This article investigates the numbers [image omitted], originally studied by Catalan. We re-confirm that they are indeed integers. Using the close relationship between them and the Catalan numbers C[subscript n], we develop some divisibility properties for C[subscript n]. In particular, we establish that [image omitted], where f[subscript k]…

Structural equation models with interaction and quadratic effects have become a standard tool for testing nonlinear hypotheses in the social sciences. Most of the current approaches assume normally distributed latent predictor variables. In this article, we present a Bayesian model for the estimation of latent nonlinear effects when the latent…

Archimedes calculated the centre of gravity of the cone but the proof of this theorem is not extant in his works. Knorr made a reconstruction of this proof utilizing geometrical arguments. This paper proves this theorem by means of a physical demonstration utilizing the law of the lever, and by adapting from Archimedes the method of mechanical…

By following learned rules rather than reasoning, students often fall into common error patterns, something every experienced teacher has observed in the classroom. In their effort to circumvent the developing common error patterns of their students, the authors decided to supplement their math text with two weeklong investigations. The first was…

This article summarizes research conducted on calculator block items from the 2007 fourth- and eighth-grade National Assessment of Educational Progress Main Mathematics. Calculator items from the assessment were categorized into two categories: problem-solving items and noncomputational mathematics concept items. A calculator has the potential to…

Nonreading tasks that share some (but not all) of the task demands of reading have often been used to make inferences about how cognition influences when the eyes move during reading. In this article, we use variants of the E-Z Reader model of eye-movement control in reading to simulate eye-movement behavior in several of these tasks, including…

This paper describes the utility of using simulated, rather than real, student solutions to problems within a peer-assessment setting and whether this approach can be used as a means of improving performance in chemical calculations. The study involved a small cohort of students, of two levels, who carried out a simulated peer-assessment as a…

Many students possess low confidence toward learning mathematics, which, in turn, may lead them to give up pursuing more mathematics knowledge. Recently, game-based learning (GBL) is regarded as a potential means in improving students' confidence. Thus, this study tried to promote students' confidence toward mathematics by using GBL. In addition,…

Critical thinking, problem solving, the use of tools, and the ability to consume and analyze information are important skills for the 21st century workforce. This article presents a qualitative case study that follows five undergraduate biology majors in a computer science course (CS0). This CS0 course teaches programming within a data-driven…

Students start to memorize arithmetic facts from early elementary school mathematics activities. Their fluency or lack of fluency with these facts could affect their efforts as they carry out mental calculations as adults. This study investigated participants' levels of brain activation and possible reasons for these levels as they solved…

We report the development and piloting of an evaluative instrument and process for monitoring the environmental literacy (EL) of undergraduate students in one large research-led university in New Zealand. The instrument addresses knowledge, affect and competencies in the general area of EL in line with this institution's adoption of EL as a…

Affirmative action in college admissions was effectively banned in Texas by the Hopwood ruling in 1997, by voter referenda in California and Washington in 1996 and 1998, and by administrative decisions in Florida in 1999. The "Hopwood" and "Johnson" rulings also had possible applicability to public colleges throughout Alabama,…

The main purpose of this study was to concurrently investigate Taiwanese high-school students' and their science teachers' conceptions of learning science (COLS) and conceptions of science assessment (COSA). A total of 1,048 Taiwanese high-school students and their 59 science teachers were invited to fill out two questionnaires assessing their…

The Organization for Economic Cooperation and Development [OECD] is a global policy organization that includes the United States and about half of the Western Europe countries. It administers international comparison tests, called Programme for International Student Assessment (PISA), for 15 year-old students in Mathematics and other subjects. I…

The College Board's recently revised curriculum for advanced placement (AP) chemistry places a strong emphasis on conceptual understanding, including representations of particle phenomena. This change in emphasis is informed by years of research showing that students could perform algorithmic calculations but not explain those calculations…