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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:

Example offense.

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When you say "write" do you mean "type"? I think this particular notational ambiguity results from the constraint of needing to type everything in-line. In handwritten work it is possible to use space in a more flexible way to disambiguate 1/3x from (1/3)x, for example by using a horizontal fraction bar and clearly placing the 3 and x directly below or to the right of the fraction bar. –  mweiss Apr 3 at 13:50
@mweiss I mean "write", e.g. on quizzes. And they definitely don't use the extra space. –  Avi Steiner Apr 3 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 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 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 at 9:32

4 Answers 4

up vote 17 down vote accepted

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.

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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.

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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.

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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."

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