How can you elicit the $\log x = {\log} \cdot x$ error?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.