II. INFLUENCE OF FACTORS IN THE SAME SENSE-FIELD AS THE OBJECTS WHOSE RELATIVE NUMEROUSNESS IS IN QUESTION.
The experiments described in the following pages are concerned with the influence exerted on the judgment of a given factor by other factors presented at the same time. The object of judgment in these studies is visual number, which is to be submitted under varying conditions of the objects whose number is in question, for example, varying conditions of form, size, distribution, with the intent to discover whether this judgment is a function of these other factors as well as of the numerical. The scope of the enquiry includes both relative and absolute number.
The objects chosen as a basis for the number-judgment were bits of paper pasted in two well-defined groups side by side upon a background of black cardboard. This card fitted into an upright frame where it was held in place by a pivoted spring, which allowed easy adjustment and removal of the card. The opening of the frame, 15×20 cm. was concealed at will from the observer by a black wooden screen that played up and down on guiding posts, when released by a cord and lever from the catch that held it in place before the card. It fell by gravity upon a cushion that deadened the sound; and it was restored to its position by the operator's thrusting his fingers beneath and lifting it till the catch above caught and held. The entire apparatus, as well as the operator's movements, was concealed from the observer by a large black cardboard screen resting upon a black-covered table. The one opening in this screen was just large enough to allow a full view of the card when the inner wooden screen fell from sight.
This apparatus which we will call the Two-Group Apparatus, admitted of simultaneous exposure of the two groups of objects, and that only. At first, to make successive exposure possible, a light wooden frame was constructed in whose grooves two leaves of black cardboard ran like sliding doors. By means of rods fastened to their outer edges these leaves could be pulled apart or thrust together till their inner edges met. When this apparatus was placed between the outer screen and the frame bearing the card, and the inner wooden screen had been dropped out of the way, this substitute divided screen was sufficient roughly to accomplish the end in view.
With this apparatus the illumination was daylight, coming through a very large window at the back of the observers. By means of a curtain, marked variations in the light could be prevented.
For the length of simultaneous exposure of the groups the following rule was adopted: Each observer was to be allowed time enough to get a satisfactory feeling of relative number, but not time enough to admit of counting. This time was kept constant during the work of any one sitting. As the weeks went on, it was found possible, under the rule laid down above, to shorten the time for some of the observers, and to use with all the same length of exposure that had sufficed for the speediest. The range of variation was from 1.2 sec. to 1.6 sec. Time was measured by the ticks of a watch. Later tests showed for the time studied that, where effective at all, the longer exposure diminished a given tendency. Often it had no apparent effect.
The method of control already described is not only rather rough but does not exclude the possibility of a space error. This possibility proved actual by experiment. So an apparatus was contrived that should present the groups in succession at approximately the same place and should shorten the exposure, if desirable, to a small fraction of a second.
This new apparatus, which we will call the One-Group Apparatus, required artificial light and a dark room. By means of a 125 cp. incandescent electric lamp, images of the groups of objects were reflected through the lens of a camera and came to a focus upon its ground-glass screen. A second screen of ground glass was placed in front of the first and as close to it as possible, that an even distribution of light might be obtained. The cards containing the objects were of the same general character as in the earlier experiments. They were held in a moveable slide whereby each group in succession could be brought before the lens. When the slide was drawn to the limit in one direction a single circle appeared in a black field. This circle was used as a signal and a means for directing the eye in the dark to that region where the groups were to appear. The exposures were made with a camera bulb, the shutter being set for instantaneous movement, with diaphragm 22 and length of exposure 126 sec. A shorter time was thought on trial to make perception too difficult. The apparatus rested upon a table of special construction and was enclosed as far as the glass screen with a wooden frame covered with denim. Double curtains of this material formed this enclosure on one side and made possible an easy adjustment of the cards between exposures, as well as the admission of the operator's hand during a given experiment for the adjustment of the shutter. This had to be set, of course, before each of the three exposures constituting one experiment. During its progress the hand was not removed at all, the curtains falling about the arm in such a way that little light escaped. The other hand managed the moveable slide from behind the enclosure.
Time was measured by watch-ticks. The three exposures—dot-signal, Group 1, Group 2—were separated from each other by intervals of 1.6 sec. This was fixed upon as the minimum for convenient operation of the apparatus.
In much of the experimentation on relative number two observers were employed at once. Their chairs were placed closely side by side on a line about 150 cm. from the plane in which the groups appeared. These groups were not very far from being on a level with the eye. Each observer recorded his own judgment, against the number of that experiment. There were three possible kinds of judgments,—equality or either group larger. If the judgment was of difference it was recorded in terms of the larger.
When the dark room was used, special arrangements were required, for convenience of the observers in making their record. After several schemes were tested the following was adopted as least trying to their eyes: A large, black-topped table was placed before them, bearing an electric lamp enclosed in a black box with a small aperture that could be closed at pleasure; or, if left open, did not let enough light escape to disturb the perception of the groups.
The absolute number of objects in the groups was determined, first, by the character of the problem, and then by convenience. If we are to learn anything about the influence exerted upon the number-judgment by other factors than the numerical, we must eliminate all influence of the latter. Correct judgments may be determined by this factor alone; erroneous judgments must have been otherwise conditioned; and these conditions it is the task of our method to isolate and study, as modifying factors. From correct judgments we learn nothing definite about our problem, but from erroneous everything. Other things being equal, it is preferable to eliminate from the results the influence of this numerical factor, just as one handles any other disturbing, unavoidable element, by equalizing the numbers in the two groups.
What may be called the standard number of objects in each is twenty. This choice was governed by the purpose of using a number large enough to make counting impossible in a brief time and yet not so large as unnecessarily to increase the labor of preparation and the difficulty, for the observer, of getting an idea of the groups as a whole. To the cards containing equal groups, 20 to 20, were added others, 20 to 19, 19 to 20, for the purpose of easy variation in arrangement, by omitting one object from a group, without making the actual numerical difference easily perceivable. In later work these small objective differences were dropped. Yet other cards, 23 to 17, 17 to 23, were added, to the end that the observers might find unmistakeable number-differences, and so not be bothered by the suspicion that the groups were all equal. The reversal of the number-relations, as indicated above, was in the interest of equalizing the influence of the actual numerical factor in the two groups.
The following proportion was kept among the numbers of observations made upon each kind of card: 12 upon groups objectively equal; 512 upon those differing by one from each other, where half each went to (20 to 19) and (19 to 20); 112 to those showing the maximum objective difference of six, where again half went to (17 to 23) and half to (23 to 17). Of course the observations upon cards of this last sort are excluded from the tables.
As to the number of cards employed for each series of experiments, it was found at first convenient to use seven,—3 (20 to 20), 1 (20 to 19), 1 (19 to 20), 1 (17 to 23), 1 (23 to 17). In each group the arrangement of objects was irregular. The use of three of the first sort was to encourage freshness of judgment, each having its particular irregularity. Cards were but rarely remembered, practically never except in the case of groups differing widely in number. So far as the observers could tell, judgment was formed afresh in all these cases. In later experiments eight cards were used. This number was in the interest of avoiding the distribution-error. At first it was thought sufficient that all the groups should be merely irregular. Later it became evident that discrimination was very fine here and so that this factor must be eliminated by the usual precise method.
The space- and time-errors, where likely to be present, were eliminated in the usual way by performing an equal number of experiments with the groups in reversed arrangement. Several methods of doing this were at first tried; but these were all abandoned in favor of the following: The experiments were arranged in sets of 24, in each of which the proportion of kinds of cards was kept as indicated above. Each set with one space- or time-order of the groups was repeated with that arrangement reversed.
A word must be added as to the arrangement of results in the tables. Judgments of equality upon objectively unequal groups are entered as overestimations of the smaller groups. The per cent of correct judgments is equally divided between the two other classes, and for this reason that interest centres, not in correctness at all, but in the difference between the tendency of error in one direction and that in the other direction. No doubtful judgments were admitted, but in such cases another trial was allowed later, usually when the observer was not aware that he was being given a new chance. The subjects are divided into three classes according as the results show a tendency to favor one or the other group or no tendency either way. A difference of 10% is arbitrarily taken as significant.
1. The Influence of Group-Area. The Two-Group Apparatus was employed. The four sets of experiments carried out with this factor differed primarily in the material upon which the observer's judgment was based, and secondarily in certain matters of method. The attempt in them all was to approximate more completely to the isolation of the factor under investigation. They are numbered in the order of approximation. As marked results were obtained from each, they have all been offered for consideration in the four parts of Table I. A description of the material used in each case follows.
A. Squares (1 cm.) Neutral Gray no. 1. (Bradley), arranged irregularly in two groups with irregular outlines on a background of black cardboard. One group was large in area, the other small, the attempt being made to fill each space homogeneously. Groups were not proportional in shape of area.
B. As above, save that circles (11 mm. approx. in diameter) were substituted for squares, in the interest of distinctness for the several objects.
C. The area of the groups was oblong and regular, and the sides were proportional. (Compact 72.5 mm.: 58 mm.; scattered 110 mm.: 88 mm. These relations were determined by the size of the frame that had already been used and by the desire to make the difference in area as marked as other necessary conditions would admit.) Each area was marked by a circle in each corner. The color of the compact group was the deepest shade of normal gray (Prang Normal Gray Darker); of scattered group the next higher shade (Normal Gray Dark). These dark grays were used in order to reduce to a minimum the tendency to produce after-images. The difference in the shades of the two groups was in the interest of avoiding the greater brightness due to the mass-effect of the compact group.
D. As in C, except that India ink outline circles ( 13 to 12 mm. line) were used on a background of granite cardboard. This change was made to avoid, as far as possible, the greater mass-stimulation due to the reënforcing effect of the compact arrangement. The size of circles remained as before.