. Perceiving at a glance how many objects are presented without estimating or counting.It takes considerable time for children to learn to count. Ordinarily they achieve this ability two or three years after they first begin to speak. It is therefore considered a rather complex and advanced mental process. Studies of counting, however, have shown that there are two other methods of determining the number of objects. We can estimate or guess the number without counting, and we can “subitize.” In subitizing, we grasp the number of objects without going through any process of counting or estimating.Experiments have turned up the interesting fact that certain animals can subitize as well as human beings. Birds and squirrels can be trained to distinguish six or seven irregularly shaped spots when they are presented for a fraction of a second, and human beings cannot do any better. In one experiment (Koehler, 1943), the birds were even able to solve a problem that involved matching groups of spots. They were first trained to raise the lid of a box to get a reward of food. After that they were shown a certain number of spots, say five, and were then presented with several boxes which had different numbers of spots printed on their lids. The birds actually picked out the five spot box, apparently by matching the two groups of spots. Moreover, they could do the same with other combinations, but again the total number of spots could not exceed seven.Birds can also learn to stop eating seeds when they have reduced the supply to a certain number, such as three or four. To train them, the experimenter repeatedly shoos them away from a small pile when they have eaten all but that particular number. It is even possible to train them to leave the same number of grains shown to them on a piece of cardboard. (Hassmann, 1952)These experiments might suggest that the talking and counting horses in vaudeville acts may be “subitizing” even if they cannot really count. However, investigators have shown that they do neither. They are actually responding to subtle cues given by their trainer, even though he may not realize he is giving them—an example of a “self-fulfilling prophecy” (Rosenthal, 1968).Research on subitizing has a much more important bearing. It has revealed that there is a stage below that of counting, and it has helped science to draw a sharper line between animal and human learning. Even though humans can subitize no better than animals, they can develop their mathematical ability to a far higher degree by using the more advanced processes of estimating and counting—not to mention all the other procedures involved in mathematics.