The process of pairing a name with representations or peculiar properties permeates many mathematics classroom situations. In school, many practices go under the label 'definition', even though they can be very different from what mathematicians conceive as a formal definition, and in fact there are substantial differences between these different…

Polynomials with rational roots and extrema may be difficult to create. Although techniques for solving cubic polynomials exist, students struggle with solutions that are in a complicated format. Presented in this article is a way instructors may wish to introduce the topics of roots and critical numbers of polynomial functions in calculus. In a…

Large gender gaps in performance on questions involving projectile motion have been observed at high school and university level, even amongst high-achieving students. This gap is particularly problematic because projectile motion is typically one of the first topics formally taught in physics, and this may give girls an inappropriately negative…

Early in their chemistry education, students learn to do empirical formula calculations by rote without an understanding of the historical context behind them or the reason why their calculations work. In these activities, students use paperclip "atoms", construct a series of simple compounds representing real molecules, and discover,…

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in…

This article describes a hands-on mathematics activity wherein students peel oranges to explore the surface area and volume of a sphere. This activity encourages students to make conjectures and hold mathematical discussions with both their peers and their teacher. Moreover, students develop formulas for the surface area and volume of a sphere…

Many workshops and meetings with the US high school mathematics teachers revealed a lack of familiarity with the use of transformations in solving equations and problems related to the roots of polynomials. This note describes two transformational approaches to the derivation of the quadratic formula as well as transformational approaches to…

An account is made of the relationship between the convergence behaviour of a sequence and the accumulation points of the underlying set of the sequence. The aim is to provide students with opportunities to contrast two types of mathematical entities through their commonalities and differences in structure. The more set-oriented perspective that…

While teaching in New York City public schools, the author began planning collaboratively with a team of mathematics teachers. Collaborative planning enabled them to consider a range of issues that may not have emerged had they been planning independently. In this article, the author first describes their work through excerpts of conversations…

In April 2009, a high school principal in a large Arizona school district met individually with 18 of his most senior teachers to inform them that they would not have a job the following year. Why didn't tenure protect them from wholesale dismissal? The answer is they all had one thing in common: they were retirees who had been leased or hired…

This article examines an innovative question taken from the 1988 Extension 1 (3 unit) Mathematics New South Wales Higher School Certificate examination. Similar questions have made regular subsequent appearances in trial examinations around NSW and many texts now devote whole chapters to the subtle and somewhat laborious process of establishing…

Proposes a scenario to describe the formation of a planetary nebula, a cloud of gas surrounding a very hot compact star. Describes the nature of a planetary nebula, the number observed to date in the Milky Way Galaxy, and the results of research on a specific nebula. (MDH)

Suggests use of the quadratic formula to build understanding that connections between factors and solutions to equations work both ways. Making use of natural connections among concepts allows students to work more efficiently. Presents four sample problems showing the roots of equations. Messy quadratic equations with rational roots can be solved…

Looks at one phase of the water cycle; the formation of drops in cooling water vapor. Examines the influence of surface shape on the equilibrium of the liquid and gas phases. Discusses the mathematical formulas that model the phenomenon. (MDH)

Introduces the method of ray tracing to analyze the refraction or reflection of real or virtual images from multiple optical devices. Discusses ray-tracing techniques for locating images using convex and concave lenses or mirrors. (MDH)