In classical calculus textbooks, the existence of primitive functions of continuous functions is proved by using Riemann integrals. Recently, Patrik Lundström gave a proof via polynomials, based on the Weierstrass approximation theorem. In this note, it is shown that the proof will be easy by using continuous piecewise linear functions.

A fascinating and catchy method for proving that a number of special lines concur is using the concept of locus. This is now the classical method for proving the concurrency of the internal angle bisectors and perpendicular side bisectors of a triangle. In this paper, we prove the concurrency of the altitudes and the medians by showing that they…

Competencies have been a common tool for talent management operations for decades. In an attempt to standardize and streamline competency modelling and assessment in the varied and evolving workplace, this paper presents a measurement architecture consisting of a modular but comprehensive construct framework and a technology-enabled assessment…

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…

We present a method to compute the power series expansions of e[superscript x] ln (1 + x), sin x, and cos x without relying on mathematical analysis. Using the properties of elementary functions, we determine the coefficients of each series through the method of undetermined coefficients. We have validated our formulae through the use of…

In this article, we state an interesting geometric conservation property between the three angle bisectors of three similar right triangles and provide a proof without words for its justification. A GeoGebra applet is also presented to help with the understanding of the progression of the proof from inductive to deductive stage.

For a given rational number r, a classical theorem of Niven asserts that if cos(rp) is rational, then cos(rp) [element-of] {0,±1,±1/2}. In this note, we extend Niven's theorem to quadratic irrationalities and present an elementary proof of that.

Across the world, teacher quality has come to be recognised as one of the most important variables affecting student outcomes; consequently, the regulation of entry into the profession is the subject of iterative review. The traditional 'one-off' interview, involving an interviewee and two or more interviewers, is a common, but not unproblematic,…

In high school geometry, students are expected to deepen their understanding of geometric shapes and their properties, as well as construct formal mathematical proofs of theorems and geometric relationships. The process of helping students learn to construct a geometric proof can be challenging given the multiple competencies involved (Cirillo…

Survey-based research is vital in education and social sciences, offering insights into human behaviors and perceptions. The prevalence of such studies in medical education has risen by 33% over the past decade. Despite this growth, the utility of survey findings depends on the study design quality and measure validity. Many manuscripts are…

I propose two practical advances to the argument-based approach to validity: developing a living document and incorporating preregistration. First, I present a potential structure for the living document that includes an up-to-date summary of the validity argument. As the validation process may span across multiple studies, the living document…

The notion of validity encapsulates the echo of the physical within the technical. The technical force of a language course, test or plan needs to be evaluated for its effects. On the norm side, this yields a design principle that asks whether the design is adequate, and can be validated. That kind of technical validation is perhaps most prominent…

Given decision-makers often prioritize causal research that identifies the impact of treatments on the people they serve, a key question in education research is, "Does it work?". Today, however, researchers are paying increasing attention to successive questions that are equally important from a practical standpoint--not only does it…

Validity is a key element of many forms of research--particularly surveys, which are often used in health professions education research. A survey must accurately measure what it is intended to measure to be considered valid. This is becoming increasingly difficult in the age of artificial intelligence (AI), where "bots" (short for…

Inquiry-based instruction in undergraduate mathematics is commonly characterized by students producing mathematical content (e.g., definitions, theorems, or proofs) rather than consuming it. In this paper, we argue that a production-centered view of inquiry is unnecessarily restrictive and limits the full range of disciplinary activity available…