In secondary math classes we often say "Math is a language", but we really
don't teach it that way.
The closest we get to that is calling the comparison operators 'verbs' and
the various kinds of values that can be combined into expressions 'nouns'.
So, as far as contemporary secondary math thinking is concerned, the only
'verbs' in 'math' are '=', '<', and '>', forms of 'to be'.
However, I'd bet that a lot of students, and even teachers, would categorize
'+', '-', '*', and '/' as 'verbs'.
After all, they seem to suggest 'action'.
But is '2 + 3' a sentence or a complete thought?
No, it is an expression which is equivalent in value to another expression,
namely, '5'.
I keep thinking that, mathematically, it makes more sense to say that '2 +
3' is one of the already-existing partitions of '5'.
Interestingly, if '2 + 3' were to be taken as a sentence, it would be
imperative:
"Hey you! Yeah, you! Find the value of 2 + 3, now!"
Mathematically speaking, it would be more accurate to read '2 + 3' as 'the
sum of 2 and 3'.
Mathematically speaking, '2 + 3' is a value that does not need to emerge in
time. It already exists.
But, computationally speaking, it actually does make sense to think of '2 +
3' as a process.
We start with a '2', then we do something with some registers or whatever,
and we increment '2' by '3', and we end up with '5'.
And then this leads into the question of functions - are they 'nouns' or
'verbs'?
In OO terms, functions are how objects relate to other objects.
Mathematically speaking, at least according to contemporary secondary
textbooks, a function is a set of ordered pairs.
The expression 'f(x)' is a VALUE. So, 'f(x)' is a 'noun'.
But again, computationally speaking, a function is a process. It takes in
some information, does something to it, and then yields some new
information.
In computational terms, a function is something we 'do' to a value to
produce a new value.
I do apologize if this is not the correct venue for raising such questions,
as it is not language specific, but I find this really interesting. I think
it might have a lot to do with contemporary students, and teachers!, not
'getting' what math really is, and it might shed a lot of useful light on
the disconnect between the current secondary math curriculum and the current
state of computational understanding.
This semester, at a high school level, I do intend to teach math as a
'language', and I'd like to get really clear about these kinds of things.
Thanks very much for any feedback,
Michel Paul