The V-E+R follows the pattern of zero-dimensional, one-dimensional, two-dimensional, and this alternating sum formulation continues for the Euler characteristic in higher dimensions. So that is one reason to keep V-E+R.

Another reason, however, is that although some negative numbers show up, the numbers are all very small, and this could be a good opportunity to introduce some exposure to negative numbers.

So it turns out that this formula generalizes to an alternating sum over many dimensions [1] (although it can be stated more simply than Wikipedia does). From that perspective, it makes more sense to put the minus sign in the middle.

In either case, I don't think it is that big of a conceptual stumbling block to have a minus sign in the middle...

[1]: http://en.wikipedia.org/wiki/Euler_characteristic#Generaliza...