The Unapologetic Mathematician

Mathematics for the interested outsider

Deconstructing XKCD

Okay, evidently I need to flex my Critical Theory muscles, atrophied from years of disuse, and bring them to bear on yesterday’s offhand remarks.

To recall, we’re looking at the XKCD comic from Monday, February 18. The title is given as “How It Works”. This is where the ambiguity begins. The phrase “how it works” can either indicate either an observational or a normative description. Observationally, we might catalogue the operations of a certain system. Here, those operations are the ways in which the system works — “how it works” as an entity isolated from the reader. Normatively, we might take our knowledge of a system and give instruction on proper interactions with the system. Such instructions tell how to achieve such results as the author finds worthy — “how it works” to the benefit of the reader, as seen by the author.

The ambiguity is important here because of the different connotations of the two readings. The observational reading is emotionally neutral with respect to its content. The system simply is, and the author renders an image of the system as it is, with no inherent judgement or commentary. On the other hand, the normative statement is inherently an endorsement. The author instructs the reader to interact thusly.

It should be noted here that with slight modifications, the observational mode can be turned to a critical mode. For example, instead of merely describing the human condition as he saw it, Nietzsche entitled his book, Menschliches, Allzumenschliches. In doing so, he explained that he was describing what it was to be “human-like”, but emphasized his disapproval by picking it out as “all too human-like” — something to be escaped rather than merely documented, let alone embraced.

Now, as to the content itself. The comic compares two nearly-identical situations. In each case, two people stand at a chalkboard. In each, the person on the right has just finished writing out the formula

\displaystyle\int x^2=\pi

on the board. In each, the person on the left comments in response. I will refer to the person on the right as the “Writer”, and the person on the left as the “Speaker”.

First, let’s dismiss the details of the mathematical fact. Others have pointed out various flaws. The alt text for the image correctly lists one such possibility — the writers have omitted constants of integration. Another problem is that each image omits the “dx” from the integral. However, these details are actually immaterial to the setup. The expression is not meant as mathematics itself, but as an icon representing mathematics. That is, it acts as a symbol meaning “mathematical work containing a glaring error”. However, the presence of an integral sign picks out the level of the signified work: basic calculus.

In each situation, the Speaker is drawn identically, and generically. The identity is clearly meant to suggest that the two speakers are actually the same person, reacting to two slightly different situations. The difference is all in the Writer.

The author’s style is for “stick figures” with a minimum of recognizable features. However, there are a few standards to his iconography. Most important here is that almost all of the characters are bald, with the exception of a female character. These look identical to the male characters, except inasmuch as they have hair.

The difference in the situations is clearly that in the first, the Writer is male, while in the second the Writer is female. And thus the Speaker’s different reaction is solely a result of the difference in the Writer’s sex. In the first situation, the Speaker says, “Wow, you suck at math.” In the second, he says, “Wow, girls suck at math.”

So we have a significant error in calculus-level mathematics. Nothing about the Writer suggests to the Speaker that this is a one-time error by a normally-competent person. The reaction is not “that’s a mistake”, but “you suck” in both situations, indicating the glaring nature of the error. The Writer, it may be assumed, actually is bad at mathematics.

But then why is someone bad at math at the board anyhow. People with little mathematical skill don’t seek out public fora like chalkboards without provocation. If they must do calculus, it will be hidden on paper, so the numerous false starts and errors can be easily swept under the rug. This identifies the Writer as a student, and the Speaker as someone with enough sway over the Writer to force an appearance at the chalkboard — likely an instructor. Since the Writer is a student with mediocre mathematical ability, it is unlikely that the instruction is taking place in a high school setting. Far more likely, this is at a college, where calculus is often a general requirement.

But despite cultural assumptions, college calculus instructors generally don’t hold their individual students in contempt. We complain about students en masse, but each individual student is to be helped to understand the material. Yes, some instructors don’t fit this mold, but if we are to adopt the observational mode with respect to this comic, we must understand it as speaking generically. There are no identifying features about the Writer or the Speaker (other than the Writer’s sex), and so we cannot understand either of them to be established characters. They are generic placeholders, filling roles to be determined (as we have above) from the context.

And so each situation — with a male Writer or a female — rings false when interpreted generically, as an observation. And yet our prior knowledge of the author tells us that he can’t be meaning this normatively. We are left unsatisfied, with an awkward, ill-contextualized comic. However, if we did not have prior knowledge of the author (as many readers may not) then the awkward contextualization provides reason to read the comic normatively. Either way, the work surely fails to achieve its goals.

February 20, 2008 - Posted by | rants


  1. The comic also has an added comment, “It’s pi plus C of course”. As we are observing the calculation of a definite integral and this is the usual constant added to indefinite integrals, this leads to the interpretation that the speaker is the one at fault. In fact the speaker could be the student.

    I have often pulled a lecturer up on little things I think have been missed.. often I am shot down!

    Comment by ollie | February 20, 2008 | Reply

  2. The work failed to achieve its goals with you.

    Given that a number of other people got the joke, it could be that the problem lies not with the writer, but with you.

    Check out a better math blog for their take on it:

    Comment by thras | February 20, 2008 | Reply

  3. Why are you able to abstract the writing on a chalkboard to an “icon” of mathematics with a glaring error, yet not able to able to do the same with the entire comic? It is not depicting a literal situation, merely describing a stereotype the inconsistency of which is comical when pointed out by stark contrast.

    Comment by C | February 20, 2008 | Reply

  4. ollie: I noted the alt text above.

    thras: thank you ever so much for pointing out how much better a group weblog which restricts itself to their current research interests is than my own. Truly, you are a master of tact.

    If you’d read my first post about the comic from Monday, you’d see that I do get the joke, but I don’t think it’s presented well at all. This post is mainly an exercise in applying deconstructionist tools to analyze the comic.

    Comment by John Armstrong | February 20, 2008 | Reply

  5. C: the point of this post is to critique the comic as a work in and of itself. I can’t abstract away from the subject of my criticism.

    Comment by John Armstrong | February 20, 2008 | Reply

  6. I claim it’s a tutor, not an instructor.

    Comment by Walt | February 20, 2008 | Reply

  7. Like all things, the comic makes sense in context.

    Randy has a few other comics on this topic. The one that comes to mind is this one (couldn’t find the actual xkcd link). If you read his other comics (or listen to him speak) it’s pretty clear that he’s not advocating the position. But I totally agree that the comic is a bit vague, and I even remember thinking that myself when I first read it.

    And I don’t exactly identify with the comic either, since almost all of my upper math classes are full of women. My mom has her MA in sampling and statistics, even. But I realize that this situation might not be true for everyone. Randy is a few years older than me, perhaps that makes all the difference.

    Check out a better math blog for their take on it:

    Fanboy alert.

    Let me guess, you support Ron Paul?

    Comment by Jon McKenzie | February 20, 2008 | Reply

  8. We have fanboys? Yeesh. Why didn’t somebody tell me?

    John, all I’ll say is to keep in mind that XKCD is written with a somewhat broad audience in mind, some of whom didn’t necessarily take calculus, or who have repressed all memories of it. For all the weird in-jokes, it shows up a lot of places. This particular comic showed up on several political blogs I read, and on the LiveJournal of a very non-mathematical friend from college, so it seems to have struck a nerve with a number of people.

    Comment by Ben Webster | February 20, 2008 | Reply

  9. Jon, that’s my point exactly: because I know the rest of XKCD I know that’s not how he means it, but absent that context it’s vague at best.

    Ben’s point about how wide it’s been circulated actually supports this point. Since it’s struck such a nerve, many more people are seeing this comic than have heretofore been regular XKCD readers, and the vagueness might start coming into play for these other readers.

    Again, as I keep saying: I know what Randall meant, but I’m not sure that the comic in and of itself communicates that very well.

    Comment by John Armstrong | February 20, 2008 | Reply

  10. We have fanboys? Yeesh. Why didn’t somebody tell me?

    Everything has a fanboy (or maybe fanperson would be more apropos for this thread). I was talking about xkcd, though, not math blogs. 😛

    Comment by Jon McKenzie | February 20, 2008 | Reply

  11. Yeah, I figured that it might be dangerous to speak ill of XKCD, which Ben avoided doing. It’s sort of like the way you can say They Might Be Giants have gotten stale, but saying word one against Tori Amos is suicide (Tori fans scare me).

    Luckily, few enough people seem to have noticed my qualms, and I survive unscathed.

    Comment by John Armstrong | February 20, 2008 | Reply

  12. Well, when I first saw the comic, I didn’t think the author meant to send a normative message. But, after all this “deconstruction”, I admit that now the message has taken a normative overtone. Blame it on the power of suggestion!!

    Comment by Vishal | February 20, 2008 | Reply

  13. (Visiting via godplaysdice.)

    As I was reading this post, by the second paragraph, I was worried: why has he not brought up the critical perspective? Has he missed the point of the comic? After a pause, I reached the third paragraph. You note the “critical mode”. “Cool, it’s alright,” I think, “he got it.” Everything is fine, I’m in full agreement with the article, and then it all falls apart in the last paragraph. I expect a reiteration of the third paragraph: sexism still happens, it’s still wrong. He can’t be meaning this normatively, he can’t observe without commenting, so he must be commenting that it is wrong.

    Comment by Douglas | February 21, 2008 | Reply

  14. I’ll just note, John, I didn’t avoid criticism of XKCD. I don’t happen to have any at the moment. I can’t speak for anyone else, but for me, the comic made its point instantly, memorably, and unambiguously. I’m sure it could have done better somehow, but I can’t think of how (well, probably there is something better he could have chosen for the boardwork).

    I think one very important point is that the comic is aimed at your immediate reaction, before you’ve had time to think about all the stuff that you list above. And after that, you’ve gotten the point, and don’t need to think about those things.

    Comment by Ben Webster | February 21, 2008 | Reply

  15. My immediate reaction was pretty much what I said Monday, and today was just fleshing that out into words: I parsed “how it works” as “instructions”, not as “description”, and the whole comic seemed awkward from there, even though I realized that’s not what he meant.

    Comment by John Armstrong | February 21, 2008 | Reply

  16. Now that you say the way you parsed “how it works” was your immediate reaction, it makes more sense to me why you’re complaining. But it seems that your immediate reaction differed from that of most people commenting here, including myself — that reading just didn’t suggest itself at all to me.

    Comment by Todd Trimble | February 21, 2008 | Reply

  17. women do math?

    Comment by Michael | February 21, 2008 | Reply

  18. John: I can’t understand your assumption that the figure on the left is male… I interpret it as neuter, and the cartoon means that anything with a gender sucks at math, because of hormonal distractions.

    The cartoonist is referring to females “without loss of generality,” because it’s easier to differentiate them from neuters in the medium of stick figures.

    Comment by Jacob Freeze | February 21, 2008 | Reply

  19. Mulling it over (it’s more fun than actual work), I’ve tentatively concluded that “How it Works” is not a good title for this comic, partly because it has an interpretive ambiguity but mostly because it’s just not snappy. Looking at the xkcd archive (arxiv?), one sees “Future”, “Post Office Showdown”, “Loud Sex”, “Insomnia” — laconic and evocative titles, few of which I can construe in normative ways. Replacing “How it Works” with, e.g., “Sexism” would be more in line with the strip’s general character.

    Comment by Blake Stacey | February 22, 2008 | Reply

  20. Blake, that’s an excellent suggestion.

    Looking it over, I also notice that a number of websites that people have been pointing me towards (as examples of who “gets it”) have posted the comic itself and not the title. I think it may make a significant difference.

    Comment by John Armstrong | February 22, 2008 | Reply

  21. As I interpreted it, the title is meant to be interpreted normatively in order to mock the normative view (am I using these words right?). The readers of xkcd are expected to think for themselves (as always), and should come to the obvious conclusion that the entire situation is ridiculous.

    Comment by miller | February 23, 2008 | Reply

  22. John, you seem to be making two points. First, that although you yourself know better, other people might see this comic and think that the author intends us to think girls are bad at math. Second, that the comic is poorly executed because it depicts the unrealistic situation of a college calculus instructor telling a student that he, or girls in general, suck at math.

    On the first point, I think the danger of this is zero. The obvious logical inconsistency in the speaker’s reactions makes it clear that the speaker’s attitude is being mocked. Even if the comic were entitled “Girls suck at math,” the point would be clear. In my opinion you are underestimating the intelligence of potential comic readers.

    As to the second point, I also think it’s clear that this is an iconographic representation of a prejudice, rather than a depiction of a literal situation. Perhaps the iconography would be more clear if the board said “2+2=5”, but I don’t really think the presence of an integral changes the comic’s entire meaning.

    I do appreciate your observation that the males in XKCD are generic whereas the females are “marked” with long hair, similarly to how females in The Far Side are identified by their pointy glasses. The idea of the “marked female” vs the “neutral male” has a long history in feminist theory, starting with Simone de Beauvoir.

    Comment by Ben | February 24, 2008 | Reply

  23. Edit: I should have said “non-human females in The Far Side are identified by their pointy glasses.” You know, amoebas and cows and such.

    Comment by Ben | February 24, 2008 | Reply

  24. I’d like to point out that the situation depicted in the comic doesn’t have to be a student-instructor relationship. In upper-undergraduate or graduate levels of math and math-heavy sciences like physics, students often do homework collaboratively in small groups. Ideas can be broadcast to these groups more easily if one person works a problem out on a board while the others look on, providing suggestions and heckling. I interpreted the comic through that light, in which the writer and the speaker were peers.

    Comment by Flavin | February 24, 2008 | Reply

  25. Flavin: again, the “calculus” part makes that ring hollow. Calculus is introductory undergraduate, not upper-level or graduate. If the math in the error were different, then it might lend itself to different interpretations.

    As for students in calculus-level classes conspiring like that.. it’s pretty evident from my students’ performances that they don’t.

    Comment by John Armstrong | February 24, 2008 | Reply

  26. Just because they’re doing calculus doesn’t mean they’re in a calculus class. They could be in an upper-level class that happens to involve using calculus.

    Similarly, just because the writer is doing calculus wrong doesn’t mean they’re in a calculus class either. Everyone at every level is prone to mistakes.

    And I wasn’t saying students in calculus collaborate—the good ones don’t need to and the less good ones don’t care—but that students in upper-level classes do.

    Comment by Flavin | February 24, 2008 | Reply

  27. Rings false? I have been in probably almost this exact situation, and many like it. In my high school calculus and maths clasess in 6th and 7th Form (Junior and Senior year), people would often do exactly what Flavin described.

    Comment by Kadin | February 25, 2008 | Reply

  28. To add a very late comment, I’ve never been forced to the board by an instructor, but have spent a lot of time in little seminar rooms and dorm rooms doing math with my peers. Until grad school, though, my math buddies were mostly female! Maybe that’s why I’m still in math — I didn’t get that crap from my peers and I ignored my less enlightened professors all through undergrad.

    A number of people have mentioned the comic to me (being ‘the woman’ in many mathematical interactions). I think it made a point that was widely understood. Many people did skip the title, as I did.

    Comment by kt | February 28, 2008 | Reply

  29. For what it’s worth, the first time I saw the comic, I looked at it and said, “I don’t get it.” I think I was partially thrown off by the math, which strikes me as less wrong and more nonsense. After I saw lots of other people responding to it, I figured out that the “it” of the title is sexism, and began to appreciate it.

    I still say that the math (and the alt text) is misleading and the comic was not effective for me.

    Comment by Matthew Morse | March 9, 2008 | Reply

  30. John, I think you’re way off the mark on this.

    It did not even occur to me to think it could be meant normatively, and I think the only way it could be seen that way would be if it were shown in a sexist context or if the person seeing it is predisposed to look for sexist meanings or meanings implying teacher-student relations in particular or any meaning in general that some person somewhere could possibly interpret to be perhaps offensive in some situations.

    The comic has nothing at all to do with teachers, students, calculus, or chalkboards. All of that is in the comic for one purpose only: to give a visual representation of the very common situation of one person drawing invalid conclusions based on preconceived notions about women and their mathematical ability. The situation might be one person reading a comment of a female on some blog, or two students studying together, or just about anything. The actual particulars in this comic are incidental and largely irrelevant.

    The point (that people interpret what they see in terms of their preconceived notions/prejudices, and especially with relation to women and mathematical ability) is an abstraction that has to be represented in some one particular way if it is going to be represented in a comic. To deal with the abstraction in a nuanced way would require an essay, so the comic creator must deal in stereotypes and iconic situations that evoke something for practically every person who looks at the comic.

    I don’t disagree that the comic does not achieve its artistic goals for you, but it’s almost impossible to satisfy everybody.

    Comment by Joseph Knecht | March 11, 2008 | Reply

  31. Are you kidding? Your over-analysis of the comic in question is part intellectual conceit and part refusal to address the real issue being raised. Obviously, the situation isn’t meant to be as concrete as a professor humiliating a student while trying to do calculus–it’s a commentary on how women doing things that used to be considered the realm of men only are perceived as somehow standing for ALL women, while men in the same situation remain individuals. A man can fail at math, or golf, or whatever, and just suck at math, or golf, or whatever. But a woman’s success or failure is taken as indicative of ALL women’s ability.

    You focused on creating specifics for what should have been a very general context because it allowed you to dicker over the web-comic equivalent of semantics rather than addressing the idea actually being presented by the artist. You clearly prefer to make an argument you can win–a calculus professor who humiliates a student at the blackboard is probably not believable–rather than the one you can’t–that the artist’s statement about the way women are judged isn’t believable. The part I don’t understand is why. It’s not that I’ve never run into someone who hated the idea that maybe white male privilege actually exists… I just don’t get how you think anyone with the fortitude to wade through this pretension could possibly lack the mental acumen to correctly interpret the comic itself.

    Comment by Beth | April 2, 2008 | Reply

  32. Beth, If I’ve learned one thing from this whole mess it’s that someone who holds critical theory in such disdain as I do can still know more about deconstruction than most of the (usually perspicacious) readership of XKCD.

    Oh, and I never said that I “hated the idea that white male privilege actually exists”. I acknowledge that it does, and I do what I can in my small ways to adjust for, and even ameliorate it. Please don’t put words in my mouth just because I’ve touched a nerve.

    And before you jump all over that, there’s absolutely no “women are emotional” subtext there. The same goes for the legions of men who have blown their tops over this post, and its much less tongue-in-cheek predecessor.

    Comment by John Armstrong | April 2, 2008 | Reply

  33. you’re all idiots, i agree with the guy who said this displays intellectual conceit, the comic is supposed to be funny (which it is), there is no ambiguity since the idea and humor of the piece is simple and straight forward, “people unfairly criticize girls of sucking at math so i’m going to create a humorous scenario that shows this hypocrisy for what it is”, there you go, deconstruction doesn’t dissect meaning to find other hidden meaning, it just destroys meaning all together by cutting what is a unit into smaller, irrelevant pieces, xkcd has it’s own “humorous” commentary on deconstruction, deconstruct it

    Comment by Bryce H. | August 10, 2009 | Reply

  34. That’s nice, Bryce. You have fun staring it in the face and not getting it.

    Comment by John Armstrong | August 10, 2009 | Reply

  35. Repeatedly people refer to the comic as a joke. I don’t think this particular comic was meant to be funny. I am a female in engineering and I feel this pressure all the time. The guy next to me can write crappy code and its not a big deal because he is not representing his gender. I have to be very, very careful and refuse to implement “hacks” because I don’t want people to draw conclusions from my work. Its silly the conclusions about professor\tutor\student that the author of this post draws. It’s on a chalkboard so that we can see it properly in a comic 😉

    Comment by Lindsay Shapleigh | October 30, 2013 | Reply

  36. Sorry, I didn’t realize the last comment was from 2009 😉

    Comment by Lindsay Shapleigh | October 30, 2013 | Reply

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