Ball State University, Muncie, Indiana. Area and volume judgments are usually underestimates of true size. The possibility that these underestimations are more ...
Perception & Psychophysics 1983,34 (6), 593·598
The role of mental computations in judgments of area and volume DARRELL L. BUTLER and CARL OVERSHINER Ball State University, Muncie, Indiana
Area and volume judgments are usually underestimates of true size. The possibility that these underestimations are more dependent on cognitive factors than on sensory ones was investigated in two experiments. In Experiment 1, subjects judged the products of multiplication problems varying in number of multiplications and number of digits per number. Judgments were analyzed using power functions. The power function exponents for problems requiring one multiplication and two multiplications were very similar to previously obtained exponents for judgments of area and volume, respectively. In Experiment 2, subjects judged three-number multiplication problems and volume ratios of depicted, rectangular boxes. Power-functionexponents for these two kinds of tasks had similar means and standard deviations and were significantly correlated. These results support the hypothesis that underestimations of area and volumejudgments are due primarily to errors in mental multiplication. The simple power function R
nents dropped to about .8. Teghtsoonian does not provide any further discussion of the multiplication strategy, thus implying to the reader that subjects adopt a different strategy with complex shapes. A hypothesis that has not been examined is that although subjects still perform multiplications, their mental multiplications are error prone. Research on mental multiplication suggests that the exponent of the power function relating judgments to true products of multiplication problems is less than 1.0. In other words, judged products of fractions are too large and judged products of numbers larger than one are too small. Cohen and Hansel (1958) gave subjects simple probabilities and asked them to estimate compound probabilities. This task is a simple multiplication task (assuming subjects know how to "compute" compound probabilities). Judgments were much larger than the true compound probabilities. In fact, errors were so large that the researchers hypothesized that some subjects added the probabilities. Tversky and Kahneman (1974) asked subjects to multiply the integers from one to eight (i.e., 1 • 2 • ... • 8). The true product is 40320. The median judgment was 512 when the digits were presented in ascending order and 2250 when the digits were presented in descending order. Note that the product of fractions (Cohen & Hansel) is overestimated and the product of integers (Tversky & Kahneman) is underestimated.
= aSb
(where R = response, a = scaling factor, S = true stimulus size, and b = exponent) has been shown to be a reasonable descriptor of size judgments (e.g., Baird, 1970; Frayman & Dawson, 1981; Teghtsoonian, 1965). The power function exponents for area and volume judgments are typically less than the exponents for line length (e.g., Baird, 1970). The present study concerns the hypothesis that these differences are due, at least in part, to "mental computations" that accompany area and volume judgments, but not linear judgments. This approach assumes that area and volume are not directly available in the visual codes of stimuli, but must be "computed" or derived from the linear information that is available. The idea that areas and volumes cannot be directly perceived, but require some secondary judgment processes, is not new. Anastasi (1936) makes the same argument, but she provides no evidence. Teghtsoonian (1965) provided some evidence from postexperimental inquiries of subjects in an area-judgment experiment. Some of Teghtsoonian's subjects indicated that their judgments were made by estimating a linear dimension of the stimulus and then squaring that number. The subjects produced exponents near 1.0. When Teghtsoonian used more complex stimuli, the expoPart of this research was presented at the meeting of the Psychonomic Society, November 1982. The authors would like to thank J. C. Baird, Dave Hines, and Roger Humm for helpful comments on the preparation of this manuscript. This work was supported by a grant from the Ball State University Office of Research. Requests for reprints should be sent to Darrell Butler, Department of Psychological Sciences, Ball State University, Muncie, Indiana 47306.
EXPERIMENT 1: POWER FUNCTIONS OF MENTAL MULTIPLICATION One goal of this experiment was to test the hypothesis that "mental multiplications" have a power. function exponent of less than one. The second goal of this experiment was to determine if the exponent
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for mental multiplications depends on the number of multiplications. To compute the area of a square requires only one multiplication, but to compute the volume of a cube requires two multiplications. Thus, if area and volume exponents differ as a result of multiplication errors, then power-function exponents of multiplication problems should decrease as a function of the number of multiplications required in a problem. Method
Subjects. There were 2 male and 13 female volunteers from an introductory psychology course. None of the students were math majors and none had ever had a calculus course. StbnuH. Stimuli were multiplication problems differing in number of multiplications (I, 2, or 3), the number of digits in each number (2 or 3), and placement of the decimal point in each number. The first two of these variables were completely crossed, producing six kinds of problems. Four stimuli of each kind of problem were created. Each number in a stimulus was created by randomly selecting digits (between 1 and 9) and then randomly inserting a decimal point. A stimulus was eliminated and another created if another problem of the same kind had the same configuration of decimal points (e.g., a.b x c.d and e.f x g.h), This was done to assure a wide range of answers for each type of problem. The ranges of stimuli within conditions varied from 3.52 to 7.28 log units. The stimuli were created using black numbers (from a professional lettering machine) on white paper. Each problem was written horizontally, numbers separated by multiplication symbols. Then stimuli were converted to transparencies. Proeedure. All subjects were tested at the same time in a classroom. Using a transparency projector, each stimulus was projected on a I.8-m square screen at the front of the room. The time a stimulus was displayed varied with type of problem according to the following formula: T=(N + 1)· (D+ I)+(N-I), where T = display time, N = number of multiplications in problem, and D = number of digits per number. This formula was used in an attempt to give subjects about the same amount of time for each multiplication. The subjects were told that the purpose of the experiment was to study mental multiplication. They were asked not to do any of the computations on paper. Instead, they were instructed to estimate the product of each problem in their heads. However, the subjects were warned that the stimuli would not be presented for very long. The time would probably not be sufficient for them to carefully follow all of the rules of formal multiplication. The subjects were asked to adopt a strategy that would produce as accurate an answer as possible. The stimuli were presented in a random order. Intertrial interval was 5 sec. The subjects were to record answers during the ITI. The end of the IT! was announced by the researcher. The subjects were given three practice trials to familiarize them with the approximate time they would be allowed. The practice trials were not analyzed. Following the practice trials, the subjects viewedthe 24 stimuli.
Results One subject did not complete the task. This subject's data are not included below. For each subject, six power functions were computed, one for each kind of problem. The functions were computed using least squares linear regression with log 10 values. The results are summarized in Table 1. The mean correlations show that the power functions fit judgments well.
Table 1 Summary of Power Functions in Experiment 1 Stimuli
Meanr
Mean Exponent
Mean Log Scaling Factor
2 Digits 3 Digits Combined
1 Multiplication .98 .90 .98 .85 .98 .86
2 Digits 3 Digits Combined
2 Multiplication .71 .92 .95 .66 .92 .68
.72
2 Digits 3 Digits Combined
3 Multiplication .90 .60 .89 .57 .89 .59
.83 1.03 .92
.16 .21 .19 .65 .76
The exponents are significantly less than one for all six kinds of problem. More importantly, the exponents vary systematically with the number of multiplications required. The exponent for problems requiring two multiplications is significantly smaller than the exponent for problems requiring one multiplication [t(13) = 3.54, p < .01]. The exponent for problems requiring three multiplications is significantly smaller than the exponent for problems requiring two multiplications [t(13) =2.99, p < .01]. The decrease in exponents with number of multiplications does not appear to be due to increases in the range of the stimuli. The geometric mean judgments for each stimulus are shown in Figure 1. Consider problems with three-digit numbers. Although the exponent decreases with number of multiplications, the range of stimuli also decreases from 7.28 log 10 units for problems requiring two multiplications to 6.62 log 10 units for problems requiring three multiplications. This finding is inconsistent with the hypothesis that the decrease in exponents is due to increases in range. However, some of the difference in exponents may be due to differences in range. The influence is apparently very small. For each of the three kinds of multiplication problem, the range of stimuli was greater for three-digit numbers than for two-digit numbers (2.5 log 10 units, 2.3 log 10 units, and 1.3 log 10 units). While exponents for problems with three-digit numbers are slightly smaller than exponents for problems with two-digit numbers (mean = .04), the difference was not significant in any condition [for one multiplication, t(13) = 1.33; for two multiplications, t(13) = 1.04; for three multiplications, t(13) = .60]. Differences in exponents related to number of multiplications are not due simply to the rounding of numbers. Baird and Noma (1975) showed that subjects prefer to.use numbers with only one significant digit of accuracy. If subjects rounded numbers to one significant digit before multiplying, the judged prod-
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Figure 1. Mean log jadplents for stimall in Experiment 1.
