The mathematics curricula of Australia (ACARA, 2019), Scotland, England and America all require an understanding of proof by contradiction. Specifically, proof by contradiction is included as a Geometry topic in Specialist Mathematics (Version 8.4). In Specialist Mathematics, it is expected that students construct proofs in a variety of contexts…

Tychonov's 1935 solution of the heat equation, exhibiting nontrivial heat fluxes spontaneously appearing along an isolated conducting rod initially held at zero degrees, has intrigued some specialists for almost a century. No doubt those practicing heat engineers who took mathematics seriously were initially relieved to learn that the construction…

The purpose of this paper is to demonstrate and describe why alternating treatments design (ATD) researchers should consider employing a no-treatment series during the alternating treatments phase. To identify limitations associated with excluding this design element, we obtained examples of ATD studies published in five school psychology journals…

In this article, we propose a justice-oriented, antiracist validity framework designed to disrupt assessment practices that continue to (re)produce racism through the uncritical promotion of white supremist hegemonic practices. Using anti-Blackness as illustration, we highlight the ways in which racism is introduced, or ignored, in current…

When comparing relations and means of latent variables, it is important to establish measurement invariance (MI). Most methods to assess MI are based on confirmatory factor analysis (CFA). Recently, new methods have been developed based on exploratory factor analysis (EFA); most notably, as extensions of multi-group EFA, researchers introduced…

The 2014 "Standards for Educational and Psychological Testing" describe universal design as an approach that offers promise for improving the fairness of educational assessments. As the field reconsiders questions of fairness in assessments, we propose a new framework that addresses the entire assessment lifecycle: universal design of…

This paper provides a detailed account of the development of an instrument to investigate the emerging concept of presence in teaching (PiT) on a larger scale, explaining how the transition from data and findings from qualitative studies to a survey instrument was accomplished. In order to ensure relevance for teachers, the instrument needed to do…

Students who are engaged writers are likely to produce better writing and to enjoy writing more than students who are disengaged writers. Yet, we are unaware of any existing tool that validly and reliably measures writing engagement. In this article, we describe what writing engagement is and why it is important. Then, we present the Writing…

In mathematics education, the research on proof by contradiction (PBC) often claims that this activity is more difficult for students than direct proof, or simply difficult in general. Many hypotheses have been offered to support or explain this belief, yet they span a disorientingly wide swath of journal articles, conference papers,…

Validity is broadly defined as how well something measures what it's supposed to measure. The reliability and validity of scores from assessments are two concepts that are closely knit together and feed into each other.

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 article proposes the Incarnation as a theological principle that illuminates the educational task, especially for Christian-oriented schools. Initially, I develop keys to understanding the Incarnation as an integrating event that can help us understand education as a comprehensive phenomenon. Then, I explore implications: First, based on the…

A group of eighth-graders was presented with a two-day lab exploring graph theory as an enrichment experience. With the school's winter break looming, students were weary of solving linear equations, and this topic was intended to inject some new life into the classroom. In addition to learning about a completely new topic, they would be exposed…

Evaluators are often called upon to assess the effects of programs. To assess a program effect, evaluators need a clear understanding of how a program effect is defined. Arguably, the most widely used definition of a program effect is the counterfactual one. According to the counterfactual definition, a program effect is the difference between…

One of the challenges of discussing validity arguments for machine scoring of essays centers on the absence of a commonly held definition and theory of good writing. At best, the algorithms attempt to measure select attributes of writing and calibrate them against human ratings with the goal of accurate prediction of scores for new essays.…