Mathematics Teachers as Facilitators of Productive Dialogue: Designing Dialogic Situations, Resulting Dialogue Patterns, and Pathways of Professional Development

in small groups is not widely practiced in middle and high school mathematics classrooms (Schwarz & Baker, 2017). A key reason lies in teachers' difficulty in identifying and creating mathematical tasks that both stimulate meaningful dialogue and align with their pedagogical needs and classroom constraints (Blatchford et al., 2003; Crespo & Harper, 2020; Kaur & Chin, 2022).
This study positions the mathematical task as a key resource for facilitating productive dialogue, and views teacher partnership in task adaptation and design as the starting point for developing teachers' agency regarding its implementation (Jones & Pepin, 2016). A preliminary pilot study examined the potential of "Who is right?" tasks to generate productive dialogue among students, and its positive findings led to focusing on these types of tasks in the main study. The research combines examining teachers' perspectives on designing dialogic tasks with an in-depth analysis of student dialogues around "Who is right?" tasks - a format teachers acknowledged as having high dialogic potential
The research addresses three objectives:

1. examining teachers' perspectives on designing and implementing dialogic situations,
2. investigating the relationship between task characteristics and classroom dialogues, along with teacher support,
3. studying the professional development of teachers participating in a task-based intervention program promoting dialogic teaching (task-based PD).

Accordingly, three research questions were formulated:

1. Within a task-based PD program, how do teachers propose to promote "productive dialogue" through mathematical tasks?
2. How do different "Who is right?" tasks create varied dialogic situations among students in small groups, and how do teacher actions influence these situations?
3. What professional development trajectories were observed among teachers, and how did components of the PD program encourage this development?

The study was conducted within the Dialogos project - an inter-university center promoting dialogic pedagogy in schools, established through collaboration among five leading education researchers. The project spanned four annual cycles of PD programs (2018-2021), engaging teachers across multiple subject areas. The mathematics group, led by the researcher, comprised 26 middle and high school mathematics teachers.
The study employed a multi-layered qualitative research design. Data was collected from PD sessions, classroom observations, reflective interviews, and written materials including lesson plans, reflection forms, and teachers' final assignments. The analysis proceeded in several phases. First, to identify teachers' considerations in designing dialogic tasks, I analyzed 60 teacher-proposed tasks along with their justifications regarding dialogic potential and implementation strategies. The theoretical models emerging from this analysis then guided the investigation of relationships between task characteristics and classroom dialogues. This phase combined two analytical approaches: examining tasks' mathematical and pedagogical features, and analyzing student interactions in small groups. Additionally, comprehensive case studies of five teachers traced professional development trajectories throughout the program.
The findings identify three central dialogic triggers in mathematical tasks from teachers' perspectives: cognitive conflict arising from exposure to common errors and creating confrontation between contradicting mathematical ideas, multiple solutions or solution methods that invite sharing of different perspectives, and mathematical complexity requiring knowledge sharing and shared Inquiry for problem resolution. The study presents a model of teachers' considerations in designing dialogic situations, encompassing both macro-level considerations for collaborative argumentation and micro-level considerations for specific dialogic triggers and supporting actions. The study offers a theoretical model defining three prototypes of dialogic situations, describing the dialogic trigger, dialogue type, and expected learning outcomes.
The findings reveal clear connections between task characteristics, teacher actions, and resulting classroom dialogue. Notably, tasks with internal verification mechanisms, allowing students to independently validate solutions, significantly promote sustained productive dialogue. These mechanisms, combined with other task features, guide teachers in determining appropriate interventions during group work. Additionally, dialogic instructions promoting group consensus and inter-group persuasion enhanced students' handling of mathematical and dialogic challenges, creating a spiral mechanism where argument quality increases with expanding audience (self, group, class). The study concludes by proposing design principles for "Who is right?" tasks.
Analysis of teachers' professional development revealed growth in three areas: selection and design of dialogic tasks, understanding and addressing dialogue, and classroom implementation. The findings emphasize the importance of integrating various activities affecting four different developmental dimensions: orientations, goals, resources (ROG - Schoenfeld, 2011) and practices. Furthermore, iterative cycles of design, implementation, and reflection, supported by ongoing personalized guidance, proved essential. Pedagogical models developed within this framework helped teachers identify dialogic tasks, recognize dialogue patterns, and set advanced dialogic goals. The study characterizes subtle processes in teachers' professional development while participating in the task-based program.
This study contributes to dialogic mathematics education theory, methodology, and practice. Theoretically, it expands understanding of productive mathematical dialogue by integrating practitioners' perspectives with research insights. The study advances argumentative design knowledge through design principles for "Who is Right?" tasks, grounded in analysis of authentic classroom dialogues. Methodologically, it offers new analytical frameworks for examining relationships between task characteristics and resulting dialogues. Practically, it provides concrete tools supporting teachers in designing dialogic tasks and understanding their impact on learning outcomes. These resources may benefit teacher educators promoting dialogic pedagogy in PD settings. The study demonstrates the value of bridging theoretical frameworks with practical applications, showing their mutual enhancement. The insights derived from this research hold potential for adaptation across various academic disciplines beyond mathematics, with appropriate subject-specific modifications.