Proportionality as a Property of Families of Objects: A Shift in the Level of Modelling
Abstract
Proportionality is commonly understood as a relationship between quantities or their measures. We argue that this viewpoint leaves several didactical and conceptual difficulties unresolved. Building on the theory of quantities developed notably by Rouche and by Chevallard and Bosch, we propose to locate proportionality to another level of modelling: that of families of objects, situated between the phenomenological level ("real world"), and the level of quantities. We show, through the analysis of textbook definitions and of classroom situations, that several classical difficulties become more intelligible when objects and the social practices through which they are combined and compared are incorporated into the mathematical model. We then provide a mathematical foundation for this proposal and prove that the usual multiplicative characterizations of proportionality follow from this object-based definition. This shift of modelling level suggests new perspectives for the teaching and study of proportionality, which it may be fruitful to investigate in the future.
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BibTeX
@misc{bourgade2026proportionality,
title = {{Proportionality as a Property of Families of Objects: A Shift in the Level of Modelling}},
author = {Jean-Pierre Bourgade and Clément Durringer},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05748360v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Proportionality as a Property of Families of Objects: A Shift in the Level of Modelling
AU - Jean-Pierre Bourgade
AU - Clément Durringer
PY - 2026
DA - 2026/09/13
PB - HAL
UR - https://hal.science/hal-05748360v1
AB - Proportionality is commonly understood as a relationship between quantities or their measures. We argue that this viewpoint leaves several didactical and conceptual difficulties unresolved. Building on the theory of quantities developed notably by Rouche and by Chevallard and Bosch, we propose to locate proportionality to another level of modelling: that of families of objects, situated between the phenomenological level ("real world"), and the level of quantities. We show, through the analysis of textbook definitions and of classroom situations, that several classical difficulties become more intelligible when objects and the social practices through which they are combined and compared are incorporated into the mathematical model. We then provide a mathematical foundation for this proposal and prove that the usual multiplicative characterizations of proportionality follow from this object-based definition. This shift of modelling level suggests new perspectives for the teaching and study of proportionality, which it may be fruitful to investigate in the future.
ER -