What are the roles of cardinal and ordinal processing in the development of arithmetic? In the present dissertation, cardinal knowledge is defined as the ability to determine the quantitative information of number symbols whereas ordinal knowledge is defined as the ability to determine the relative relationship among number symbols. This dissertation includes two studies examining the development of the relations among cardinal, ordinal and arithmetic skills, both concurrently and predictively, for children in the early grades of elementary school. In both studies, children completed a number comparison task (e.g., which number is bigger, 4 or 5?) as an index of their cardinal knowledge. They also completed two novel order tasks: (a) missing number (e.g., which number is missing, 1 _ 3 4?), and (b) number ordering (i.e., order the three digits from the smallest to the largest, e.g., 4 5 3 or 2 7 9). Furthermore, children completed two measures of inhibitory control. Last, children's arithmetic skill (e.g., solving problems such as 4 + 5 or 7 + 6) was measured.
In Study 1, I evaluated the internal consistency and validity of the novel order measures for children entering grades 1 to 3 (n = 70). In Study 2, multi-group path analysis showed that for children in grade 1 (n = 66), number ordering was strongly predicted by number comparison, but not by the missing number task or inhibitory control. Moreover, performance on the number comparison and missing number task independently predicted addition. Further, performance on the number comparison task uniquely predicted the growth of addition. In contrast, for children in grade 2 (n = 80), variance in the number ordering task was shared among the number comparison, missing number, and inhibitory control tasks. Number ordering uniquely predicted addition concurrently and it also predicted the growth of addition.
I interpret the different patterns of results from grades 1 to 2 as reflecting different ongoing processes of integration of symbolic numerical associations. These findings suggest that development of number competence involves the integration of cardinal, ordinal, and arithmetic associations in an extensive network of relations among numbers.