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 this study, we aimed to find problematic and supportive issues in Afghan undergraduate students' proofs of the irrationality of [square root]3 and [square root]5/8 while using the indirect proof method. Collecting and analyzing produced proofs of 30 sophomore and 48 senior undergraduate students on the irrationality of [square root]3 and…

In this note we demonstrate two instances where matrix multiplication can be easily verified. In the first setting, the matrix product appears as matrix element concatenation, and in the second, the product coincides with matrix addition. General proofs for some results are provided with a more complete description for 2×2 matrices. Suggested for…

Proof comprehension is an important skill for students to develop in their proof-based courses, yet students are rarely afforded opportunities to develop this skill. In this paper, we describe two implementations of an activity structure that was developed to give students the opportunity to engage with complex proofs and to develop their proof…

We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the previous results to determine the quadratic factors of polynomials over commutative coefficient rings and, in particular, to completely factorize in Z[x] any…

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.…

"Epistemological justification" is a way of thinking that manifests itself through perturbation-resolution cycles revolving around the question "why and how was a piece of mathematical knowledge conceived?" The paper offers a conceptual framework for constituent elements of epistemological justification. The framework provides:…

This study aimed to describe students' ability to arrange proof of abstract algebra problems (Subgroups) after receiving Information Technology (IT)-assisted scaffolding materials. To achieve this objective, we conducted exploratory qualitative research with students of Mathematics Education, Universitas PGRI Ronggolawe Tuban, participating in the…

Examining the narratives of algebra content of three popular series of mathematics textbooks in China, this study explored the opportunities for students to learn about reasoning and proof (RP). In this study, we incorporated Davis's subdivision of conjecture into Stylianides's framework. Based on this, we analysed the components of RP (patterns,…

In this article we consider two triangles: one inscribed in another. We prove that the area of the central triangle is at least the harmonic mean of the areas of corner triangles. We give two proofs of this theorem. One is based on Rigby inequality and the other is based on the known algebraic inequality, to which we bring a new, geometric, proof.…

This qualitative study aims to investigate novice undergraduate mathematics students' first encounter with the First Isomorphism Theorem, which is, more often than not, the pinnacle of a typical introductory course in Group Theory. Several studies have reported on the challenges that this mathematical result poses to inexperienced mathematicians,…

Teachers need ways to efficiently assess students' cognitive understanding. One promising approach involves easily adapted and administered item types that yield quantitative scores that can be interpreted in terms of whether or not students likely possess key understandings. This study illustrates an approach to analyzing response process…

As a follow-up to their article, "Emojis and Their Place in the Mathematics Classroom" (EJ1358586), the authors examine how emojis can be used as bridging representations to support student understanding of proof and algebra in upper primary school. They take a problem from reSolve, Level 3, (AAMT, 2020), look at it from the perspective…

This study examined opportunities provided for preservice secondary mathematics teachers (PSMTs) to learn reasoning and proof in algebra from the perspective of college instructors. We analyzed interview transcripts of 15 course instructors recruited from three teacher education programs in the United States. We examined the reported opportunities…

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…