Children at this level continue to extend to larger numbers their conceptual subitizing of small groups to make a larger number, for example, I see one thumb and four fingers make my five fingers (this is part of the relation and operation core and is discussed more there). The 5-groups are particularly important and useful. These 5-groups provide a good way to understand the numbers 6, 7, 8, 9, 10 as 5 + 1, 5 + 2, 5 + 3, 5 + 4, 5 + 5 (see Figure 5-1). The convenient relationship to fingers (5 on one hand) provides a kinesthetic component as well as a visual aspect to this knowledge. Without focused experience with 5-groups, children’s notions of the numbers 6 through 10 tend to be hazy beyond a general sense that the numbers are getting larger. Knowing the 5-groups is helpful at the next level, as children add and subtract numbers 6 through 10; the patterns are problem-solving tools that can be drawn or used mentally. Children in East Asia learn and use these 5-group patterns throughout their early numerical learning (Duncan, Lee, and Fuson, 2000). Children can continue to experience and begin remembering other addends that make totals (e.g., 3 and 3 make 6, 8 is 4 and 4).

As noted, beyond the first ten words, which are arbitrary in most languages (e.g., see the extensive review in Menninger, 1958/1969), most languages begin to have patterns that make them easier to learn. English, however, has irregularities that are challenging for children. A major difficulty in understanding the meaning of the teens words is that English words do not explicitly say the ten that is in the teen number (teen does not mean ten even to many adults), so English-speaking children can benefit

FIGURE 5-1 Five groups to understand the numbers 6, 7, 8, 9, and 10.