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Cody L. Patterson; Paul Christian Dawkins; Holly Zolt; Anthony Tucci; Kristen Lew; Kathleen Melhuish – PRIMUS, 2024
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…
Descriptors: Mathematics Instruction, Algebra, Validity, Mathematical Logic
Michael D. Hicks – PRIMUS, 2024
Analogy has played an important role in developing modern mathematics. However, it is unclear to what extent students are granted opportunities to productively reason by analogy. This article proposes a set of lessons for introducing topics in ring theory that allow students to engage with the process of reasoning by analogy while exploring new…
Descriptors: Mathematics Instruction, Mathematical Logic, Logical Thinking, Algebra
Samuel B. Allan; Peter K. Dunn; Robert G. McDougall – International Journal of Mathematical Education in Science and Technology, 2024
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…
Descriptors: Mathematics Instruction, Teaching Methods, Multiplication, Addition
Geena Taite; Helene Leonard; Amanda Provost; Nicole Panorkou – Mathematics Teacher: Learning and Teaching PK-12, 2024
It has been over thirty years since the nuclear reactor meltdown at the Chernobyl nuclear power plant, but why there is still an officially designated exclusion zone? The Chernobyl Disaster Task combines the learning of exponential functions with properties of radioactive substances to help students understand the ongoing effects of the meltdown.…
Descriptors: Radiation, Nuclear Energy, Mathematical Models, Mathematical Logic
Laudano, Francesco – International Journal of Mathematical Education in Science and Technology, 2021
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…
Descriptors: Mathematics Instruction, Division, Algebra, Mathematical Logic
Andriunas, R.; Boyle, B.; Lazowski, A. – PRIMUS, 2022
This paper discusses a project for linear algebra instructors interested in a concrete, geometric application of matrix diagonalization. The project provides a theorem concerning a nested sequence of tetrahedrons and scaffolded questions for students to work through a proof. Along the way students learn content from three-dimensional geometry and…
Descriptors: Algebra, Geometry, Matrices, Mathematics Instruction
Margaret Walton; Janet Walkoe – Mathematics Teacher: Learning and Teaching PK-12, 2025
Seeds of Algebraic Thinking comes from the Knowledge in Pieces (KiP) perspective of learning. KiP is a systems approach to learning that stems from the constructivist idea that people learn by building on prior knowledge. As people experience the world, they acquire small, sub-conceptual knowledge elements. When people engage in a particular…
Descriptors: Mathematics Instruction, Prior Learning, Knowledge Level, Algebra
Oxman, Victor; Sigler, Avi – International Journal of Mathematical Education in Science and Technology, 2021
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.…
Descriptors: Geometry, Mathematics Instruction, Validity, Mathematical Logic
Holton, Derek; Symons, Duncan – Australian Primary Mathematics Classroom, 2021
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…
Descriptors: Computer Mediated Communication, Mathematical Logic, Validity, Algebra
Hawthorne, Casey; Gruver, John – Mathematics Teacher: Learning and Teaching PK-12, 2023
The ability to interpret mathematical symbols and understand how they capture contextual relationships is a critical element of algebraic thinking. More often than not, students see algebra as merely a list of rules for manipulating abstract symbols, with limited to no meaning. Instead, for students to see algebra as a powerful tool and rich way…
Descriptors: Algebra, Symbols (Mathematics), Mathematical Logic, Mathematics Skills
Gkioulekas, Eleftherios – International Journal of Mathematical Education in Science and Technology, 2020
We review the history and previous literature on radical equations and present the rigorous solution theory for radical equations of depth 2, continuing a previous study of radical equations of depth 1. Radical equations of depth 2 are equations where the unknown variable appears under at least one square root and where two steps are needed to…
Descriptors: Problem Solving, Equations (Mathematics), Mathematical Concepts, Mathematical Logic
Melhuish, K.; Lew, K.; Hicks, M. – PRIMUS, 2022
Connecting and comparing across student strategies has been shown to be productive for students in elementary and secondary classrooms. We have recently been working on a project converting such practices from the K-12 level to the undergraduate classroom. In this paper, we share a particular instantiation of this practice in an abstract algebra…
Descriptors: Mathematics Instruction, Teaching Methods, Best Practices, Algebra
Sandefur, James; Manaster, Alfred B. – ZDM: Mathematics Education, 2022
Recursive reasoning is a powerful tool used extensively in problem solving. For us, recursive reasoning includes iteration, sequences, difference equations, discrete dynamical systems, pattern identification, and mathematical induction; all of these can represent how things change, but in discrete jumps. Given the school mathematics curriculum's…
Descriptors: Abstract Reasoning, Problem Solving, Mathematical Logic, Logical Thinking
Gabour, Manal – International Journal of Mathematical Education in Science and Technology, 2022
In this article special sequences involving the Butterfly theorem are defined. The Butterfly theorem states that if M is the midpoint of a chord PQ of a circle, then following some definite instructions, it is possible to get two other points X and Y on PQ, such that M is also the midpoint of the segment XY. The convergence investigation of those…
Descriptors: Mathematics Instruction, Computer Software, Secondary School Mathematics, College Mathematics
Melhuish, Kathleen – International Journal of Research in Undergraduate Mathematics Education, 2019
Introductory group theory provides the foundational course on algebraic structures. Yet, we know little about students' underlying conceptual understandings. In this paper, I introduce the Group Theory Concept Assessment (GTCA), a measure created for the purpose of conducting large-scale studies of student conceptual understanding in group theory.…
Descriptors: Algebra, Mathematical Concepts, Mathematical Logic, Knowledge Level