This article describes how to multiply binomials so that middle school students produce the equivalence (a + b)[superscript 2] = a[superscript 2] + 2ab + b[superscript 2] after reasoning about and representing problems. First, a two-dimensional array representation is used so that students can become comfortable with multiplication embedded in…

Few direct-assessment procedures are designed to identify potential negative reinforcers (e.g., including demands in the escape condition of functional analyses). Two participants were systematically exposed to a series of demands nominated by caregivers as potential negative reinforcers. Sessions ended following the first instance of problem…

This article describes an instructional model that science teachers can use to promote and support student engagement in scientific argumentation. This model is called the evaluate-alternatives instructional model and it is grounded in current research on argumentation in science education (e.g., Berland and Reiser 2009; McNeill and Krajcik 2006;…

In this commentary, I, a research participant in Hilary Conklin's study, respond to her article by relating honesty to compassion and mindfulness. In addition to endorsing the intrapersonal honesty that Conklin supports in her article, I assert that interpersonal honesty can benefit the researcher, research participants, and research findings. I…

Parent volunteers can be an important asset to a well-run school library. Parent volunteers are that extra pair of hands and extra eyes. Monotonous and even tedious tasks can be accomplished quickly by people searching for ways to spend a little time with adult conversation while providing a benefit to their children. And eventually they can…

The best artistic challenges open students' eyes, hearts, and minds by combining both formal and conceptual concerns. In this article, the author describes a project inspired by a temporary exhibition of African Shona sculpture entitled "Mutambo! (Celebrate!)" at the Norfolk Botanical Gardens in Norfolk, Virginia. (Contains 2 online…

In this note, we present an improved Heaviside approach to compute the partial fraction expansions of proper rational functions. This method uses synthetic divisions to determine the unknown partial fraction coefficients successively, without the need to use differentiation or to solve a system of linear equations. Examples of its applications in…

The Picture Exchange Communication System (PECS) is a communication program that has become widely used, especially with children with autism. This paper reports the results of a review of the empirical literature on PECS. A descriptive review is provided of the 27 studies identified, which included randomized controlled trials (RCTs), other group…

During the period 1729-1826 Bernoulli, Euler, Goldbach and Legendre developed expressions for defining and evaluating "n"! and the related gamma function. Expressions related to "n"! and the gamma function are a common feature in computer science and engineering applications. In the modern computer age people live in now, two…

In this study, preliminary evidence of the potential for Direct Behavior Rating (DBR) in screening assessment of school social behavior is provided through evaluation of the concurrent validity of DBR with a commonly used criterion measure. The teacher-completed form of the "Social Skills Rating System" (SSRS) was selected as the…

The practical applicability of a semiconductor quantum dot with spin-orbit interaction gives an impetus to study analytical solutions to one- and two-electron quantum dots with or without a magnetic field.

In this paper, we consider solutions to the three-dimensional Schrodinger equation of the form [psi](r) = u(r)/r, where u(0) [is not equal to] 0. The expectation value of the kinetic energy operator for such wavefunctions diverges. We show that it is possible to introduce a potential energy with an expectation value that also diverges, exactly…

In this article, the classic problem of finding three ways to determine the probability that the pieces of a stick randomly broken in two places will form a triangle is analyzed anew. To be useful in the classroom, an application must be incorporated into an often-crowded curriculum. What is nice about this triangle problem is that it fits in…

Recognizing schemes, which are different from strategies, can help teachers understand their students' thinking about fractions. Using Steffe's advanced fraction schemes, the authors describe a progression of development that upper elementary and middle school students might follow in understanding fractions. Each scheme can be viewed as a…

The problem of obtaining the apparent equation of motion and shape of a moving body from its arbitrary given equation of motion in special relativity is considered. Also the inverse problem of obtaining the body's equation of motion from a known equation of motion of its image is discussed. Some examples of this problem solution are considered. As…