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This article reports on a study designed to estimate the effect of high school coursetaking in the algebra-calculus pipeline on the likelihood of placing out of postsecondary remedial mathematics. A nonparametric variant of propensity score analysis was used on a nationally representative data set to remove selection bias and test for an effect among all comparable students and among students least likely to enroll in precalculus, trigonometry, or calculus. The effect failed to reach practical significance among all comparable students and was particularly weak among students least likely to take precalculus, trigonometry, or calculus. These findings suggest that the regression-based results that factor strongly into discussions on high school graduation requirements for mathematics coursetaking contain substantial selection bias. [ABSTRACT FROM PUBLISHER]

This note presents demonstrations of quadratic expressions that come about when particular problems are posed with respect to matchsticks that form regular triangles, squares, pentagons and so on. Usually when such ‘matchstick’ problems are used as ways to foster algebraic thinking, the expressions for the number of matchstick quantities are linear and not quadratic. It will be shown that a pedagogy of ‘summing all the matchsticks’ is central to the emergence of quadratic expressions. This pedagogy involves generational and transformational activities which are considered as some of the main activities of algebra. Key elements to these activities are processes such as recognizing and extending patterns, and specializing and generalizing particular functional relationships. Implications of these processes in terms of algebraic thinking are considered. [ABSTRACT FROM PUBLISHER]

An important component of intermediate and college algebra courses involves teaching students methods to factor a trinomial with integer coefficients over the integers. The aim of this article is to present a theoretical justification of that which is often taught, but really never explained as to why it works. The theory is presented, and a suggestion for an inquiry-based learning project is given. [ABSTRACT FROM AUTHOR]

We describe a simple photographic assignment appropriate for an abstract algebra (or other) course. Students take digital pictures around campus of various examples of symmetry. They then classify these pictures according to which of the 17 plane symmetry groups they belong. [ABSTRACT FROM AUTHOR]

This note presents demonstrations of mathematics that emerges when problems are posed with respect to a combined 12 × 12 multiplication table showing multiplier and multiplicand. Through processes such as recognizing and extending patterns, specializing and generalizing particular functional relationships between the diagonal and row sequences are compressed. Insights obtained from the various methods can be used to deepen teachers' and students' understandings of possible ways to bridge arithmetic and algebra. [ABSTRACT FROM AUTHOR]

Following an introduction to the diagonalization of matrices, one of the more difficult topics for students to grasp in linear algebra is the concept of Jordan normal form. In this note, we show how the important notions of diagonalization and Jordan normal form can be introduced and developed through the use of the computer algebra package Maple®. [ABSTRACT FROM AUTHOR]

A functional Menger system is a set of n-place functions containing n projections and closed under the so-called Menger's composition of n-place functions. We give the abstract characterization for subsets of these functional systems which contain functions having one common fixed point. [ABSTRACT FROM AUTHOR]

Let D be a Noetherian domain containing a field, a ∈ D a nonzero nonunit, and z an indeterminate over D. We prove that the generic fiber of the extension D[[z]]↪ D[1/a][[z]] has dimension ≥ dim(D/aD). [ABSTRACT FROM AUTHOR]

Left-symmetric algebras have close relations with many important fields in mathematics and mathematical physics. Their classification is very complicated due to the nonassociativity. In this article, we re-study the correspondence between left-symmetric algebras and the bijective 1-cocycles. Then a procedure is provided to classify left-symmetric algebras in terms of classification of equivalent classes of bijective 1-cocycles. As an example, the 3-dimensional complex left-symmetric algebras are classified. [ABSTRACT FROM AUTHOR]

Let I be a regular proper ideal in a Noetherian ring R, and let P(I) be the set of all integrally closed ideals J of R that are projectively equivalent to I. It is known that there is naturally associated to I a positive integer d and an additive subsemigroup {ci|i ∈ +} of + such that [image omitted], where [image omitted] is the integrally closed ideal {x ∈ R|xd is in the integral closure (Ici)a of Ici}. It is shown here that the sequences of ideals [image omitted] and [image omitted] are filtrations on R, and that we have subfiltrations {(Ii)a}i≥0 = f ⊆ f* ⊆ e. Associated with these filtrations, we have graded rings [image omitted], and these inclusion maps induce isomorphisms on the homogeneous prime spectra of these graded rings. This raises the question of when the homogeneous prime spectrum HSpec(R[u,tf*]) of R[u,tf*] is isomorphic to either HSpec(R[u,tf]) or HSpec(R[u,te]). It is shown that HSpec(R[u,tf*]) ≅ HSpec(R[u,tf]) if and only if c1 = 1 if and only if P(I) is projectively full. [ABSTRACT FROM AUTHOR]

The structure of weak Hopf algebras corresponding to Uq(sl2) are classified by their algebra structure and coalgebra structure. The algebra structure of weak Hopf algebras corresponding to Uq(sl2) can be written as the direct sum of Uq(sl2) and an algebra of polynomials. The coalgebra structure of weak Hopf algebras corresponding to Uq(sl2) are classified by their Ext quiver. There are four types of such structures. [ABSTRACT FROM AUTHOR]

Generalized down-up algebras were first introduced in Cassidy and Shelton (2004). Their simple weight modules were classified in Cassidy and Shelton (2004) in the noetherian case, and in Praton (2007) in the non-noetherian case. Here we concentrate on non-noetherian down-up algebras. We show that almost all simple modules are weight modules. We also classify the corresponding primitive ideals. [ABSTRACT FROM AUTHOR]

In this article, we describe all group gradings by a finite abelian group Γ of a simple Lie algebra of type G2 over an algebraically closed field F of characteristic zero. [ABSTRACT FROM AUTHOR]

In this article, the author explores the "algebra for all" issue to raise awareness about the many facets of this dilemma facing educators at the middle and high school levels. She discusses both sides of this controversial issue, especially regarding its impact on students' futures relative to higher education and employment. The author concludes that unless dramatic reform occurs in the delivery of algebra so that all students benefit from such instruction, students must make the decision to take algebra based on individual needs, interests, and desires. [ABSTRACT FROM AUTHOR]

The article discusses the expansion of the list of divisor methods to include those associated with the logarithmic and identric means. It shows that the two methods provided in the article stem from mirror-image objective functions in terms of optimization, in the same sense that the Hill and Webster methods have mirror-image objective functions. It explores the connections of the various objective functions to information theory and statistics. Finally, it compares optimal congressional allocations associated with the four means- geometric, logarithmic, identric, and arithmetic.

This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS Maxima, is taken as a case study with which to test the implementation of the ideas developed in this paper. The general characteristics needed from a CAS for the application of computer aided assessment, or computer based learning is discussed. In particular the need for consistently implemented noun forms of elementary arithmetic operations, together with their traditional verb forms is identified. [ABSTRACT FROM AUTHOR]

Student enrollment in K-12 online learning programs showed a tenfold expansion in the years between 2002 and 2005. Despite increased implementation to fulfill critical local needs, there is very little evidence-based research available to inform education leaders' decisions relating to these initiatives. To address the important question of whether online learning can be as effective as traditional face-to-face learning, this research presents the findings from a quasi-experimental design implemented to examine the effect of the Louisiana Algebra I Online initiative on student outcomes. The findings presented suggest that the Louisiana Algebra I Online model is a viable online model for providing effective Algebra I instruction. [ABSTRACT FROM AUTHOR]

This study reports on basic, mostly pre-senior mathematical skills, as measured by a multiple choice questionnaire, of 566 students enrolled in a first year introductory data analysis subject within a science or broadly scientific degree programme. Most students had previously undertaken a senior high-school algebra and calculus based mathematics subject. Rasch analysis was used to validate the questionnaire and assist in the description of levels of skill. General linear models demonstrated that a student's skills score depended on the result obtained in senior mathematics, whether or not the student was a mathematics student, gender, whether or not higher level mathematics had been studied, self-efficacy and year. It is concluded that in order to have full and confident use of basic, pre-senior mathematical skills, algebra-based mathematics needs to be studied beyond this level. [ABSTRACT FROM AUTHOR]

Let Λ be an algebra over an algebraically closed field K of infinite transcendence degree over its prime subfield. We prove in particular that if every s-dimensional indecomposable nonprojective Λ-module is Ω-periodic (resp. DTr-periodic) then there exists a common bound for the Ω-periods (resp. DTr-periods) of s-dimensional indecomposable nonprojective Λ-modules. [ABSTRACT FROM AUTHOR]

Let (A,-): = ℳ(J) be the 2 × 2-matrix algebra determined by Jordan algebra J: = H3(𝔸) of hermitian 3 × 3-matrices over a real composition algebra 𝔸, where (-) is the standard involution on A. We show that the triple systems BA(x,∼⃒,z), x,y,z ∈ A, are models of simple compact Kantor triple systems satisfying the condition (A), where BA(x,y,z) is the triple system obtained from the algebra (A,-) and (∼⃒) denotes a certain involution on A. In addition, we obtain an explicit formula for the canonical trace form for the triple systems BA(x,∼⃒,z). [ABSTRACT FROM AUTHOR]

The purpose of this study was to investigate the effect of integrating Microsoft Excel into a high school algebra class. The results indicate a slight increase in student achievement when Excel was used. A teacher-created final exam and two Criterion Referenced Tests measured success. One of the Criterion Referenced Tests indicated that the variability of the students scores were reduced considerably in the class using Excel, indicating that possibly an interest in the course was generated and that students’ interests were spurred by the use of the software. An opinion survey indicates an overall improved feeling about algebra when using the software to supplement instruction. [ABSTRACT FROM AUTHOR]

The article focuses on the research project Increasing Student Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS), aimed at finding ways to enhance students' attainment and participation in various fields including mathematics, science, and engineering. The research is being done in England and it consists of two phases. The first phase involves a survey on the accomplishment of multiplicative reasoning and algebra in students of 11-14 years of age while the second phase focuses on a collaborative study to improve their attainment. The project also compares children's current attainment with that during the 1970s.