Limitations

A number of limitations in this research are worthy of consideration. This section identifies limitations with the potential to impact on the quality and interpretation of the findings in relation to the research questions. Through reflection on the research design and methods applied, limitations related to the aim of the study, the content and stage level that were the focus of the study, the tasks selected as measures of teacher knowledge and the selection of items in the Problem Solving Task are important to take into account.

This research set out to investigate how teachers’ understandings of content influence their knowledge for teaching it. The two measures of knowledge in the pedagogical domain focused on teacher knowledge in the curriculum mapping arena, including how teachers plan for student learning and how teachers respond to students’ written work samples. Each of these measures is an authentic measure of pedagogical knowledge related to the work that teachers do in the classroom. However, the study is limited to the impact of subject matter knowledge on two pedagogical aspects of knowledge related to planning and evaluating student learning. It does not set out to evaluate the translation of this pedagogical knowledge into the classroom or the impact of this pedagogical knowledge on students’ learning gains. Evidence of the relationship between pedagogical knowledge and students’ learning gains relies on the findings of previous studies in the literature review.

In this research, the measures of, and relationships between, teachers’ understandings of mathematics and their pedagogical knowledge, pertain to Stage 3 primary school teachers in New

South Wales. The results were identified specifically in relation to understanding content taught in the final two years of primary school. Consequently, the results cannot be generalised to all teachers or teaching all levels of schooling. All measures of teacher knowledge were related to content from the same content strand. This decision was made to provide a deep investigation and thorough investigation of mathematical topics that are recognised as problematic for students at this stage of learning. Therefore the results reflect the relationship between three aspects of teacher knowledge in relation to the teaching of measurement, rather than the teaching of all mathematics. It is possible that different results might be attained by selecting a different strand of mathematics or different outcomes. Future research should be undertaken to verify whether examination of the same aspects of teacher knowledge, using similar measures in relation to different content, result in the identification of similar relationships. As teacher knowledge can be finely grained, the strong relationships identified between teachers’ understandings of content and their pedagogical knowledge do not necessitate that these relationships would be similar for other content or outcomes. For example, teachers’ understandings of Addition and Subtraction, the tasks they design and their noticing of student thinking might exhibit either a stronger or weaker relationship than that identified using area, perimeter and volume content. There is a need for future studies to verify whether examination of the same aspects of teacher knowledge identified similar relationships using a range of content.

The tasks that teachers engaged in as measures of their mathematical knowledge for teaching were selected to authentically represent the work that teachers engage in when planning and evaluating learning. In this research, measures of teachers’ pedagogical knowledge were limited to the design of mathematical tasks and their noticing of student thinking in written work samples. The study did not collect evidence of the levels at which tasks were implemented in the classroom or teachers’ abilities to question students to gain further insights into their thinking using face-to-face interactions. As teachers apply their knowledge for teaching in many ways, both in and out of the classroom, it is possible that, should teacher knowledge be exemplified through other tasks, then different results might be produced.

The major statistical strength of this study the focus on correlational analysis. The strength of correlations between data gathered as measures of aspects of teacher knowledge provided evidence of the extent to which one aspect of teacher knowledge was predictive of others. While the results identified teachers’ subject matter knowledge as being significantly predictive of two aspects of pedagogical knowledge, this does not prove that high levels of subject matter cause high levels of cognitive demand in tasks and noticing higher levels of student thinking. The study did not examine

how changes in knowledge over time for the same group of participants were related to changes in other aspects of their knowledge for teaching mathematics. The findings suggested that teachers with stronger subject matter knowledge generally designed more challenging tasks and also were more likely to notice higher levels of student thinking. The regression equations suggested that increases in understandings of mathematics when solving problems would be predictive of increases in pedagogical knowledge. However, the study was not a longitudinal study and therefore did not measure the effect of increasing the knowledge of teachers with lower levels of subject matter on their pedagogical knowledge. Consequently, the study is limited to finding that these aspects of knowledge are related and that increases in one aspect of teacher knowledge were predictive of increases in others.