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…

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…

Cheating attracts a significant amount of attention in conversations about assessment, and with good reason: if students cheat, we cannot be sure they have met the learning outcomes of their course. In this conceptual article we question the attention given to cheating as a concept and argue that the broader concept of validity is a more important…

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…

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…

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…

Assessment literacy is an essential competency area for professional counselors who administer tests and interpret the results of participants' scores. Using factor analysis to demonstrate internal structure validity of test scores is a key element of assessment literacy. The underuse of psychometrically sound instrumentation in professional…

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…

This is the second article in a series of three in this special issue on establishing a boundary object to foster network health and development. The first article laid out the theoretical rationale for an Improvement Network Health and Development Framework. This article details the efforts to develop a set of practical measures tied to this…

Educational researchers often need to construct precise and reliable measurement scales of complex and varied representations such as participants' written work, videoed lesson segments and policy documents. Developing such scales using can be resource-intensive and time-consuming, and the outcomes are not always reliable. Here we present…