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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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Brain activity is probably markedly different if you did some testing...probably not very helpful for most uses though – Ben Brocka May 23 '12 at 4:02
I would accept an answer that relied on differences on imaging. – fgregg May 23 '12 at 13:05

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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.:

Add 4+2 (6)
Add 9+6 (15)
Add 3+1+1 (carrying the one-- 5)

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 4+4+4+4+4+4

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:

As I have suggested elsewhere (Ashcraft, 1992), a broad consensus has developed across models for issues of basic memory representation and processing. Current theories (e.g., Ashcraft 1992; Campbell & Oliphant, 1992; Sigel & Jenkins, 1989) propose that the simple whole-number facts (i.e. single-digit operands 0 + / x 0 through 9 + /x 9 are stored in an associative or network representation in long-term memory,with the strength of the network connections among operands and answers reflecting the degree of learning and mastery, and probably the degree to which the representations can be accessed via automatic processing. […] Processing strategies other than retrieval are prominent only in Siegler's model […]

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.

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Great answer! So, is latency the only way that would know whether we retrieved the entire answer to the problem or whether we broke it into subproblems? Would imaging help? – fgregg May 23 '12 at 13:09
No, latency is not necessarily the only way, just the most obvious. Imaging might help-- as I mentioned, there are ACT-R models of arithmetic, and a corresponding ACT-R "brain hypothesis", which translates a model into fMRI predictions, so you could use that, or another free-standing experiment. not sure if the temporal resolution of fMRI is good enough or not. you could also look at error rate, if there's some reason to believe certain operations are more error-prone than others. there are probably other possibilities as well... – Jeff May 23 '12 at 13:46
@fgregg sometimes the types/frequency of mistakes can be used to understand how individuals represent/process information (apart from timing data). The most famous example of this is the study of how beginner versus expert chess players memorize and recall chess positions. On real-game boards: beginners make simple swap mistakes and experts make formation mistakes --- suggesting that experts chunk and memorize on the level of formation, while beginners do so on the level of individual pieces. Note that when a random non-real-game position is given, both experts and beginners do equally poorly. – Artem Kaznatcheev May 24 '12 at 2:50
@Jeff What exactly do you mean when you say that there is "no computation"? If what we do is break problems down into smaller ones to retrieve their solutions from memory in order to combine them into a single answer, doesn't that constitute "computation" too? – Pavel Jul 8 '12 at 16:59
@Pavel by computation i meant calculation (as opposed to recall), whose meaning i inferred from fgregg's question. – Jeff Jul 8 '12 at 19:01

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