---
_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. Proceedings of the Eleventh Congress of the European Society
for Research in Mathematics Education (CERME11, February 6 – 10, 2019). Utrecht,
the Netherlands: Freudenthal Group & Freudenthal Institute, Utrecht University
and ERME.'
apa: 'Schumacher, J., & 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, & M. Veldhuis (Eds.), Proceedings of the Eleventh
Congress of the European Society for Research in Mathematics Education (CERME11,
February 6 – 10, 2019). Utrecht, the Netherlands: Freudenthal Group &
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 & 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 Proceedings of
the Eleventh Congress of the European Society for Research in Mathematics Education
(CERME11, February 6 – 10, 2019), edited by Uffe Thomas Jankvist, Marja Van
den Heuvel-Panhuizen, and Michiel Veldhuis. Utrecht, the Netherlands: Freudenthal
Group & 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 Proceedings of the Eleventh Congress
of the European Society for Research in Mathematics Education (CERME11, February
6 – 10, 2019), Utrecht.
mla: Schumacher, Jan, and Sebastian Rezat. “A Hypothetical Learning Trajectory for
the Learning of the Rules for Manipulating Integers.” Proceedings of the Eleventh
Congress of the European Society for Research in Mathematics Education (CERME11,
February 6 – 10, 2019), edited by Uffe Thomas Jankvist et al., Freudenthal
Group & 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 & 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'
...