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Functions and mappings are usually introduced late in the curriculum, and functions of arity two or more are considered "advanced" (many don't even see them before college).

On the other hand, the concept of a mapping ("the dogie gets a bone, the kitten gets a fish, the bear gets an apple") is intuitive and many of my non-math friends told me when asked, that mapping from pairs or triplets isn't really a hard thing, just that they never thought about it from this perspective.

Hence, the questions: why it is not feasible to teach multiple-argument functions earlier, perhaps even before the addition is introduced in a more systematic way (the moment where one learns that it is one of many possible operators, and that multiplication is one too, and that there is some order to apply those operators, etc.)?

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One must draw a line somewhere. I mean, if binary functions were taught early on, would we ask for ternary functions too? what about functions of infinitely many variables? What about relations? The question is what is one trying to teach about functions. Which aspects of functions are the focus. Functions of a single variable are somewhat simpler than their binary or ternary friends, even though everybody is familiar with binary functions (like addition and multiplication) very early on.

It is also quite simple to draw graphs for single variable functions than it is for other functions, a very useful visual aid. One can argue that once one is comfortable with single variable functions, the passage to multi-variables is quite trivial and one looses basically no generality by only introducing multi-variables later on. That while introducing multi-variables at the get-go serves little purpose, and may confuse the student (higher dimensions etc.).

So, there seems to be no loss in taking the small jump to multi-variables only at a later stage, after the ideas of functions are firm.

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    $\begingroup$ What's wrong with talking about binary functions (resp. $n$-ary) as taking pairs (resp. $n$-tuples) as an input? BTW I think there is too much attachment to graphs and it causes problems when higher-order functions appear. I had to draw a lot of silly examples for them to repeat the foundations and only then continue to the actual topic. $\endgroup$ – dtldarek Mar 18 '14 at 15:48

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