For n = 1, 2, ... , we give a solution (x[subscript 1], ... , x[subscript n], N) to the Diophantine integer equation [image omitted]. Our solution has N of the form n!, in contrast to other solutions in the literature that are extensions of Euler's solution for N, a sum of squares. More generally, for given n and given integer weights m[subscript…

The current study investigated college students' content knowledge and cognitive abilities as factors associated with their algebra performance, and examined how combinations of content knowledge and cognitive abilities related to their algebra performance. Specifically, the investigation examined the content knowledge factors of computational…

A challenge: Can you create a stable top from a single paper clip? Several interesting solutions to this problem were provided by Takao Sakai from Japan, the requirement of each being that the center of gravity be located on the vertical y-axis at the center of the top. In the simplest configuration, we see that there exists a single angle ?…

This article outlines a problem-centered approach to the topic of canonical matrix forms in a second linear algebra course. In this approach, abstract theory, including such topics as eigenvalues, generalized eigenspaces, invariant subspaces, independent subspaces, nilpotency, and cyclic spaces, is developed in response to the patterns discovered…

The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…

Little is known about how students learn the basic ideas of ring theory. While the literature addressing student learning of group theory is certainly relevant, the concepts of zero-divisor and, more generally, elements with no multiplicative inverse are among those for which group theory has no analogue. In order to better understand how students…

The use of multiple forms of collaborative instruction to teach integrated algebra makes it difficult for teachers to determine which collaborative form is best suited for the curriculum. An inconsistent approach to integrated algebra instruction at the study school needed to be addressed for the benefit of teacher effectiveness and student…

The purpose of this study was to develop and validate an assessment tool entitled the Algebra Curriculum Based Measure (ACBM) with the intent of measuring preschool children's sorting and classifying skills based on one attribute (color, shape, and size) and two attributes (color and shape) simultaneously; and their ability to explain their…

Anecdotal observations reveal that most students with strong arithmetic skills will succeed in the Beginning Algebra course even if they have no previous experience with algebra. In trying to quantify this with an initial teacher-created survey of arithmetic skills, it was observed, for three consecutive semesters, that students who scored in the…

In most Intelligent Tutoring Systems, the help messages (hints) are not very clear for students as they are only presented textually and have little connection with the task elements. This can lead to students' undesired behaviors, like gaming the system, associated with low performance. In this paper, the authors aim at evaluating if the gestures…

When we administered the Test of Understanding of Vectors (TUV) to students who were completing a physics university remedial course (which covers subjects of a traditional high school physics course), we observed that they had considerable difficulties in interpreting the dot product as a projection. As a result of this, we decided to design a…

Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic…

The Rural Math Excel Partnership (RMEP) identified math competencies used by technicians in the workplace compared to standards of learning required in the public school curriculum. A modified DACUM process revealed 39 math competencies used by technicians in STEM-related occupations of the rural region. Group interviews with faculty in three…

This quantitative study examined the effect of a student's entry point into a school district and his/her performance on the New York State English Language Arts and Integrated Algebras Regents Exams. The subjects of this study were students assigned to the 2010 and 2011 cohorts from one Long Island school district. Three entry points were…

Math homework is transforming at a rapid rate with the continuous advances in educational technology. Computer-based homework, in particular, is gaining popularity across a range of schools, with little empirical evidence on how to optimize student learning. In the current study, middle school students (N = 143) solved a set of challenging…