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A mixed-methods quasi-experimental methodology was used to identify relationships between primary-school students' beliefs about intelligence, mathematics self-efficacy, and achievement, by investigating the effects of two interventions. One intervention aimed to strengthen students' mathematics self-efficacy, and the other aimed to develop in students' an incremental theory-of-intelligence – a belief that intelligence is malleable. In one group, teachers implemented both interventions with their students; in a second group, teachers implemented only the mathematics self-efficacy intervention, and the third (control) group were involved in no intervention. Year 4 and 5 students (n = 152) completed a questionnaire on three occasions, at intervals of about 7 months, to measure their theory-of-intelligence and their mathematics self-efficacy. Students made self-efficacy judgments in relation to specific number problems, which they were subsequently required to solve for the mathematics achievement measure. Both achievement and self-efficacy were then calibrated for each participant using the difficulty parameters for test items. Teachers completed questionnaires about their theory-of-intelligence and self-efficacy for teaching mathematics. Sub-samples of teachers and students were interviewed to develop a deeper understanding of what their questionnaire responses signified. The combined interventions had no significant effect on students' beliefs about the malleability of intelligence, mathematics self-efficacy, or achievement. In contrast, positive effects on students' mathematics self-efficacy and achievement were evident for students who experienced only the self-efficacy intervention. Teachers in this intervention group reported increased use of three strategies aimed at building students' mathematics self-efficacy: providing students with strategies for coping when learning became difficult; increasing their use of descriptive teacher-student feedback; and increasing their use of similar peers as models. For the self-efficacy intervention group, increases in students' mathematics achievement and self-efficacy appeared to be reciprocally related. The combined quantitative and qualitative evidence from the study showed that the complexity of some students' and teachers' beliefs about increasing intelligence was not reflected in their total scores on the theory-of-intelligence items used widely in earlier studies. In interviews, all students and most teachers described intelligence as malleable to varying degrees, which did not support previous dichotomous interpretations of theory-of-intelligence data. From students' definitions of intelligence, two related dimensions were established, one a fairly stable capacity for acquiring knowledge and skill in a given domain, and the second, the more malleable rate at which such knowledge and skill can be acquired. A variety of beliefs were expressed by students about which of these dimensions intelligence includes, and about how malleable the dimensions are. The findings raise questions about the value of advocating an incremental theory-of-intelligence for all students, regardless of their ability and how they conceptualise intelligence.

The Effects of Primary Students' Mathematics Self-efficacy and Beliefs about Intelligence on Their Mathematics Achievement: A Mixed-methods Intervention Study