# Students using ambiguous notation

I've noticed that many of my calculus students (all college students) will write, e.g., $1/3x$ to mean $(1/3)x$. This is an inherently ambiguous notation which I'd like them to avoid. Is simply pointing this out in class effective?

Edit: Here's a handwritten version of the ambiguous notation:

• @mweiss I mean "write", e.g. on quizzes. And they definitely don't use the extra space. – Avi Steiner Apr 3 '14 at 14:03
• Start using such notation yourself for a while :-) Then they will experience first hand how much trouble it is. This is of course meant as a joke as written, but I could imagine something along these lines could be an eye-opener. However, I would not know how to implement it effectively. Thus just a comment. – quid Apr 3 '14 at 15:52
• I think this is a really good question, and maybe speaks to the more general and challenging issue "How can we take points off for something that is not incorrect, but is very bad for some other reason?" I've seen "style points" given in abstract courses and have considered something similar for my earlier courses. – Chris Cunningham Apr 3 '14 at 16:54
• There are two object in the handwriting. A fraction and an x. That is because the line in the fraction only extends over the 3. We write objects horizontally and they are separated by a vertical whitespace, which you could draw in this example. The only possibility is (1/3) * x. That is because if you assume that the x is in the fraction, you can also assume another possibility that 3 is not in the fraction. So the only non-ambiguous version is the (1/3)*x. – this Apr 4 '14 at 9:32
• I don't think there's anything ambiguous about it. $1/3x$ is the same as $(1/3)x$. Have your students play around with a computer algebra system; they will quickly learn how to be precise from the unforgiving syntax parsing. – DumpsterDoofus Apr 7 '14 at 21:06

A colleague of mine includes on all his tests a line that reads (something like) "you will be graded on what you actually write down, not what I think you may have meant by what you wrote." He spends time carefully discussing what this means (examples like the one you gave are among them) before and after testing times. Normally after the second round (of four), students have developed the ability to reflect on how their own writing will look to others, and can avoid this type of mistake.

Many people simply do not realize that written math has notational conventions, and despite the fact that the algorithms and proofs are logical and complete, the notation is just a language and has holes and ambiguities like anything else.

As plenty of troll questions on Facebook show, many people will defend their interpretation of "what is the value of 5 + 1/2(6)" to the death, acting like PEMDAS is a complete mathematical rule handed down by the gods unto man and not just an incomplete (doesn't include juxtaposition!), arbitrary convention that makes it easier to figure out what some other person wrote.

The best way I've ever seen this solved is just to write a bunch of really easy but ambiguous expressions on the board. Tell the class to solve them on a sheet of paper, have them hand it forward, and then take a tally of how many completely different but still correct answers you get for each problem. Then use this as a learning experience about how math requires clear communication just like everything else, and that despite it being a logical field, not everything is an invariant theorem so you really don't know what they mean when they write confusing and ambiguous expressions.

I would take off one percent if I thought the wrong meaning was unlikely, and more if I thought the wrong meaning was possibly what they meant. I would write: "This looks like 1/(3x). You may have meant 1/3 * x, but it looks like your x is in the denominator."

• I'm not sure your answer actually answers the question. I respect your decision to mark students the way you want to mark them. I don't think it's a great way to mark them but it's your decision how to mark them and there's no problem with it. I just think that because the way you do it isn't that great a way, giving an answer that says how you would do it doesn't actually provide a useful answer to the question and there may be no need to write any answer at all. I'm not experienced on this site. I'm just giving a possible suggestion that might be valid. I think there is very little harm in – Timothy Apr 3 '19 at 20:17
• writing such an answer and it's probably fine to write an answer taking a risk of it being of a tiny bit of harm in case it ends up being a useful answer. – Timothy Apr 3 '19 at 20:19

At Warwick University then there were explicitly given 5 points per homework (out of 25, I think) for "Clarity and Style". By really emphasising this in the first year, I found that with my students (I was a graduate student at the time, so working as a sort of TA) then this sort of problem soon went away.

So I don't have an answer to the specific question ("Is simply pointing this out in class effective?"), but I can say that putting it in the rubric for homeworks and backing that up in tutorial time is effective.

I don't think the other answers answer what you have appeared to be asking so I will write my own answer. It looks like your question was whether pointing out the mistake is useful.

I think it depends on what you mean by pointing it out. Do you mean just stating that that's not how it's supposed to be written or do you mean giving a reason why that's not how it's supposed to be written? I think for some students, just pointing out that that's not how it's done might not actually work. I think some students when they're taught to do things a certain way will keep insisting on understanding why it's supposed to be done that way and of those students, the ones who wrote 1/3x to mean (1/3)x, they might need to have it explained that until other people are told how to do things, to them, 1/3x could either mean 1/(3x) or (1/3)x. I don't know if it's true or not but if it is true that by convention, 1/3x means 1/(3x), then they could be given that reason. It could also be explained to them that when they write the other notation for 1/3, others may take it as writing that notation and be confused because they haven't been taught to use the other notation to indicate that they mean (1/3)x and not 1/(3x).