American Journal of Undergraduate Research www.ajuronline.org Volume 15 | Issue 3 | December 2018 23 Enhancing Number System Knowledge to Promote Number Sense and Adaptive Expertise: A Case Study of a Second-Grade Mathematics Student Cami C. Player* & Jessica F. Shumway School of Teacher Education and Leadership, Utah State University, Logan, UT Student: cami.crump9@aggiemail.usu.edu* Mentor: jessica.shumway@usu.edu ABSTRACT 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—systematic relations among numerals and the use of number relations to solve arithmetic problems—has been identified as a key cognitive mechanism in number sense development. Number system knowledge is a component of number sense, and the researchers of this study hypothesize 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 eight-year-old student’s number system knowledge learning as she participated in an instructional treatment over nine weeks. The main research question of this study was: In what ways does a student struggling in mathematics develop number system knowledge during a nine-week period in her second-grade classroom as she engages in a number system knowledge instructional treatment? The case in this study was selected based on her low pretest score combined with her desire for making sense of mathematics. The data sources for this study were a number system knowledge assessment and student interviews. The analysis involved a multiple-cycle coding process that resulted in themes of adaptive expertise and the union of procedural and conceptual knowledge in mathematics instruction. The results suggest that this number system knowledge instructional treatment provided this case-study student to develop more pronounced adaptive expertise in solving mathematics problems. An in-depth analysis of how and why one struggling student develops number system knowledge during a nine-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. KEYWORDS Number Sense; Number System Knowledge; Mathematics Education; Whole Numbers and Operations; Elementary Education; Teaching and Learning; Case Study Research INTRODUCTION In mathematics education, number sense is defined in multiple ways because it is a complex construct of interconnected knowledge. 1,2 Overall, number sense includes components such as understandings of quantities, numbers, and number system knowledge. The importance of number sense is reflected in the Common Core State Standards of Mathematics, in which connected understandings of foundational number concepts, such as number sense are an important theme in the document. 3,4 The focus of this article is on number system knowledge, which is one component of number sense. This article is about one eight-year-old student’s development of number system knowledge, which is defined as knowledge of systematic relationships among numbers and the ability to use number relationships to solve arithmetic problems. 5 This student participated in nine weeks of a number system knowledge instructional treatment and completed pretest, benchmark, and posttest assessments and assessment interviews targeting number system knowledge understandings and skills. The purpose of this exploratory case study was to investigate the variations of the student’s number system knowledge learning during the instructional treatments. THEORETICAL FRAMEWORK Number sense view Number sense requires a theoretical analysis because of the complexity of the construct. 6 The number sense view provides a theoretical lens for understanding students’ number sense development. The number sense view is the perspective that students’ development of interconnected number concepts (as opposed to isolated skills) leads to more meaningful mathematical understandings. The number sense view explains that as reasoning with numbers improves (i.e., number system knowledge concepts), computational fluency with arithmetic problems improves. 7,8 It provides a lens for understanding how number system knowledge influences, and is connected to, computational fluency, number line estimation, and problem solving.