# Notation Conflict between Teachers and Textbooks

In mathematics notation plays an important role in clarifying the subject. A bad notation could be confusing. Recently I use a logic textbook which has a very nice approach and content but an inappropriate old notation which I never use personally. I decided to teach its content using my own notation but I see the students are a bit confused about this double notation method. They cannot translate between lecture and textbook notations quickly and this is somehow annoying for them.

Question. What should I do about this notation conflict? Does passing time help them and solve the problem or I should change my notation or my textbook? Is this problem serious at all? What happens if we continue in this situation?

• Very useful and interesting question. +1 for bringing this up. Mar 28 '14 at 7:25
• I think it is a bad idea to use your own notation. Mar 28 '14 at 13:57
• @GeraldEdgar, why? Obviously a completely idiosyncratic notation is undesireable, but sometimes books include notation that is out-of-date, or inappropriate for the audience (for example, mathematician-specific in a class for engineers, or vice versa). In such a case, I think that the teacher has a responsibility at the very least to discuss the notation, and possibly also to use it. Sep 3 '14 at 15:53
• @LSpice: The reason is explained in the first paragraph of the question. But the question is not about whether using different notation is desirable (it's not), but rather what to do about it after the damage has been done. Sep 3 '14 at 15:58
• @GeraldEdgar, if "the students are a bit confused" is an argument against a topic, then I think that rules out most of the curriculum! Although you're right that the question is not about whether or not to use the different notation, I think that such a question (if it were asked) could be re-phrased as "Is it better to confuse students during the class, or in the remainder of their mathematical careers?" (a phrasing which makes apparent my bias). Sep 3 '14 at 16:51

When I teach introductory logic, I usually show my class a list of alternative notations. Even if the textbook and I agree on notation, students who look at literature beyond the textbook are likely to see other notations, and it's good for them to know that this can happen before it actually happens and confuses them.

An especially bad source of confusion is that the same symbol can be used by different authors with entirely different meanings. For example, one can find the symbol $\sim$ used for equivalence and also for negation. One can find $|$ used for disjunction and for Sheffer's stroke.

– user230
Mar 28 '14 at 14:32

In any case, you should warn them about the different notation in the textbook.

This question is related to Should students be asked to use more than one notation for the derivative in an introductory calculus class? - In my answer, I wrote "I think, it is important that students should be flexible and open-minded to notation" and went into detail there (It will not be the first time that they will be confused with notation, once they read a textbook on their own or they have a different teacher, a similar problem arises).

However, in your situation, the problem is that the textbook is older and the notation there is out of date. But, also this will not be the last time that students will have a textbook or an article with outdated notation!

I think, in general it depends on the "type" students you have. If this a class of engineers who will only have a few courses in mathematics, what was said above might not be true - and in this case, you should probably change your textbook; but if you have a class of math majors who will eventually do their master or even more in mathematics (which I think you have since you mentioned logic), at some point you have to talk with about notation. Most textbook have some pages where they explain the notation needed to read the book and the rest of the notation is explained when intoducing. Explain your students how to use that and explain them the (dis)advantages of your notation and the textbook's notation. At least, they will learn to be more flexible (which is good for them in future as mentioned above) and they will also learn that mathematics is a historical progress and notation is part of this progress as well. If you have the feeling that your message does not arrive, you can give them examples from fields and notations they already know, e.g., Bourbaki introduced some important notation like $\varnothing$.

It is okay to deviate from a notation in a course textbook as long as

1. The notation you introduce is still a standard notation.
2. You clearly demonstrate and emphasize notational differences.
3. You explain where/why (and when, if one has perhaps fallen out of style) each notation is used.
4. You remind students every now and then about your notation vs. the book's notation.

For example, in a class field theory class learning about Tate cohomology, notation can be all aflutter. Neukirch, Lang, Milne, and Serre all use different notations for the cohomology (including $H^0$, $H_T^0, \hat{H}^0$ - and that's not even mentioning the names for the various groups and modules).

I remember this because the professor who taught me class field theory was careful to point out that the notation he uses differs from Neukirch (our text) because Neukirch's notation was out of date, and he told us about the other notations and who used which. Interestingly, I don't remember which one Neukirch uses nor which one is actually used now, which I think is an indication of good, notation-independent teaching (and a bad memory on my part).

I think the damage has been done because the notations have already been introduced and you can't really do away with one or the other at this point. I think you can make a lot of progress by motivating what the different notations mean and why one notation has become outdated. If the students understand the underlying principles well, it should help with translating quickly. There was a question about notation for differentiation here and one can surely use the lessons learned there in your situation. For example, the $\dot{y}$ notation is very similar to the $y'$ notation but conveys a slightly different meaning. Ideally one would not teach it right away and introduce it once the core idea is understood well-enough. However, if both have been introduced from the start it would help to differentiate (sorry) between the two so as to introduce a different point of view that will go along with said notation.

Unfortunately I don't know the particulars of your situation with the logic textbook so perhaps I'm way off.

Perhaps you should write up (the most important) results of the text in your notation as a complement to the text. Perhaps "translate" salient exercises too, and do a "dictionary."

But as other answers say, it very much depends on your students. First years, I'd never consider doing something like this (and I take care to use the same notation used in other courses, as far as possible). Advanced/graduate students, specially ones for whom the subject is a specialty course, must be able to read alternative notations, so there it is much less critical.

As a graduating college student, I have gone through countless variations of logic notation since middle school. The only thing that ever bothered me was being forced to use a specific notation. Certain teachers would mark answers incorrect for being in a different, but otherwise completely valid, format.

Tell them various notations exist, let them use what they want, and everything will be fine.

• The student's opinion: let the student use what they want. Not the instructor's opinion, however. Mar 29 '14 at 1:38
• If they want to use non-standard notation, explain it beforehand. Apr 10 '14 at 8:24

A thought from Steven Krantz's How to Teach Mathematics, 3rd Ed., published by the American Mathematical Society (sec. 2.12):

If a textbook uses notation or other conventions that you do not like, then don't use that book. You really are obliged to follow the notation and definitions and other paradigms in the text you have chosen. Otherwise all but the gifted students will be lost. If you repeatedly criticize the text as the course proceeds, then you will be sending a confusing message to the students: Why did you choose this book if it is obviously so full of flaws? Isn't it your job to select a text that you can teach from?

• This might be true for basic areas of math for which there is a plenty of books to choose from, but not for more advanced, specialized courses. Sometimes you just don't have time to write your own book or lecture notes. Oct 25 '16 at 8:57