{"title":"A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers","editor":[{"last_name":"Jankvist","full_name":"Jankvist, Uffe Thomas","first_name":"Uffe Thomas"},{"last_name":"Van den Heuvel-Panhuizen","full_name":"Van den Heuvel-Panhuizen, Marja","first_name":"Marja"},{"first_name":"Michiel","full_name":"Veldhuis, Michiel","last_name":"Veldhuis"}],"conference":{"start_date":"2019-02-06","location":"Utrecht","end_date":"2019-02-10","name":"CERME 11"},"publication":"Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)","abstract":[{"text":"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.","lang":"eng"}],"language":[{"iso":"eng"}],"place":"Utrecht, the Netherlands","quality_controlled":"1","date_updated":"2022-01-06T06:51:28Z","publisher":"Freudenthal Group & Freudenthal Institute, Utrecht University and ERME","type":"conference","keyword":["diagrammatic reasoning","hypothetical learning trajectory","induction extrapolatory method","integers","negative numbers","permanence principle","semiotics"],"citation":{"short":"J. Schumacher, S. Rezat, in: U.T. Jankvist, M. Van den Heuvel-Panhuizen, M. Veldhuis (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019), Freudenthal Group &#38; Freudenthal Institute, Utrecht University and ERME, Utrecht, the Netherlands, n.d.","ama":"Schumacher J, Rezat S. A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers. In: Jankvist UT, Van den Heuvel-Panhuizen M, Veldhuis M, eds. <i>Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)</i>. Utrecht, the Netherlands: Freudenthal Group &#38; Freudenthal Institute, Utrecht University and ERME.","ieee":"J. Schumacher and S. Rezat, “A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers,” in <i>Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)</i>, Utrecht.","bibtex":"@inproceedings{Schumacher_Rezat, place={Utrecht, the Netherlands}, title={A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers}, booktitle={Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)}, publisher={Freudenthal Group &#38; Freudenthal Institute, Utrecht University and ERME}, author={Schumacher, Jan and Rezat, Sebastian}, editor={Jankvist, Uffe Thomas and Van den Heuvel-Panhuizen, Marja and Veldhuis, MichielEditors} }","chicago":"Schumacher, Jan, and Sebastian Rezat. “A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers.” In <i>Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)</i>, edited by Uffe Thomas Jankvist, Marja Van den Heuvel-Panhuizen, and Michiel Veldhuis. Utrecht, the Netherlands: Freudenthal Group &#38; Freudenthal Institute, Utrecht University and ERME, n.d.","mla":"Schumacher, Jan, and Sebastian Rezat. “A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers.” <i>Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)</i>, edited by Uffe Thomas Jankvist et al., Freudenthal Group &#38; Freudenthal Institute, Utrecht University and ERME.","apa":"Schumacher, J., &#38; Rezat, S. (n.d.). A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers. In U. T. Jankvist, M. Van den Heuvel-Panhuizen, &#38; M. Veldhuis (Eds.), <i>Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)</i>. Utrecht, the Netherlands: Freudenthal Group &#38; Freudenthal Institute, Utrecht University and ERME."},"department":[{"_id":"360"},{"_id":"98"}],"date_created":"2019-08-30T12:00:50Z","publication_status":"accepted","_id":"13107","status":"public","author":[{"orcid":"0000-0002-6676-9774","last_name":"Schumacher","full_name":"Schumacher, Jan","first_name":"Jan","id":"26809"},{"first_name":"Sebastian","full_name":"Rezat, Sebastian","last_name":"Rezat"}],"year":"2019","user_id":"26809"}