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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 article reports on an exploratory study designed to investigate the reasoning behind algebraists’ selection of examples. Variation theory provided a lens to analyze their collections of examples. Our findings include the classes of examples of groups and rings that algebraists believe to be most pedagogically useful. Chief among their selection criteria was that these examples illustrate not only the concept at hand but also lay the foundation for more abstract constructions. Additionally, we found that the algebraists, tending to think in terms of classes of examples, used a relatively small number in their own teaching and research. [ABSTRACT FROM AUTHOR]

This paper presents the partial results of a research on the professional knowledge of mathematics olympiad evaluators teachers put into action when analyzing errors relating to algebra. By means of two tests on errors interpretation and by using analysis inspired by the grounded theory, we obtained dimensions for the characterization of knowledge in understanding about errors. In the results section, we describe a teacher's knowledge related to practice and knowledge about students' learning. [ABSTRACT FROM AUTHOR]

We consider the so-called covariance set of Moore-Penrose inverses in rings with an involution. We deduce some new results concerning covariance set. We will show that if a is a regular element in a C*-algebra, then the covariance set of a is closed in the set of invertible elements (with relative topology) of C*-algebra and is a cone in the C*-algebra [ABSTRACT FROM AUTHOR]

This paper is a case study of the teaching of an undergraduate abstract algebra course, in particular the way the instructor presented proofs. It describes a framework for proof writing based on Selden and Selden () and the work of Alcock () on modes of thought that support proof writing. The paper offers a case study of the teaching of a traditionally-taught abstract algebra course, including showing the range of practice as larger than previously described in research literature. This study describes the aspects of proof writing and modes of thought the instructor modeled for the students. The study finds that she frequently modeled the aspects of hierarchical structure and formal-rhetorical skills, and structural, critical, and instantiation modes of thought. This study also examines the instructor's attempts to involve the students in the proof writing process during class by asking questions and expecting responses. Finally, the study describes how those questions and responses were part of her proof presentation. The funneling pattern of Steinbring () describes most of the question and answer discussions enacted in the class with most questions requiring a factual response. Yet, the instructional sequence can be also understood as modeling the way an expert in the discipline thinks and, as such, offering a different type of opportunity for student learning. [ABSTRACT FROM AUTHOR]

Classical Koszul duality sets up an adjoint pair of functors, establishing an equivalence F : Db(A) ⇆ Db(A!) : G, where A is a quadratic algebra, A! is the quadratic dual, and Db refers to the bounded derived category of complexes of graded modules over the graded algebra (i.e., A or A!). This duality can be extended in many ways. We consider here two extensions: first we wish to allow a ∧-graded algebra, where ∧ is any abelian group (not just Z). Second, we will allow filtered algebras. In fact we are considering filtered quadratic algebras with an (internal) ∧-grading. [ABSTRACT FROM AUTHOR]

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]

The article focuses on the conceptual approach to the practice of algebraic procedures which integrates procedures and learning process. It mentions that the nine processes of the cognitive dimension include direct application of procedures, reverse thinking, and global comprehension. Moreover, the approach attempts to maintain an appropriate balance between the skills needed by students to learn and the type of cognitive demands for them.

Abstract: Taking as the point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following five pairs of 7th grade students as they progress in algebraic discourse during 24 months, from their informal algebraic talk to the formal algebraic discourse, as taught in school. Our analysis follows changes that occur along time in the discourse produced by the students when they deal with specific types of algebraic problems. In this paper we take a look at the earliest stage in this process. Our data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses. This finding prompted us to extend the study and take a look at spontaneous meta-arithmetic of younger children. Our data collected among 5th graders have confirmed that informal algebraic discourse emerges early, and they shed light on the development of meta-arithmetic in the span of two years prior to the formal introduction of algebra in school. [Copyright &y& Elsevier]

It is proved that an algebra over a Dedekind ring whose maximal ideals have finite indices is hereditarily pure iff it is representable as a direct sum of an elementary Abelian algebra and an elementary Jacobsonian algebra. [ABSTRACT FROM AUTHOR]

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]

In this article the author has used 19th-century arithmetic and algebra textbooks as a way to reflect on current practices in developmental mathematics education. Five areas of special interest were found: motivation, relevance, depth, pedagogy, and textbooks. Philosophic and practical statements from vintage textbook authors remind educators of a number of questions and issues within each of those areas of interest. In some respects, little has changed over the years and many issues remain unresolved or little progress has been made. One hundred years from now will things be the same, or is it time for a change, a rethinking? [ABSTRACT FROM AUTHOR]

Abstract: In discussion-oriented classrooms, students create mathematical ideas through conversations that reflect growing collective knowledge. Linguistic forms known as indexicals assist in the analysis of this collective, negotiated understanding. Indexical words and phrases create meaning through reference to the physical, verbal and ideational context. While some indexicals such as pronouns and demonstratives (e.g. this, that) are fairly well-known in mathematics education research, other structures play significant roles in math discussions as well. We describe students’ use of entailing and presupposing indexicality, verbs of motion, and poetic structures to express and negotiate mathematical ideas and classroom norms including pedagogical responsibility, conjecturing, evaluating and expressing reified mathematical knowledge. The multiple forms and functions of indexical language help describe the dynamic and emergent nature of mathematical classroom discussions. Because interactive learning depends on linguistically established connections among ideas, indexical language may prove to be a communicative resource that makes collaborative mathematical learning possible. [Copyright &y& Elsevier]

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]

The article provides information on different factors that affect the strategies used by students when doing algebraic tasks. It notes the three techniques commonly used by students in solving numeric tasks and the factors influencing them such as structure of the mathematical task and students' mathematical knowledge. Moreover, it discusses the elements involved in the goals of classroom instruction that include exploring various strategies to solve problems.

The article presents two methods for teaching middle school students algebraic reasoning: one focuses on the use of number patterns, another on the measurement of real-world qualities. Specific topics discussed include the importance of quantitative reasoning, the creation of an educational unit on linear quantities and suggestions for using number patterns in the classroom. Multiple sample classroom exercises are presented.

We present a research methodology used in a classroom study in which two different learning methods of additive problems with negative numbers are compared. With the "writing method" students write the problems, learn their structures and solve problems that have been written by their partners. With the "solving method" students practice additive problems in a given sequence, according to the problems level of difficulty. The knowledge acquired was compared with the one obtained by other students who solved additive problems from the textbook, as application of operational rules. We have used a mixed methodology: A statistical treatment to contrast the effectiveness of the methods for solving problems and a qualitative study of certain aspects of the "writing method". [ABSTRACT FROM AUTHOR]

This paper contributes to the research strand concerning early algebra and focuses on the distributive law. It reports on a study involving pupils aged 8 to 10, engaging in the solution of purposefully designed problem situations. These situations are organized to favor specifying the students' solutions and to motivate a collective comparison of the arithmetic expressions that codify the solution processes. The study focuses on ways in which perception leads to different mental images that influence the choice of either the (a + b) × c or the (a × c) + (b × c) representation. It highlights that understanding these dynamics is a fundamental step for a meaningful learning of the property [ABSTRACT FROM AUTHOR]

The article provides information on the role of computer algebra systems (CAS) in developing new ideas in mathematics. It states that CAS, a multirepresentational tool with symbolic, graphical and numeric capabilities, are becoming essential tools for teaching and learning high school mathematics. In connection, the article illustrates several useful features of CAS, especially when it is incorporated as part of a classroom tool set. The author also notes that a CAS-inclusive environment is conducive to expand the idea of function beyond learner's usual experiences.

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]

The article reports on the teaching of algebra through the use of courses focusing on functions. It argues that algebra should be viewed as a means to providing solutions to actual problems rather than a skill to be memorized. It stresses the importance of educating students on mathematical modeling and its interdisciplinary uses. The article mentions the publication "A Collective Vision: Voice of Partner Disciplines" from the Mathematical Association of America which provides guidelines on teaching courses on the application of algebra. The article also provides examples of courses in algebra including those offered in the Institute for Mathematics Learning in West Virginia University.

The article investigates mathematics tasks that required students to make connections to the Pythagorean theorem, congruence and combinations by stretching a rubber band to form an oblique segment between any two pegs of a geoboard. It also demonstrates how the activity can be extended into algebra by having students find a generalization on how many oblique segments of different lengths can be made on the geoboard.

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 presents an abbreviated version of the methodologies and musings experienced by mathematics educators through a genuine problem-solving investigation. It investigates dimensions of an algebraic concept known to many high school students. It also shows how these dimensions lead to an intuitive understanding of the limit in calculus. It is stated that the readers of the article will experience and come to a deeper understanding of the Process Standards and experience the necessity of using multiple representations to make and solve mathematical conjectures. The article also describes a chronological development of mathematical ideas.

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.

The article presents a Tangent Line problem which is designed to give students an opportunity to think more deeply about the meaning of the concept of velocity. It reports that students tends to overgeneralize the rule d=rt, which is applicable to problems relating to motion at constant velocity. It suggests that the Tangent Line problem presented in the article is designed to allow students to explore accelerated motion and instantaneous velocity by using only familiar tools of algebra. It notes that the purpose of the problem is to engage students in thinking about the concepts of velocity and tangency and how they are represented and computed in a mathematical description. It concludes with some pedagogical comments on the problem and its uses in the classroom.

We present a study with secondary students about three specific processes of algebraic language: Formal substitution, generalization, and modelling. Using a test, we develop a students' errors classifications, and we analyze its possible origins. Finally we present some didactical conclusions from the results. [ABSTRACT FROM AUTHOR]

How do students think about algebra? Here we consider a theoretical framework which builds from natural human functioning in terms of embodiment – perceiving the world, acting on it and reflecting on the effect of the actions – to shift to the use of symbolism to solve linear equations. In the main, the students involved in this study do not encapsulate algebraic expressions from process to object, they do not solve ‘evaluation equations’ such as $$ 3x + 2 = 8 $$ by ‘undoing’ the operations on the left, they do not find such equations easier to solve than $$ 3x - 1 = 3 + x $$ , and they do not use general principles of ‘do the same thing to both sides.’ Instead they build their own ways of working based on the embodied actions they perform on the symbols, mentally picking them up and moving them around, with the added ‘magic’ of rules such as ‘change sides, change signs.’ We consider the need for a theoretical framework that includes both embodiment and process-object encapsulation of symbolism and the need for communication of theoretical insights to address the practical problems of teachers and students. [ABSTRACT FROM AUTHOR]

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]

Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics, and comparison is emerging as a fundamental learning mechanism. To experimentally evaluate the effects of comparison for mathematics learning, the authors randomly assigned 70 seventh-grade students to learn about algebra equation solving by either (a) comparing and contrasting alternative solution methods or (b) reflecting on the same solution methods one at a time. At posttest, students in the compare group had made greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom. [ABSTRACT FROM AUTHOR]

Abstract: We examine the rhetorical methods of Leonardo of Pisa in his exposition of single false position in Liber Abbaci. For example, Leonardo makes extensive use of formulaic phrases in his solutions. Some of these formulas also seem to indicate whether a particular solution needs further justification. Although he prefers proofs in terms of the pseudo-Euclidean canon of al-Khwārizmī, sometimes such proof eludes Leonardo and he resorts instead to justification by experiment. We also look at the extent to which using symbolic representations might distort our view of Leonardo''s thinking. [Copyright &y& Elsevier]

Research investigating algebra students' abilities to generalize and justify suggests that they experience difficulty in creating and using appropriate generalizations and proofs. Although the field has documented students' errors, less is known about what students do understand to be general and convincing. This study examines the ways in which seven middle school students generalized and justified while exploring linear functions. Students' generalizations and proof schemes were identified and categorized in order to establish connections between types of generalizations and types of justifications. These connections led to the identification of four mechanisms for change that supported students' engagement in increasingly sophisticated forms of algebraic reasoning: (a) iterative action/reflection cycles, (b) mathematical focus, (c) generalizations that promote deductive reasoning, and (d) influence of deductive reasoning on generalizing. [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]

The article presents the argument about what and how algebra and geometry are taught. The author emphasizes that algebra and the geometry are not necessarily the algebra and geometry of importance and most useful to students. An essay is presented which suggested the deletion of the traditional word problems, trinomial factoring, and complicated fractional expressions from the algebra curriculum. It is also recommended to cut the length of time in teaching geometry and that theorems which are not that important be deleted.

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]

Explores on some research about teaching and learning algebra and related classroom issues. Diagnostic instruments that may be used by senior secondary teachers in teaching algebra to senior classes; Strategies for remediating algebraic difficulties and misconceptions; Impact of technology on the algebra curriculum; Usefulness of copying algebraic expressions while using Computer Algebra Systems or mathematics processing software in a calculus class.

Information processing researchers have assumed that algebra symbol skills depend on mastery of the abstract rules presented in the curriculum (Matz, 1980; Sleeman, 1986). Thus, students' ubiquitous algebra errors have been taken as indicating the need to embed algebra in rich contextual settings (Kaput, 1995; National Council of Teachers of Mathematics [NCTM] Algebra Working Group, 1998). This study explored a nonrepresentational account of symbolic algebra skills as feature correlation within the visual field. We present evidence that algebra students respond spontaneously to the visual patterns of the notational display apart from engagement with the declarative content of the rules. Thus, persistent algebra errors may reflect disengagement from declarative content rather than an inability to deal with it. We sketch a Lexical Support System designed to sustain students' engagement with the declarative content of algebraic rules and processes, thus complementing the exciting curricular possibilities being developed for referentially rich algebra. [ABSTRACT FROM AUTHOR]

We describe the inherently non-dualisable finite algebras from some semigroup related classes. The classes for which this problem is solved include the variety of bands, the pseudovariety of aperiodic monoids, commutative monoids, and (assuming a reasonable conjecture in the literature) the varieties of all finite monoids and finite inverse semigroups. The first example of an inherently non-dualisable entropic algebra is also presented. [ABSTRACT FROM AUTHOR]