Opening tasks, opening minds – a rediscovery of the open-ended approach

Open-ended tasks are designed such that they may have more than one correct solution or may be solved in more than one way. Although such tasks constitute an irreplaceable tool for prompting young learners to be actively and creatively involved in mathematical discourse, their implementation poses a challenge. Primary school students in Poland are usually offered algorithmic and rote teaching methods and are thus very often deprived of important elements of successful mathematics learning. The ubiquitous teacher-centred approach dedicates little time to any contribution from learners. The aim of this design research was to implement a change in early childhood mathematics education. The change comprised students creating and/or solving open-ended tasks in groups, thus promoting dialogic teaching. The results confirmed that students who are challenged with openended tasks through dialogic teaching not only genuinely engage in their activities, develop a better number sense and flexibility of thinking, but also help each other gain a deeper understanding of new concepts. Captured in this research were the synergistic images of the beauty of children’s reasoning and the beauty of mathematics as an open subject – an incentive for others to begin their journey with freedom of speech for young mathematicians.

Supporting All Students: Productive Mathematical Discourse in Online Environments

Classroom instruction focused on discussion-based learning opportunities can provide productive and inclusive learning experiences for all students, including students with learning disabilities in mathematics and those without learning disabilities. Mathematical discourse allows students to share their ideas, justify their thinking, critique the reasoning of others, and refine their thought processes. While one might typically envision mathematical discourse happening during face-to-face instruction, meaningful discourse can also occur in online learning environments. This article presents a blended format of both synchronous and asynchronous learning opportunities, coupled with Smith and Stein’s (2018) “5 Practices” for productive mathematical discourse to support teachers in designing and facilitating lessons in which all students are actively engaged in the learning processes both for themselves and their classmates.

Exploring a Model to Develop Critical Reflective Thought Among Elementary School Math Preservice Teachers

This study focuses on the challenge of promoting significant mathematical discourse among preservice teachers. The study was conducted as part of an academic course that accompanies their practical training. Twenty-three math preservice teachers’ learning process was examined as a result of using an analytic model designed for discourse protocols’ analysis. Findings revealed that an active and dynamic process occurred, modifying teacher practice and developing critical reflective thinking among preservice teachers. The change occurred in two “ripples of influence”: (1) improving discourse to one promoting learning by demonstrating hypothetical scenarios and (2) perception of the role of teachers and class management.

University Mathematics Lecturing as Modelling Mathematical Discourse

The lecture format, while being the subject of much criticism, is still one of the most common formats of university mathematics teaching. This paper investigates lecturing as a means of modelling mathematical discourse, sometimes highlighted in the literature as one of its most important functions. The data analysed in the paper are taken from first-semester lectures given by seven mathematics lecturers at three Swedish universities, all concerning various aspects of the function concept. Analysis was carried out from a commognitive perspective, which distinguishes between object-level and meta-level discourse. Here I focus on two aspects of meta-level discourse: introducing new mathematical objects; and what counts as valid endorsement of a narrative. The analysis reveals a number of metarules concerning the modelling of mathematical reasoning and behaviour, both more general rules such as precision and consensus, and rules more specifically concerning construction and endorsement of narratives. The paper contributes to a small but growing body of empirical research on university mathematics teaching, and also lends empirical support to previous claims about the modelling aspect of mathematics lecturing, thus contributing to a deepened understanding of the lecture format and its potential role in future university mathematics teaching.