Professor Jo Boaler of Stanford University is quoted as saying 'Teachers are stopping children using their fingers at a ridiculous age - four or five - so that has to change.' She point out that when we work something out mathematically, the brain maps this onto fingers - and better maths achievement goes hand-in-hand with better finger perception.
I think she possibly stretches this a little too far by saying 'It explains why musicians, particularly pianists, typically have a higher level of understanding of mathematics' - but apart from anything else, to discourage the use of a readily available resource seems crazy. I am happy to admit that if I am asked by a website to input the sixth character of a password I don't use regularly, I will count the letters off on my fingers - it just makes a good outcome more likely.
This whole business was also interesting as I open my new book Are Numbers Real? with a fictional account of numbers being 'invented'. It is perfectly possible to count using set theory without having numbers, and initially I suggest, the fingers might have been used as a tally - so folding fingers over as you count goats (say) out and repeating the process as you count them in. You don't need numbers to realise your hands are different if you've lost a goat. But then it's not a huge step to realise that rather than describing the problem by showing someone your hands, you can give a name to the finger configurations - referring, say, to a 'hand' of goats. And the tally has started the transition into numbers.
We don't know how it actually happened, but it's hard to believe that fingers weren't involved in the development of numbers - making this particular dispute poignant. You can read more about my goat system (and the weird way that the maths managed to become detached from reality yet remain useful) in Are Numbers Real. And you can read the Stanford paper here.