Within the Threshold Concepts Framework, 'postliminal variation' has been defined as the variation in the point and state of exit into a new conceptual space, and the epistemological and ontological terrain encountered from that point onwards. However, in the extensive published literature on threshold concepts, we find many cases in which its…

In this article, I argue that the common practice across many school mathematics curricula of using a variety of different representations of number may diminish the coherence of mathematics for students. Instead, I advocate prioritising a single representation of number (the number line) and applying this repeatedly across diverse content areas.…

The laser pointer has been widely used to demonstrate some simple optics phenomena, like reflection, refraction, total reflection, and diffraction. However, the rays of laser light cannot be seen in the air because the scattered light is too weak. Many physics teachers use milk or smoke to visualize rays of laser light in physics labs, but it is…

This article aims at examining the relationship between students' expressed notions of infinity and strategies used to compare the cardinalities of infinite sets. The notions of infinity were classified into two categorizations of potential and actual infinity, and the strategies were classified as part-whole, single infinity, incomparability and…

The paper is the result of extensive research carried out among Czech students and focuses on a conception of infinity. The questionnaire survey was taken by 861 students ranging from grades 7 to 13. The aim of the research was to describe the development of students' conceptions of infinity. These conceptions are built on the intuitive phenomenon…

This study describes the integration of citizenship education into the K to 12 Junior High School Economics curriculum in the Philippines. Content analysis of the 2016 Curriculum Guide Learning Competencies (CGLCs) and 2020 Most Essential Learning Competencies (MELCs) for Social Studies was conducted to determine the alignment of the Economics…

The dialogues, debates, and even arguments resulting from misunderstanding, misinterpretation, and mishandling of programs and people associated with the term "giftedness" are longstanding. Despite many years of efforts to assuage concerns of equity and need--many of which are legitimized by research evidence, the terminology continues…

Preparing engineering undergraduate students for the workforce is a goal of engineering programs. Engineering educators arguably provide students with conceptual understanding of engineering fundamentals; however, few studies focus on how knowledge of these concepts transitions into the engineering field. Concept inventories have been used in…

Contribution: This article describes the implementation, assessment, and evaluation of conceptual-based writing exercises in an introductory course on electric circuit analysis. Background: Students' struggles in gateway courses such as circuit analysis are often traced to inadequate metacognitive skills on the part of the student as well their…

In the multivariable calculus course, a standard application of triple integration is to find volumes of bounded regions in R[superscript 3]. In this article, we consider the problem of computing volumes, by way of examples, of regions bounded by planes and quadric surfaces. For illustration, we present the solution to two basic but non-standard…

Refutation texts are resurfacing as a suggested antidote to misconceptions and "fake news." In a previous critical review of the refutation text literature, we, the author team, suggested further interrogation of the assumptions these texts imply: that learners share the same common misunderstanding about a topic, and that there is one…

In this article, we discuss the process of understanding continuity, which is one of the most fundamental concepts in mathematics. The continuity of mathematical functions is formally defined in terms of abstract symbols and operations. This representation of continuity is very abstract or dis-embodied. Therefore, it is difficult to acquire a…

This book aims to advance ongoing debates in the field of mathematics and mathematics education regarding conceptions of argumentation, justification, and proof and the consequences for research and practice when applying particular conceptions of each construct. Through analyses of classroom practice across grade levels using different lenses -…

The study of engagement has become recognized by engagement researchers, educational practitioners, and administrative policy makers as having the potential for offering solutions to many of the concerns associated with aspects of modern formal learning environments. Engagement may afford protective aspects for students who have (or are in danger…

The premiere book in the STEM for Our Youngest Learners Series introduces the Infant Toddler Inquiry Learning Model, a new way to think about how young children (birth-age 3) explore, think, and learn STEM concepts. The book also demonstrates how the Inquiry Teaching Model can guide teachers in implementing STEM experiences for this age group.…