I get where your teacher was coming from in thinking it should be made as obvious as possible, but the primary problem we contend with in making it obvious is one of ambiguity, not recognizability. Commas are frequently used for denoting entries in a series in a sentence, and numbers oftentimes appear in a series. Consider the following:
A) 123,456, 789,0
B) 123, 456, 789, 0
C) 123, 456,789, 0
Effectively, we're relying on the spaces to provide necessary meaning. (A) represents two real numbers, (B) represents four integers, and (C) represents two integers and a real number, but at a quick glance, it isn't necessarily apparent which is which since the only difference between them is where the spaces are located. Moreover, had a space been forgotten due to a typo, it would have substantially altered the meaning of the series, and unlike words that may be affected in a similar way (e.g. "good one" vs. "goo done"), which are relatively easy to recognize as typos within context, we rarely have useful context clues with numbers from which to recognize that a simple typo has occurred.
Contrast that with the use of the decimal point:
A) 123.456, 789.0
B) 123, 456, 789, 0
C) 123, 456.789, 0
It's clear where each number begins and ends, and what quantity it represents. That said, decimal points have the potential to become ambiguous when dealing with the ends of sentences, but even there, they are unlikely to cause confusion, given that it's rather rare that we have back-to-back sentences with the first ending in a number and the second beginning with one. Besides which, even when we do, we generally have ample context clues in the text that can help us to recognize that one sentence has ended and another has begun.
Just as I don't see how most of* my fellow Americans can keep arguing for using Imperial units, I don't understand how some Europeans can continue to argue for using commas instead of decimal points. Using an entirely different punctuation mark may be a better option than either the comma or the point, but if we're constrained to choose between the two, I have yet to hear a great case for why the comma is the superior choice.
* I say "most of", because I actually have had several of my engineering friends, particularly those in petroleum engineering, provide specific examples of situations in which they greatly prefer to use Imperial, rather than metric, units. Apparently it's one of those situations like weight vs. mass where the two units aren't actually analogous, and working with the metric unit ends up making the computations significantly more convoluted. In most other cases though, they, and I, tend to prefer metric (even if I don't necessarily think in terms of metric on a daily basis).