This study sought to explore what sort of relationships exist between students' number sense and their problem solving ability, and the contribution of the teacher's teaching style and the students' learning style towards students' performance in these two respective areas. The problem solving ability and number sense proficiency of three classes of Year 7 students, from three metropolitan primary schools, were compared to their learning style, and their mathematics teacher's teaching style. Sixty-eight students, comprised of twenty-six males and forty-two females, and their three Year 7 teachers, identified as "effective teachers of mathematics", were involved in this study. A mixed methods design was employed through which a combination of the ethnographic approach, the case study, framework approach and grounded theory was applied, to investigate the relationships between students' number sense and problem solving abilities, and the teaching and learning style compatibility which promote such abilities. Hence, the method followed was both quantitative by scoring of test results and quantification of qualitative data and qualitative through observations and tape-recorded interviews. Results indicate that although these three teachers tended to use different teaching approaches, their focus was more on getting students to understand the rationale behind any concept and process under discussion; they taught to the ability of the students first, and in so doing they considered individual learning preferences, although the former was given a lot more prominence than the latter. Although all three teachers expressed a strong preference for receiving information through the verbal learning modality, they taught largely through the visual mode and employed the verbal mode mainly for discussions, with very little teacher exposition. This could be one reason why a large majority of students showed a preference for receiving information through the visual learning modality. This interpretation was supported by the results obtained from the Number Sense (NS) and Problem Solving (PS) tests. Many of the students with lower number sense seemed to identify and solve an imaginary problem different to the one proposed. On the other hand the ability to solve both NSIP and DNSP seemed to increase with level of proficiency in number sense. Both the pre-tests and post-tests results revealed that there is significant correlation between students' number sense and problem solving ability. Performance gain analysis indicated that most students' number sense and problem solving performance improved, and the teaching style of the teacher could be one of the main factors responsible for such an improvement. Teachers should find ways and means of combining both contemporary and traditional teaching theories and methods so as to enhance the quality of their students' learning experiences. Preparation of lessons should not only shift from a focus on content but also pay equal attention to catering for individual learning ability, which is closely tied to learning style. Teachers need to find ways and means of identifying the number sense proficiency level, problem solving ability and also learning preference of their students in order to be able to mathematically empower the latter. Appended are: (1) Letters and Consent Forms; (2) Year 7 Number Sense Tests; (3) Year 7 Problem Solving Test; (4) Student and Teacher Learning Style Inventories; (5) Teaching Style Inventory; (6) Formal Teacher Interviews; (7) Student Interviews; and (8) Sample Worksheets. (Contains 63 tables and 33 figures.) [Ph.D. Dissertation, Edith Cowan University.]