We investigate proving activities from a monistic embodied perspective, namely the functional dynamic systems (FDS) approach. Using proofs without words, we focus on proving activities that can be conceptualized as identifying and filling gaps. Using a microethnographic methodology enhanced by eye-tracking, we investigate sensory-motor processes…

This article offers the construct "unitizing predicates" to name mental actions important for students' reasoning about logic. To unitize a predicate is to conceptualize (possibly complex or multipart) conditions as a single property that every example has or does not have, thereby partitioning a universal set into examples and…

Many mathematics educators believe a goal of instruction is for students to obtain conviction and certainty in mathematical statements using the same types of evidence that mathematicians do. However, few empirical studies have examined how mathematicians use proofs to obtain conviction and certainty. We report on a study in which 16 advanced…

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were…

To design and improve instruction in mathematical proof, mathematics educators require an adequate definition of proof that is faithful to mathematical practice and relevant to pedagogical situations. In both mathematics education and the philosophy of mathematics, mathematical proof is typically defined as a type of justification that satisfies a…

We report on an innovative design of algorithmic analysis that supports automatic online assessment of students' exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic…

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration, serving as an effective tool for solving counting problems and underlying many key combinatorial formulas. In this study, we used guided reinvention to investigate 2 undergraduate students' reasoning about the MP, and we sought to answer the following research…

Supporting students in making mathematical arguments is an important part of discourse practices in mathematics classrooms. Differences in teachers' support for collective argumentation have been observed and documented, and the importance of the teacher's role in supporting collective argumentation is well established. This article seeks to…

This study examined the genre of undergraduate mathematical proof writing by asking mathematicians and undergraduate students to read 7 partial proofs and identify and discuss uses of mathematical language that were out of the ordinary with respect to what they considered conventional mathematical proof writing. Three main themes emerged: First,…

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…

Research has demonstrated that textbooks exert a considerable influence on students' learning opportunities and that technology has the potential to transform mathematics instruction. This brief report provides a systematic analysis of how technology tasks are integrated into secondary mathematics curricula by analyzing a sample of 20 textbooks.…

This article documents differences between novice and experienced undergraduate students' processes of reading mathematical proofs as revealed by moment-by-moment, think-aloud protocols. We found three key reading behaviors that describe how novices' reading differed from that of their experienced peers: alternative task models, accrual of…

We present the evolving fraction conceptions of two elementary school children with mathematics learning disabilities (MLD). We use qualitative analyses to capture the mathematical knowledge and experiences of each child and show how teaching was used to support advancement of their fractional reasoning. Results illustrate two viable pathways of…

Responsiveness to students' mathematical thinking is a characteristic of classroom discourse that reflects the extent to which students' mathematical ideas are present, attended to, and taken up as the basis for instruction. Using the Mathematically Responsive Interaction (MRI) Framework and data from 11 middle-grades classrooms, we illustrate…

We investigated the extent to which one elementary school child with working-memory differences made sense of number as a composite unit and advanced her reasoning. Through ongoing and retrospective analysis of eight teaching-experiment sessions, we uncovered four shifts in the child's real-time negotiation of number over time: (a) initial…