Algebra, Algebraic Thinking and Number Concepts In: Sacristán, A.I., Cortés-Zavala, J.C. & Ruiz-Arias, P.M. (Eds.). (2020). Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mexico. Cinvestav / AMIUTEM / PME-NA. https:/doi.org/10.51272/pmena.42.2020 296 INVESTIGATING THE LEARNING SEQUENCE OF DECIMAL MAGNITUDE AND DECIMAL OPERATIONS Erik Jacobson Indiana University erdajaco@indiana.edu Patricia A. Walsh Indiana University pawalsh@iu.edu Pavneet Kaur Bharaj Indiana University pkbharaj@iu.edu Research is mixed on whether understanding decimal magnitude supports operations with decimals or whether operations can be learned before and while students develop understanding of decimal magnitude. In the present study, we used a large scale, longitudinal design to investigate students’ knowledge of decimal comparison and operation before and after decimal comparison alone was introduced in the curriculum. Student performance on a decimal comparison task did not increase, but there was an increase in performance on decimal subtraction and decimal multiplication tasks, topics which were not part of the mandated curriculum during the relevant period of instruction. Keywords: Number Concepts and Operations, Rational Numbers, Elementary School Education, Cognition Perspectives Researchers examined various conceptual hurdles involved in meaningful interpretation and use of the notational system involving decimals (Resnick et al., 1989). Hiebert (1992) proposed three types of knowledge which are important to comprehend the decimal system: knowledge of the notation, knowledge of the symbol rules and knowledge of quantities and actions on quantities. The knowledge of notation comprises of “how the symbols are positioned on paper” (ibid, p. 290) rather than understanding of what ‘.’ means or what quantities it represents. For instance, a student can compare two decimals correctly, but can have incorrect reasoning to explain their answers (see Resnick et al., 1989 for details on erroneous rules while comparing decimal numbers). The knowledge of the symbol rules prescribes on “how to manipulate the written symbols to produce correct answers” (Hiebert, 1992, p. 290). For instance, while adding and subtracting two or more decimals, the numbers need to be lined up systematically (Lai & Murray, 2015). This knowledge is analogous to Skemp’s (1976) idea of instrumental understanding where an individual can manipulate mathematical syntactic symbols using appropriate rules, procedures, algorithms, etc. to produce the correct answer, even when without understanding the underlying reasons. Knowledge of quantities and actions includes the understanding of decimal numbers are representing quantities, i.e., measures of objects “…by units, tenths of units, hundredths of units, and so on” and comprehending the reasons that explain “what happens when the quantities are moved, partitioned, combined, or acted upon in other ways” (Hiebert, 1992, p. 291). Lai and Murray (2015) related the knowledge of quantities and actions on quantities with developing a comprehensive understanding of the decimal topics. Decimal Comparison Students build on whole number ideas when they engage with decimals, and this both helps and hinders learning. Lee and colleagues (2016) argued that due to the representational nature of decimal numbers, which is virtually indistinguishable from that of whole numbers, the students find decimal magnitude comparison tasks easier as compared to the fraction magnitude (see also, DeWolf et al., 2014; Iuculano & Butterworth, 2011). Researchers claim that students often perform well on the decimals comparison tasks by following syntactical rules (Lachance & Confrey, 2002), rather than developing a conceptual understanding of it. However, the common practice of teaching decimals as