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## Ashcraft, M. H. & E. H. Stazyk (1981). Mental addition: A test of three verification models. Memory & Cognition. v9 pp 185-196

### Author of the summary: Jim Davies, 1999, jim@jimdavies.org

#### Cite this paper for:

• MIN Accounts for 80% of variance in 1st graders.(Groen and Parkman 1972)
• The MIN model cooresponds to a counting strategy.
• Sum-squared predicts well-- there is an exponential growth in RT
• Verifying a false problem is faster if the answer presented is way off.
• Verifying true ties is constant RT, but false shows effect of split size.
• split 13's were 200 ms faster than their true counterparts
• If facts are retrieved, they are not retrieved at the same speed.
• The difference between the addends (addend split) is not very informative.
Models of arithmetic:
• MIN model (Groen and Parkman 1972).
People count up from the larger number. Takes MIN time. Accounts for 80% of variance in 1st graders. 73% in adults (Parkman and Groen 1971) .
• SUM model (Parkman and Groen 1971).
71% of variance in adults. However, this implies a 20ms increment when counting, which seems unlikely. (Note that in the present study 72.9 was found rather than 20ms.) Ties (x + x) are at a constant value.
• SUM-squared (Ashcraft & Battaglia 1978)
Shown to be better than MIN. An exponentially increasing RT is difficult to reconcile with an increment model.
• Fast-access (Groen and Parkman 1972).
In adults, most facts are retrieved. Those not retrieved (about 5%) need to be counted at a 400ms/increment rate. The exponential increase in RT found with the SUM-squared seems problematic for this model.
• Four-stage retrieval/decision (Ashcraft & Battaglia 1978)
Facts are functionally represented as a table. RT is proportional to distance traveled during the search. Table is "stretched" for larger sums (post-hoc to account for exponential time fitting). Next a decision is made (comparing it to the stimulus in a verification task). This decision takes constant time for positives, but for negatives time proportional to the difference between correct and incorrect. In answer production tasks, the decision part would not take place.
• Revised fast-access
Accounts for the previous results by making the following modification: Retrieval failure prob is a function of size of min. With this change you can account for the exponential increase.
• Concatenation (Restle 1970)
When shown lengths, they are faster at "adding" a shorter to a longer than adding similar lengths. That is, an increase in addend split results in an increase in RT (However there were no statistical analyses performed). The theory is that it's easier to move the small piece to the end of the large.
The next experiment is to test the rival explanations: fast access vs. search through a network representation.

Experiment: 1
Number of Subjects: 20
Kind of Subjects: undergrads in intro psych
Method: After 20 practice trials, Ss saw 100 true addition problems, and 100 false ones, mixed up. Addends were digits. For the false ones, the split (the difference between the presented and correct answer) was either plus or minus 1, 5, 9 or 13. All answers, correct and incorrect, ranged from between 0 and 18. Thus the +- 13 could not be completely randomly assigned. Ss answered with buttons true or false for each trial.

• All findings were significant, p is less than .01
• Hypothesis: Split related to RT
• Result: 4.1% errors
RT declined as split increaced
• Hypothesis: True faster than false
• Result: Confirmed. 100 ms faster p is less than .01
• Hypothesis: Small faster than Large
• Result: Confirmed. 175 ms faster p is less than .01
• Hypothesis: Ties will be constant
• Result: Confirmed. They all appear to be retrieved.
Interestingly,the split effect was here too.
• Hypothesis: sum, sum squared, min will predict
• Result: confirmed.
• Hypothesis: none stated
• Result: The split results suggest a decision stage which is not predicted by the fast-access/retrieval model. You could add this post-hoc, but it still wouldn't be able to explain why split 13's were 200 ms faster than their true counterparts.
• Hypothesis: fast-access/retrieval predicts retrieval for the bottom 95% RT responses. There should be a nonincreasing RT for these across problem size
• Result: Not confirmed. Even for the ones that should have been retrieved, there was an effect of problem size.
• Hypothesis: none
• Result: Slope was found to be 72.9 ms for true, rather than the 20ms found in previous studies
Conclusions:
• Direct-access/retrieval is disconfirmed on two counts: 1) No decision stage, and 2) no account for different RT's for true fast trials.
• It can't be that you get the answer and then decide if the presented is true because for large problems, false problems with big splits are found faster than their true counterparts!

How can the 4 stage model be altered to account for these problematic findings? Well, imagine that in parallel with the retrieval there is a process that finds the sum through gradual refinement of an answer. With this model, large splits in the false problems will be caught sooner than small splits. (p191)

Experiment: 2
This experiment used some facts with double digits. The retrieval model might predict that adding 12 and 14 would be retrieving 1+1 and 2+4.
Number of Subjects: 20
Method: 46 problems were basic 100 facts.
54 contained at least one 2 digit addend.br> Answers ranged between 0 and 30.
For the false ones, the split was either plus or minus 1, 5, 9 or 13.

• Hypothesis:none
• Result:
error rate was 2.8%
All effects were like those in exp 1, and significant (p is less than .05).
• Hypothesis: Carry problems would be slower
• Result:confirmed.
• Hypothesis: Addend split will be an important factor
• Result: Not confirmed.
Conclusion: In general the revised 4 stage model was supported.

The model being proposed is a network one, so predictions can be borrowed from the network semantic literature:

• priming effects for similar addends
• decay of priming effects
• lateral inhibition (this has been supported in multiplication facts, see Stayzk 1980)

### Summary author's notes:

• There's an interesting finding here. Bigger split means faster verificaiton that a fact is false. How do you know faster that 2 numbers are very different?

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