Sharing a unit, measuring a length: a didactic analysis of a teaching progression for fractions from Grade 1 to Grade 8
Vincent Tariel-Adam · arXiv:2610.14735 [math.HO] · 10 October 2026 · CC BY 4.0
Abstract
Fractions require learners to revise properties associated with whole numbers. This paper presents a design rationale and a critical analysis of an eight-level source progression for fractions (Grades 1-8, nominal ages 6-14), from which the French-curriculum and International Baccalaureate chapters published at https://commeunjeu.org are assembled. Its central hypothesis is a bridge from sharing to numerical measure: a share of one unit is represented by a bar, read as a length relative to a specified unit, and located on the number line. The reference unit is kept stable in the initial tasks to focus attention on fraction magnitude, equivalence and operations; changing the unit is deliberately postponed, without treating this as a mathematical requirement or an established reduction in cognitive load. Three principles organise the meanings: sharing one unit, coordinating bars and number lines, and establishing fractions as numbers before explicitly treating them as scalar operators. Two supporting principles concern valid local justifications before formal generalisation and chains of equalities, introduced in Grades 2-3, with written reasons in lessons from Grade 4. The analysis relates these choices to the research literature and selected curricula, examines task strategies and feedback, and distinguishes realised features from discrepancies and proposed revisions. The source documents and their curriculum projections do not establish the order actually experienced by pupils. No empirical evaluation is reported. Primary hypotheses address transfer between representations, coordination of units and the numerical status of fractions before operators; the audit also identifies limits to the reproducibility of its findings.
Keywords: fractions; rational numbers; didactics of mathematics; fair sharing; measure; number line; curriculum design; learning trajectory
For the classroom
The guide How to Teach Fractions 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 1: Halves and Quarters
- PYP 2: Fractions
- PYP 3: Fractions
- PYP 4: Fractions
- PYP 5: Fractions: Equivalence and Comparison
- PYP 5: Operations with Fractions
- MYP 1: Fractions
- MYP 2: Fractions
- MYP 3: Rational Numbers
- MYP 3: Algebraic Expressions and Fractions
- MYP 4: Algebraic Fractions
How to cite
Tariel-Adam, V. (2026). Sharing a unit, measuring a length: a didactic analysis of a teaching progression for fractions from Grade 1 to Grade 8. arXiv:2610.14735 [math.HO]. https://doi.org/10.48550/arXiv.2610.14735
@misc{tarieladam2026,
author = {Tariel-Adam, Vincent},
title = {Sharing a unit, measuring a length: a didactic analysis of a teaching progression for fractions from Grade 1 to Grade 8},
year = {2026},
eprint = {2610.14735},
archivePrefix = {arXiv},
primaryClass = {math.HO},
doi = {10.48550/arXiv.2610.14735},
url = {https://arxiv.org/abs/2610.14735}
}