Copying from Calculus Made Easy by Silvanus Thompson (2nd ed., 1914):
CHAPTER I:TO DELIVER YOU FROM THE PRELIMINARY TERRORS
The preliminary terror, which chokes off most fifth-form boys from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning -in common-sense terms-of the two principal symbols that are used in calculating. These dreadful symbols are: (1) $d$ which merely means "a little bit of." Thus $dx$ means a little bit of $x$; or $du$ means a little bit of $u$. Ordinary mathematicians think it more polite to say "an element of," instead of "a little bit of." Just as you please. But you will find that these little bits (or elements) may be considered to be indefinitely small. (2), $\int$ which is merely a long $S$, and may be called (if you like) "the sum of." Thus $\int dx$ means the sum of all the little bits of $x$; or $\int dt$ means the sum of all the little bits of t. Ordinary mathematicians call this symbol "the integral of." Now any fool can see that if $x$ is considered as made up of a lot of little bits, each of which is called $dx$, if you add them all up together you get the sum of all the $dx$'s, (which is the same thing as the whole of $x$). The word "integral" simply means "the whole." If you think of the duration of time for one hour, you may (if you like) think of it as cut up into $3600$ little bits called seconds. The whole of the $3600$ little bits added up together make one hour. When you see an expression that begins with this terrifying symbol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow.
I was wondering whether there has been (specific) research as regards the educational merits (or demerits) of such an approach to calculus.
Considering that basic computational and calculation skills are necessary for the reproduction of society, I can understand why one could approach arithmetic like that (in an attempt to make all citizens literate on arithmetic). But once we are into differential and integral calculus, it is not about basic computational skills anymore.
So does such an approach make for perhaps a fabulous ride in the beginning, only to be regretted later when symbols become more abstract in content and their manipulation must become more rigorous?
Or does it instill in the students an attitude like:
What? The integral should not necessarily be thought of as a sum? Great, because we were getting bored – let’s see what else we can do with it
Again, I am not asking for opinions (I find opinions interesting and useful, since they usually come with arguments – at least on this site –, but they are not allowed here as far as I know), but for any research results on the matter.