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You know the error, when you're watching a student work through an algebraic calculation to solve for a variable trapped in the argument of a function, usually $\log$ or a trig function, and you watch them write this: $$ \log x = 42 \qquad\text{so}\qquad x = \frac{42}{\log} $$ This error signals a distressing misunderstanding of what a function is, but I only seen to catch students making this error weeks after we've talked about function fundamentals. My goal is to identify students who'd make this error early. What sort of questions can I ask to elicit this error in reasoning so I can intervene early?

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    $\begingroup$ Have you asked students who make this error what they were thinking when they wrote this down? $\endgroup$
    – Steve
    Apr 25, 2022 at 16:40
  • $\begingroup$ @Steve Maybe a long time ago. The instance that inspired this post, I just corrected them. But I suppose it was obvious: they're going into autopilot mode when they want to isolate $x$, and seeing the $\log$ next to the $x$ they divide by it, just like they would a constant. Autopilot happens though. The real concern of mine is the fact that, after having written $x = 42/\log$ there are no red-flags of oooh that looks weird that can't be right thrown. So I suppose my goal is to identify students who would see something like $x = 42/\log$, the $\log$ having not argument, and not bat an eye. $\endgroup$ Apr 25, 2022 at 23:34
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    $\begingroup$ My impression when I see something like this is that the student has no clue what they are looking at and is throwing up a hail mary to see if they can get any partial credit. I would imagine that they understand what they wrote is complete gibberish but is the best they can come up with in a minimal amount of time, but that’s why I think it would be best to hear from them. Do you see mistakes like this interspersed on “A” papers? $\endgroup$
    – Steve
    Apr 25, 2022 at 23:45
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    $\begingroup$ I don't think adopting careful notation on your end is going to fix the problem by itself. Even if you try to use just one of the notations $\log(x)$ or $\log x$ consistently, or one of $(2)(3)$, $2 \times 3$, or $2 \cdot 3$ in your teaching, the fact remains that all of this is very common notation and your students will see it in their later classes. That $f(x)$ looks like multiplication but isn't is something that students need to learn not to get confused about rather than to avoid showing them the various notations that are used in math. Maybe discuss this error early and why it is wrong. $\endgroup$
    – KCd
    Apr 26, 2022 at 14:44
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    $\begingroup$ The amusing thing is that if they wrote $x = \log^{-1} 42$, you'd be forced to give them at least partial credit. $\endgroup$ Apr 28, 2022 at 13:10

3 Answers 3

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Writing from a software engineer's point of view, it's a fact that mathematics uses a notation that's highly ambiguous.

If you don't know that $log$ is used to denote some logarithm function, then seeing a term like $log x$ might as well mean the product of $l$, $o$, $g$ and $x$. Even writing $log(x)$ doesn't really solve the problem, it can still be the product of $l$, $o$ and $g(x)$. That is because adjacency of symbols can mean a lot of different things:

  • it can mean multiplication (e.g. $2ab$ for $2 \cdot a \cdot b$),
  • it can form a longer word (e.g. $log$),
  • it can mean function application (e.g. $logx$ for $log(x)$.

That makes it hard for the math teacher. When introducing functions, the students not only have to grasp that new concept, but also the peculiarities of the new notation:

  • There are multi-letter words now (e.g. $log$), whereas up to that point, mathematical entities were denoted by single letters (sometimes with some decoration).
  • A series of some letters no longer necessarily means multiplication, but can denote function application, if the initial letters name a function.

As a teacher, you can't change the mathematics world to use a less ambiguous notation, so you can only teach your students formula understanding.

I'd have them translate terms from the standard shorthand math notation into the most explicit possible one, e.g. that

$$3\log x \log y - \log z$$

effectively is

$$\bigl[3 \cdot \log(x) \cdot \log(y) \bigr] - \log(z)$$

By the way, when tutoring (weaker) math students, I often found that they lacked a thorough understanding of a mathematical term, so doing exercises on that topic from time to time might help many of them.

EDIT:

One more problem comes with the order of operations for function names, as that isn't obvious: $sin 2 \alpha$ is meant to be $sin(2 \cdot \alpha)$ and not $sin(2) \cdot \alpha$, but where are the rules defining that? And surely $sin \alpha + sin \beta$ is $sin(\alpha) + sin(\beta)$ and not $sin(\alpha + sin(\beta))$, but why?

From these examples, function application seems to be higher in the order of operations than addition, but lower than multiplication. But there are counter-examples as well, e.g. $\log x \log y$ isn't $\log (x \cdot \log (y))$.

Is this ever taught explicitly, like the rule that multiplication is higher than addition? I doubt that it's even possible to give a rigorous definition how to read a math expression, without ambiguities and guesswork.

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  • $\begingroup$ These are good points. I think of this often when the suggestion "we should teach students programming at a younger age" comes up. Doing this would get them used to descriptive multi-letter function names, and the compiler/interpreter is strict enough to not work if their syntax/notation is incorrect; it gives immediate feedback like an online homework system. $\endgroup$ Apr 28, 2022 at 14:43
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    $\begingroup$ If it weren't started so early, I think mathematicians' choice to use single-letter function names $f$ would be seen as weird. I've heard legends that philosophy/logic researchers drop the parenthesis even in this case, and instead of $f(x)$ write either $f.x$ or just $fx$. $\endgroup$ Apr 28, 2022 at 14:44
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    $\begingroup$ Conventions - as a software engineer you should know that they are used in software development as well. In math abc is a×b×c, but log is a logarithm. If you are writing syntactic parser you would check for known combinations of letters first, then you would treat letters as separate variables. After all, there are predefined keywords like "var" or "private" and then there are variables that can have any name. It all comes to knowing the dictionary and the syntax. $\endgroup$
    – Rusty Core
    Apr 29, 2022 at 1:30
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    $\begingroup$ @RustyCore Valid point. Only: I don't know a definitive source where to find a list of all "known combinations of letters", and without such a list, a parser won't work correctly. $\endgroup$ Apr 29, 2022 at 8:59
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    $\begingroup$ Although not necessarily helpful for students (especially when reading handwriting), there is a distinction between $l\cdot o \cdot g$ and the function $\log$ in typed text: named functions are written in upright roman characters, but variables are italicized. I'm sure there are some exceptions to this rule, though. $\endgroup$
    – Ben Jones
    May 1, 2022 at 1:45
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Ask the student to critique this work:

Solve for $x$: $\sqrt{x} = 3$

Easy: $x = \frac{3}{\sqrt{\phantom{x}}}$.

I have tried this a small number of times, and it has worked so far. The students recognize that the $\sqrt{\phantom{x}}$ symbol on its own is not a valid mathematical object, like the log in your question. This gives you a way to connect your issue to something they already know.

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    $\begingroup$ Does this really work in the sense that they stop this nonsense in new settings? I think when a student imagines such "algebra" is possible, they will make these kinds of mistakes in the future (e.g., sin(x)/x = sin) because they whole conception of functions is so poor. $\endgroup$
    – KCd
    Apr 26, 2022 at 14:38
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    $\begingroup$ But log and $\sqrt$ are valid mathematical objects. (I like the question though. I guess the horizontal bar of the root makes it more obvious that something is wrong?) $\endgroup$ Apr 26, 2022 at 18:35
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    $\begingroup$ I agree that this somehow is a less clear explanation the more mathematics you know, but for some reason this has always worked for me. I think you might be right that the notation more obviously demands an input. $\endgroup$ Apr 26, 2022 at 19:37
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    $\begingroup$ Bridging this example to something closer to sin(x), log(x) etc but still intuitive: height(Chris Cunningham) = 180cm => Chris Cunningham = 180cm / height. $\endgroup$
    – NiloCK
    Apr 27, 2022 at 17:04
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    $\begingroup$ I think your example "worked so far" because for them it provides more of a jolt, along with a revelation, than (what is for them) a humdrum discussion about numbers and function/operation symbols. If they really get a kick out of your example, you might try following it with something even sillier, such as: $3+9=12,$ so $+=\frac{12}{39}=\frac{4}{13}$ The idea is to get something that sticks with them enough (or sufficiently resonances with them) so that, if later, they find themselves thinking about dividing by $\log$ or $\sin,$ then the silly examples you gave pop out from their memory. $\endgroup$ Apr 28, 2022 at 11:37
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Function concept and various types of symbols and terms (e.g. log) are new to the students. It is not uncommon for weaker ones to struggle. The solution is not some secret aha revelation, not some change to convention, log(x) vice logx. But rather drill and repetition and corrective feedback for mistakes and praise for success. These things are new and strange to the kids. It's like learning a language. You need practice.

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    $\begingroup$ This, to me, makes it sound like they will ultimately learn that it is Yet Another Arbitrary Rule to be memorized, because "practice" in this sense simply drills the fact in until they stop questioning it. $\endgroup$ Apr 26, 2022 at 3:45
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    $\begingroup$ @The_Sympathizer Some things in life just have to be memorized, like the conjugation of the verb "to be" and the fact that sin, cos, tan, and log are functions. $\endgroup$
    – shoover
    Apr 26, 2022 at 6:14
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    $\begingroup$ -1, What corrective feedback is to be given? What language-learning technique is recommended? What drills are recommended? $\endgroup$ Apr 26, 2022 at 13:22

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