Investigating connections between fractional competence and algebraic reasoning in the middle years

To succeed in mathematics, middle-years’ students must move from additive to multiplicative thinking and from arithmetic calculations to generalised algebraic reasoning. This thesis’ focus on fractions gives insight into 607 middle-years students’ solution strategies and the potential to prompt algebraic thinking and reasoning as they attempted two paper and pencil assessments: The Fraction Screening Test (FST) and the Algebraic Thinking Questionnaire (ATQ). Analysis of the results has revealed that students exhibit many previously reported misconceptions. For example, students ordered fractions by size of denominator or by the difference between the numerator and denominator (gap thinkers). While many students struggled with the ATQ, which focussed on multiplication and division of whole numbers, fractions and decimals, others demonstrated their understanding about equivalence which is an important component of algebraic reasoning. A revised version of the FST included 'reverse fraction' tasks, so named because they require students to find the number of objects in the whole collection when given the number of objects representing a particular fraction. Analysis of student responses to both tests, in conjunction with these additional items, provided new insights revealing evidence of emerging algebraic reasoning. A semi-structured interview was constructed using reverse fraction tasks similar to those in the FST.