This paper introduces a new matrix tool for the sowing game Tchoukaillon, which establishes a relationship between board vectors and move vectors that does not depend on actually playing the game. This allows for simpler proofs than currently appear in the literature for two key theorems, as well as a new method for constructing move vectors.We…

Solving equations in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial equation of the form (1). We specify that this equation has been…

Can one infinity be more than another infinity? Ask students this question, and many will be puzzled; others will insist that "infinity is infinity." The question seems to pique their interest and provides an opportunity to present the beautifully simple but counterintuitive proofs concerning the size of infinity first constructed by…

In 2012, Edward Haertel received the NCME Career Contributions Award. The focus article for this issue emerged from his address on the topic "How Is Testing Supposed to Improve Schooling?" His focus article provides a discussion of the relationships between testing and schooling in which he issues a call to action to the measurement community to…

In his article, Haertel (this issue) asks a fundamental question about how use of a test is expected to cause improvements in the educational system and in learning. He also considers how test validity should be investigated and argues for a more expansive view of validity that does not stop with scoring or generalization (the more technical and…

In his thoughtful focus article, Haertel (this issue) pushes testing experts to broaden the scope of their validation efforts and to invite scholars from other disciplines to join them. He credits existing validation frameworks for helping the measurement community to identify incomplete or nonexistent validity arguments. However, he notes his…

The author is deeply gratified by the commentators' thoughtful responses and finds almost nothing to disagree with in any of them. Each offers additional insights prompting further reflection. In drawing out just a few common themes, this brief rejoinder omits many important ideas from the individual contributions. As stated in his title, the…

Even when relatively infrequent, careless and random responding (C/RR) can have robust effects on individual and group data and thereby distort clinical evaluations and research outcomes. Given such potential adverse impacts and the broad use of self-report measures when appraising addictions and addictive behavior, the detection of C/RR can…

Any quadruple of natural numbers {a, b, c, d} is called a "Pythagorean quadruple" if it satisfies the relationship "a[superscript 2] + b[superscript 2] + c[superscript 2]". This "Pythagorean quadruple" can always be identified with a rectangular box of dimensions "a greater than 0," "b greater than…

A new and promising area of research has recently emerged in the field of career development: career competencies. The present article provides a framework of career competencies that integrates several perspectives from the literature. The framework distinguishes between reflective, communicative, and behavioral career competencies. Six career…

The Arnett Caregiver Interaction Scale (CIS) has been widely used in research studies to measure the quality of caregiver-child interactions. The scale was modeled on a well-established theory of parenting, but there are few psychometric studies of its validity. We applied factor analyses and item response theory methods to assess the psychometric…

This article describes the author's pedagogical transformation from "traditional" lecture-based instruction to Inquiry Based Learning (IBL) instruction of an introductory proofs class for sophomore mathematics majors. The story of the course overhaul follows from inception, through implementation, and ultimately to reflection.…

Data integration is a critical component of enterprise system integration, and XML data validation is the foundation for sound data integration of XML-based information systems. Since B2B e-commerce relies on data validation as one of the critical components for enterprise integration, it is imperative for financial industries and e-commerce…

The role of the body, particularly gesture, in supporting mathematical reasoning is an emerging area of research in mathematics education. In the present study, we examine undergraduate students providing a justification for a task about a system of alternating gears, which involves concepts of number relating to even/odd patterns. Some…

Agile software development has become very popular around the world in recent years, with methods such as Scrum and Extreme Programming (XP). Literature suggests that functionality is the primary focus in Agile processes while non-functional requirements (NFR) are either ignored or ill-defined. However, for software to be of good quality both…