1. The ability to verbally count using number words. This ability is initially developed as a sequencing of words (one, two, three …) without an understanding of the specific meaning attached to the words. Quantity is still understood nonnumerically as more or less, big or small.

  2. The ability to count with one-to-one correspondence. When this ability develops, children are able to point at objects as they count, mapping the counting words onto the objects so that each is tagged once and only once. This ability is initially developed as a sensorimotor activity, with an understanding of quantity still absent. Children who can successfully count four objects and five objects cannot answer the question, “Which is more, four or five?”

  3. The ability to recognize quantity as set size. With development of this ability, children do understand that “three” refers to a set with three members. Initially this understanding is concrete, and children will often use their fingers as indicators of set membership.

  4. The ability to “mentally simulate” the sensorimotor counting. When this ability is in place, children can carry out counting tasks as though they were operating with a mental number line. They understand that movement from one set size to the next involves the addition or subtraction of one unit.

While children with middle and higher socioeconomic status generally come to school with the central conceptual structure in place, many children from disadvantaged backgrounds do not. When first grade math instruction assumes that knowledge, these children are less likely to succeed.

Sharon Griffin and Robbie Case designed a curriculum called Number Worlds that deliberately puts the central conceptual structure for whole number in place in kindergarten (Griffin and Case, 1997). Additional activities extend the knowledge base through second grade. Developed and tested with classroom teachers and children, the program consists primarily of 78 games that provide children with ample opportunity for hands-on, inquiry-based learning. Number is represented in a variety of forms—on dice, with chips, as spaces on a board, as written numerals. An important component of the program is the Number Knowledge Test, which allows teachers to quickly assess each individual student’s current level of understanding, and to choose individual or class activities that will solidify fragile knowledge and take students the next step.

The Number Worlds program has been tested with disadvantaged populations in numerous controlled trials in both the United States and Canada with positive results. One longitudinal study charted the progress of three groups of children attending school in an urban community in Massachusetts for three years: from the beginning of kindergarten to the end of second grade. Children in both the Number Worlds treatment group and in the control group were from schools in low-income, high-risk communities where about 79 percent of children were eligible for free or reduced-price lunch. A third normative group was drawn from a magnet school in the urban center that had attracted a large number of majority students. The student body was predominantly middle income, with 37 percent eligible for free or reduced-price lunch.

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