In this paper, I discuss one of the most distinguishing features of Jamshid al-Kashi's pedagogy from his "Key to Arithmetic", a well-known Arabic mathematics textbook from the fifteenth century. This feature is the multiple paths that he includes to find a desired result. In the first section light is shed on al-Kashi's life…

For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…

How do students think about an angle measure of ninety degrees? How do they think about ratios and values on the unit circle? How might angle measure be used to connect right-triangle trigonometry and circular functions? And why might asking these questions be important when introducing trigonometric functions to students? When teaching…

Providing students with an opportunity to explore mathematical content through games allows teachers to include tasks that: (1) present alternative representations of the content; (2) welcome various expressions of mathematical reasoning; and (3) incorporate variations that empower all students to engage in the problem solving process. Games not…

With regard to the growing interest in developing teacher education to match the twenty-first-century skills, while many assumptions have been made, there has been less theoretical elaboration and empirical research on this topic. The aim of this article is to present our pedagogical framework for the twenty-first-century learning practices in…

A strong and vibrant economy is reliant on a strong and vibrant workforce equipped with the necessary skills, competencies, and knowledge. Afterschool and summer learning programs play an integral role readying students for their career pathways; not only helping them develop the foundational skills that are critical across the different contexts…

This article describes what is often referred to as the dog, beetle, mice, ant, or turtle problem. Solutions to this problem exist, some being variations of each other, which involve mathematics of a wide range of complexity. Herein, the authors describe the intuitive solution and the calculus solution and then offer a completely new solution…

"Instant Insanity II" is a sliding mechanical puzzle whose solution requires the special alignment of 16 colored tiles. We count the number of solutions of the puzzle's classic challenge and show that the more difficult ultimate challenge has, up to row permutation, exactly two solutions, and further show that no…

In an era of unprecedented volatility, ambiguity, impassioned conflict, and intractable problems that affect the basic living conditions and prosperity of many, education has never been more important or more in need of purpose, meaning, and applicability. The solutions to the dilemmas that define our world will be created and implemented in the…

In their years of teaching geometry, Wong and Bukalov realized that the greatest challenge has been getting students to improve their reasoning. Many students have difficulty writing formal proofs--a task that requires a good deal of reasoning. Wong and Bukalov reasoned that the solution was to divide the lessons into parallel tasks, allowing…

Pythagoras Theorem is an old mathematical treatise that has traversed the school curricula from secondary to tertiary levels. The patterns it produced are quite interesting that many researchers have tried to generate a kind of predictive approach to identifying triples. Two attempts, namely Diophantine equation and Brahmagupta trapezium presented…

The power of 3D printing technology has grown exponentially in just the past few years--people around the world are using 3D printers to prepare food, create tailored clothing, build cars and homes, and advance the medical field in ways that never seemed possible. In classrooms across the nation, 3D printers have become increasingly common because…

While there is disagreement among mathematics educators about some aspects of its meaning, mathematical modeling generally involves taking a real-world scenario and translating it into the mathematical world (Niss, Blum, and Galbraith 2007). The complete modeling process involves describing situations posed in problems with mathematical concepts,…

Increasing student participation in science is an ongoing challenge for many universities. In this active learning workshop, centered on inquiry and teamwork, we introduce high-school students to biochemistry and molecular biology techniques using a murder mystery activity. During this intensive 3 hr workshop, we engage students in a murder…

Engineering design is known as an answer to an ill-defined problem. As any answer to an ill-defined problem, it can never be completely right or absolutely wrong. The methods that universities use to teach engineering design, as a consequence of this, suffer from the same fate. However, the accumulated experience with the "chalk and…