Using teaching representations is one of the common means for preparing pre-service mathematics teachers (PSTs) to learn how to teach mathematics meaningfully. However, turning PSTs' attention to the mathematical objects that students engage with during the lesson is a challenging pursuit. This paper suggests using the Realization Tree Mediator…

This article uses a cultural-developmental framework to illuminate the interplay between collective and individual activity in the mathematical reasoning displayed in a university Masters level lesson on fractals. During whole class and small group discussions, eleven students, guided by an instructor, engage in inductive reasoning about the area…

What makes a productive mathematics discussion? Who does the talking? Whose ideas are taken up? Can a student question take the lesson in an unexpected but valuable direction? What does it mean to be responsive to students' mathematical thinking? These questions intrigued the authors as teachers, teacher educators, and researchers, and they…

A discussion forum is a valuable tool to support student learning in online contexts. However, interactions in online discussion forums are sparse, leading to other issues such as low engagement and dropping out. Recent educational studies have examined the affordances of conversational agents (CA) powered by artificial intelligence (AI) to…

Learning to facilitate whole-class mathematical discussions is a complex task. Teachers must become adept at using discourse moves to elicit and extend student thinking and move discussions towards a mathematical point. Practice-based mathematics methods courses give pre-service teachers opportunities to rehearse these moves in a controlled…

Issues of equity in mathematics classrooms existed prior to COVID-19. For many students, however, meaningful participation in mathematical discussions became nearly impossible in online settings during the pandemic. In this study, we note the diversity in and nature of participation in mathematical discourse in an online course for preservice…

A discussion forum is a valuable tool to support student learning in online contexts. However, interactions in online discussion forums are sparse, leading to other issues such as low engagement and dropping out. Recent educational studies have examined the affordances of conversational agents (CA) powered by artificial intelligence (AI) to…

This study explores the relationship between students' perceptions and teachers' discourse practices in mathematics classrooms. It reframes the sequence of Initiate-Response-Follow-up (IRF) with a renewed discourse structure that focuses on teachers' follow-up actions including listening, thoughtful questioning, and effective talk moves.…

Language, which plays a special role for the learning of mathematics, is investigated in this article for the specific discourse practice of explaining during whole-class discussions: On the one hand, explaining is a medium for learning since school cannot be thought of without communication. On the other hand, students at the beginning of…

One of the challenges of understanding the complexity of so-called reform mathematics instruction lies in the observational tools used to capture it. This article introduces a unique tool, drawing from commognitive theory, for describing classroom discussions. The Realization Tree Assessment tool provides an image of a classroom discussion,…

From a commognitive approach, this article focuses on the discourse generated by preservice primary teachers who are solving didactic-mathematical tasks. Our aims are to study the characteristics of the aforementioned discourse and, through these characteristics, identify whether a discourse close to the one of primary teachers is beginning to…

This study employs interviews and observations to investigate instructional rationales of two purposefully sampled teachers with divergent classroom discourse practices in Swedish-speaking Finnish lower secondary mathematics classrooms. Studies on classroom discourse often point to beliefs and contextual factors shaping teachers' discourse…

Learning by invention is an alternative approach to teaching statistics where students are tasked with attempting to solve a problem before being taught the canonical formula for solving it, often resulting in increased understanding of material compared with traditional instruction. The first study, conducted in a college statistics classroom…

Extensive research has been published on the nature of classroom mathematical discourse and on the impact of technology tools, such as virtual manipulatives (VM), on students' learning, while less research has focused on how technology tools facilitate that mathematical discourse. This paper presents an emerging construct, the Techno-Mathematical…

This study investigates the nature and role of common ground in group learning of mathematics by means of the analytical constructs of "focal projects" and "contextualization." The analysis investigates two students (12-13 years old) playing a dice game, where their task is to distribute a set of markers based on the total of…