\[ \boldsymbol{U_{g_{l_y}}}\]

This blog is fuelled, obviously, by annoyance. There are the large annoyances, such as ACARA and VCAA and VIT, and sundry medium-sized annoyances. Seven hundred blog posts, and counting, is a lot of being annoyed. Then there are the small annoyances, and this post is about, pun intended, a small annoyance: subscripts.

We hate subscripts. They are intrinsically tricky to read and in school mathematics they are typically pointless and pretentious, simply the adornment of pedestrian considerations with pretend rigour. As a typical and not particularly egregious example, consider the following line from Q5 of the 2023 NHT Specialist Mathematics Exam 1:

Let z1 and z2 be two of the solutions of the equation z3 = -8i.

Are these subscripts horrible? No. But the subscripts also serve no purpose, and the sentence is undeniably clearer to read if instead the two solutions are simply named u and v. Exactly as was done, for example, in QB(2) of the 2022 Specialist Mathematics Exam 2.

In the same vein, QB(5) of 2021 Specialist Mathematics Exam 2 refers to two masses being m1 and m2. Similarly, and more weirdly, QB(4) of the 2022 NHT Specialist Mathematics Exam 1 refers to the masses of the cyclists Yuna and Paul as m1 and m2. These masses could be more clearly referred to as Y and P, or at least as the once-standard m and M. (Or if, you’re really gonna go with the subscripts, then still don’t; but if you really really insist, then use mY and mP.)

There is no shortage of examples such as with Yuna and Paul above, where the subscripts are misdirected as well as pointless. As just one more example, Cambridge‘s Year 12 Specialist Mathematics text introduces the addition of complex numbers with z1 = a +bi and z2 = c + di, which is neither Arthur nor Martha. In general, textbooks are very bad subscript abusers, and Cambridge is one of the worst.

There are instances where subscripts or similar are helpful, or pretty much mandatory, but this is less common than one might suspect. Take, for example, QB(2) of the 2023 NHT Methods Mathematics Exam 2. This question considers the amount of caffeine, C(t), in a body, which is then adjusted to a new “model”, C2(t). It is obviously clunky and confusing to refer to a C2 when there has been no prior C1, but what is the alternative? Referring to the first function as C1, with no C2 yet on the horizon, is no less confusing. And, C(t) and c(t) doesn’t work, since by happenstance the upper and lower case letters are too similar. So, just leave it? No, you fix it. Maybe use C and C*, or C and K, or K and k. Or C and S, for “Second model”. Or, change the whole question to be a concentration of barium, and use B and b. After all, it’s not like you really care about the caffeine. But, in any case, you do something.

A more borderline case is QB(1) of the 2023 NHT Specialist Mathematics Exam 2. This question is concerned with a function \boldsymbol{f(x) = \frac{4x^2 + 2x +7}{2x^2 - 3x -2}}, which is then generalised to a family, \boldsymbol{g_k(x) = \frac{4x^2 + 2x +7}{2x^2 - 3x +k}} with k real. Then, later, \boldsymbol{g_2(x)}} is specifically considered. Here, one can appreciate the argument for using the subscripted function names, and it may even be considered natural (although fk may be more natural). That does not make it a good decision to do so, however. At any one time, there is really just one function gk being considered in this question, with k either being known or unknown. As such, nothing is lost by referring to this function as g(x) and the later, specified function as h(x). And, as always, the unsubscripted version would be easier to read.

Of course there are cases where the argument for subscripts is much stronger, and/or strongly conventional. The nth term of a sequence will almost always be written as sn or similar, even if s(n) is easier to read and more helpfully indicative of the sequence being a function. Even, however, where some n-signifying is mandatory, it may not be as mandatory as all that. A classical Greek example is Euclid’s familiar proof of the infinity of primes. However one frames it, the essence of Euclid’s proof is that given any collection of found primes, we can always find another one. So, the proof is standardly set up with p1, p2, …, pn, and the students’ eyes immediately glaze over. How to avoid it? Just do what Euclid did: show that given three primes, A and B and C say, we can always find a fourth, and then declare the proof done. Or, even do a Mathologer and choose three specific and suitably general primes. That’s plenty. Most kids will then understand the n-generalisation readily enough, or at least as much as they are likely to at that stage.

To sum up, the moral to all teachers (and textbook writers and examiners) out there: before employing subscripts in a test or handout or class example, ask yourself: “Do I really need them?” Then, if the answer is “Yes”, ask yourself: “Am I sure?” Then, maybe, it’s ok. But probably not.

20 Replies to “Substandard”

  1. Reading this post reminded me of certain moments in my teaching experience where the concept of dummy variables didn’t click for students and they were confused because they were very firmly focused/fixed on certain letters having specific definitions.

    Usually in publishing they have very specific style guides that explicitly dictate details like these, I guess whoever writes these exams can take a leaf from their books!

    1. Thanks, Matt. I’m mostly fine with such conventions. Yes, of course one wants students to learn to be flexible, to focus upon the meaning rather than the form. But in normal usage, having standard forms means your mind is free to think about whatever else is going on. What makes no sense, of course, is having no clear or good style at all.

  2. Don’t get me started on the pretentiousness and distraction of Fraktur. And how should you pronounce those curly letters?

  3. I mostly agree, but partially disagree. I do not think subscripts are a significant issue facing student comprehension. I become much more irritated by poor or non-standard ambiguous wording.

    Anyway, on the topic of the post: I like a subscript that has a function, as in the g_k(x). But I would personally have just graduated the k to a variable and written g(x,k) in that example, especially since you said k was real. If I was worried that students would be confused by a function of two variables, I’d reconsider the whole exercise, since clearly g_k(x) *is* a function of two variables.

    I agree about the primes and about the complex solutions. What about the components of a vector? I like using (x,y,z) and (x,y) but for higher dimensions, just like sequences, I use subscripts. (I won’t write u(n) instead of u_n though, contrary to what I wrote above.)

    In the end I agree completely with the motivation behind this peeve: Being mindfully consistent with notation will free up brain cycles to use on actually doing mathematics. Notation is a vehicle and the clearer or more standard it is, the better.

    1. Glen, this is high school. There are no higher dimensions.

      With the gk(x) example, there is no value in the context of considering the function to be of two variables, either subscripted or more explicitly functiony. If I consider a linear function y = mx + c, I don’t think of this as a function y(x,m,c), unless there’s a clear and good purpose to do so.

    2. My 85 cents (cost of living pressures):

      My concern with using the notation g(x, k) is that the x is a variable and k is a parameter (yes, k is also a ‘variable’ but a variable in a different ‘sense’ to the x) and this could create confusion for some students who think that we’re dealing with a multi-variable function (which we kind of are but we’re not). I disagree that \displaystyle g_k(x) is a function of two variables. I’d say you have multiple functions of a single variable.

      1. BiB, a variable is a variable is a variable. It’s whatever you choose to vary.

        The only point of labelling g with k in any manner is because the k is changing. In which case Glen’s notation is correct and, at higher academic levels, more standard, albeit inappropriate for a school audience. But the issue is not of being correct, but of being clear. For the purposes of the question, g(x) suffices and is clearer than both gk(x) and g(x,k).

        This is really the point. Again, subscripts are a really, really minor issue compared to the mountain of writing sins in our maths ed (and maths) world. But the subscripts are a really good indicator of a more general perversion of intent. The vast majority of maths ed writers are either so unsure of themselves, or so sure of themselves, or both, that they fuss endlessly in attempting some weird notion of correctness, and then it is simply forgotten to consider whether what is written is remotely comprehensible.

  4. Usually I’m fine with using subscripts, in fact I tend to prefer them for sequences (might be from over exposure), but especially when teaching General Maths, the students tend to understand/use t_n and t_{n+1} better than t(n) and t(n+1), where some think it’s a multiplication.

    But lord, VCAA is testing me using the subscript as a place to hold a parameter they are using in a function. Think 2017 MM2 Q4 was where it started to become common. Students left looking for g_1 when g_k was given.

    1. Thanks, Alex. I’m not really disagreeing on the sequence notation. When you really have n things around, you need something, and I agree that the function notation there is clumsy.

  5. As you say Marty, it’s a minor complaint, but anything we can do to ease the cognitive load of the reader…

    At some stage the secondary school student needs to be exposed to subscripts. In my case the first such was when considering straight lines, formulas such as m=\frac{\displaystyle y_2-y_1}{\displaystyle x_2-x_1} and y-y_1 = m(x-x_1). [I would prefer m=\frac{\displaystyle \delta y}{\displaystyle \delta x}, but students who have seen the earlier formula seem to prefer that.] Here we have 3 symbols x_1, x_2 and x; it makes sense to call them all variants of x. And there seems to be a theme starting: the subscripted symbols x_1 and x_2 refer to “data points” while the naked x is a more general “variable”.

    This distinction becomes obvious in statistics with \overline{x} = (x_1+x_2+\cdots +x_n)/n. I think it’s worth a full lesson to teach the idea of a dummy variable in say \sum_{i=1}^n x_i. But if teaching \sqrt{\sum_{i=0}^n (x_i-\overline{x})^2 /(n-1)} to a university audience of Psychology students then be prepared for mass hysteria.

    1. Thanks, tom. Yes, the point isn’t to ban subscripts, but to realise that they come with a cost. So, they should be used sparingly and intelligently.

      Re the subscripty linear formulas you raise, I don’t like them and try to avoid them. For gradient, like you, I prefer to hammer “change in y over change in x”, verbally as well as with notation. (I prefer Δ to δ.) Weaker students are masters at screwing up the formula, particularly if there are negatives, but the change from -3 to 4, for example, is pretty easy, even for those guys. (Which can then be used to shore up their negatives.) Similarly, I hate the distance formula, and I’ll just keep yelling “Pythagoras” until the cops come.

      You say that it makes sense to label variants of x with subscripts, and that is true, but it also makes sense not to. The variables x and y play a critical role in the way that x-based and y-based constants do not, and the subscripting muddies the distinction. Which is clearer: y = mx + c, or y = mx + y0?

      But you also make a good point that, whatever we might argue is clearer, if convention has meant students are familiar with some particular notation, you don’t run roughshod over that.

      1. It seems we are similar; I too prefer y=mx+c to y-y_1=m(x-x_1) and I too prefer to go back to Pythagoras. In fact I try to avoid formulas at least as a first approach to a process. So that frightening formula for standard deviation would only be presented after many calculations of the thing using instructions in plain English. So my version of the quotient rule is “the bottom [touch your bum] by derivative of the top – minus the top, by derivative of the bottom. All divided by the square of the bottom”. I find students learn and remember better this way.

        1. I also prefer point-slope, but I go with y=m(x-h)+k. Avoids the subscripts and looks closer to a quadratic in TP-form or other transformed functions. Makes the jump easier.
          But that’s ridiculous that they only accept y=mx+c when they don’t specify the form they want for a line half the time when other questions will accept any reasonably simplified form of an expression.

              1. Nice one Ron. The magic of rhymes. But I would still include the physical component. Students will remember me looking ridiculous – again.

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