Enacting mathematical content knowledge in the classroom: the preservice teacher experience of lower secondary algebra

Effective secondary mathematics teachers possess particular forms of mathematical content knowledge (MCK) which they purposefully enact in the classroom. Secondary mathematics preservice teachers are in the process of developing their MCK and their instructional decision making skills regarding the MCK they teach. However, the quality of secondary mathematics preservice teachers' MCK has been found lacking, both nationally in Australia and internationally. Arguably even more problematic, is the challenge of finding an accurate measure of preservice teacher MCK. In contrast to the common "paper test" approach used, this interpretive Australian study sought to describe the nature of secondary mathematics preservice teacher MCK by investigating what they enact "live" in their teaching practice. Because enacted MCK results from a decision making process, the study also aimed to identify the influences impacting preservice teachers' consideration of goals and of the MCK chosen to achieve the retained goals. The study limited itself to the context of lower secondary algebra lessons, mainly up to and including linear equations. Observation data (video footage, field notes, and lesson artefacts) pertaining to a total of six Year 8 and four Year 10 algebra lessons, taught by six 3rd and 4th year secondary preservice teachers, were collected during the preservice teachers' practicum phases. Within 48 hours of each lesson, a follow-up interview was completed with each preservice teacher. The semi-structured interview featured stimulated recall procedures using edited lesson footage. The interviews generated data concerning the decisions made by the preservice teachers that led to their enacting or withholding particular MCK during instruction. Teaching actions that attracted comment from the preservice teachers in the interviews were sorted into 137 "episodes", defined by the goal(s) pursued by the preservice teachers when performing those actions. The researcher coded the type and quality of MCK that manifested in each episode. Corresponding interview reflections were coded for evidence of decision making influences that impacted MCK related decisions. General lesson reflections, observation field notes, and lesson artefacts were also analysed for influencing elements. Lesson and interview data were analysed using pattern-seeking techniques and cross-variable analyses to identify the type and quality of MCK enacted and the influencing elements on MCK related decisions. The results of the study suggest that five categories of influencing elements, referred to in the study as influences, impacted the MCK related decisions. The first influence was the practicum context, comprising the advice from the supervising teacher, information provided in term overviews, school perceptions of the mathematical ability of student cohorts, and content presented in the class textbook. The second influence was the preservice teachers' pedagogical intentions, evidenced in the goals they formed at the macro, meso, and micro levels of their lessons. The third influence was the classroom circumstances that the preservice teachers considered as they made MCK related decisions. This influence comprised classroom events that captured the preservice teachers' attention and the instructional setting of classroom interactions, such as small group or whole of class instruction. The fourth influence was the preservice teachers' own MCK, MCK which they rarely sought to develop further when they prepared for their lessons. The fifth and final influence was the judgements that preservice teachers made about students. These judgements applied to how students develop mathematical understandings and to their mathematical needs, including exposure to appropriate mathematical content. The MCK that the preservice teachers enacted showed a preponderance of procedural knowledge emphasising mathematical rules and automated sequences of procedural steps. Occasionally, there was evidence of a specialised knowledge of algebraic procedures needed for teaching lower secondary mathematics, including connections involving conceptual knowledge and algebraic ways of thinking. However, the preservice teachers only sporadically enacted conceptual knowledge and algebraic ways of thinking to supplement their presentation of rules, steps, and algebraic manipulations. The superficial treatment or notable absence of conceptual knowledge and algebraic ways of thinking in the majority of the preservice teachers' teaching episodes reduced the overall quality of the content delivered. A lack of verbal precision and a lack of attention to the limitations of the procedures demonstrated also characterised the MCK that manifested in the classroom. The quality of the MCK delivered appeared to be associated with particular influences on the decisions the preservice teachers made. The preservice teachers tended to enact automated, imprecise, and contextually limited MCK when their own MCK was inadequate or when they made questionable judgements regarding the mathematical content they believed that students should be exposed to or the ways that students develop mathematical understandings. The preservice teachers enacted better quality MCK, which included connections involving conceptual knowledge, algebraic ways of thinking, and specialised knowledge of procedures, when their goals focussed on highlighting mathematical connections or on addressing student confusion. Stronger MCK was also evident when preservice teachers were responding to a particular student query rather than enacting MCK that they had planned to share before the lesson began. Finally, small group rather than whole of class instructional settings were associated with better quality MCK. The study highlights the significance of the preservice teachers' own prior mathematical experiences, of their understanding of how students learn, and of their live classroom interactions with students on the MCK related decisions they make. Preservice teachers' most recent university mathematics experiences may lead to a compressed knowledge of secondary algebra procedures and an automated treatment of algebraic manipulations which are evident in their teaching actions. Their lack of experience with school learners causes them to make MCK related decisions based on their own past observations of mathematics teachers and learners, which are inevitably limited by the student vantage point from which they were observed. The live classroom context in which preservice teachers interact with students positively impacts the mathematical content delivered. By sharing mathematical ideas with students, preservice teachers refine their knowledge of students' mathematical needs and begin to unpack their own MCK to accommodate those needs, improving the quality of the MCK they subsequently enact as interactions unfold. This study shows that preservice teacher MCK enacted in live classroom situations is not easily measured. Even when visible, it may not be a true indication of the MCK the preservice teacher possesses. The MCK that is enacted may indicate the mathematical knowledge they possess but it may also merely reflect the choices they have made, the quality of which are determined by the knowledge that preservice teachers bring to the decision making process. Hence, developing the preservice teacher MCK that manifests in their teaching actions requires attention not only to the MCK that preservice teachers hold but also to their evidence-based knowledge of how students learn mathematics. The findings of the study may improve the design and delivery of both the university-based component and the school-based component of secondary mathematics teacher education programs. Stronger partnerships between university and school-based educators are needed to (a) provide more opportunities for preservice teachers to develop evidence-based knowledge of how students learn mathematics, (b) privilege conceptual knowledge, algebraic ways of thinking, and associated connections to procedures in algebra, (c) explicitly highlight specific aspects of MCK, including precise use of mathematical terminology, that preservice teachers should be attending to in practicum lessons, and (d) provide opportunities beyond the practicum context for preservice teachers to be involved in MCK related interactions, ideally with secondary mathematics students.