In this paper we analyze the mathematical treatment of numerically and geometrically defined patterns in a variety of middle school curricula. We provide several examples that illustrate a need for rethinking how the study of patterns and generalizations are presented in middle grade mathematics curricula and discuss some natural implications to…

This study examines how students solve problems that involve the concept of absolute value. One hundred seventeen students participated in the first (written) part of the study, 24 students participated in the second part (interview). Particular attention was paid to the actual process of problem solving and to the students' ability to establish…

The worked example effect within cognitive load theory is a very well-established finding. The concrete effectiveness of worked examples in a learning situation, however, heavily depends on further moderating factors. For example, if learners improve their processing of worked examples by actively explaining the worked examples to themselves, they…

Recent work in embodied cognition has proposed that representations and actions are inextricably linked. The current study examines a developmental account of this relationship. Specifically, we propose that children's actions are foundational for novel representations. Thirty-two preschoolers, aged 3.4 to 5.7 years, were asked to solve a set of…

This article describes some problems of the maximum/minimum type, which are generally solved using calculus at secondary school, but which here are solved algebraically. We prove six algebraic properties and then apply them to this kind of problem. This didactic approach allows pupils to solve these problems even at the beginning of secondary…

In this article, we discuss how to use a diagrammatic approach to solve the classic sailors and the coconuts problem. It provides us an insight on how to tackle this type of problem in a novel and intuitive way. This problem-solving approach will be found useful to mathematics teachers or lecturers involved in teaching elementary number theory,…

Fermat's Last Theorem says that for integers n greater than 2, there are no solutions to x[superscript n] + y[superscript n] = z[superscript n] among positive integers. What about rational exponents? Irrational n? Negative n? See what an undergraduate senior seminar discovered.

This article analyzes the challenge of counting the number of rectangles of all sizes in the n-tableau and to provide a combinatorial reason for the answer. The authors present a solution on enumerating rectangles in the n-tableau using Grassl and Mingus results. The authors demonstrate their conjecture for the n-tableau and attempt to apply their…

The development of flexible, transferable problem-solving skills is an important aim of contemporary educational systems. Since processing limitations of our mind represent a major factor influencing any meaningful learning, the acquisition of flexible problem-solving skills needs to be based on known characteristics of our cognitive architecture…

In this article, we derive one-parameter family of Schroder's method based on Gupta et al.'s (K.C. Gupta, V. Kanwar, and S. Kumar, "A family of ellipse methods for solving non-linear equations", Int. J. Math. Educ. Sci. Technol. 40 (2009), pp. 571-575) family of ellipse methods for the solution of nonlinear equations. Further, we introduce new…

The trade-off between exploration and exploitation is common to a wide variety of problems involving search in space and mind. The prevalence of this trade-off and its neurological underpinnings led us to propose domain-general cognitive search processes (Hills, Todd, & Goldstone, 2008). We propose further that these are consistent with the…

This paper addresses several objectives of the special issue on universal screening by addressing gaps in the current research base concerning universal screening for mental, emotional, and behavioral health and by providing a framework for addressing the limitations of extant approaches. Specifically, an adaptive model of behavioral assessment…

"Research studies show that when students discover mathematical ideas and invent mathematical procedures, they have a stronger conceptual understanding of connections between mathematical ideas." Flewelling and Higginson state that inquiry, investigations, and problem solving "give students the opportunity to use their imagination…

When given the opportunity to work with mentor programs in schools, this author has recommended a global prescription for mentors to help new teachers make the transition into the world of teaching: encourage them to follow the practices that are most like those of good, experienced teachers. Experienced teachers are different from new teachers;…

Librarians can--and should--be on the lookout for new and unique ways to improve their instructional skill sets and service. One out-of-the-box opportunity to consider is to study--by reading, watching, or participating in--improvisational theater. Improv games and theater are like life: performers who may or may not know each other are given…