I recall my music teacher once put a quote on the board which I shall now adjust to the problem: Take 2 piles of sand and 2 more piles of sand and add them together. What do you get? 1 or more piles of sand.
Not directly applicable to the general understanding of integers, but amusing to me. You could also do similar quibbles with musical tones or beats.
Then again it could all be rubbish...for I don't think I could argue any of the points argued so far, though I do find my attempt at understanding it enjoyable if not complete.