Children's quantitative sense of fractions

Learning the meaning of common fractions and how to operate with them is a traditionally difficult aspect of learning mathematics. The symbol system used to represent fractions, one whole number written above another whole number, is not transparent to the meaning of fractions. This difficulty of interpreting fractions contributes to traditional practice in teaching fractions emphasising the syntax of fractions over their semantics. Without a sense of the size of fractions students must rely solely on learning the fraction syntax, as there is no feasible way of checking the reasonableness of an answer. To investigate how well students had developed a sense of the size of fractions, a large cross-section of students from Years 4-8 (over 300 students in each grade) completed 37 tasks designed to draw upon a quantitative sense of fractions. The students' answers provided data for both a conceptual analysis and a related Rasch item analysis. The conceptual analysis found that although some studnets have a strong sense of the size of fractions, many students have developed pseudo fraction concepts. Rather than seeing fractions as relational numbers, more than 10% of the students responded as if fractions corresponded to a count of the number of parts in fraction representation. Questions involving the fraction notation increased the variety of incorrect interpretations of fractions. The Rasch analysis confirmed that the items related to the same trait and provided an ordering of the difficulty of the questions. The Rasch item map provided backing for a description of how students develop a sense of the size of fractions. The current methods of developing students' understanding of fractions have resulted in a large number of pseudo concepts being formed. The basis of making meaning from models used to introduce fractions, needs to be the focus of teaching fractions. As well as using counter-example to limit the number of unintended features of models students associate with fractions, comparison of length rather than area should be used to introduce fractions.