretionary jurisdiction. They’re the mandatory jurisdiction. We will have to decide in every case whether the Democrats win or the Republicans win. So it’s going to be a problem here across the board.

And if you’re the intelligent man on the street and the Court issues a decision, and let’s say, okay, the Democrats win, and that person will say: “Well, why did the Democrats win?” And the answer is going to be because EG was greater than 7 percent, where EG is the sigma of party X wasted votes minus the sigma of party Y wasted votes over the sigma of party X votes plus party Y votes.

And the intelligent man on the street is going to say that’s a bunch of baloney. It must be because the Supreme Court preferred the Democrats over the Republicans. And that’s going to come out one case after another as these cases are brought in every state.

And that is going to cause very serious harm to the status and integrity of the decisions of this Court in the eyes of the country.

Now, there’s a lot here one could react to.[1] But the main thing I reacted to was this:

The Chief Justice of the highest court in the land thinks Americans don’t feel empowered to judge an argument on the merits if there’s math involved.

You know what? He’s probably right about that.

But this situation is very, very wrong.

Math is being used increasingly to make decisions governing our lives, for good or ill. Increasingly sophisticated math.[2] The instance most familiar to readers of this blog is probably teacher value-added scores, but the many various uses share this: they are not accountable to the public.

One reason the Wisconsin case is so hot is because the process that led to the map currently being challenged included a lot of fancy mathematical modeling intended to make the Republican legislative majority as bomb-proof as possible — an effort that appears to have worked really well. That the map was drawn with this goal and these tools is not a controversial point in the case. This was a use of math by legislators aimed at becoming less accountable to the public.

What I’m getting at: math is a species of power, and it’s a species that multiple antidemocratic forces are using, very effectively. And it’s a kind of power that citizens, by and large, totally lack.

So, the game is unfair. We the People are supposed to be able to participate in public decision-making. That’s the heart of democracy. But math is increasingly becoming a kind of secret key to power that, if the Chief Justice is right, We the People mostly don’t have. As soon as there’s math involved, we can’t even participate in debates about the very consequential choices that are being made. In which case, nobody who wants to use the power of math (for good or ill!) needs to be accountable to us.

I mean, this was true before the explosion of data-science driven business and governmental practices Cathy writes about, or the computer-assisted 2010-11 legislative redistricting.[3] But now it is more intensely true than ever.

What this leaves me with is that doing our jobs well as math educators is completely urgent for democracy. Every kid we leave traumatized and alienated from formulas and data analysis is a citizen that doesn’t have a voice.

Don’t let anybody tell you it doesn’t matter.

Notes:

[1] For example: This author at ThinkProgress thinks Roberts has a lot of nerve claiming to be concerned with the perception that the court is partisan when he has so consistently voted along partisan lines in landmark cases. This author at WaPo thinks it’s not legitimate for the Court to be considering its public perception in the first place. I am personally inclined to believe that Roberts is earnestly concerned about the court’s reputation and that his question was earnest (mostly because of his surprising and apparently similarly-motivated vote in NFIB v Sebelius), although I do think that the fact that he doesn’t appear to be equally concerned with the perception of partisanship if the court does not “allow the claim to proceed” reflects a rather striking partisan limitation in his image of the “intelligent man on the street.” I know plenty of intelligent men, and women, who would be inclined to conclude that he himself is a partisan hack on the basis of the above quotation alone.

[2] Shout out to Cathy O’Neil.

[3] This seems like a good moment to acknowledge the deep debt of my thinking here to Bob Moses, who has been on this tip for a long time. Also, there is some relationship to the work of math educators in the Freirean tradition such as Marilyn Frankenstein and Rico Gutstein, though I can’t take the time now to figure out exactly what it is.

# Hidden Figures: Visibility / Invisibility of Brown Brilliance, Part I

Has everybody seen Hidden Figures yet?

It’s delightful: a tight, well-acted, gripping drama, based on a true story about an exciting chapter in national history. You can just go to have a good time. You don’t need to feel like you are going to some kind of Important Movie About Race or whatever. It is totally kid friendly, and as long as they know the most basic facts about the history of racial discrimination, it doesn’t force you to have any kind of conversation you aren’t up for / have every day and don’t need another… / etc. Just go and enjoy yourself.

THAT SAID.

Everybody, parents especially, and white parents especially, please go see this film and take your kids.

I was actually fighting back tears inside of 5 minutes.

Long-time readers of this blog know that I am strongly critical of the widespread notion of innate mathematical talent. I’ve written about this before, and plan on doing a great deal more of this writing in the future. The TL;DR version is that I think our cultural consensus, only recently beginning to be challenged, that the capacity for mathematical accomplishment is predestined, is both factually false and toxic. My views on the subject can make me a bit of a wet blanket when it comes to the representation of mathematical achievement in film – the Hollywood formula for communicating to the audience that “this one is a special one” usually feels to me like it’s feeding the monster, and that can get between me and an otherwise totally lovely film experience.

In spite of all of this, when Hidden Figures opened by giving the full Hollywood math genius treatment to little Katherine Johnson (nee Coleman), kicking a stone through the woods while she counted “fourteen, fifteen, sixteen, prime, eighteen, prime, twenty, twenty-one, …,” I choked up. I had never seen this before. The full Good Will Hunting / Little Man Tate / Beautiful Mind / Searching for Bobby Fischer / Imitation Game / etc. child-genius set of signifiers, except for a black girl!

What hit me so hard was that it hit me so hard. For all the brilliant minds we as a society have imagined over the years, how could we never have imagined this one before now? And she’s not even imaginary, she’s real! And not only real, but has been real for ninety-eight years! And yet this is something that, as measured by mainstream film, we haven’t even been able to imagine.

You’ll do with this what you will, but for me it’s an object-lesson in the depth and power of our racial cultural programming, as well as a step toward the light. I am a white person who has had intellectually powerful black women around me, whom I greatly admired, my whole life, starting with my preschool and kindergarten teachers, and including close friends and members of my own family, as well of course as many of my students. And yet the type of representation that opened Hidden Figures is something that only fairly recently did it begin to dawn on me how starkly it was missing.

So, go see this movie! Take your kids to see it! Let them grow up easily imagining something that the American collective consciousness has hidden from itself for so long.

# The History of Calculus / Honor Your Dissatisfaction

I was just rereading an email exchange with a friend (actually the O of this post), and found that I had summarized the history of calculus from the 17th to 20th centuries, up through and including Abraham Robinson’s invention of nonstandard analysis, in the form of a short play! I’m sharing it with you.

Mainly this is for fun, but it’s also part of my ongoing campaign promoting the value of honoring your dissatisfaction. The dialectic between honoring our impulse to invent ideas to understand the world better and honoring our dissatisfaction with these ideas is where mathematics comes from.

Here’s the play!

# The History of Calculus, in 4 Extremely Short Acts

Featuring a lot of oversimplification and a certain amount of harmless cursing

Act I

Late 17th century

Leibniz, Newton: Look everybody, we can calculate instantaeous speed!

Everybody: How??

Leibniz: well, you consider the distance traveled during an infinitesimal interval of time, and you divide distance/time.

Everybody: Leibniz, what do you mean, “infinitesimal”? Like, a millisecond?

Leibniz: No, way smaller than that.

Everybody: A nanosecond?

Leibniz: Nah, dude, you’re missing the point. Smaller than any finite amount.

Everybody: So, zero time?

Leibniz: No, bigger than that.

Some people: Oh, cool! Look we can use this idea to accurately calculate planetary motion and stuff!

Other people: WTF are you talking about Leibniz? That makes no effing sense.

Act II

18th century

Bernoullis, Euler, Lagrange, Laplace, and everybody else: Whee, look at everything we can calculate with Newton and Leibniz’s crazy infinitesimals! This is awesome!

Bishop George Berkeley: But nobody answered the question of WTF they are even talking about. “What are these [infinitesimals]? May we not call them the ghosts of departed quantities?”

Lagrange: Hold on, let me try to rebuild this theory from scratch, I will make no mention of spooky infinitesimals, and will do the whole thing using the algebra of power series.

Everybody: Cool, good luck with that.

Act III

19th century

Cauchy: Lagrange, homie, it’s not gonna work. $e^{-1/x^2}$ doesn’t match its power series at zero.

Lagrange: Sh*t.

Everybody: I think we don’t actually understand this as well as we thought we did.

Ghost of departed Bishop Berkeley: OMG I HAVE BEEN TRYING TO TELL YOU THIS.

Cauchy: How about we forget the whole “infinitesimal” thing and just say that the average speeds are approaching a certain limit to whatever desired degree of accuracy. As long as we can identify the limit and prove that it gets as close as we want it to, we can call that limit the “instantaneous speed” without ever trying to divide some spooky infinitesimals by each other.

Everybody: Awesome.

Weierstrass: I have an even better idea. Let’s formalize Cauchy’s thinking into some tight symbols and quantifiers. “Let us say that the limit of a function $f(x)$ at $c$ is a number $L$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that whenever $0 <|x-c|<\delta$, it follows that $|f(x)-L|<\varepsilon$…”

All the mathematicians: AWESOME. Down with spooky infinitesimals! Calculus can be built soundly on the firm footing of “for any $\varepsilon>0$ there exists a $\delta>0$ such that…” and you never have to talk about any spooky sh*t!

All the mathematicians, in private: … but thinking about infinitesimals sure streamlines some of these calculations…

[Meanwhile all the physicists and engineers miss this whole episode and continue blithely using infinitesimals.]

Act IV

20th century

Scene i

Mathematicians: Infinitesimals are satanic voodoo!

Physicists and engineers: What are you talking about, what about CALCULUS?

Mathematicians: Whatever dude, don’t you know about Weierstrass and $\varepsilon$ and $\delta$?

Physicists and engineers: Um, no, and I don’t care either! What’s the point when everything already works fine?

Mathematicians, in public: No, dude, there are all these tricky convergence issues and you will F*CK UP EVERYTHING IF YOU’RE NOT CAREFUL!

Mathematicians, in private: … but those infinitesimals are indispensible as a heuristic guide…

Scene ii

Abraham Robinson: Um, whatever happened to infinitesimals?

Mathematicians: I mean we rejected them as satanic voodoo because nobody was ever able to tell us WTF THEY ARE.

Robinson: I have a proposal. How about we consider them to be [fancy-*ss definition based on formal logic and other fancy sh*t]. Would you say that constitutes an answer to “wtf they are?”

Mathematicians: … why, yes!

Some mathematicians: omg awesome I can now RESPECTABLY use infinitesimals in calculations, I don’t have to hide anymore!

Other mathematicians: Whatever, I have no need to do the work to master this fancy sh*t. It doesn’t do anything good ole’ Weierstrass $\varepsilon$ and $\delta$ couldn’t do.

Physicists and engineers: wow, you guys are way over-concerned with the little stuff. Literally.

# End

(Long-time readers of this blog will recognize the bit of dialogue with Leibniz from something I shared long ago.)

The point is that the whole episode is driven by uncertainty about what is even being discussed. The early developers of calculus shared the conviction that there was something there when they talked about “infinitesimals”, but none of them (not even Euler) gave a definition that was satisfying to everybody at the time (let alone to a modern audience). But this encounter, between the intuition that there’s something there and the insistence of the world to honor its dissatisfaction until a really satisfying account was given, was a generative encounter, resulting in several hundred years’ worth of powerful math progress.

So. Honor your dissatisfaction.

# Math is like…

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

# Hard Problems and Hints

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) –

Dear Ben,

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 $\tan 54$ (or something horrible like that) and the formula for the volume of a dodecahedron involves $15 + 7\sqrt{5}$ 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.

Best, O

Dear O,

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.
Learner: Yeah.
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