Convexity-based measures of shape compactness provide an effective way to identify irregularities in congressional district boundaries. A low convexity coefficient may suggest that a district has been gerrymandered, or it may simply reflect irregularities in the corresponding state boundary. Furthermore, the distribution of population within a…

In this essay Angelo Caranfa maintains that what education should be engaging in is a ceaseless effort of cultivating in the students attention to the things of the spirit--that is, the world of aesthetic apprehension as described by such figures as Plato and Simone Weil. Caranfa attempts to show that in Plato and in Weil, we receive a vision of…

We have developed a two-week guided-inquiry laboratory in which first-semester general chemistry students investigate a suspected arson using gas chromatography--mass spectrometry and paper chromatography. In the process of evaluating evidence from the crime scene, students develop and test hypotheses and learn the fundamentals of chromatography,…

The jigsaw technique has been used in a fourth-year biochemistry course to increase problem-solving abilities of the students. The jigsaw method is a cooperative-learning technique that involves a group structure. Students start with a "home" group. That group is responsible for learning an assigned portion of a task. Then the instructor separates…

This geometrical account of primitive Pythagorean triples was stimulated by a remark of Douglas Rogers on a recent paper by Roger Alperin (Alperin, 2005). Rogers, in commenting on this paper, noted that Fermat in the 17th century had posed a challenge problem on Pythagorean triples that suggested he knew how to construct a sequence of them,…

Taking part in the United States Department of Energy's Real World Design Challenge (RWDC), the author describes how four technology education students at Cumberland Valley High School in Mechanicsburg, Pennsylvania, took on the challenge to redesign an aircraft wing, attempting to make it more efficient than one currently in operation. The…

Blended learning strategies can be employed in innovative ways to solve real-world academic problems across all academic disciplines. This article can provide administrators and faculty with specific examples to guide them when making decisions about academic planning or institutional strategies for any discipline at all levels of higher…

This paper presents a problem-solving test that deals with the regulation of the "trp" operon of "Escherichia coli." Two mutants of this operon are described: in mutant A, the operator region of the operon carries a point mutation so that it is unable to carry out its function; mutant B expresses a "trp" repressor protein unable to bind…

In this paper, canonical transformations generated by constants of motion in the case of the Kepler problem are discussed. It is shown that canonical transformations generated by angular momentum are rotations of the trajectory. Particular attention is paid to canonical transformations generated by the Runge-Lenz vector. It is shown that these…

Novices to the design process often struggle at first to understand the various stages of design. Learning to design is a process not easily mastered, and therefore requires multiple levels of exposure to the design process. It is helpful if teachers are able to implement various entry-level design assignments such as reverse-engineering…

In this article we show how the Sudoku puzzle and the three simple rules determining its solution can be used as an introduction to proof-based mathematics. In the completion of the puzzle, students can construct multi-step solutions that involve sequencing of steps, use methods such as backtracking and proof by cases, and proof by contradiction…

Quilt investigations, such as the Barn quilt problem in the December 2008/January 2009 issue of "Teaching Children Mathematics" and its solutions in last month's issue, can spark interdisciplinary pursuits for teachers and exciting connections for the full range of elementary school students. This month, North Dakota's centennial quilt…

Colleen Vale makes the case for professional learning teams collaborating together to improve their teaching and hence children's achievement. In this article she describes how this may be done. Along the way the teachers explored the idea of equivalence and the common conceptions and misconceptions held by children in their classes.

In order to be successful in problem-or project-based learning (PBL), students must take responsibility for the learning process by setting goals, monitoring, reflecting, and sustaining their motivation from the beginning of the project until the end. However, for many students, these processes do not occur naturally or easily. Therefore, the…

This paper is a descriptive and preliminary report on recent efforts to address two questions: 1) Can school mathematics be used to enhance our students' ability to understand their changing world? and 2) What role might computer technology play in this regard? After recounting some of the mathematical tools that led to a better understanding of…