Say I ask subjects what $20 \cdot 20$ is. Is there any way, besides introspection, to evaluate whether subjects recalled the answer or calculated the answer? In general, I would expect that the recalled answer would be presented faster, however some calculations can be done quickly and sometimes it can take a while to recall information.
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A consensus within cognitive psychology is that there really is no such thing as "computation", though that depends how you define it. It seems all arithmetic ( + - * / ) is simply fact retrieval from long-term memory. More complex problems simply rely on the fact that we can break them down into simpler problems. For instance, when you learn to solve 364+192, you're really doing a set of simpler retrieval operations, e.g.:
The only true "computation" aside from fact-retrieval is counting, which we resort to when we don't have a fact required... e.g., If I forget 4x6, then I'll add
There's still the question of whether someone memorizes 20x20=400, or whether they break this down into smaller components (2x2=4, add two 0s). It would be hard to tell what strategy someone is using here, but reaction time may give a clue. Retrieval time is going to be influenced by the number of steps in the problem, and the strength of the memory for each retrieval. John Anderson has used his ACT-R (http://act-r.psy.cmu.edu/) model explicitly to model reaction time in arithmetic which can give you quantitative predictions. Still, I think in most cases it will be difficult to distinguish between such a small difference in reaction time.
Re: cognitive arithmetic as mental fact-retrieval, see Ashcraft, 1992. Or, this quote from Ashcraft, 1995:
M.H. Ashcraft (1992). Cognitive arithmetic: A review of data and theory. Cognition, 44, 75-106.
M.H. Ashcraft (1995). cognitive psychology and simple arithmetic: a review of summary and new directions. mathematical cognition.