Subitize and subitization - cardinality without counting


Subitizing is being able to identify the value of a group of objects by looking at it without actually counting the objects. Children can do this at a very young age and seem to be able to understand one, two, and three as a perceptual magnitude not cardinality. Being able to perceive two or three as a whole without doing mathematical thinking can be done by birds and some other animals. Therefore, young children can label small groups, as two, three, or four, accurately by subitizing, but no cardinality maybe recognized for the group. However, as children mature and gain experience it can also be a confident judgement of cardinality for a small number of objects.

Subitize, comes from the Latin subitus meaning sudden, which represents the sudden dawning of that's three or four. It was suggested in 1949 by E. L. Kaufman, M. W. Reese, T. W. Volkmann, and J. Volkmann.

It is possible for children to identify two fingers as two fingers. Or say they have five fingers. Or that you are holding up three fingers, without knowing the value of the the numbers. Particularly for five and above. They can recognize the value of a group through perceptual recognition of visual patterns of the objects' relative positions (subitizing) or through procedural counting without understanding the group of object's cardinality.



image of stars to subitize


People can learn to subitize and develop skill with practice. The benefit of drilling students with subitization exercises, rather than counting or basic fact operations, will increase the student's ability in number sense. Relationships of number quantity, cardinality, conservation of numbers, and proportion. Also being able to recognize and apply those abilities in different instances is a first step toward being able to use mathematics in a flexible and creative way. With practice students will be able to quickly subitize values of subgroups within a larger group and mentally join the subgroups to find the total value of the larger group. Unitized for subitized...


SubQuan and friends, SubQuan, from Latin meaning sudden quantity, is organized subitizing.

Examples show how objects can be placed into containers, subitizing by digits is possible, which allows the human eye to see very large quantities very quickly and to also see their shape and segments: squares, cubes, segments of cubes, square of cubes, and so on.

In addition, the container size can change, permitting individuals to see quantities in various bases. This leads to recognizing identical patterns between bases, otherwise known as metapatterns, which reveal polynomic expressions. So far we have found that children as young as 10 years old can "see" polynomials.

The visual nature of subQuanning transforms mathematics, removes the stumbling block students have been hidden behind, and leaves many rote memorization techniques in math education in question.

Incorporating two more steps entitled differences and polynomial derivation and any polynomial equation can be found from a set of data points.


Dr. Robert Sweetland's notes &