One machine, three drawings: a didactic analysis of a teaching progression for the function concept from Grade 4 to Grade 10
Vincent Tariel-Adam · preprint [math.HO] · 3 November 2026 · CC BY 4.0
Awaiting arXiv: the version prepared for its submission. Its arXiv identifier will appear on this page.
Abstract
The function concept can be entered through a correspondence between sets, a relation that is single-valued, or through a process that turns an input into an output. This paper presents the design rationale and a critical analysis of a source progression for functions (Grades 4-10, nominal ages 9-16, with two levels on relations between quantities), from which French-curriculum and International Baccalaureate chapters published at https://commeunjeu.org are assembled. One schema is presented first and kept throughout: a function is a machine that takes an input, applies a rule and gives exactly one output. Its drawing changes with age (a robot, then a box, then the arrow f : x ↦ f(x)), and each change is shown once, the two drawings side by side; after 15 the box returns only in the lessons, to explain composition and the inverse. The properties that research identifies as difficult (the uniqueness of the output, the constant function, the function without a formula, two rules for one function, preimages) are posed on the machine from Grade 4; the word relation is kept for the link between two quantities, where covariation is taught; the set of ordered pairs comes back as the graph. The choices are related to the function machine as a cognitive root and to its authors' own warnings, to process-object accounts, to studies of concept images and of covariation, and to the French and IB curricula. A block-level audit of the dated source documents found the machine in only 2 of the 177 exercise blocks of Grades 7 to 10, no exercise asking pupils to draw a machine, and public chapters that opened on the relation; the revisions made after the audit are recorded with their counts before and after. No empirical evaluation is reported; hypotheses for one are stated.
Keywords: function concept; function machine; correspondence; covariation; concreteness fading; didactics of mathematics; curriculum design
For the classroom
The guide How to Teach Functions takes from the article what is useful in class: what to show, which exercises to give, which errors to watch for.
The chapters studied
The article analyses the progression from which these IB chapters are assembled:
- PYP 4: Patterns and functions
- PYP 4: Data Handling
- PYP 5: Number sequences and rules
- MYP 1: Line Graphs
- MYP 2: Coordinates and Straight-Line Graphs
- MYP 2: Formulae
- MYP 5: Functions
- MYP 5: Further Functions
How to cite
Tariel-Adam, V. (2026). One machine, three drawings: a didactic analysis of a teaching progression for the function concept from Grade 4 to Grade 10. Preprint. https://commeunjeu.org/enseignants/fonctions-article/
@misc{tarieladam2026functions,
author = {Tariel-Adam, Vincent},
title = {One machine, three drawings: a didactic analysis of a teaching progression for the function concept from Grade 4 to Grade 10},
year = {2026},
note = {Preprint},
url = {https://commeunjeu.org/enseignants/fonctions-article/}
}