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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, firstname.lastname@example.org
Cite this paper for:
Models of arithmetic:
- 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
The next experiment is to test the rival explanations: fast access vs.
search through a network representation.
- MIN model (Groen and Parkman 1972).
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).
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
- 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
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.
Number of Subjects: 20
Kind of Subjects: undergrads in intro psych
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
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
- 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
- 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)
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
Kind of Subjects: fundergrads
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.
In general the revised 4 stage model was supported.
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
- Hypothesis: Addend split will be an important factor
- Result: Not confirmed.
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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