Cami Player, Utah State University
Instruction for developing students’ number sense is a critical area of research in mathematics education due to the role number sense plays in early mathematics learning. Specifically, number system knowledge— knowledge of the systematic relations among Arabic numerals and the skills in using this knowledge to solve arithmetic problems—has been identified as a key cognitive mechanism in number sense development. We view number system knowledge as a component of number sense and theorize that it plays a critical role in second-grade students’ understanding of relationships among numbers and adaptive expertise with mathematics problems. The purpose of this exploratory case study was to investigate the variations of an 8-year-old student’s number system knowledge learning as she participated in an instructional treatment over 9 weeks. Our main research questions were: 1) In what ways does a student struggling in mathematics develop number system knowledge during a 9-week period in her second-grade classroom? 2) How does the instructional treatment influence her number system knowledge and number sense development during the 9-week period? The case in our study was selected based on her low pretest score combined with her desire for making sense of mathematics. Data were collected with a number sense assessment (which included a targeted number system knowledge assessment), student interviews, and classroom observations. Data were triangulated using these multiple sources. The analysis involved a multiple-cycle coding process that resulted in themes regarding the development of adaptive expertise and the union of procedural and conceptual knowledge in mathematics instruction. The results suggest that this instructional treatment provided this case-study student to develop more pronounced adaptive expertise in mathematical problem solving. Additionally, it revealed the importance of mathematics instruction that includes numerals linked to quantities in order to help students to conceptualize basic mathematical practices. Typical elementary school teachers are challenged by reaching the needs of students struggling in mathematics. Number system knowledge is a predictor of later mathematics achievement, and an in-depth analysis of how and why one struggling student develops number system knowledge during a 9-week instructional treatment within the context of her mathematics class provides exploratory evidence to help researchers and teachers develop and implement similar practices in elementary mathematics instruction.