TY - CONF AB - In this paper, we first outline a Hypothetical Learning Trajectory (HLT), which aims at a formal understanding of the rules for manipulating integers. The HLT is based on task formats, which promote algebraic thinking in terms of generalizing rules from the analysis of patterns and should be familiar to students from their mathematics education experiences in elementary school. Second, we analyze two students' actual learning process based on Peircean semiotics. The analysis shows that the actual learning process diverges from the hypothesized learning process in that the students do not relate the different levels of the diagrams in a way that allows them to extrapolate the rule for the subtraction of negative numbers. Based on this finding, we point out consequences for the design of the tasks. AU - Schumacher, Jan AU - Rezat, Sebastian ED - Jankvist, Uffe Thomas ED - Van den Heuvel-Panhuizen, Marja ED - Veldhuis, Michiel ID - 13107 KW - diagrammatic reasoning KW - hypothetical learning trajectory KW - induction extrapolatory method KW - integers KW - negative numbers KW - permanence principle KW - semiotics T2 - Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019) TI - A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers ER -