Assessing students' knowledge and conceptual understanding of fraction equivalence

Abstract

Teachers recognise that many students of all ages experience difficulties understanding fractions, especially fraction equivalence. These students often resort to the application of procedures without understanding or invent their own based on their misconceptions. To advance students’ learning with understanding, teachers must gain insight into student thinking. By knowing what students can and can’t do, teachers can adjust their instruction by focusing tasks and lessons on particular areas of concern. Considerable research into fraction understanding was undertaken during the 1980’s. Few if any studies, rigorously tested, pencil and paper based assessments that measure students” knowledge of fraction equivalence. General fraction assessments which do incorporate fraction equivalence items are limited in their scope and focused primarily on what students can do, while the diagnostic potential of such items has received minimal consideration. When assessment tasks are valid and reliable, the adequacy of students’ responses should reflect the quality of their conceptual understanding, and alert the teacher to any misconceptions their students may possess. This study addresses a gap in assessment of students’ knowledge and conceptual understanding of fraction equivalence. The purpose of this study was threefold. Firstly, to design and develop a pencil and paper based instrument to assist in the diagnosis of students” misconceptions, and to establish their level of knowledge and understanding of fraction equivalence. Secondly, review responses to individual items to identify students’ correct and incorrect calculation strategies, and assist with the identification of common fraction equivalence misconceptions. Finally, develop a pathway for understanding fraction equivalence to assist teachers focus their instruction to extend students’ knowledge and address their misconceptions.

The Assessment of Fraction Understanding (AFU) developed as part of this study, underwent rigorous testing using a two phase design with each phase comprising a quantitative and smaller qualitative component. During the first phase, Initial Instrument Testing, the Assessment of Fraction Understanding version 1 (AFle) was administered to 297 students in grades 3 to 6 attending three Sydney primary schools. The reliability of the instrument was examined by the application Rasch analysis. Results showed that AFUv] could be improved. Consequently, the instrument was reviewed and revised, and a second phase, Confirmatory Instrument Testing was conducted. The Assessment of Fraction Understanding version 2 was administered to 349 students in grades 3 to 6 attending another three Sydney primary schools. Rasch analysis confirmed the Assessment of Fraction Understanding version 2 as valid and reliable. During each stage of testing, interview data were also collected to confirm and extend the calculation strategies hypothesised from the quantitative assessment data. Significant results of the study include the development and confirmation of the Levels of Understanding of Fraction Equivalence table which can be used by teachers to identify at a summary level, key components of understanding for each student. Three empirically based learning pathways for area models, number—line models and written language were also established and confirmed. These pathways enable teachers to the identify four aspects of learning for each student: (a) knowledge that has been mastered, (b) likely misconceptions that will be exhibited, (0) knowledge required to further conceptual understanding, and (d) instructional focus to overcome misconceptions and enable acquisition of further knowledge and understanding.