Is using different notations in one course a good idea?

Question

From aesthetical point of view, using two symbols for the same concept during the same course is obviously a very bad idea.

However, especially when I teach freshmen, I often deliberately mix different notations. The rationale is that the ability to cope with such a situation is an important part of mathematical maturity.

Is that a good practice? Are there any other ways to teach students that they should be flexible wrt. notations, e.g. when studying from different sources? (A similar case can be made for different words, like "increasing" vs. "weakly increasing" vs. "nondecreasing", all three of them meaning the same thing in different books.)

Answer 1

There is a certain benefit to "confusing" students; I alluded to the ideas of disequilibrium and the resulting equilibration in an earlier MESE post. More comments about Piaget can be found on this site.

In the context about which you are asking: I think that if you want to introduce multiple notations, then you should, at least, abide by two principles:

1. Make it very clear which notation you will use and be consistent.

2. Help students to understand why other notation styles exist.

With regard to the second principle, this could be as simple as the $\div$ sign, but you might also consider an example such as how to denote the cyclic group of order three, e.g., $C_3, \mathbb{Z}_3,$ or $\mathbb{Z}/3\mathbb{Z}$.

Each of these has its own advantages and disadvantages. For example:

$C_3$ is easy to remember because the $C$ can stand for cyclic, but its use is not so widespread.

$\mathbb{Z}_3$ is succinct and shows the similarity with $\mathbb{Z}$, but it can also be read as the $3$-adic integers.

$\mathbb{Z}/3\mathbb{Z}$ emphasizes the quotient group aspect, but is often introduced before quotient groups. (Plus, the notation is a bit more cumbersome than either of the previous two.)

Similar remarks could be made about increasing and nondecreasing, or countable and enumerative.

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Your question reminds me of a paradigm-shift in theories about language learning. In the past, many thought that teaching a child two different languages would confuse him or her, and, therefore, be disadvantageous. Today, I believe a more common interpretation is that while this can be confusing, it is the resolution of this confusion that can be cognitively beneficial.

With regard to the mathematics classroom: I don't think that varying the use of notation is the best way to induce the sort of productive struggle that leads to learning. Instead, I think discussing why different mathematicians use different notations (as described above) and helping students practice the "bilingualism" involved in moving between, e.g., set theoretical notation and paragraph style proofs are both better ways to raise students' mathematical maturity.

Answer 2

I think that there is enough notation chaos between courses/books/papers that you don't need to make special effort in this direction. Surely, it would be beneficial to mention that there are different notations representing the same thing, so the students won't be surprised when reading a book by yet another author, but I suggest consistence, i.e. stick to one way during a single course.

On the other hand, there are times when you want to use different notation for the same concept, the most important reason being a different perspective or emphasis.

For example, during a programming course I used four different notations for the same concept, in this case function.

• The standard notation, mostly for dealing with mathematics, named and simple functions; the advantage is the familiarity, the disadvantage can be seen easily in the second example; \begin{align} \mathrm{id}(x) &= x \\ \mathrm{apply}(f)(x) &= f(x) \end{align}
• An alternate notation for high-order or unnamed functions; one has to get used to this notation, but afterwards it just feels good; \begin{align} \mathrm{id} &= x \mapsto x \\ \mathrm{apply}(f) &= x \mapsto f(x) \\ \mathrm{apply} &= f \mapsto \big(x \mapsto f\ x\big) \end{align}
• $\lambda$-notation for terms of lambda-calculus (which are also functions); used to stress lambdiness, also for historic reasons; it's awkward; \begin{align} \mathrm{id} &= \lambda x.\ x\\ \mathrm{apply} &= \lambda f.\ \lambda x. f\ x \end{align}
• Programming notation, when dealing with code (that would also include functions); to emphasize that students could type it into an interpreter and it would run; uses the syntax of some programming language, here Haskell \begin{align} \mathtt{id\ x}&\mathtt{= x} \\ \mathtt{apply\ f\ x}&\mathtt{= f\ x} \end{align}

Another example could be semantics of programming languages, where there are four different notations that describe essentially the same thing (i.e. the semantic of some collection of expressions), e.g. the meaning of if-then-else could be written down as

• in operational semantics with small steps $$\frac{\Gamma \mid b \leadsto \mathtt{true}}{\Gamma \mid \verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2, s \leadsto e_1, s}, \frac{\Gamma \mid b \leadsto \mathtt{false}}{\Gamma \mid \verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2, s \leadsto e_2, s}$$
• in operational semantics with large steps $$\frac{\Gamma \mid b \leadsto \mathtt{true}, \Gamma \mid e_1,s \leadsto e_3, s'}{\Gamma \mid \verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2, s \leadsto e_3, s'}, \frac{\Gamma \mid b \leadsto \mathtt{false}, \Gamma \mid e_2,s \leadsto e_3, s'}{\Gamma \mid \verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2, s \leadsto e_3, s'}$$
• in denotational semantics (there are actually two versions here, with or without fix-point operator, but it doesn't matter for if-then-else) $$ \newcommand{llb}{\big[\hspace{-2.5pt}\big[} \newcommand{rrb}{\big]\hspace{-2.5pt}\big]} \mathcal{E}\llb\verb`if`\ b\ \verb`then`\ e_1\ \verb`else`\ e_2\rrb\ \varrho\ s = \begin{cases} \mathcal{E}\llb e_1\rrb\ \varrho\ s &\text{ if } \mathcal{B}\llb b \rrb\ \varrho\ s = \mathtt{true}\\ \mathcal{E}\llb e_2\rrb\ \varrho\ s &\text{ if } \mathcal{B}\llb b \rrb\ \varrho\ s = \mathtt{false} \end{cases} $$

Note, that in each of those cases there are very specific uses for introduced notations. Each time there is a class that draws parallels between such concepts, how they are similar and why we treat them differently.

Concluding, there are uses for different notations during the same course or same class, but those are rare. Introduce a different notation only when you have a concrete reason.

Answer 3

I always try to use several notations. If you just mention once that another notation for something exists, they will forget it. Then in later courses, or when reading, when they come across the other notation, they will be confused; they might vaguely recall that they should know what it means, and have a tentative idea of how to interpret it, but it will always be surrounded with a fog of uncertainty and doubt, which is harder to dispel after the fact than to prevent in the first place. Instead, you should strive to use various notations freely enough so that they're all instinctive.

I fell prey to this back when I was taking undergraduate classes, and I wish my professors had made a point to use more notations, so that they would be natural to me.

When I'm teaching, this is primarily about using both Newton and Leibniz notation for differentiation. There are good reasons why we still have both: each is more convenient for certain cases, and the students should know both.

Answer 4

My perspective is teaching at an open-admissions community college, and I have to say that it's an almost impossible challenge to get students to properly read and write even one elementary notation. In the rare cases that we want them to know a secondary notation (e.g., radicals and rational exponents), then we have to devote an hour-long lecture & practice period on the particular topic.

I must say that even as a student I would not want arbitrarily mixed notation in a given course or lecture series. Given a particular context I would prefer to not be distracted by that switching cost. A quick comment, dictionary, or footnote on what you may see in other texts is fine and helpful. If the student is advanced and expects to be reading multiple texts in the future, then they will have the maturity to look it up or reference it as needed (i.e., they will already be past the point of needing the productive struggle of discovering that different languages exist in the world). Don't create notational Spanglish.

A quote from Steven Krantz in How to Teach Mathematics (Sec. 2.12):

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.

Answer 5

Different notations show their strengths in different contexts. In equations of motion, for example, Newton's (dot) notation is compact, Leibniz' is less so but has advantages elsewhere, for example when using a chain rule.

In vector mechanics, for example, I hop between Gibbs', matrix and subscript/index notations depending on what aspect of the vector(s) I'm trying to emphasis. Gibbs' notation is good, for example, when considering the relationship between vectors, matrix notation is good when doing computations with components, subscript notation (given a few rules) shows explicitly how to do the computation. I expect students to be(come) familiar with each notation and so be able to read different textbooks, each of which might show a different preference.

The same is true with technical words. I can't count the number of times I've come across an unfamiliar technical term ("the Trefftz stress") and only realised later on that it was something I knew by another name ("the 2nd Piola-Kirchhoff stress").

Answer 6

Don't do it. It just helps confusing them. In case several alternative notations are common (apear in textbooks, ...) do point them out, but stick to one.

Graduate classes ("more (mathemathical) maturity") it might be fine, but I'd prefer to stick to one (there must be a "best" one for you, or pick "arbitrarily").