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I'm learning about groups and $a^n$ suddenly doesn't mean exponentiation anymore, but repetition of $\underbrace{a\circ a\circ \cdots\circ a}_n$. In some sense I think it would be useful to learn $a\!\overset{n}{\circ}$ first so it's clear that it's not exponentiation, but repetition of $a\!\circ$, and then learn $a^n$ later when I've become accustomed to the meaning. It's somewhat confusing to think of $4^3$ to mean $4+4+4$ in $(\mathbb{N},+)$...

I'm just a student, so I'm just viewing this from my perspective, but in general, is it useful to use non-standard notation to teach new concepts?

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    $\begingroup$ This doesn't answer the question, but is worth mentioning: when I learned group theory, we used two different notations depending on what sort of operation we used. When it was multiplicationish, we used $a^n$, and when it was additionish we used $na$. This did help when we got to rings and fields with two operations. $\endgroup$ – DavidButlerUofA Feb 7 '15 at 4:58
  • $\begingroup$ I've often seen the notation $a^{\otimes n}$ used when $\otimes$ is the notation for the operation we're repeating. I'm not sure if I've seen this used for other operator symbols. $\endgroup$ – user797 Feb 10 '15 at 5:57
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At the algebra level you are at, you must toss out many of your old ideas of what the symbols mean and use what they are currently defined as. Actually, I've seen many introductory algebra texts that start out with the operator $\star$ to rule out this old intuition.

It is common at a more mathematically mature level to abstract the idea of some concept with a new notation, symbol, or even a name. This allows for the manipulation of concepts at a level where prior knowledge might get in the way or to have students practice in thinking more abstractly. Just one brief example might be defining $\frac{a}{b} = (a,b)$ so that $(a,b) + (c,d) = (ad+bc,bd)$, highlighting a slightly different area of the concept (and allowing it to be on a ring product too e.g. $\mathbb{Z}\times\mathbb{Z}$). I have also seen professor's use uncommon notation or terminology when they want to discourage students from looking up the solutions to problems online.

That said, I don't think it would look at any lower maturity levels. Many students would grow flustered when you convert back to the more accustomed notations, thinking of it as one more thing to keep track of. They usually don't have the practice in abstraction to see that they are the same object. You can see this frustration when presenting a problem to beginning calculus students with just relabeling indices on a summation. They are the same object, but the look is just enough off than many might not realize it at first.

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Indeed, there is nothing wrong to use personal non-standard notations to learn new concepts providing that you know how to change them to the standard ones; otherwise, as a student you don't get the mark that you deserve since your lecturer could hardly understand what you mean.

But, the example you mentioned is more about teaching a general concept and then specific ones than using non-standard notations to teach. That one, I don't think has any clear-cut answer. For example, probably years ago you were taught first about some special parallelograms, say square and rectangles, before the general notion of parallelograms. Some people think the other way round is better whenever possible (for example when the general notion is not that much abstract). It depends on the teaching philosophy one advocate.

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