The Bernoulli brothers, Jacob and Johann, and Leibniz: Any of these might have been first to solve what is called the Bernoulli differential equation. We explore their ideas and the chronology of their work, finding out, among other things, that variation of parameters was used in 1697, 78 years before 1775, when Lagrange introduced it in general.

Most educators claim that problem solving is important, but they take very different perspective on it and there is little agreement on how it should be taught. This article aims to sort out the different perspectives and discusses problem solving as a goal, a method, and a skill. As a goal, problem solving should not be limited to well-structured…

Technological advances have led to the generation of large databases of organizational data recognized as an information-rich, strategic asset for internal analysis and sharing with trading partners. Data mining techniques can discover patterns in large databases including relationships considered strategically relevant to the owner of the data.…

Teachers often look for good sources to find problems motivating students and stiffen the concepts they are going to teach and use in the classroom. This causes the question: "Where can we find good problems that can be used in classrooms?". History of mathematics presents many historical problems which can enrich the classroom…

Although the theory that creativity requires blind variation and selective retention (BVSR) is now more than a half-century old, only recently has BVSR theory undergone appreciable conceptual development, including formal three-parameter definitions of both creativity and sightedness. In this article, these new developments are for the first time…

In this paper, I comment on the set of papers in this special issue on mathematical problem posing. I offer some observations about the papers in relation to several key issues, and I suggest some productive directions for continued research inquiry on mathematical problem posing.

Framing design problems and solutions has been recognised in design studies as a central designerly activity. Some recent findings with expert designers relate framing practices to problem-solution co-evolution and analogy use, two further widely recognised design strategies. We wanted to understand if interaction design novices also use…

The development of problem solving skills is a shared goal in science, engineering, mathematics and technology education. In the applied sciences, problems are often open-ended and complex, requiring a multidisciplinary approach as well as new designs. In such cases, problem solving requires not only analytical capabilities, but also creativity…

The discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike. One of the most prominent applications is signal processing. Usually, it is important that the signals be transmitted fast, for example, when transmitting images over large distances such as between the moon and the earth or when generating…

This paper reports on a conversation analytic study of problem-solving interactions during shared reading at three kindergartens in the Netherlands. It illustrates how teachers and pupils discuss book characters' problems that arise in the events in the picture books. A close analysis of the data demonstrates that problem-solving interactions do…

Even mentioning data analysis puts many educators on edge; they fear that in data discussions, their performance will be judged. And, the authors note, it's a human trait to look for the source of a problem in the behavior of people involved rather than the system surrounding those people--what some call the Fundamental Attribution Error. When…

This task asks readers to figure out why when you stir a cup of hot liquid and tap on the side of the cup with a spoon, the pitch of sound starts low and ends up high. The solution to last month's tasks relating to the circumference of the Earth and how many stars are in the (visible) sky is also presented.

Separation of variables can be a powerful technique for solving many of the partial differential equations that arise in physics contexts. Upper-division physics students encounter this technique in multiple topical areas including electrostatics and quantum mechanics. To better understand the difficulties students encounter when utilizing the…

The purpose of this article is to contribute to the growing body of research that focuses on active learning techniques. Active learning techniques require students to consider a given set of information, analyze, process, and prepare to restate what has been learned--all strategies are confirmed to improve higher order thinking skills. Active…

We investigate teachers' practices in a whole-class context when they scaffold students' learning in situations where students use technologies that facilitate group learning to solve mathematical problems in small groups. We describe teachers' practices in order to evaluate their contribution to "Whole-Class Scaffolding" in the context…