Numbers: Their Tales, Types, and Treasures.

Chapter 2: Numbers and Psychology



The ability to simultaneously track objects through space and time is certainly essential for survival. When you try to cross a street, you will probably have to observe the motion of several cars at the same time. Animals would need the same mental skill when tracking the motion of several predators. Obviously, in a potentially hostile environment, individuals would have diminished chances for survival if they lacked this particular ability. Evolution would thus have selected individuals with this skill over those without it. Therefore, neuroscientists regard the “object-tracking system” as a basic functionality of the brain, an inherited cerebral mechanism. It is a mental device that keeps perceptual information of up to three or four individual objects in the working memory. Our consciousness seems to know automatically the number of objects stored in the object-tracking system. Thus, it seems that this system is responsible for the following important manifestation of our inborn “number sense”:

Subitizing (from the Latin word subitus, meaning “sudden”) is the mental ability to enumerate small sets rapidly without counting. When we see a small group of not more than three or four objects, we often know their number immediately. This perception of number appears to be automatic, effortless, and exact, and it occurs without conscious counting. For example, look at the fields in the first row of figure 2.1. Even if there is no regularity whatsoever in the placement of stars, we see immediately if a field contains two or three stars (some people also have no difficulty with four objects). This feeling of instantaneous recognition is absent when we look at the fields in the second row, where determining the number of stars is much more difficult and requires actual counting.


Figure 2.1: Subitizing—Which numbers do you recognize without counting?

Like the object-tracking system, subitizing is limited to three or four objects. Within that range, perception of numbers “at a glance” is not only fast but also highly accurate, and errors seldom occur. It is quite different from the counting described in chapter 1, and it does not require that we direct our attention from one object to another. Eye-tracking experiments have indeed shown that during subitizing, one doesn't look at the objects individually. Instead, a single glance at the whole group is sufficient to know the number. When the number of objects reaches four or five, eyes start moving around in order to scan the collection, either for counting the objects or to search for familiar arrangement patterns.

With training, one can learn to subitize sets with a slightly higher number of objects, like six or seven, but this remains different from the subitizing of small sets. The effect can be measured exactly through the reaction time of test persons. Consider a person who is asked to determine as fast as possible the number of dots presented on a computer screen. Within the subitizing range, it takes only a reaction time of about one-half of a second until the test person starts giving the answer. This reaction time increases only a little from one to three dots, but after that it starts to increase significantly by roughly a quarter of a second for each additional dot on the screen. At the same time, the error rate increases with the increasing number of dots. This indicates that beyond three or four, one obviously relies on other mechanisms for determining the number, such as finding familiar patterns or explicit counting.


Figure 2.2: Pattern recognition may help to enumerate objects.

One might think that subitizing also has to do with pattern recognition, because two objects are always arranged in a line and three objects either form a line or a triangular shape, which is easily recognizable. You can see in figure 2.2 that pattern recognition indeed facilitates the task of determining the number. Here the stars are arranged either in familiar patterns or in quickly recognizable subgroups, so that it becomes much easier to tell the number without counting. Because there are so many possibilities for different spatial arrangements with higher numbers of objects, the probability to encounter familiar patterns decreases. Subitizing, however, does not need static patterns; it also works if the objects are moving and change places, but it seems not to work as well for sequentially presented stimuli (like drum beats). All this indicates its connection with the object-tracking system, which has the purpose to track simultaneously perceived objects through space and time.