---
_id: '13107'
abstract:
- lang: eng
  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.
author:
- first_name: Jan
  full_name: Schumacher, Jan
  id: '26809'
  last_name: Schumacher
  orcid: 0000-0002-6676-9774
- first_name: Sebastian
  full_name: Rezat, Sebastian
  last_name: Rezat
citation:
  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.'
  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.'
  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.'
  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.
  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.
  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.'
conference:
  end_date: 2019-02-10
  location: Utrecht
  name: CERME 11
  start_date: 2019-02-06
date_created: 2019-08-30T12:00:50Z
date_updated: 2022-01-06T06:51:28Z
department:
- _id: '360'
- _id: '98'
editor:
- first_name: Uffe Thomas
  full_name: Jankvist, Uffe Thomas
  last_name: Jankvist
- first_name: Marja
  full_name: Van den Heuvel-Panhuizen, Marja
  last_name: Van den Heuvel-Panhuizen
- first_name: Michiel
  full_name: Veldhuis, Michiel
  last_name: Veldhuis
keyword:
- diagrammatic reasoning
- hypothetical learning trajectory
- induction extrapolatory method
- integers
- negative numbers
- permanence principle
- semiotics
language:
- iso: eng
place: Utrecht, the Netherlands
publication: Proceedings of the Eleventh Congress of the European Society for Research
  in Mathematics Education (CERME11, February 6 – 10, 2019)
publication_status: accepted
publisher: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME
quality_controlled: '1'
status: public
title: A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating
  Integers
type: conference
user_id: '26809'
year: '2019'
...