Knowledge of Number Integration

Abstract

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 quantity 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 iii 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. iv ACKNOWLEDGEMENTS First, I would like to thank the one and only Dr. Jo-Anne LeFevre for being my role model. I have been extremely lucky to have her as my mentor for eight years. She has encouraged me gently to come outside my comfort zone to achieve more than I could have imagined. I will be forever thankful to her from the core of my heart.

Children’s Knowledge of Symbolic Number in Grades 1 and 2: Integration of Associations

“…Only the measures that were relevant to the current research goals are presented here. Other measures that were collected but not analyzed included quantity‐digit mapping, rapid automatic naming, spatial span, mental rotation, problem solving, and number line estimation (see Xu, 2018 for more details about these tasks). Children’s performance on the number line task was analyzed in detail in Xu (2019).…”

“…This article is based on the dissertation completed by Xu (2018). Thanks to the members of Xu’s thesis committee for their comments: Dr. Daniel Ansari, Dr. Kevin Cheung, Dr. Deepthi Kamawar, and Dr. Monique Sénéchal.…”

Evaluating the Psychometric Properties of the Early Literacy and Numeracy Observation Tool (ELNOT)

“…Gradually, children link number words to other representations such as quantities and digits. Verbal counting thus forms the foundation of numerical abilities such as the cardinality principle, that is, mapping number words to quantities (Jiménez Lira et al, 2017) and, more generally, counting aids children's understanding of the link between quantitative information of numbers and their symbolic representations (Xu, 2018).…”