I would say there is a big difference between arithmetical theorems that you can derive from the Peano axioms, and the algorithms (i.e. computations) that we are taught in elementary school to do things like multiplication or long division.
I suppose you could still show that those algorithms are correct using the Peano axioms, but that would go a good bit beyond just a proof that $5 \times 4 = 20$. Indeed, to prove that the general method works for numbers of any length (written in decimal notation), we'd need to prove things about arbitrary length sequences of decimal digits, which is not an easy thing to do in Peano Artihmetic; see Godel's Beta function.
Of course, I am talking about the addition and multiplication of multiple multi-digit numbers here ... when it comes to just two single digit numbers, the basic addition and multiplication table is pretty much drilled into our heads ... so I don't think there is much 'computation' going on there. So for something like $5 \times 4 = 20$, maybe a Peano axiom based proof that this is in fact the case would in fact capture that aspect of our arithmetical 'belief'.