New technology-enabled assessments offer the potential to understand more than just whether a student answered a test question right or wrong. Using multiple forms of media that enable both visual and graphical representations, these assessments present complex, multistep problems for students to solve and collect detailed information about an…

Mathematical investigations are loosely-defined, engaging problem-solving tasks that allow students to ask their own questions, explore their own interests and set their own goals. The value of investigations for students lies in their complexity. Scaffolding plays an important role in supporting students' high-level engagement by encouraging…

While some students have the opportunity to engage in the kinds of structured inquiry and real-world problem solving called for in the science education reform literature, many other students receive only a daily grind of note taking, end-of-chapter questions and sample test items from state assessments. The result is an engagement gap whereby…

An attempt to model the physical process of annealing lead to the development of a type of combinatorial optimization algorithm that takes on the problem of getting trapped in a local minimum. The author presents a Microsoft Excel spreadsheet that illustrates how this works.

This article presents some identities on the sum of the entries in the first half of a row in Pascal's triangle. The results were discovered while the author was working on a problem involving Laplace transforms, which are used in proving of the identities.

This article considers an old version of the classical pursuit problem as posed by Francis Godwin in 1599, who imagines a wedge of swans flying from the earth to the moon in twelve days, always flying at constant speed toward the moon. The return trip, during which the swans always fly toward the earth at the same speed, takes eight days. A little…

Abstract. The conditions under which one can gain mechanical advantage by pulling with a force F perpendicular to a cable (or rope) that is fixed at both ends are examined. While this is a commonly discussed example in introductory physics classes, its solution in terms of fundamental properties of the cable requires one to model the elasticity of…

This text has been centered on two main ideas: the specifications of a good problem to be introduced in a classroom; and according to Freudethal's view, the importance of teaching the students how to apply mathematics in their own real life problems. Putting these two ideas together, we may conclude that historical real world problems fit the…

New methodologies for determining the pressure inside an unopened carbonated beverage are presented. Such investigations also help the students to think deeply about a subject, giving them better understanding.

In this note, it is shown that in a symmetric topological space, the pairs of sets separated by the topology determine the topology itself. It is then shown that when the codomain is symmetric, functions which separate only those pairs of sets that are already separated are continuous, generalizing a result found by M. Lynch.

The purpose of this note is to outline further enrichment exercises with rhotrices and thus to stimulate ideas on mathematical arrays and structures. Rhotrices provide students with examples on how to imagine new forms of arrays and their classifications as abstract structures.

This article describes how all students can apply the algebra skills they have learned to problem solving through investigation, along with different levels of proving the conjectures algebraically.

Participatory action research advocates for teachers, parents and others to engage in practical inquiry as part of their everyday work for the purpose of improvement. Findings from this project affirm that a collaborative community of researchers, one in which the participants can critically analyse and transform their own situations, can have a…

The author states that the U.S. is at risk of falling behind in the global economic competition because too few of the young people are choosing to study fields like engineering and science. A 2005-06 Lemelson-MIT Invention Index survey found that while American teenagers are comfortable with rapid technological change and optimistic about the…

For many years, the author has been involving his students in classroom teaching of their own classes. The day-to-day practice is described, and the advantages and disadvantages for both the instructor and the students are discussed. Comparisons with the Moore Method of teaching are made.