PoSWW 21: Des is Mos’ Disturbing

Once upon a time, we were invited to publicly debate the use of “technology” in mathematics education. The Lord of the Meeting, however, decided that we were not the right kind of person, we were disinvited and plans for the debate ended. Instead, our would-be debating opponent and their mate were granted the platform to spruik to their heart’s content, unchallenged. A shame.

41 Replies to “PoSWW 21: Des is Mos’ Disturbing”

  1. I remember my differential equations lecturer saying how Mathematica found an analytic solution to an integral known to not be integrabale (or something along those lines) and that it is very important to know what you’re doing when using digital tools. Otherwise, you’ll espouse crap.

    On a side note, he also said it doesn’t matter what teaching strategies you use, if students are not doing the work they won’t learn mathematics. Took me a while to understand that one fully.

    1. Thanks, Potii. Whenever I read “digital tools” I immediately think of Roy Slaven talking about “Dick Pound”.

      Re Mathematica, yes, no Black Box will be completely trustworthy. Long ago, when I was an active mathematician, i had colleagues in the same field who used Mathematica to compute very painful complex line integrals. They were aware that Mathematica was prone to error, tending to get confused by principal values. But seriously. Deeply hidden principal values is one thing, but zero to the bloody zero?

    1. Or perhaps the other way around…?

      Looking at the pictures again, it looks as though the software has the discontinuity at x=0 which means someone in the programming team has at least considered this situation, perhaps.

      1. Nah, the dot at the point (0, 1) is just indicating the y-intercept and telling users that you can click there to see the coordinate.

          1. Hi RF. As far as I can tell, your original guess was correct. Simon, I’m not sure what you’re trying to say precisely, but if you just put in “0^x” to desmos, it does indicate that the value at zero is 1, which would be a discontinuity.

            The upshot is that desmos is internally consistent with saying that 0^0 = 1. It doesn’t change anything though, the important thing is that giving it any value at all is rubbish.

            1. I was saying that the “dot” in the Desmos graph was not indicating a discontinuity, but rather just the axis intercept and telling a student that you can click there to find the intercept.

              Try plotting y=x+1 in Desmos, you see the same small grey dot.

  2. I use technology in teaching. I turn on the lights in the classroom. I turn on the heater/air-conditioner if required. I mark the roll with a computer. I prepare my lessons with LaTeX. I store the lessons on the school’s learning management system for students and my colleagues to see. I send electronic messages to students and parents and colleagues. I prepare assessments on the computer and store them on the school’s learning management system for the students and parents to see. I write reports on the computer. I have been presenting lessons virtually, and participating in meetings as well as parent/teacher interviews virtually. And, I never use chalk for writing on the board. Gee whiz – What else can a man do?

    1. The way I see it, there are four broad categories:

      1) The use of “technology” in mathematics education by teachers.
      2) The use of “technology” in mathematics education by students.
      3) The use of “technology” in mathematics education by the educational institute (schools, Universities, TAFE’s etc.)
      4) The use of “technology” in mathematics education by Govt bodies (VCAA, ACARA, VCAA, VCAA etc.)

      The harm most of us think of when technology in education is discussed is the harm in 2). Which is sometimes facilitated by 1) and nearly always facilitated by 4).

      The COVID pandemic has reinforced and strengthened certain harms from 3) and created embryonic harms from 3) that will become very powerful as time passes.

      Terry, you are 1), and most likely are not facilitating the harm happening in 2). So your point is not really a point.

  3. Marty, this is probably just a consequence of Desmos being built on top of Javascript and using the default behaviour of Math.pow(a, b).

    The Math library in JS (like that in most languages) provides the IEEE-754 recommended behaviours for floating point number calculations and pow(0, 0) is defined to be 1.0. There is a powr function that returns NaN, but the default pow function returns 1 as that is often more useful… ref: https://stackoverflow.com/a/19956365

    I don’t think that this is a big deal in the graphing calculator (although, maybe should be thought about more carefully in the scientific calculator). Desmos is a useful tool that gives quick and easy explorations of graphs and data – there’s not much like it out there in terms of speed and simplicity. Desmos is not intended to be a reliable tool for professional mathematics, which is why it can sacrifice some care for speed.

    1. Thanks, Simon. In response:

      1) “this is probably just a consequence of Desmos being built on top of Javascript …”

      I cannot tell you how much I do not care.

      2) “Desmos is not intended to be a reliable tool for professional mathematics, …”.

      Or, it seems, teachers and students. Seriously, Simon, that is a ridiculous attempt to wave away this idiocy.

      3) “Desmos is a useful tool that gives quick and easy explorations of graphs and data …”

      You may be correct, and I agree that this is open for debate. Oh, wait …

      1. ‘ 1) “this is probably just a consequence of Desmos being built on top of Javascript …”

        I cannot tell you how much I do not care. ‘

        What likely happened is that the person who created Desmos probably just borrowed some library for expression evaluation, that happened to contain the error of defining a value for 0^0. They probably just blindly used this library without testing for edge cases like 0^0. I don’t believe that Desmos’ error was a result of willful ignorance, but merely a result of not adequately testing their software.

        What I find to be a bigger crime is the fact that Google themselves return the result of 0^0 as being 1: https://www.google.com/search?q=0^0

        Desmos is just some small company, but Google should know better, and have the adequate staff and resources to notice and fix such an error.

        1. Thanks, A. Yes, I remember now seeing \boldsymbol{0^0=1} in Google, but had forgotten about it. I’m not sure the monster corporation getting it wrong is worse than a small but specifically education-focussed company getting it wrong, but both are clearly awful. And, Desmos isn’t *that* small.

          As for Desmos not testing for edge cases or whatever, that is an empty argument. Sure, maybe they plugged in some library/program at some point, without being fully aware of the content. But any suggestion that they haven’t been long aware of this error is absurd. They’ve had ten years to fix it and they haven’t. It may not have been their wilful ignorance that introduced the error, but it is their wilful indifference that stops the error being corrected.

          Clearly some people greatly appreciate Desmos. I’m not convinced that it’s all that great, and not actively damaging, as a classroom tool, but I agree the point is up for debate, Lords permitting. But, whatever value Desmos has, that is is no excuse for people making these lame defenses when Desmos stuffs up.

          A stuff up is a stuff up. Why is this hard?

          1. Huh. For some reason I was under the impression that Desmos was just a small web application (like the grapher here https://www.mathsisfun.com/data/function-grapher.php, which appears to make the same mistake of defining 0^0 to be 1) that wasn’t that popular, and that wasn’t actively maintained. Seeing that they are a big(ger) company, I guess making the mistake of defining a value for 0^0 is inexcusable…

            I wonder if upon hearing about the complaint that their calculator gives 0^0=1, they searched 0^0 on Google, and decided that their calculator is right because Google is certainly right.

            1. Nah. They must be fully aware of the error, and that the error is an error.

              Possibly it is difficult to remove the error. Maybe, as you and Simon suggested, they plugged in something substantial, and to replace or modify that thing is a major undertaking. Nonetheless, to maintain that error is, or at least should be, seriously embarrassing.

              1. Ah, the joys of programming… Of all the indeterminate forms, it’s only 0^0 that gets fucked up due to the prior artefact outlined. This is a major issue if you’re looking at some indeterminate form where the limit is not 1. And yes, it’s possibly an issue with maintenance of code, where for some unusual reason, changing the result would cause a massive issue, that a developer may have once found, and decided that it wasn’t worth unravelling the possible monster to fix the issue.

                1. Thanks, Sai. I have some sympathy, and I have no sympathy. The error may be very difficult to fix, but to not fix a classroom black box that spits out a wrong answer for a very standard question is absurd and obscene. It is anti-educational poison.

      2. Thanks Marty – always happy to debate (when time permits!)

        It all comes down to the right tool for the job. Interactive calculators like Desmos can help to build understanding before diving into the details and doing things by hand – or to automate tedious tasks (or sanity check your work) after you know how to do things by hand.

        You *should* care. Knowing the strengths & limitations of your tools is part of being a responsible teacher and user. Spreadsheets, calculators, CAS are useful in the right places and students will encounter them. But spreadsheets have floating point rounding errors – try adding 0.1 repeatedly; spreadsheets also have row limitations, which affected the UK’s Covid data last year; etc. Scientific calculators use BCD that leads to less surprising rounding, but can still overflow. Students should see their calculators fail to teach them to think for themselves. Had a nice example of this from the Insight Spec Maths exam in class today!

        Desmos uses default floating point behaviour for the sake of speed – it’s a design decision that helps make the tool more useable. Consequences are the 0^0 issue and also other floating point errors eg https://mrhonner.com/archives/11643.

        Final point, the 0^0 debate is not complete idiocy:
        – eg Donald Knuth talks about it and gives some history in this note (which is often slightly misused to say that he argues for 0^0=1) https://arxiv.org/abs/math/9205211
        – The IEE 754 standards for the three pow functions follow the basic idea that if you’re talking about discrete maths then 0^0 = 1 is ok. If you’re talking about real numbers and limits, then 0^0 = NaN (undefined). Seems reasonable!

        1. Thanks, Simon.

          0) No one is suggesting that the \boldsymbol{0^0} debate is complete idiocy. People are, or at least I am, suggesting that this debate is entirely irrelevant to Desmos having screwed up.

          1) Maybe I should care *why* Desmos gives wrong answers, but that is way, way, way, way second to caring about the *fact* that it gives wrong answers. That’s what I meant.

          2) In fact, I *don’t* care about why Desmos gives wrong answers. At all. I am not teaching now and, presumably, will never teach again. But, if I do, I’ll be damned if I worry, or make my students worry, about such trivialities.

          Students are missing out on so, so much now, largely because of the techno-ising and the de-mathsing of maths, the triumph and glorification of zombifying black boxes, and the fetishistic stuffing around with aimless explorations of nothings. And I should spend any time worrying about which idiot box stuffs up when and why? Forget it.

          3) I am not at all convinced that Desmos, at least in a secondary classroom, “helps build understanding”. All the time people tell me this or that electronic gizmo “helps build understanding”, but I never see it. All I ever see at the time or emerge is a passive, thoughtless, ignorant acceptance of whatever passed by the TV screen. I see the exact opposite of understanding. I see the absence of even an understanding of what understanding is.

          The medium is the message. Or Neil Postman. Take your pick.

  4. Here’s some fuel for the fire:

    Donald Knuth has said that “Anybody who wants the binomial theorem \displaystyle (x + y)^n = \sum_{k=0}^n {n \choose k} x^k y^{n-k} to hold for at least one non-negative integer must believe that \displaystyle 0^0 = 1, for we can plug in \displaystyle x = 0 and \displaystyle y = 1 to get 1 on the left and \displaystyle 0^0 on the right.”

    And as has been remarked elswhere:
    “The fact that people can maintain that \displaystyle 0^0 should be left undefined depends on the circumstance that instances of the expression almost never occur when doing mathematics. This state of affairs is a consequence of the habit of systematically suppressing multiplicative factors of the form \displaystyle x^0 in writing formulas. This notwithstanding the fact that the equation \displaystyle x^0 = 1 implicitly applied during this suppression can only be justified if \displaystyle x^0 is always defined, in particular for \displaystyle x = 0.”

      1. lol!

        Re: “– eg Donald Knuth talks about it and gives some history in this note (which is often slightly misused to say that he argues for 0^0=1)”

        Simon, I’m glad you posted (and included the reference, which I should have). A Likeable post. Indeed, Knuth’s whole article has to be read to appreciate what he’s saying. And what he’s saying makes good sense (unsurprisingly).

  5. Knuth’s argument here is just a blind or ignorant need for convenience. The extreme programmer’s aversion to special cases. You want a formula to hold in a case for which it hasn’t been proved, and so you use that desire — pure desire, nothing logical or mathematical — to not only demand that a convention be followed for that formula in that case, but that it should be followed for all formulae and in all contexts.

    It’s the death of thinking. Shame, Knuth.

    Fact is, we have discussed this issue already quite a bit and

        \[x^y\]

    is a function with important, wild properties. It is an important fact that we can approach the origin in a variety of ways to see a variety of values. That is true, and provable. And it means that 0^0 is NOT DEFINED.

    1. The point of mathematical notation is to aid in thinking. Knuth was arguing for the use of Iverson’s notation as (in his opinion) it makes things simpler to express and thus allows you to think about other things, even if it does have edge cases you need to be aware of. 0^0 is the same. https://arxiv.org/abs/math/9205211

      Another example is Dirac’s “ket” notation – it makes things simple, which is why physicists like it. But has a few sharp corners that need to be avoided. https://arxiv.org/pdf/quant-ph/9907069.pdf

      1. How is 0^0 “the same”? As I’ve explained, it is an important mathematical lesson for a student to think specifically about the so-called (here) “edge cases”. By pretending as though 0^0 = 1, the teacher is lying (an unforgivable sin) and killing this important lesson before it can even be considered.

        1. I’m not advocating telling a student that 0^0 = 1 and ending the discussion. I agree it is an important mathematical lesson for students to think about this and similar indeterminate forms – and in the end, you make sure that they know that in general it is undefined and why.

          However, adults know how and when to break rules for notational convenience and clarity of expression. Provided you know to avoid the cases where it is trouble, 0^0 = 1 can be a useful definition – as argued by many current and historical mathematicians, and as used in many aspects of computing.

          Finally, “lies” are necessary when learning – for children, adults, scientists, etc – the world is complicated and we can’t comprehend everything we need in one go. See Terry Pratchett and the idea of “Lies to children”. The trick is making sure the lies are useful and not misleading. And simply stating 0^0 = 1 would be misleading!

          https://en.wikipedia.org/wiki/Lie-to-children
          https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero
          https://grouper.ieee.org/groups/msc/ANSI_IEEE-Std-754-2019/background/power.txt

  6. Hi Marty — I use Desmos, also in teaching. I am careful in what I say to students about it.

    It is not perfect, and it is not great. It is however better than many alternatives. I will give it that.

    That said: I unreservedly condemn them for this mistake. It needs to be fixed. It needed to be fixed as soon as they found it.

    1. Which is the only point that matters here.

      Thanks, Glen. I have no idea why others are failing to acknowledge, and seemingly even to recognise, the insidiousness of a faulty black box.

      1. Marty, I agree that it’s diabolical and wrong. And I’m sure every poster agrees. Many posts have focussed on \displaystyle 0^0 per se rather than its appearance/context in Desmos, which probably gave the wrong impression. In passing, I note that Mathematica says that it’s Indeterminate.

        And yes, I agree (without checking) that \displaystyle \lim_{(x,y) \rightarrow (0,0)} x^y can be calculated along different paths to get different values and so the limit does not exist. But this does not prove that \displaystyle 0^0 is not defined. It proves that \displaystyle \lim_{(x,y) \rightarrow (0,0)} x^y does not exist. To claim the former is equivalent to claiming that \displaystyle \lim_{x \rightarrow 0} \frac{\sin(x)}{x^2} does not exist therefore limits of the form \displaystyle \frac{0}{0} do not exist.

        1. John, I am significantly less than sure every poster agrees.

          I am generally happy for posters to let the comments follow whichever tangent is of interest to the commenters, and definitely I’m not hoping for ritualistic “wow, this is so important” responses to whatever nonsense I happen to post. And yes, this crap, courtesy of King Edward, has been hammered before. But there seemed to me to be a definite “meh” tone to the discussion, which I found astonishing. This is much worse than Eddie’s garbage. Eddie’s idiotic argument is there and can be critiqued, but the very purpose of a black box is that it’s a black box.

          The general, slothlike acceptance of black boxes inside classrooms is deplorable, even when the damn things work. To have them not work negates the sliver of an argument for their presence. I don’t sense the commenters on this post get it sufficiently, I am guessing the wider readership of this post don’t get it sufficiently, and I am 100% sure that the wider maths teacher community simply does not give a stuff. The Treachery of Images.

          1. A related issue is the fact that most students will never see an example of a limit with the indeterminant form \displaystyle 0^0 that does not approach a value of 1 …

            Picking up from Simon’s first post – the Desmos plot of the graph of \displaystyle 0^x shows a point at (0, 1), which is consistent with its default \displaystyle 0^0 = 1 but is wrong since f(0) is not defined for the function \displaystyle f(x) = 0^x. Desmos has ‘arbitrarily’ defined \displaystyle f(x) = 0^x as a piecewise function by assuming \displaystyle f(0) = \lim_{x \rightarrow 0} 0^x = 1.

        2. Hi John,

          I did not claim in this comment thread what you say at the end of your post. I might have written something along those lines earlier, but I’ve forgotten if I clearly framed it or not. If you’re thinking of the Eddie Woo post, I was quite annoyed then and probably wasn’t clear.

          Let me try again.

          If something does not hold, the burden of proof is not on the party claiming it does not hold. The burden of proof is on the party (if there is one) that claims it does hold.

          A definition of 0^0 can be made. I can say that 0^0 = 1/2. Here’s me satisfying my own imposed “burden of proof”: I like zero and 1 equally and halfway in between is 1/2. So I am defining it to be 1/2. Also, I really like the idea of infinity to the power of seven, so I’m just going to say also that infinity to the power of seven equals a half. I might add more cool definitions that I enjoy later.

          Alright, hopefully that’s clearly ridiculous. Why is it ridiculous? Does that logic apply to the case of 0^0, or not? Let’s keep to the context of high school students learning mathematics for definiteness.

  7. I followed up with Desmos about the 0^0 issue.

    It turns out, I was wrong. It is not an artifact of them just using the default floating point pow(a, n) function or some other convenience.

    To quote: they have “gone back and forth on this point a bit (we reported 0^0 as undefined for a short period), but ultimately we think that 0^0=1 is the most lovely and useful convention, and the one most favored by mathematicians.” Then they supply this article: http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to-the-zeroth-power-equal-why-do-mathematicians-and-high-school-teachers-disagree/

    So, it is a conscious choice they made 🤷🏽‍♀️

  8. Oof. I was going to chime in with “has anyone asked Desmos?” And … oof. “… the most lovely and useful convention.” Aneurysm territory.

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