The Significance of the Research

This study contributes to research on teacher knowledge for teaching mathematics. It is significant at the level of the individual student, for teachers as major stakeholders in education and in terms of its methodological innovation and practical and applied significance. Overarching these contributions is the importance of Mathematics education research and the need for Australia to invest in quality Mathematics learning and teaching. At a time of increasing global competition, mathematics education plays a critical role in Australia’s social compact. The extent to which improvement in mathematics education underpins economic, technological and scientific progress is evident in its prioritisation on the national agenda. A challenge that remains unaddressed is how teachers who did not experience a mathematics education founded on deep conceptual understandings, communication, problem solving and reasoning, might teach students to achieve these objectives. This study is significant in its intent. It is focused on investigating aspects of teacher knowledge that are fundamental to providing the quality of mathematics education that we aspire to as a nation. The foundation for almost every mathematical concept and principle is laid in the primary years of schooling. If we aspire to increase student achievement in mathematics, then this must start from the outset of schooling. Understanding the nature of primary teachers’ mathematical knowledge for teaching is therefore of the utmost importance. Improved educational outcomes can only result from increases in teacher knowledge. Hence, providing the means for developing the type, depth, understanding and organisation of primary teachers’ mathematical knowledge for teaching is fundamental to increasing students’ attainment in mathematics. Until the role and nature of teachers’ understandings of mathematics in their development of knowledge for teaching it are understood, the mathematical reasoning, problem solving and communication of students may continue to limit their future opportunities as competitors in the global workforce. At a time when maintaining proficiency in the professional standards is a requirement for all teachers, the need to understand and support primary teachers’ mathematical knowledge for teaching has never been greater. The study is significant because, while it seems reasonable that teachers’ understandings of mathematics influence the effectiveness of their teaching, proof of this phenomenon has remained elusive (Kilpatrick et al., 2001). This study contributes to the national interest, to the interests of primary teachers as major stakeholders in education and to ensuring the provision of quality mathematics education for students. The methodological innovation used in the development of this study also makes a significant contribution to mathematics education research.

6.2.1 Significance for Students

This study is significant because it seeks to understand the knowledge required by teachers to make high quality mathematics education a reality for every student. Education is the vehicle for equipping students with knowledge, understanding and skills by providing access to high quality schooling that is free from discrimination (MCEETYA, 2008). This research is concerned not only with overall performance in mathematics education, but with maximising the educational opportunities afforded to each individual. If every student in Australia is entitled to a quality mathematics education, then every teacher of mathematics must provide quality mathematics teaching. In primary schooling this means that teachers require the type of knowledge demanded by proficient mathematics teaching. Australia’s overall educational performance comprises disparate individual performances. In the TIMSS (2011) thirty-seven per cent of Australian Year 8 students did not reach the intermediate benchmark. A study of how teachers’ understandings of subject matter are associated with the level of challenge available in their classrooms and their noticing of student thinking is therefore significant. The study of teachers’ understandings of content, and their skill and confidence in acting upon this knowledge to maximise the learning of all students, is paramount in providing equity and excellence in mathematics education.

6.2.2 Significance for Teachers

This study focuses on practising primary classroom teachers as stakeholders who carry the responsibility for translating the intentions of the syllabus into the outcomes attained by students in the final two years of primary schooling. Teachers’ knowledge implementing the curriculum in ways that improve students’ mathematical thinking each day are the conduit for this transformation. The research is unique in its investigation of how teachers know and understand mathematics and the alignment between their own knowledge and their endeavour of providing quality mathematics teaching for students. Teachers are committed to teaching mathematics in ways that develop deep conceptual understandings that lay a foundation for more advanced mathematical ideas. Despite this, research has focused more on the effectiveness teaching than on investigating how teachers’ understand the mathematics and how their understandings support them in teaching students. Primary teachers teach across all key learning areas. At a time of ongoing curriculum change the amount of new knowledge and skills that primary teachers have been asked to learn and teach has been constant and overwhelming. This study is significant because it focuses on relationships between aspects of teacher knowledge that can reduce how much teachers need to know by investigating how knowledge is connected and organised. By evaluating the strength of relationships between aspects of teacher knowledge, the correlational research in this study provides a starting point for considering

how increases in one aspect of teacher knowledge might be used to support efficient, effective increases in others.

6.2.3 Methodological Innovation

The review of research into conceptualisations of teacher knowledge and its impact on student learning illuminated the absence of measures of teacher knowledge that authentically reflect the work of quality teaching. No existing measures of teacher knowledge were identified as sufficiently aligned with the syllabus outcomes, professional standards and grade levels that were the focus of this study in the Australian context. Overcoming this challenge demanded methodological innovation. Comprehensive evaluations of teacher knowledge based on a number of different, yet appropriate and authentic, measures were required to study relationships among aspects of teacher knowledge. This study required the design, testing and development of multiple data gathering instruments that could capture important information about teachers’ knowledge in relation to practical daily aspects of their work, while also taking into account the costs of collecting accurate data. The research design was mindful that validity lay not only in the data gathering instruments themselves, but in how well they related to the work that teachers do.

The methods applied in this study make a significant contribution by responding to the need to develop and validate measures of teacher knowledge that are replicas of the tasks that teachers engage in. The study is innovative in using different measures for each aspect of teacher knowledge being studied and connecting these measures by keeping the content, grade level and day of data collection constant for the same group of participants. In particular, the study responds to a gap in the research literature on teachers’ work in the curriculum mapping arena which has become increasingly important. Data gathered in this research recognises that the work that teachers do outside the classroom enables quality learning in the classroom. Two structured artefacts, including protocols for analysing the quality of designs for learning and an instrument for evaluating teacher noticing when analysing written student work samples, were developed, trialled and implemented. While it seems logical that teachers’ knowledge of the content they teach must be related to the effectiveness of their teaching, quantitative research evidence of this relationship has been elusive. The design of a problem solving instrument closely aligned with assessments of student attainment to gain insight into teachers’ understandings of content was a significant methodological innovation. It facilitated the design of research uncovering relationships between aspects of teacher knowledge by starting with teachers ‘doing’ the type of mathematics that is used to measure their impact on student learning.

6.2.4 Applied and Practical Significance

Increases in teacher knowledge are vital to improving the quality of mathematics education in Australia. A major challenge to the improvement agenda is the juxtaposition between the substantial amount of knowledge needed and the limited time and resources available. Teachers, schools, educational systems and governments need to be pragmatic in the focus and design of opportunities for teachers to increase their professional knowledge in ways that impact directly on students’ learning outcomes. There is a need to better understand ways in which different aspects of teacher knowledge are related so that the quality and impact of teacher professional learning in mathematics education can be increased. If thresholds exist in the extent to which teachers need to understand mathematics in order to teach it effectively, then it may be inefficient to focus on teaching mathematics prior to increasing teachers’ understandings of mathematics. This study is significant in identifying and testing the strength of relationships between three aspects of teacher knowledge each of which has been associated with quality mathematics teaching. By identifying the strength of correlations between aspects of teacher knowledge, this study has the potential to inform the design of professional learning that builds foundational knowledge which can then efficiently support and increase other aspects of knowledge for teaching mathematics.