How subitizing actually works remains something of a mystery.

One interesting clue, however, is that contrary to what we once thought, it is not independent of our attention. Subjectively, subitizing seems to be automatic: one glance at a set seems enough to effortlessly recognize that it contains 1, 2, or 3 objects. This is an illusion, however. Sets that are presented when our mind is temporarily occupied elsewhere are no longer accurately perceived, even when they comprise only 2 or 3 items. Far from being "pre-attentive" and effortless, subitizing requires attention. We can select a small number of items, and even track them through time and space, but this taxes our attention.

So how does subitizing work ? Current research suggests that we have 3 or 4 memory slots where we can temporarily stock a pointer to virtually any mental representation. In other words we can hold about 3 or 4 pieces of information (or "pointers" to information) in our conscious mind at a time. This memory store is called "working memory" -- a transient supply that keeps the objects of thought on-line for a brief moment. We use it, for instance, to remember which shapes appear on a flash card: three or four objects can be neatly stored in this mental filing cabinet, each with all of its perceptual properties. When we keep information in this way, we also get their number for free, because the system implicitly encodes the number of slots that are occupied at a given moment. To understand this, imagine that you have three shoe boxes, a green, a red and a blue one, that you use in a set order when packing your running shoes before going on trips. Because the boxes are used in a fixed order, a glance at their colors allows you to determine the number of pairs you have taken. If only the green box is used, it means that you took only one pair, green + red means two, and green + red + blue means three. Such a filing system is a good metaphor for how subitizing might work: when we attend to objects, our perceptual system immediately places their properties in the available slots of an object-tracking device. To subitize, all we have to do is link the contents of this mental file to the names of numbers one, two or three.

What is unique about the subitizing code is that it provides a discrete cipher for each of the small numbers 1, 2, and 3. Each addition of a new object opens a new memory slot -- an additional notch in the mind that clearly indicates the move to a new number. This coding principle is radically different from the way that numbers are encoded on the approximate mental number line. The approximate number system represents larger numbers through distributions of neural activation. These distributions have significant overlap, particularly for adjacent numbers. For example, the mental representations of seven and eight share considerable overlap in their activation patterns, making them more difficult to distinguish quickly and accurately. However, numbers that are further apart, like two and eight, have less overlap, making them easier to differentiate. There is nothing in the approximate number system to support a system of exact arithmetic with discrete numbers.

With the object file system, however, we can track each object precisely (as long as their number does not exceed three). The key difference lies in the precision and discreteness of these systems. The subitizing system, with its discrete slots, supports exact number recognition and precise arithmetic operations for small quantities. Each number has a unique "code" or "cipher" that is distinctly different from all others. In contrast, the approximate number system represents quantities as overlapping patterns of activation, making exact arithmetic operations impossible and leading to estimation rather than precise recognition.

At this point, we can state that, young children possess two independent numerical processing systems initially: 1. The subitizing system that allows precise tracking of small quantities (1-3 objects). 2. The approximate number sense that provides an intuitive understanding that all sets have a cardinal number, even if that number isn't precisely known.

Around ages 3-4, a remarkable cognitive development occurs when these two systems merge. This integration represents a fundamental shift in mathematical thinking. children begin to understand that the precise, discrete nature they experience with small numbers through subitizing extends to all quantities, no matter how large.

This integration leads to a transformative realization: every number is a distinct, precise concept. Children grasp that 13, for instance, is not merely an approximation between 12 and 14, but rather a specific, unique quantity. This understanding forms the foundation of the natural number concept - the idea that numbers are discrete, sequential entities that can be counted and manipulated with exact precision.

This cognitive breakthrough is uniquely human and marks the beginning of higher mathematical thinking. It enables children to: - understand exact counting for any quantity - grasp the concept of succession (that each number has a precise next number) - begin developing more sophisticated arithmetic operations

This mental revolution, unique to homo sapiens, is the first step on the way to higher mathematics.