Subitizing, a quick apprehension of the numerosity of a small set of items, is inconsistently utilized by preschool educators to support early number understandings (Sarama & Clements, 2009). The purpose of this qualitative study is to investigate the relationship between children’s number understanding and subitizing activity. Sarama and Clements (2009) consider students’ subitizing activity as shifting from reliance upon perceptual processes to conceptual processes. Hypothesized mental actions carried into subitizing activity by children have not yet been empirically investigated (Sarama & Clements, 2009). Drawing upon Piaget’s (1968/1970) three mother structures of mathematical thinking, the theoretical implications of this study consider expanding the scope of Piaget’s (1968/1970) definition of topological thinking structures to include patterned orientations. Increasing the scope of this definition would allow for the investigation of the development of topological thinking structures and subitizing activity. An 11-week teaching experiment was conducted with six preschool aged children in order to analyze student engagement with subitizing tasks (Steffe & Ulrich, in press). To infer what perceptual and conceptual processes students relied upon when subitizing, tasks were designed to either assess or provoke cognitive changes. Analysis of interactions between students and the teacher-researcher informed this teacher-researcher of cognitive changes relative to each student’s thinking structure. Results indicated that students rely upon the space between items, symmetrical aspects of items, and color of items when perceptually subitizing. Seven different types of subitizing activity were documented and used to more explicitly describe student reliance upon perceptual or conceptual processes. Conceptual subitizing activity was redefined in this study, as depending upon mental reversibility and sophisticated number schemes. Students capable of conceptual subitizing were also able to conserve number. Students capable of conserving number were not always capable of conceptual subitizing. The symmetrical aspects of an item’s arrangement elicited students’ attention towards subgroups and transitioning students’ perceptual subitizing to conceptual subitizing. Combinations of counting and subitizing activity explained students’ reliance upon serial and classification thinking structures when transitioning from perceptual subitizing to conceptual subitizing. Implications of this study suggest effectively designed subitizing activity can both assess students’ number understandings, and appropriately differentiate preschool curriculum. / Ph. D.

This longitudinal study tracks how 5- to 7-year-olds perform with early number skills. The aim of this study is to diagnose at-risk mathematics students by distinguishing the skills that, if not mastered by the end of Kindergarten, lead to greater difficulty in mathematics in 1st grade. This study’s methodology is mixed as it follows an exploratory and inductive path in light of its use of a hypothesis, an interpretive path in light of its interest in the individual student, and a positivist path in light of its focus on developing rules from analyzed data. An oral diagnostic test based on a comprehensive collection of early number skills was used to test students as Kindergarteners and again as 1st graders. The test results created benchmarks, revealing how the majority of the students performed with early number skills. The test results also revealed that each early number skill is highly, moderately, or minimally predictive in terms of student placement by the end of 1st grade. When comparing the individual skill scores of each Kindergarten student to his/her total test results of 1st grade, the predictive power of each skill emerged. Performing poorly with skills that are minimally predictive did not seem to have an impact on how the Kindergarten student finished in 1st grade; performing poorly with moderately predictive skills had a greater impact on 1st grade placement; performing poorly with highly predictive skills in Kindergarten increased the likelihood that the student would finish in the lower attaining group in 1st grade. A third result of the test showed that certain skills serve as preconditions for other skills; success with certain skills usually meant success with other skills. These connections between skills point to a learning model called in this study “simultaneous pathways,” indicating that there are connections between certain skills, and that students can be learning on several pathways simultaneously. The impact of the predictive power of early number skills is that diagnosis becomes more effective. Early diagnosis means early remediation which may prevent at-risk students from falling further behind their peers. The benchmarks developed by this research will help teachers assess their students because they will know the general skill level of Kindergarteners and 1st graders. This oral diagnostic test informs curriculum development. If test results show that students are missing the skills that are highly predictive, teachers can address those gaps in order to insure mastery.