This article presents the authors' response to Asha K. Jitendra's Response to Xin and Zhang ("The Journal of Educational Research," 2009, Vol. 102, No. 6). The claim of "factual errors, including inappropriate quotes, misrepresentation of information, and inadequate scholarship in Xin and Zhang's text" are not justified. In this rejoinder, the…

"Solving the Unknown with Algebra" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards and designed to help students practice pre-algebra skills including using formulas, solving for unknowns, and manipulating equations. Developed by The Actuarial Foundation with Scholastic, this program provides…

In this study, differential item functioning (DIF) methods utilizing 14 different matching variables were applied to assess DIF in the constructed-response (CR) items from 6 forms of 3 mixed-format tests. Results suggested that the methods might produce distinct patterns of DIF results for different tests and testing programs, in that the DIF…

An anharmonic solution to the differential equation describing the oscillations of a simple pendulum at large angles is discussed. The solution is expressed in terms of functions not involving the Jacobi elliptic functions. In the derivation, a sinusoidal expression, including a linear and a Fourier sine series in the argument, has been applied.…

The mathematical topic of inverse functions is an important element of algebra courses at the high school and college levels. The inverse function concept is best understood by students when it is presented in a familiar, real-world context. In this article, the authors discuss some misconceptions about inverse functions and suggest some…

Nonroutine function tasks are more challenging than most typical high school mathematics tasks. Nonroutine tasks encourage students to expand their thinking about functions and their approaches to problem solving. As a result, they gain greater appreciation for the power of multiple representations and a richer understanding of functions. This…

This study investigated students' activating positive affective experiences in mathematics lessons, their antecedents, their cognitive and motivational consequences, as well as their effect on achievement. The participants were 682 Grade 8 and 9 students from 37 classes from Germany and Switzerland who participated in a video study of lessons on…

This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos--satisfying an axiom sin[superscript 2] + cos[superscript 2] = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two…

The typical profit-maximization solution for the joint-production problem found in intermediate texts, managerial texts, and other texts concerned with optimal pricing is oversimplified and inconsistent with profit maximization, unless there is either no excess of any of the joint products or no costs associated with dumping. However, it is an…

The idea of what it means to understand mathematics has changed throughout history. Throughout, the function concept has remained a central theme. A conceptual understanding of function includes connections among multiple representations: (1) graphical; (2) verbal; (3) numerical; and (4) analytical. The idea of a function as a rule that describes…

Exploring and deriving proofs of closed-form expressions for series can be fun for students. However, for some students, a physical representation of such problems is more meaningful. Various approaches have been designed to help students visualize squares of sums and sums of squares; these approaches may be arithmetic-algebraic or combinatorial…

It seems rather surprising that any given polynomial p(x) with nonnegative integer coefficients can be determined by just the two values p(1) and p(a), where a is any integer greater than p(1). This result has become known as the "perplexing polynomial puzzle." Here, we address the natural question of what might be required to determine a…

This paper is concerned with improvements in some exact formulae for the period of the simple pendulum problem. Two recently presented formulae are re-examined and refined rationally, yielding more accurate approximate periods. Based on the improved expressions here, a particular new formula is proposed for the period. It is shown that the derived…

The boiling point of a monofunctional organic compound is expressed as the sum of two parts: a contribution to the boiling point due to the R group and a contribution due to the functional group. The boiling point in absolute temperature of the corresponding RH hydrocarbon is chosen for the contribution to the boiling point of the R group and is a…

A group of mathematicians and mathematics educators are collaborating in the fine-grained examination of selected "slices" of video recordings of lectures, drawing on Schoenfeld's Resources, Orientations and Goals framework of teaching-in-context. In the larger project, we are exploring ways in which this model can be extended to examine…