We provide a gentle alert that the standard definitions of basic algebraic structures--such as that of a group--contain aspects that may be questionable to students, and discuss what instructors could do to minimise the questionability.

Using the basic properties of the base-b representation of rational numbers, we will give an elementary proof of Gauss's lemma: "Every real root of a monic polynomial with integer coefficients is either an integer or irrational." The paper offers a new perspective in understanding the meaning of 'irrational numbers' from a deeper…

Eigentheory concepts are central in mathematics and physics; they serve multiple functions, such as symbolizing physical phenomena and facilitating mathematical computations. Words and meanings associated with eigentheory develop and vary over time, as do their associated symbols. In this study, we investigate how "eigen" develops over…

The concept of function is critical in mathematics in general and abstract algebra in particular. We observe, however, that much of the research on functions in abstract algebra (1) reports widespread student difficulties, and (2) focuses on specific types of functions, including binary operation, homomorphism, and isomorphism. Direct, detailed…

The number line has been proposed as a central construct used by students to solve a range of mathematics problems. Given the capacity of number lines to represent all real numbers and to be used in a variety of contexts, there have been calls to increase the use of number lines in mathematics instruction. However, due to the recency of these…

Understanding the concept of completeness for an ordered field is known to be difficult for many university mathematics students. We hypothesise that the variety of possible axioms of completeness for the set of real numbers is one of the sources of difficulties as is the lack of understanding of the "raison d'être" of these axioms. In…

This paper initiates a teaching sequence that focuses on building up equivalent definitions to the standard ones for the limit concept in Real Analysis. It comprises two parts: The first provides a classroom assignment where students, guided by Analysis lecturers, are led to develop an alternative definition to the [epsilon] - [delta] one for…

We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier…

The purpose of this study is to conduct a bibliometric analysis of the research published in the field of mathematics misconception from 1947 to 2023, to determine the general knowledge structure and participation in research publication. An analytical approach was used based on Scopus database data. This study used mixed methods; quantitative…

This paper reports on students' conceptions of minima points. Written assignments and individual interviews uncovered salient, concept images, as well as erroneous "mis-out examples" that mistakenly regard examples as non-examples and "mis-in examples" that mistakenly grant non-examples the status of examples. We used Tall and…

Number lines can benefit students in learning an array of mathematical concepts. An area of mathematics where number lines are visibly underused is in teaching measurement concepts. For students in upper elementary grades, accurate measurements require the use of mathematical precision and coordination, including skills in fractions and decimals,…

Purpose: This research lies in the domain of Vedic mathematics, and it explores the application of the related Vedic sutras in different branches of mathematics, science, education, and engineering across Asia and Europe. Design/methodology/approach: The study embraced a qualitative research design followed by a systematic literature review (SLR)…

Beyond mathematical complexity, a proof's length may, in and of itself, impede its comprehension. The same would apply to constructions, calculations and other mathematical expositions. Today's technology provides readers websites and electronic documents with hyperlinks, giving readers direct access from one location of the exposition to a…

Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of…

This article presents a theoretical-didactic perspective on the formation and development of concepts concerning mathematical objects as a result of research on didactic difficulties in dealing with concepts in school. The theoretical foundation that supports the research is based on the contributions of the materialist approach to the theory of…