The use of carefully planned board work to support the productive discussion of multiple student responses in a Japanese problem-solving lesson

In this paper, we analyse a grade 8 (age 13–14) Japanese problem-solving lesson involving angles associated with parallel lines, taught by a highly regarded, expert Japanese mathematics teacher. The focus of our observation was on how the teacher used carefully planned board work to support a rich and extensive plenary discussion (neriage) in which he shifted the focus from individual mathematical solutions to generalised properties. By comparing the teacher’s detailed prior planning of the board work (bansho) with that which he produced during the lesson, we distinguish between aspects of the lesson that he considered essential and those he treated as contingent. Our analysis reveals how the careful planning of the board work enabled the teacher to be free to explore with the students the multiple alternative solution methods that they had produced, while at the same time having a clear overall purpose relating to how angle properties can be used to find additional solution methods. We outline how these findings from within the strong tradition of the Japanese problem-solving lesson might inform research and teaching practice outside of Japan, where a deep heritage of bansho and neriage is not present. In particular, we highlight three prominent features of this teacher’s practice: the detailed lesson planning in which particular solutions were prioritised for discussion; the considerable amount of time given over to student generation and comparison of alternative solutions; and the ways in which the teacher’s use of the board was seen to support the richness of the mathematical discussions.

Compare and Discuss Multiple Strategies

CDMS is a routine that allows teachers to organize instruction around students’ mathematical discussions and multiple problem-solving methods.

Enacting Number Talks in a Simulated Classroom Environment: What Do Preservice Teachers Notice About Students?

Number talks are short mathematical discussions offering sensemaking opportunities for students. Aside from bolstering students’ mathematical learning, this instructional routine may also support preservice teachers (PSTs) in investigating how to facilitate discussion-focused instruction. In this study, PSTs engage in a learning cycle to explore, plan and rehearse two separate number talks during human-in-the-loop simulations, and then reflect on these experiences. During the first simulation, PSTs focus on understanding the routine’s components while positioning avatar-students as sensemakers as they elicit their participation. In the second simulation, PSTs build their instructional skills as they record representations of students’ mathematical thinking, probe students’ thinking in order to make mathematics visible, as well as notice missed opportunities to support students’ mathematical reasoning during reflections of their experiences. Implications of this study suggest that simulations, when embedded within a cycle of enactment and reflection, support PSTs in developing professional noticing skills.

Promoting Discourse: Fractions on Number Lines

Examine fourth graders’ thinking about the unit, partitioning, order, and equivalence on the number line and consider ways to orchestrate mathematical discussions through the Five Practices.

Combinatory thinking: an approach from the password game

The purpose of this article is to present the paths outlined for the development of combinatorial reasoning and the fundamental principle of counting through problem situations arising from the game. The conception of this work arose from the contact with games at the Mathematics Teaching Laboratory (LEM) and from discussions about new teaching methodologies in the Supervised Internship IV discipline of the Full Mathematics Degree course at the State University of Maringá. In this way, in partnership with LEM, this activity proposal was developed, whose target audience was students from the 8th and 9th years of Elementary School. The Password Game was used as a methodological resource, in order to work the multiplicative principle in a playful way and different from the traditional one. The realization of this activity with the students made us realize that it is possible, through the use of games, to trigger very rich mathematical discussions, as well as to promote learning situations in the classroom.

A Framework for Investigating Qualities of Procedural and Conceptual Knowledge in Mathematics—An Inferentialist Perspective

This study introduces inferentialism and, particularly, the Game of Giving and Asking for Reasons (GoGAR), as a new theoretical perspective for investigating qualities of procedural and conceptual knowledge in mathematics. The study develops a framework in which procedural knowledge and conceptual knowledge are connected to limited and rich qualities of GoGARs. General characteristics of limited GoGARs are their atomistic, implicit, and noninferential nature, as opposed to rich GoGARs, which are holistic, explicit, and inferential. The mathematical discussions of a Grade 6 class serve the case to show how the framework of procedural and conceptual GoGARs can be used to give an account of qualitative differences in procedural and conceptual knowledge in the teaching of mathematics.

UNIVERSISM AND EXTENSIONS OF V

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favor of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in ‘ideal’ outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist.