Unit Fractions as Superheroes for Instruction

The complexity of understanding unit fractions is often underappreciated in instruction. We introduce a continuum of children's understanding of unit fractions to explore this complexity and to help teachers make sense of children's strategies and recognize milestones in the development of unit-fraction understanding. Suggestions for developing this understanding are provided.

Iteration: Unit Fraction Knowledge and the French Fry Tasks

Using these tasks can help nurture children's multiplicative notions of unit fractions beyond part-whole understanding.

Bidimensional expressions of fractions in Chinese

A generic pattern of expression for fractions accounts for numerators and denominators, thus generally producing bidimensional numerical expressions. In Qin–Han mathematical texts, fractions were constructed as predicative phrases: the monodimensional expression ‘denominator’s name + fēn’ of a unit fraction 1/n acted as subject, and the numerator’s name acted as predicate. The morpheme zhī could be used as an optional marker of this predicative relation. Later evolutions were not linear, and reveal the effects of language planning and of free linguistic invention, finally giving rise to the inseparable fraction names of Contemporary Chinese. Un schéma générique pour dire les fractions rend compte des numérateurs et dénominateurs produisant des expressions numériques bidimensionnelles. Dans les textes mathématiques Qin–Han, elles étaient construites comme des énoncés prédicatifs où la désignation monodimensionnelle « nom du dénominateur + fēn » d’une fraction unitaire 1/n servait de sujet et le numérateur de prédicat. Le morphème zhī était utilisé facultativement comme marqueur de cette relation prédicative. Les évolutions de ces expressions n’ont pas été linéaires et révèlent l’action d’inventions et de normalisations linguistiques pour aboutir aux appellations insécables des fractions en chinois contemporain.

The Splitting Loope

Teaching experiments have generated several hypotheses concerning the construction of fraction schemes and operations and relationships among them. In particular, researchers have hypothesized that children's construction of splitting operations is crucial to their construction of more advanced fractions concepts (Steffe, 2002). The authors propose that splitting constitutes a psychological structure similar to that of a mathematical group (Piaget, 1970b): a structure that introduces mutual reversibility of students' partitioning and iterating operations that the authors refer to as the splitting loope. Data consisted of 66 sixth–grade students' written performance on 20 tasks designed to provoke responses that would indicate particular fractions schemes and operations. Findings are consistent with hypotheses from related teaching experiments. In particular, they demonstrate–consistent with the notion of the splitting loope—that equipartitioning and the partitive unit fraction scheme mediate the construction of splitting from partitioning and iterating operations.