Towards more meaningful math notation

Students struggle a lot with the way mathematics is written.

For example, most students don't have too much of a problem with:

5(a + b) = 5a + 5b

Then they see this and it is also OK:

5(ab) = 5ab

In most cases you can substitute various values of a and b and the students can see that it works. Fair enough. Then the student does twenty (mind-numbing) examples of such bracket expansion and they feel they have got it.

Later, they come across things like:

sin(a + b)

And then their math teacher goes ape when the student expands it like:

sin(a + b) = sin a + sin b

Perfectly logical, in the minds of the student.

Similarly, it is logical to have the following, isn't it?

log(a + b) = log a + log b

Oh, and then we have functions. You know, like this:

f(x)

Is that the same as

f × x? (That is, f multiplied by x?)

Why not?

I wish to propose an alternative notation for concepts where you cannot expand in the way you do with simple algebra. It might look something like this:

This would send a much clearer message to students that the particular function or operation does not work in the same way as simple algebra works.

Now, the proposed rectangle would be a nightmare given that we need to type mathematics (actually, everything is a nightmare when you are trying to type mathematics...).

So a more computer friendly option would be to (exclusively) use [ ] - square brackets - for such concepts, like this:

sin[x + y]

log[x + y]

f[x]

Would this work? Would it confuse everyone even more? I feel that if such a notation were to be universally adopted, then less confusion would arise.

[I wrote about notation before in Phase shift or Phase Angle?].

See the 66 Comments below.