For a suitably nice, real-valued function "f" defined on an open interval containing [a,b], f(b) can be expressed as p[subscript n](b) (the nth Taylor polynomial of f centered at a) plus an error term of the (Lagrange) form f[superscript (n+1)](c)(b-a)[superscript (n+1)]/(n+1)! for some c in (a,b). This article is for those who think that not…

The title question has at least two natural answers, one "global" and one "local." Global: "when they can be made to coincide by a rigid motion of the whole plane;" local: "when there is a one-to-one distance preserving mapping of one onto the other." Self-evidently global implies local. We show that in fact these different notions lead to…

When reasoning about infinite sets, children seem to activate four categories of conceptual structures: geometric (g-structures), arithmetic (a-structures), fractal-type (f-structures), and density-type (d-structures). Students select different problem-solving strategies depending on the structure they recognize within the problem domain. They…

Separate a three-digit number into its component digits. After raising each digit to the third power and computing the sum of the cubes, determine how often the original number reappears. Modular arithmetic is used to reduce the number of potential solutions to a more manageable quantity. (Contains 4 tables.)

A particle that moves along a smooth track in a vertical plane is influenced by two forces: gravity and normal force. The force experienced by roller coaster riders is the normal force, so a natural question to ask is, what shape of the track gives a normal force of constant magnitude? Here we solve this problem. It turns out that the solution is…

In this note, we present an interesting approach to sum subseries in closed form. This approach seems to be not widely known and remains under-appreciated. Our study will lead to the discovery of results some of which have been known for a long time, some which were found only recently, as well as those which appear to be new.

One of the most difficult tasks that classroom teachers face is finding ways to reach all their students and match each student's level of mathematical readiness and performance to the skills they are required to teach. In classrooms and schools, current federal and state requirements have increased the emphasis on accountability for improved…

How do we measure students inquiry, problem-solving, and critical thinking abilities so that we know they are prepared to meet the challenges of the 21st century? John Barell explains how inquiry leads to problem-solving and provides specific steps for formative assessment that informs instruction of 21st century skills. Included are examples that…

During the last 20 years, there has been an increasing awareness that the U.S. is falling behind in its mathematics education of primary and secondary school students. For those who teach mathematics at the community college level, the deficiencies in mathematics education are painfully evident by the number of students requiring remedial math…

The dorsal striatum plays a key role in the learning and expression of instrumental reward associations that are acquired through direct experience. However, not all learning about instrumental actions require direct experience. Instead, humans and other animals are also capable of acquiring instrumental actions by observing the experiences of…

The use of a classroom observation tool to monitor differentiation strategies is described, and relevant research findings using the form are reported. The advantages for using this approach to document differentiation are discussed as are the reasons teachers may question its intent. Applications for practice include its use as a self-assessment…

In looking at successful inquiry activities, patterns in pedagogical approach emerge (Meyer et al. 2011). This article discusses one such approach--the design challenge. A design challenge can be defined as an activity in which students are given an explicit task to create a product that meets a defined goal. However, simply asking students to…

One of the biggest obstacles to overcome in creating and sustaining an administrative professional learning community (PLC) is time. Administrators are constantly deluged by the tyranny of the urgent. It is a Herculean task to carve out time for PLCs, but it is imperative to do so. In this article, the authors describe how an administrative PLC…

This article outlines how a Brisbane independent school, Clayfield College, improved the ability of its Year 3 students to solve addition and subtraction word problems by utilising a schematic approach. It was observed that while students could read the words in the text of a written problem, many had difficulty identifying the core information…

This article will describe two activities in which students conduct experiments with random numbers so they can see that having at least one repeated birthday in a group of 40 is not unusual. The first empirical approach was conducted by author Cauto in a secondary school methods course. The second empirical approach was used by author Flores with…