Multicomponent solution calculations can be complicated for students and practiced chemists alike. This article describes how to simplify the calculations by representing a solution's composition as a point in a "concentration space," whose axes are the concentrations of each solute. The graphical representation of mixing processes in a…

The use of approximations in solving problems in sciences can be vital for students. Order of magnitude estimation also helps with physical understanding of the quantities involved in a calculation. For example, in the first exam in a first-year class in chemistry the student must find the mass in grams of an atom; a couple of students will…

Number fact knowledge spans far beyond conventional automatic mastery practices involving rote learning. Having the ability to automatically recall number facts fluently does not mean students will have strong conceptual understanding. This article provides suggestions for how number fact thinking and fluency might be developed in primary…

The COVID-19 crisis has demonstrated the importance of being able to understand complex computational models for everyday life. To make sense of the evolving predictive models of the COVID-19 pandemic, global citizens need to have a firm grasp of both systems thinking (ST) and computational thinking (CT). ST is the ability to understand a problem…

To meet the demands of the 21st century, adult learners must display a range of sophisticated strategies for dealing with the digital world. With the switch to online instruction over the last months this need is only greater. In this article, I describe specific activities I implement with adult learners in a math and science class to help them…

Representations are used throughout school mathematics for students to both think through a problem and communicate their ideas. Students also often come to mathematics class with a repertoire of representations. This article presents the ideas of different and multiple representations with conceptual connections as their underlying distinguishing…

We present action research of a problem posed as part of a multi-participant national (Israeli) test checking the mathematical knowledge of high school students at the ages of 16-17, where some of those who solved this problem made an error by using the converse to a well-known theorem, where the converse is not true. In order to examine the…

Since formal mathematics is discussed in terms of abstract symbols, many students face difficulties to acquire a clear understanding of mathematical concepts and ideas. Transforming abstract or dis-embodied representations of mathematical concepts and ideas into embodied representations is a strategy to make mathematics more tangible and…

In this paper, we consider mathematical competitions for pre-university students, such as the "International Mathematical Olympiad" (IMO) and many national and regional Olympiads following a similar model. The problems proposed in these contests must be solvable by 'elementary' methods (i.e., without using calculus) and belong…

In this paper, we present an overview of a series of 1-credit applied problem solving courses. The goal of these courses was to help students develop their oral and written communication skills, ability to work as a team, and general problem solving skills through preparation and participation in the Mathematical Contest in Modeling (MCM) and…

The static equilibrium properties of a spool, resting on an incline and subject to the tension exerted by a string wrapped around the core cylinder, are studied by means of Newtonian mechanics. The overall behaviour of this system is imagined to be similar to that of a doggie kept on a leash. Starting from the well-known mechanical properties of…

Challenging behavior can impact the learning environment and prevent educators and students from demonstrating behavioral and academic success. Schools and districts may implore a framework of tiered systems of support to respond to students struggling behaviorally or academically. These systems provide a decision-making process to aid educators'…

The adoption of blended and hybrid Flipped Classroom (FC) models increased dramatically during COVID-19 and is still highly valued and recommended for enhancing the quality, flexibility, and effectiveness of teaching and learning post-pandemic. While meta-analyses indicate a small yet meaningful effect size of the FC approach, examined studies…

There is a mismatch between established practice in Learning Design and the increasingly complex nature of the challenges the world is facing. This article connects Learning Design to complexity science so learners can be better equipped to create a thriving future. Learning Design methods have traditionally leant heavily upon the reduction of…

Escape rooms have been receiving more attention for use in education. Escape rooms are games in which teams solve multiple puzzles and challenges using clues, hints, and strategy to determine how to escape from a locked room. The implementation of escape rooms has the potential for increasing students' mathematical understanding and engagement.…