A main feature of mathematics teachers’ professional practice is their actions in the classroom, namely while conducting whole class discussions. These classroom moments often take place after students’ work on challenging tasks and may include students’ presentation of unexpected answers and strategies. The role of the teacher is to articulate those answers and conduct a discussion that lead students to a deeper understanding of mathematical ideas (Stein, Engle, Smith, & Hughes, 2008). The study discussed in this paper aims to understand teacher’s actions while conducting a whole class discussion regarding mathematical sequences.
With Jaworski and Potari (2009), we consider teachers’ practice to be the activity developed by the teacher, which unfolds in teachers’ actions defined according to an action plan (Schoenfeld, 2000). A particularly challenging feature of teachers’ practice are their actions in conducting whole class discussions and this has been addressed by several authors with various lenses. For example, Wood (1999) focuses on students’ participation, highlighting the relevance of exploiting students’ disagreements by making them to present their thinking and ideas and also to listen to others and to discuss their ideas. These “instances of disagreement arise from the diverse ideas generated by children” (p. 172), enhancing their learning potential as students intertwine their knowledge with the knowledge of their colleagues. Potari and Jaworski (2002) emphasize how teachers’ questions are posed and distinguish different degrees of challenge in questions during discussions. Stein et al. (2008) center their ideas on pulling together students’ thinking and ideas and shaping those into powerful and precise mathematical ideas. They also present a model to prepare and conduct mathematical discussions which include five subsequent elements: (i) anticipating likely students’ responses; (ii) monitoring students’ responses; (iii) selecting students to present their responses; (iv) sequencing students’ responses; and (v) making connections between students’ responses and key mathematical ideas. Cengiz, Kline, and Grant (2011) argue that there three main types of teachers’ actions: eliciting students to present their methods, supporting students’ global conceptual understanding, and extending students’ thinking. Similarly, Scherrer and Stein (2013) analyze teachers’ actions (moves) after proposing a cognitively demanding task, according to four categories: (i) beginning a discussion, (ii) elaborating or deepen students’ knowledge, (iii) eliciting information, and (iv) making other moves, as thinking aloud or providing information.
In a previous study, Ponte, Mata-Pereira and Quaresma (2013) distinguish between teachers’ actions during whole class discussions concerned with the management of learning or addressing mathematical topics and processes. In this last group they consider inviting actions which lead students to engage in the discussion. During the discussion they distinguish three main categories: (i) guiding or supporting actions, that conduct students in an implicit or explicit way in order to continue the discussion, (ii) informing or suggesting actions, that introduce information or validate students’ interventions, and (iii) challenging actions, that lead students to add information, provide an argument or evaluate an argument or a solution. Informing/suggesting actions and challenging actions are quite similar in the information they provide, nevertheless, the latest is the pith of exploratory classes while the first, despite its important role in those classes, is more teacher-centered. These three kinds of actions support the development of whole class mathematical discussions and encompass key mathematical processes, such as (a) representing, that include providing, revoicing, using and changing a representation (including procedures), (b) interpreting, that include interpreting a statement or idea and making connections, (c) reasoning, that include raising a question about a claim or justification, generalizing a procedure, a concept or a property, justifying and providing an argument, and (d) evaluating, that include making judgments about a method or solution and comparing different methods.
