Restructuring mathematics teaching | learning: Co-teaching as a designed approach to teachers' collaborative inquiry

<p>This study explores how participation in collaborative inquiry opens space for an expanded set of understandings and practices for mathematics teaching | learning. It examines the affordances of collaborative inquiry to promote, or constrain, teacher learning in the context of teachers’ day to day work. Sociocultural perspectives underpin the study whereby professional learning is presumed to be situated in the social and cultural contexts of teachers’ work. A survey of the literature supports the assumption that persistent underachievement in mathematics for some groups of learners requires shifts in what teachers know and can do and reveals the importance of collaboration and inquiry for teacher learning. The study involved a participatory, design-based approach underpinned by an authentic and appreciative inquiry stance. Design-based research was chosen for its proximity to practice and its focus on connections between the enactment of learning designs and outcomes of interest. The research was iterative and cyclical whereby the researcher worked with a group of four teachers in one New Zealand primary school to design, implement and refine an approach to teachers’ collaborative inquiry. A range of data were gathered during a 6-month collaboration, including from teacher interviews, classroom observations and three-weekly group meetings. The analysis took a pragmatic and multi-theoretical approach to examine what it meant to design and enact teachers’ collaborative inquiry. Cultural-historical activity theory (CHAT) was employed to capture the complexity of the teachers’ collaborative inquiry activity and to analyse and interpret the contradictions that arose. A key finding was that a co-teaching inquiry approach fostered conditions that afforded teachers’ expanded access to and depth of engagement with new, and often dissonant, practice ideas. Through co-teaching, mathematics teaching | learning was restructured within three interconnected fields of practice: the teachers’ enacted practice, their talk about practice, and their noticing of student thinking within practice. The co-teaching inquiry activity was increasingly directed at a collective purpose; involved an interplay of risk and trust; supported shifts in teachers’ roles and responsibilities; and allowed teachers to constantly renegotiate the goals of their shared activity. The co-teaching arrangement disrupted practice whereby teachers’ actions served as minor interruptions to each other’s practice and thus became a resource for teacher learning. Opportunities to engage deeply with one another’s practice opened space for an expanded set of actions for each of the teachers in their own practice. This thesis adds nuanced understandings of the interrelated roles of collaboration and inquiry in improving teaching. It contributes to the growing body of literature exploring co-teaching arrangements for teacher learning, in this case in the previously under-examined context of teachers’ collaborative inquiry for their ongoing professional learning. It offers insights into how co-teaching might support teachers to enact new and challenging pedagogies aimed at addressing the persistent and considerable challenges posed by an ethical imperative to promote mathematics learning for diverse (all) students. Participating in the co-construction of a design for their collaborative inquiry enabled teachers to restructure their work and expand the possibilities for their individual and collective practice. It allowed teachers to reconstruct their identities from the lone operator whose professional reputation needs protection from exposure of any weaknesses in their mathematics knowledge or practice, to a learner whose naïve questions and gaps in practice served as a resource for all in their learning.</p>

Restructuring mathematics teaching | learning: Co-teaching as a designed approach to teachers' collaborative inquiry

<p>This study explores how participation in collaborative inquiry opens space for an expanded set of understandings and practices for mathematics teaching | learning. It examines the affordances of collaborative inquiry to promote, or constrain, teacher learning in the context of teachers’ day to day work. Sociocultural perspectives underpin the study whereby professional learning is presumed to be situated in the social and cultural contexts of teachers’ work. A survey of the literature supports the assumption that persistent underachievement in mathematics for some groups of learners requires shifts in what teachers know and can do and reveals the importance of collaboration and inquiry for teacher learning. The study involved a participatory, design-based approach underpinned by an authentic and appreciative inquiry stance. Design-based research was chosen for its proximity to practice and its focus on connections between the enactment of learning designs and outcomes of interest. The research was iterative and cyclical whereby the researcher worked with a group of four teachers in one New Zealand primary school to design, implement and refine an approach to teachers’ collaborative inquiry. A range of data were gathered during a 6-month collaboration, including from teacher interviews, classroom observations and three-weekly group meetings. The analysis took a pragmatic and multi-theoretical approach to examine what it meant to design and enact teachers’ collaborative inquiry. Cultural-historical activity theory (CHAT) was employed to capture the complexity of the teachers’ collaborative inquiry activity and to analyse and interpret the contradictions that arose. A key finding was that a co-teaching inquiry approach fostered conditions that afforded teachers’ expanded access to and depth of engagement with new, and often dissonant, practice ideas. Through co-teaching, mathematics teaching | learning was restructured within three interconnected fields of practice: the teachers’ enacted practice, their talk about practice, and their noticing of student thinking within practice. The co-teaching inquiry activity was increasingly directed at a collective purpose; involved an interplay of risk and trust; supported shifts in teachers’ roles and responsibilities; and allowed teachers to constantly renegotiate the goals of their shared activity. The co-teaching arrangement disrupted practice whereby teachers’ actions served as minor interruptions to each other’s practice and thus became a resource for teacher learning. Opportunities to engage deeply with one another’s practice opened space for an expanded set of actions for each of the teachers in their own practice. This thesis adds nuanced understandings of the interrelated roles of collaboration and inquiry in improving teaching. It contributes to the growing body of literature exploring co-teaching arrangements for teacher learning, in this case in the previously under-examined context of teachers’ collaborative inquiry for their ongoing professional learning. It offers insights into how co-teaching might support teachers to enact new and challenging pedagogies aimed at addressing the persistent and considerable challenges posed by an ethical imperative to promote mathematics learning for diverse (all) students. Participating in the co-construction of a design for their collaborative inquiry enabled teachers to restructure their work and expand the possibilities for their individual and collective practice. It allowed teachers to reconstruct their identities from the lone operator whose professional reputation needs protection from exposure of any weaknesses in their mathematics knowledge or practice, to a learner whose naïve questions and gaps in practice served as a resource for all in their learning.</p>

Philosophy for All Children: enhancing knowledge

<p>Building students’ critical thinking has been a focus in Education around the world in recent years. The New Zealand Curriculum (2007) is no exception, with an emphasis on critical and creative thinking in its vision statement. However, no advice is offered on how to teach critical thinking. This study uses a qualitative case study approach to explore how teachers teach critical thinking through Philosophy for Children (P4C) and offers some guidance on this for teachers. One of the contributions of this thesis is the claim that the P4C approach enhances social justice by supporting diverse learners – children who need support to achieve at school – to develop critical thinking and language capability. The theoretical framework underpinning my research draws on Bernstein’s (2000) ‘democratic pedagogic rights’ to enhancement, inclusion, and participation, and Wheelahan’s (2007), Young’s (2009) and Young and Muller’s (2013) conceptions of powerful knowledge. While much attention has been given to the theory of powerful knowledge in tertiary and secondary education contexts, to date, very little research has explored what powerful knowledge might look like, in practice, at primary school. Therefore this study makes an original contribution by investigating this unexplored area. Participants in this research included 104 primary school students aged between 8 and 11 years old and their teachers (n = 4) from four diverse primary schools in New Zealand. The research data is drawn from four main sources: audio recordings of classroom discussions, semi-structured focus group interviews with sample students, open-ended interviews with teachers, and student thinking journals. Over a period of 6 months, classes held weekly hour-long P4C dialogues with a focus on both philosophical content and developing critical thinking language and skills. Students were encouraged to evaluate their philosophical and critical thinking both verbally and in written form. The methodological approach incorporates sociocultural perspectives highlighting the key role of language as a mediator and tool for thinking (Mercer, 2000; Vygotsky, 1962, 1978). Transcripts of classroom discussions were analysed both inductively and deductively, using Hennessy et al.’s (2016) coding system for analysing classroom dialogue across educational contexts and Daniel et al.’s (2005) matrix outlining the development of the dialogical critical thinking process. Observation of students’ everyday learning in the classroom, together with their and their teachers’ reflections, revealed that teachers played a significant role in shaping students’ language and critical thinking through modelling, facilitating dialogue, establishing a democratic classroom culture, making thinking visible, and innovating and personalising teaching according to their students’ particular needs. Both students and teachers developed a range of capabilities through participating in (or, in the case of the teachers, leading) P4C dialogue: resilience, receptivity, intersubjectivity. This thesis presents one of the first investigations into the practice of teaching powerful knowledge in primary schools and suggests that explicit teaching of disciplinary language enhances students’ critical thinking capability and constitutes powerful knowledge. A new conception of powerful knowledge is advanced, termed enhancing knowledge, which emphasises how disciplinary knowledge enhances and is complementary to other knowledges, such as social or cultural knowledge, without devaluing their importance. This term also seeks to avoid the potential dominant connotations of the word ‘powerful’. The findings of this study suggest that P4C can offer diverse learners access to enhancing knowledge which has the potential to improve equity in education and enhance social justice.</p>

Philosophy for All Children: enhancing knowledge

<p>Building students’ critical thinking has been a focus in Education around the world in recent years. The New Zealand Curriculum (2007) is no exception, with an emphasis on critical and creative thinking in its vision statement. However, no advice is offered on how to teach critical thinking. This study uses a qualitative case study approach to explore how teachers teach critical thinking through Philosophy for Children (P4C) and offers some guidance on this for teachers. One of the contributions of this thesis is the claim that the P4C approach enhances social justice by supporting diverse learners – children who need support to achieve at school – to develop critical thinking and language capability. The theoretical framework underpinning my research draws on Bernstein’s (2000) ‘democratic pedagogic rights’ to enhancement, inclusion, and participation, and Wheelahan’s (2007), Young’s (2009) and Young and Muller’s (2013) conceptions of powerful knowledge. While much attention has been given to the theory of powerful knowledge in tertiary and secondary education contexts, to date, very little research has explored what powerful knowledge might look like, in practice, at primary school. Therefore this study makes an original contribution by investigating this unexplored area. Participants in this research included 104 primary school students aged between 8 and 11 years old and their teachers (n = 4) from four diverse primary schools in New Zealand. The research data is drawn from four main sources: audio recordings of classroom discussions, semi-structured focus group interviews with sample students, open-ended interviews with teachers, and student thinking journals. Over a period of 6 months, classes held weekly hour-long P4C dialogues with a focus on both philosophical content and developing critical thinking language and skills. Students were encouraged to evaluate their philosophical and critical thinking both verbally and in written form. The methodological approach incorporates sociocultural perspectives highlighting the key role of language as a mediator and tool for thinking (Mercer, 2000; Vygotsky, 1962, 1978). Transcripts of classroom discussions were analysed both inductively and deductively, using Hennessy et al.’s (2016) coding system for analysing classroom dialogue across educational contexts and Daniel et al.’s (2005) matrix outlining the development of the dialogical critical thinking process. Observation of students’ everyday learning in the classroom, together with their and their teachers’ reflections, revealed that teachers played a significant role in shaping students’ language and critical thinking through modelling, facilitating dialogue, establishing a democratic classroom culture, making thinking visible, and innovating and personalising teaching according to their students’ particular needs. Both students and teachers developed a range of capabilities through participating in (or, in the case of the teachers, leading) P4C dialogue: resilience, receptivity, intersubjectivity. This thesis presents one of the first investigations into the practice of teaching powerful knowledge in primary schools and suggests that explicit teaching of disciplinary language enhances students’ critical thinking capability and constitutes powerful knowledge. A new conception of powerful knowledge is advanced, termed enhancing knowledge, which emphasises how disciplinary knowledge enhances and is complementary to other knowledges, such as social or cultural knowledge, without devaluing their importance. This term also seeks to avoid the potential dominant connotations of the word ‘powerful’. The findings of this study suggest that P4C can offer diverse learners access to enhancing knowledge which has the potential to improve equity in education and enhance social justice.</p>

Recording Student Thinking in a Mathematics Discussion

Representing and recording student thinking in public spaces during mathematics discussions is challenging work. We share principles for recording student thinking in the moment and share an activity for improving your recording practice.

Designing for Framing in Online Teacher Education: Supporting Teachers’ Attending to Student Thinking in Video Discussions of Classroom Engineering

Participating in discussions of classroom video can support teachers to attend to student thinking. Central to the success of these discussions is how teachers interpret the activity they are engaged in—how teachers frame what they are doing. In asynchronous online environments, negotiating framing poses challenges, given that interactions are not in real time and often require written text. We present findings from an online course designed to support teachers to frame video discussions as making sense of student thinking. In an engineering pedagogy course designed to emphasize responsiveness to students’ thinking, we documented shifts in teachers’ framing, with teachers more frequently making sense of, rather than evaluating, student thinking later in the course. These findings show that it is possible to design an asynchronous online course to productively engage teachers in video discussions and inform theory development in online teacher education.

Preservice Teachers’ Eliciting and Responding to Student Thinking in Lesson Plays

This study presents an analysis of 95 lesson play scripts—hypothetical dialogues between the teacher and a student—written by 32 preservice teachers (PSTs). Writing lesson scripts was part of the assessment design activities to elicit and respond to students’ thinking. The findings present the types and frequencies of teacher talks/moves in fraction-related tasks during a stage of lesson plays, such as launch, active elicitation, and closure. Our analysis indicates a wide range in the number of turns taken by the PSTs, while there is little correlation between the number of turns and effectiveness at eliciting and responding to student thinking. The study also confirmed that some unproductive talk moves were still present in the lesson play context, although the PSTs had plenty of time to craft a script. This study drew implications of PSTs’ prior perceptions, experiences, knowledge, and needs in mathematics teacher education regarding the ways to create learning opportunities for them to elicit and respond to student thinking.

A Framework of Construct-Irrelevant Variance for Contextualized Constructed Response Assessment

Estimating and monitoring the construct-irrelevant variance (CIV) is of significant importance to validity, especially for constructed response assessments with rich contextualized information. To examine CIV in contextualized constructed response assessments, we developed a framework including a model accounting for CIV and a measurement that could differentiate the CIV. Specifically, the model includes CIV due to three factors: the variability of assessment item scenarios, judging severity, and rater scoring sensitivity to the scenarios in tasks. We proposed using the many-facet Rasch measurement (MFRM) to examine the CIV because this measurement model can compare different CIV factors on a shared scale. To demonstrate how to apply this framework, we applied the framework to a video-based science teacher pedagogical content knowledge (PCK) assessment, including two tasks, each with three scenarios. Results for task I, which assessed teachers’ analysis of student thinking, indicate that the CIV due to the variability of the scenarios was substantial, while the CIV due to judging severity and rater scoring sensitivity of the scenarios in teacher responses was not. For task II, which assessed teachers’ analysis of responsive teaching, results showed that the CIV due to the three proposed factors was all substantial. We discuss the conceptual and methodological contributions, and how the results inform item development.

The Skills of Mathematics Teachers with Different Professional Experiences to Notice the Evidence of Student Thinking

There are different types of evidence that reflect students' thinking in classroom interactions. Student discourse, gestures, actions can be shown among these. The aim of the current study is to reveal the skills of secondary mathematics teachers with different professional experiences to notice different types of evidence of student thinking. For this aim, the study was designed within the context of a case study, one of the qualitative research methods. The data of the study were obtained through video-based interviews with five secondary mathematics teachers. Six video episodes containing different types of evidence for video-based interviews were shown to teachers. In the interviews, targeted questions were asked to reveal what types of evidence the participating teachers took into account in the videos they watched, how they interpret these types of evidence, and what kind of instructional decisions they suggested. The types of evidence that teachers noticed in the videos and how they made sense of the evidence were analyzed qualitatively. Various findings have been revealed depending on the different professional experiences of the teachers. First, experienced (5 years and above) teachers paid more attention to the types of verbal evidence of student thinking. Student statements and questions were more visible to experienced teachers, especially among oral evidence. Second, the evidence-based comments described by the experienced teachers were aimed at drawing more conclusions. However, it is noteworthy that teachers who make inferences based on student's cognitive thinking also attend postgraduate education or professional experience courses. Third, the instructional decisions of experienced teachers who attended postgraduate education and professional experience courses, based on student thinking, were mostly related to specific mathematical subjects.