rather p r i v i l e g e d v i e w o f the s o u r c e s o f i n f o r m a t i o n p r o v i d e d b y these two c o m p o - nents o f the l e x i c a l p r o c e s s i n g s y s t e m .
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Some Quick Arithmetic
Vesna Mildner
UniversiO' of Zagreb, Croatia
Mathematical abilities for the four simple arithmetic operations were studied on a sample of 53 female right-handers. True and false statements were presented auditorily and a manual response was required as to the trueness of the statements. Response times, accuracy, and laterality index showed no significant ear advantage, but the responses to true statements were significantly faster than to the false ones. The shortest response times were found for addition problems, and the longest for subtractions. The correlation between the size of the difference between the two operands and response time was not conclusive but the trend was unexpectedly positive. © 2001 Academic Press
Introduction
206 TENNET XI
With respect to differential vulnerability of numerical abilities to deficits of various etiologies, including degenerative deterioration, there have been reports of patients who could do additions and multiplications, but not subtractions (Dehaene & Cohen, 1998), those with initial impairment limited to multiplication and division (Grafman et al., 1989) or to multiplication and complex subtraction (Girelli et al., 1999).
Furthermore, it has been argued (Geary, 1996) that addition and subtraction are biologically primary mathematical abilities and multiplication and division secondary ones.
The aims of this work were (1) to check for possible ear advantage as an indicator of functional cerebral asymmetry in solving simple arithmetic problems and (2) to determine which arithmetic operations were the most difficult or the easiest.
Me~od
The subjects were 53 healthy right-handed and right-footed female university stu- dents, between 18 and 25 years of age (mean 20), without familial sinistrality, with normal and symmetrical hearing, and with no history of neurological disorders, native speakers of Croatian.
Recording and reproduction were done on a Sony MiniDisc recorder. Test material was prepared on a 486 PC (Sound Blaster 16, Creative Wave Studio 2.01). A n a l o g - digital conversion was at the l l,025-Hz (8-bit) sampling frequency. The apparatus and software for computer-aided RT recording and measurement were designed at the University of Zagreb. Rona Kern Type G stereophonic headphones were used.
Testing was performed individually, in a quiet but not specially sound-treated room (ambient noise less than 40 dB).
The test material consisted of 32 arithmetic problems in the form of statements, using the four basic operations: addition, multiplication, subtraction, and division on integers from 1 to 10. The statements were pronounced at the rate of approximately three syllables per second by a male speaker and presented at 75 dB. Half of the solutions were correct (true statements) and the other half were incorrect (false state- ments) (e.g., 1 + 3 = 4 or 6 + 1 = 8). Half of the statements were presented to the right ear, and the other half to the left, with the opposite ear masked by a murmur -+ 5 dB representation level. This yielded four stimuli categories of eight statements (two for each operation): true statements presented to the right ear, false statements presented to the right ear, true statements presented to the left ear, and false state- ments presented to the left ear. The order and side of stimuli presentation were quasi- random. Three-hundred milliseconds before each statement a 1000-Hz pure tone was presented simultaneously to both ears for 250 ms in order to bring the subject's atten- tion back to the middle. The language of testing was Croatian.
Oral instructions and a training session were given before the test. The task was to answer whether the statements were true or false. The responses were given manu- ally, by pressing the response plate positioned on the table in the midline in front of the subject, with both hands--thumbs for a positive answer (true statement) and index fingers for a negative answer (false statement). The subjects were encouraged to respond as fast as possible.
Response times (to the nearest ms) and accuracy were measured, recorded, and calculated by a personal computer. Laterality index (LI) was calculated from the accuracy measures: LI = (Rc - Lc/Re + Le) * 100. R and L stand for right and left ear, respectively; c is correct responses, and e is incorrect responses.
RT (ms)
FIG. 1. 600
500
400
300
200
I00
0 - -
Left Right
EAR
[] true [] false 1
Response times for true and false statements for both ears.
Results and Discussion
Overall accuracy was very high: 99.32%. Mean response time for the entire sample was 295 ms. LI for the whole test was positive (5.66), indicating right-ear advantage (REA), i.e., left-hemisphere dominance. However, as many as 71.70% subjects, showed no laterality (LI = 0), only 17% exhibited REA, and 11.30% actually exhib- ited left-ear advantage (LEA), indicative of right-hemisphere dominance.
Two-way analysis of variance has shown that the stimulated ear had no effect on the accuracy or response time (p > .05) (Fig. 1). Mean RTs for left and right ear were identical: 295 ms. Accuracy of responses to the stimuli presented to the left ear (98.91%) was insignificantly lower than to the stimuli presented to the right ear (99.73%).
The responses to true statements were significantly faster than those to the false ones (p = .00), without significant interactions between the two independent vari- ables (Fig. 1). Faster responses to true statements have been found in earlier studies as well (e.g., Lemaire, Abdi, & Fayol, 1996) (but not in De Rammelaere, Stuyven, & Vandierendonck, 1999) and the present results may be, in a broader sense, interpreted along the lines of the findings that yes responses are in general faster than the no responses.
The responses were further analyzed by type of operation. All results broken down by type of operation (addition, subtraction, multiplication, division) and trueness of statement (true, false) are summarized in Table 1.
Analysis of variance for response accuracy showed only main effect of type of operation (p = .03) and no significant interactions (p > .05). As can be seen from Table 1, the differences were very small, with subtraction problems yielding the least accurate responses.
TABLE 1
Summary of Results for Response Times in ms (RT) and Accuracy (%)
Addition Subtraction Multiplication Division
Statement RT % RT % RT % RT %
True 159 100 272 96.74 170 100 190 100
[image:3.540.148.397.74.229.2]208 TENNET XI
Analysis of variance for response times showed main effects of trueness-of-state- ment and type-of-operation variables (p = .00), but no main effect of stimulated ear or any significant interactions (p > .05).
If biologically primary mathematical abilities (addition and subtraction) are as- sumed to be automatic as opposed to the secondary ones (multiplication, division) that are learned later in life, it would be reasonable to expect a REA. On the other hand, simple addition and multiplication of single-digit numbers are considered to be performed by rote (Dehaene & Cohen, 1998), such performance being also charac- teristic of the left hemisphere. However, the absence of any significant ear advantage indicates lack of functional cerebral asymmetry, regardless of the operation type.
It was also reasonable to expect the biologically primary mathematical abilities to have shorter RTs than the secondary ones. Furthermore, based on Grafman et al.'s (1989) report on a patient whose dementia, with respect to numerical abilities, was manifested initially as impairment in division and multiplication, it could be expected that the 'robustness' of addition and subtraction would be manifested as yielding better results than multiplication and division. The RTs for multiplication and divi- sion problems are indeed very similar, but it is not clear why subtraction should be so much more difficult as indicated both by RTs and accuracy. Such results were found within both the subset of true statements and the subset of false statements-- the addition in both cases being on the average the fastest and the most accurate and subtraction the slowest and the least accurate. This warrants further study with a greater number of stimuli belonging to each type of operation, including better control of the other factors that may influence responses, such as the odd-even effect or the associative-confusion/interference effect (De Rammelaere, Stuyven, & Vandieren- donck, 1999).
The combination of significant main effects showed that the shortest RTs were to true statements involving addition problems (159 ms) and the longest RTs were to false statements involving subtraction problems (450 ms).
In the subset of the fastest responses there was no effect of the size of the difference between the two operands in the problem (p > .05). The analyzed differences were 2, 5, and 8. This result is contrary to that of De Rammelaere, Stuyven, & Vandieren- donck (1999), who found a significant effect of difference size on response times.
In the subset of the slowest responses the effect of the size of the difference was significant (p = .00). The analyzed differences were 2, 4, 5 and 9. However, Scheffe test showed that the differences of 4, 5, and 9 belonged to a homogeneous group, with only the difference of 2 eliciting significantly faster responses. This is an interest- ing result in itself, because it is contrary to the literature data. Namely, previous studies have shown that the smaller the difference, the slower the response (Posner & Raichle, 1997; De Rammelaere, Stuyven, & Vandierendonck, 1999), whereas in this study the data for the 'slow' subgroup suggest a positive correlation of the between- operand difference and RT, i.e., the greater the difference, the slower the response (with inconclusive data for the 'fast' subgroup).
Similarly to the results of De Rammelaere, Stuyven, and Vandierendonck (1999), there was no tradeoff between accuracy and response time. In fact, there was a trend for the fastest subjects to be the more accurate ones.
Conclusions
ments were significantly faster than to the false ones. The shortest response times were found for addition problems and the longest for subtractions. The correlation between the size of the difference between the two operands and response time was not conclusive but the trend was unexpectedly positive.
ACKNOWLEDGMENT
This research was supported by Grant 130721 of the Croatian Ministry for science and technology.
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Geary, D. C. (1996). Sexual selection and sex differences in mathematical abilities. Behavioral and Brain Sciences, 19, 229-284.
Girelli, L., Luzzatti, C., Annoni, G., & Vecchi, T. (1999). Progressive decline of numerical skills in Alzheimer-type dementia: A case study. Brain and Cognition, 40, 132-136.
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H a n d e d n e s s a n d I m m u n e Function
N. S. Morfit and N. Y. Weekes Department of Psychology, Pomona College
