Counting is one thing and I'm not sure whether this is possible to do or not (my understanding is that it's "not"), but the example in the original article is specifically trying to learn to reproduce strings of exactly n a's followed by exactly n b's, for arbitrary positive n.

That's learning the CFG a^nb^n for n > 0 in a nutshell and is therefore impossible to learn, and indeed the LSTM in the article falters after test-time n exceeds training-time n by a few units.

The paper you link to discusses subject matter I'm not familiar with (bootsrapping as in psychology, say) so I'm not really qualified to opine on it. However, from a very quick look it seems like they're learning to associate counting words with numbers- sorry if I'm wrong about that. Still, that's not "learning to count", where you have to be able to make the inductive leap that n + 1 > n, for all n in your target set of numbers in addition to learning the recursive structure of number strings.

Finally, from the little I've seen of attempts to model counting with machine learning, there hasn't yet been much success, probably because it's impossible to learn a language with both structure and meaning from examples only of its structure and no access to their meaning.