This article presents an inquiry-oriented lesson for teaching Lagrange's theorem in abstract algebra. This lesson was developed and refined as part of a larger grant project focused on how to "Orchestrate Discussions Around Proof" (ODAP, the name of the project). The lesson components were developed and refined with attention to how well…

In recent years, professional organizations in the United States have suggested undergraduate mathematics shift away from pure lecture format. Transitioning to a student-centered class is a complex instructional undertaking especially in the proof-based context. In this paper, we share lessons learned from a design-based research project centering…

Recent research in university mathematics education has moved beyond the traditional focus on the transition from secondary to tertiary education and students' understanding of introductory courses such as pre-calculus and calculus. There is growing interest in the challenges students face as they move into more advanced mathematics courses that…

In this paper we develop a case for introducing a new teaching tool to undergraduate mathematics. Lean is an interactive theorem prover that instantly checks the correctness of every step and provides immediate feedback. Teaching with Lean might present a challenge, in that students must write their proofs in a formal way using a specific syntax.…

This research article is about "Introducing a Supportive Framework to Address Students' Misconceptions and Difficulties in Learning Mathematical proof techniques (MPT): A Case of Debark University". This study aims to develop, introduce, and implement a supportive framework to overcome students' misconceptions and difficulties in MPT.…

The purpose of this study is to examine the performances of university students' using dynamic mathematics software GeoGebra in argumentations and proving processes. A task related to the limit involving "sinx/x" was designed and 18 university students worked on the task during the collaborative learning, scientific debate, and…

Many mathematics departments have transition to proof (TTP) courses, which prepare undergraduate students for proof-oriented mathematics. Here we discuss how common TTP textbooks connect three topics ubiquitous to such courses: logic, proof techniques, and sets. In particular, we were motivated by recent research showing that focusing on sets is…

This theoretical paper sets forth two "aspects of predication," which describe how students perceive the relationship between a property and an object. We argue these are consequential for how students make sense of discrete mathematics proofs related to the properties and how they construct a logical structure. These aspects of…

In this position paper, we propose commognition for the study of proof teaching at university lectures through an integrative literature review. We critically examine studies that focused on proof teaching but did not use the commognitive framework. Through this examination, we gain an understanding of the pedagogical aspects of proof teaching and…

Coming from the commognitive standpoint, we consider proof-based mathematics as a distinct discourse, the transition to which requires special rules for endorsement and rejection of mathematical statements. In this study, we investigate newcomers' learning of these rules when being taught them explicitly. Our data come from academically motivated…

The aim of this study is to characterize ways of reasoning and arguing that first year university mathematics students exhibit in problem-solving activities from a course that emphasizes the importance of formulating conjectures and the search for different ways to support or validate them. The use of a Dynamic Geometry System in the…

The ability to distinguish between exact and inexact differentials is an important part of solving first-order differential equations of the form Adx + Bdy = 0, where A(x,y) [not equal to] 0 and B(x,y) [not equal to] 0 are functions of x and y However, although most undergraduate textbooks motivate the necessary condition for exactness, i.e. the…

Knowledge assessments in undergraduate mathematics education commonly evaluate response correctness to determine learner proficiency. However, simultaneous evaluation of learner metacognition more accurately assesses the multiple dimensions of knowledge and has been shown to increase assessment validity and reliability. Research into…

While proof has been studied from different perspectives in the mathematics education literature for decades, students continue to struggle to build proof comprehension. Complicating this, the manner in which proof comprehension is assessed largely remains to be the definition-theorem-proof format in which students are asked to reproduce proofs or…

The aim of the study was to investigate the validity level of Socrates-based linear algebra e-module, both material validity and design. This is a research and "development" (R&D) with ADDIE procedure: analysis, design, develop, implement, and evaluate. The participants were 30 students and 2 lecturers who supervised linear algebra…