The title of this study shows the aims on which the research questions were based. These included the areas in which the student teachers felt confident as well as those they perceived to be more difficult. This study adopted a\ud phenomenographic approach in order to provide further insight into each student teacher’s subject knowledge. The purpose of this study was to discover the individual and distinct ways in which student teachers understood fractions.\ud The study was undertaken in two universities with a group of thirteen undergraduate and postgraduate Initial Teacher Education students. The creation of a comfortable, supportive working atmosphere and the use of self-selected small groups, enabled a range of rich and honestly reflective data to be collected. Observations were made of groups working on two collaborative tasks involving the sequencing of fractions by magnitude and finding, followed by diagnostic interviews. Each interview was structured by the student’s individual selections from a range of questions where they indicated which they felt were most and least accessible. A constructivist perspective was adopted where the students had the opportunity to reconstruct their own understanding of fractions through the explanation and discussion of their existing ideas.\ud A range of successful strategies was demonstrated, especially the use of mathematical anchors as a means of comparison and the use of residual or gap thinking to consider differences in magnitude. Improper fractions and\ud reunitising were difficulties cited by many in the group. A certain level of anxiety and lack of flexibility in their chosen approaches was evident with the common assumption that there was a particular method which should be adopted; this was usually based on their “secondary school” experiences
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