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I wonder if a more complex example would help clarify what happens in your head. E.g. how do you approach 2484+5839? Or 349753+935731?
Very differently, though still with some overlap: multiple instinctive approximate models, but then detailed step-by-step models for exact computation. Though the exact model will tend to be somewhat made up on-the-fly.
The first one, I immediately think 8200 plus the sum of 84 & 39. I then work that out (123), and there we are 8323...
That's interesting. I do roughly the same thing: basically add digits left to right, and occasionally correct where I'm at when there's something to carry. But if the numbers were on paper I'd probably go right to left. 
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