Textbooks on Dynamical Systems Theory usually present only three types of codimension-one bifurcations occurring in autonomous first-order nonlinear differential equations. These three types are called saddle-node, transcritical, and pitchfork. However, the number of bifurcations is virtually infinite. In this didactic work, the bifurcation…

Frost diagrams provide convenient illustrations of the aqueous reduction potentials and thermodynamic tendencies of different oxidation states of an element. Undergraduate textbooks often describe the lowest point on a Frost diagram as the most stable oxidation state of the element, but this interpretation is misleading because the thermodynamic…

Textbook models of externalities tacitly assume that those externalities fall upon individuals "outside" of the market. In many contexts--including common undergraduate examples--externalities fall "inside" the market instead. Positive externalities associated with vaccination, for instance, accrue to other individuals who…

The mechanisms of dissemination of the counting frames in the teaching of mathematics in Spain during the nineteenth century are studied. José Mariano Vallejo and Pablo Montesino proposed them in order to facilitate an arithmetic initiation based on intuition, following Pestalozzi's proposals. The diffusion channels considered were pedagogical…

"Representational transformation diagrams" are used to compare and contrast standard textbook presentations of vector line integrals in undergraduate courses in both mathematics and physics. These presentations are taken as the lower anchor in a learning trajectory. Two principal approaches in the lower division are identified, roughly…

Over the past 20 years, root cause analysis (RCA) has become one of the most widely used retrospective methods for detecting safety hazards in medicine and healthcare. Despite its wide use in management practice and growing popularity in academic research, there is currently a dearth of coverage of RCA in popular healthcare management textbooks…

This research explored the effectiveness of using the word network technique to teach English vocabulary to first graders in Thai Primary schools. The 150 top-frequency words used in three nationally-used textbooks were inventoried. The list of the words then was compared with the list proposed by the Ministry of Education, Thailand. The combined…

Students often use incoherent strategies in their problem solving involving force and motion, as revealed, e.g. when they are asked to draw force diagrams for amusement rides involving circular motion, whether in horizontal or vertical planes. Depending on the questions asked, assignments involving circular motion can reveal different types of…

Much like one's experiences with the wonders of the world, concrete and multiperspective representations of mathematical concepts are key to understanding. Ideally, "learning environments should give students the opportunity to get to know different representations of a mathematical object in order for them to develop an appropriate…

The analysis of a historical mathematical textbook constitutes a particular challenge when the original had been composed before the invention of the printing press and an even greater one when the very original might even not have been fixed as a manuscript. The analysis of such cases affords a focus on the relation between orality and written…

Two foundational concepts in biology education are 1) offspring are not identical to their parents, and 2) organisms undergo changes throughout their lives. These concepts are included in both international and U.S. curricular standards. Research in psychology has shown that children often have difficulty understanding these concepts, as they are…

Using multiple representations (MR) such as graphs, symbols, diagrams, and text, is central to teaching and learning in physics classrooms. While different studies have provided evidence of the positive impact of the use of MR on physics learning, a comprehensive overview of existing literature on the use of MR in physics education, especially at…

Learning about components and processes of complex biological systems is challenging and requires abstraction through mediational tools such as text, visuals, models and visualisation experiences. The 'systems concepts' necessitate building coherent relations and anticipating consequences, requiring greater mediational support. This article…

Emoji are commonly used in social media to convey attitudes and emotions. While popular, their use in educational contexts has been sparsely studied. This paper reports on the students' use of emoji in an online course forum in which students annotate and discuss course material in the margins of the online textbook. For this study, instructors…

The rising prevalence of data visualizations in the political and economic spheres requires an increasingly data literate population. Data are used to persuade people how to vote, support policies, adopt arguments or agendas, and buy products. Furthermore, data literacy is a specialized literacy skill that students need to employ to make meaning…