Algebraic Ideas Research Articles

The pop-switch planar algebra and the Jones–Wenzl idempotents

The Jones–Wenzl idempotents are elements of the Temperley–Lieb (TL) planar algebra that are important, but complicated to write down. We will present a new planar algebra, the pop-switch planar algebra (PSPA), which contains the TL planar algebra. It is motivated by Jones' idea of the graph planar algebra (GPA) of type An. In the tensor category of idempotents of the PSPA, the nth Jones–Wenzl idempotent is isomorphic to a direct sum of n + 1 diagrams consisting of only vertical strands.

Ariel's cycles of problem solving: An adolescent acquires the mathematics register

This analysis illustrates the interdependency between developing mathematical understanding and using the specialized language unique to the discipline to express those understandings employing the mathematics register. Our focus is on Ariel, a 7th grade bilingual student from an urban district, over an 18-month period. He develops knowledge of both mathematics and the language (both oral and written) required for his successful problem solving and communication of this process. Three questions are addressed: (1) How does Ariel display his mathematical understanding during problem solving? (2) How does Ariel use the mathematics register while solving the problem? (3) What systems of representation, including language (oral and written), does Ariel bring to solving the problem? The findings show that initially, Ariel displays his mathematical understanding during his attempt to solve a specific algebraic problem; in this exploration of understanding algebraic ideas, he seeks to identify patterns. We follow his mathematical and language learning from this initial conjecture, to his reformulation and generalization of a rule. Eighteen months later, after studying algebra, Ariel attends to the systems of representation, including both the oral and written expression of the mathematics register. The findings underscore the assertion that students may not be in full command—for production and comprehension—of the mathematics register until they understand the underlying mathematics. Teachers should be encouraged to create situations in which students deploy all of their diverse repertories of mathematics knowledge and skills to the instructional tasks at hand, and to refine their use of the mathematics register.

Unifying the Algebra for All Movement

There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the Algebra for All movement. In light of this movement, there must be a clear consensus about what is taught in the name of algebra. Yet, researchers documented this is not the case. The present research proposes to unify the leading algebra standards and assessment framework documents to identify the key ideas of algebra. The analysis resulted in six key ideas: (a) Variables, (b) Functions, (c) Patterns, (d) Modeling, (e) Technology, and (f) Multiple Representations. Outlined is the research process and resulting unification of existing algebra framework documents, and consideration is given for its uses in educational policy regarding algebra and potential directions for future research.

Children’s Use of Variables and Variable Notation to Represent Their Algebraic Ideas

In this article, we analyze a first grade classroom episode and individual interviews with students who participated in that classroom event to provide evidence of the variety of understandings about variable and variable notation held by first grade children approximately six years of age. Our findings illustrate that given the opportunity, children as young as six years of age can use variable notation in meaningful ways to express relationships between co-varying quantities. In this article, we argue that the early introduction of variable notation in children’s mathematical experiences can offer them opportunities to develop familiarity and fluency with this convention as groundwork for ultimately powerful means of representing general mathematical relationships.

List-Decoding Multiplicity Codes

We study the list-decodability of multiplicity codes. These codes, which are based on evaluations of high-degree polynomials and their derivatives, have rate approaching 1 while simultaneously allowing for sublinear-time error correction. In this paper, we show that multiplicity codes also admit powerful and local algorithms that work even in the presence of a large error fraction. In other words, we give algorithms for recovering a polynomial given several evaluations of it and its derivatives, where possibly many of the given evaluations are incorrect. Our first main result shows that univariate multiplicity codes over fields of prime order can be list-decoded up to the so-called list-decoding capacity. Specifically, we show that univariate multiplicity codes of rate R over fields of prime order can be list-decoded from a (1 R e) fraction of errors in polynomial time (for constant R;e). This resembles the behavior of the Folded Reed-Solomon Codes of Guruswami and Rudra (Trans. Info. Theory 2008). The algorithm is based on constructing a differential equation of which the desired codeword is a solution; this differential equation is then solved using a power-series approach (a variation of Hensel lifting) along with other algebraic ideas. Our second main result is a algorithm for decoding multivariate multiplicity codes up to their Johnson radius. The key ingredient of this algorithm is the construction of a special family of algebraically-repelling curves passing through the points of F m ; no moderate-degree multivariate polynomial over F m can simultaneously vanish on all these A version of this paper was posted online as an Electronic Colloq. on Computational Complexity Tech. Report (20). Supported in part by a Sloan Fellowship and NSF grant CCF-1253886.

The Development of Children's Algebraic Thinking: The Impact of a Comprehensive Early Algebra Intervention in Third Grade

This article reports results from a study investigating the impact of a sustained, comprehensive early algebra intervention in third grade. Participants included 106 students; 39 received the early algebra intervention, and 67 received their district's regularly planned mathematics instruction. We share and discuss students' responses to a written pre- and post-assessment that addressed their understanding of several big ideas in the area of early algebra, including mathematical equivalence and equations, generalizing arithmetic, and functional thinking. We found that the intervention group significantly outperformed the nonintervention group and was more apt by posttest to use algebraic strategies to solve problems. Given the multitude of studies among adolescents documenting students' difficulties with algebra and the serious consequences of these difficulties, an important contribution of this research is the finding that—provided the appropriate instruction—children are capable of engaging successfully with a broad and diverse set of big algebraic ideas.

Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Algebra

Background and IntroductionIn recent decades, algebra has become infamous as a gatekeeper of success in school (Cai et al., 2005; Jacobs, Franke, Carpenter, Levi, & Battey, 2007; Stephens, 2008). Moses and Cobb (2001) underscored importance of algebra by making a comparison between people who lack an education in algebra today to the people who couldn't read and write in Industrial Age (p. 14). Unfortunately, though, it has been documented repeatedly many students struggle when they reach algebra in middle school or high school (e.g., Kenney & Silver, 1997).In response to phenomenon, members of education community have called for inclusion of algebra content in elementary school curriculum, with goal of removing abrupt, often derailing transition from arithmetic to algebra by infusing algebraic ideas into in elementary and intermediate grades (Kaput, 1998). For example, National Council of Teachers of Mathematics (NCTM, 2000) suggests incorporating algebra into elementary level curricula:By viewing algebra as a strand in curriculum from prekindergarten on, teachers can help students build a solid foundation of understanding and experience as a preparation for more sophisticated work in algebra in middle grades and high school, (p. 37)Research also lends support for notion of including algebraic ideas in elementary school curricula (e.g., Britt & Irwin, 2008; Schliemann et al., 2003). In her research brief on algebra, Kieran (2007) concludes current body of research emphasizes arithmetic can be conceptualized in algebraic ways and this emphasis can be capitalized on to encourage young students to make algebraic generalizations without necessarily using algebraic notation (p. 1). Thus, algebra can be infused into arithmetic in a way is appropriate for elementary-aged children.Accordingly, algebra topics have been included in recent standards documents as an essential component of elementary curriculum. The NCTM's Principles and Standards for School Mathematics (2000) states students in all grades should develop their understanding of following algebraic ideas:* understanding patterns, relations, and functions;* representing and analyzing mathematical situations and structures using algebraic symbols;* using mathematical models to represent and understand quantitative relationships; and* analyzing change in various contexts, (p. 37)More recently, in Common Core State Standards (CCSS; National Governors Association Center for Best Practices, Council of Chief State School Officers [NGA & CCSSO], 2010), Operations and Algebraic Thinking content domain begins in kindergarten and continues through fifth grade, progressing from a focus on understanding properties of, and having flexibility with, four basic operations, toward a focus on generalizing, describing, and justifying patterns and relationships, and interpreting symbolic expressions.In light of these standards, it is evident elementary school teachers are responsible for facilitating their students' development in algebraic concepts, and, therefore, they need to have a deep understanding of foundations of algebra themselves (Hill, Rowan, & Ball, 2005; Ma, 1999). Moreover, members of education community support notion that there is a powerful relationship between what a teacher knows, how she knows it, and what she can do in context of instruction (Hill, Blunk, et al., 2008, p. 498].Thus, mathematical education of PTs in algebra is of critical importance to quality of mathematical education of children. This is reflected in recently updated recommendations of Mathematical Education of Teachers II (METII; Conference Board of Mathematical Sciences, 2012] report, which states kindergarten through Grade 5 teachers need to be able to [recognize] foundations of algebra in elementary mathematics (p. …

An example of analysis of arithmetic and early algebraic thinking

This text is our contribution to conceptualization of the domain early algebra and the term early algebraic thinking in our primary school education. The paper describes arithmetic - early algebraic thinking about early numeric operations of graduated students from the faculties of education. According to this curriculum, students are expected to deal with algebraic ideas referring to many concepts of natural numbers, and we can form the hypothesis that students can connect what they have learnt at the course Teaching Methodology of Mathematics with their knowledge about arithmetic concepts within the semi circle of natural numbers. We justified this hypothesis by interviews with candidates at teachers' exams, and we dealt with algebraic contents incorporated into same arithmetic concepts. This text about involving algebraic ideas into algebraic concept of 'subtracting the number from the sum' is a partial review about these interviews. Based on our knowledge about the mentioned talks with candidates, and on this occasion they were given the possibility of realization of this teaching unit accepting the concept of early algebra, so we can estimate that candidates are capable of making a certain connection, but that there are individual differences in achievement in these connections.

A REPRESENTATION THEOREM FOR LYAPUNOV-LIKE TRANSFORMATIONS ON EUCLIDEAN JORDAN ALGEBRAS

A Lyapunov-like (linear) transformation L on a Euclidean Jordan algebra V is defined by the condition [Formula: see text]where K is the symmetric cone of V. In this paper, we give an elementary proof (avoiding Lie algebraic ideas and results) of the fact that Lyapunov-like transformations on V are of the form La+ D, where a ∈ V, D is a derivation, and La(x) = a ◦ x for all x ∈ V.

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.

Linear algebra for everyone

This book provides students with the rudiments of Linear Algebra, a fundamental subject for students in all areas of science and technology. The book would also be good for statistics students studying linear algebra. It is the translation of a successful textbook currently being used in Italy. The author is a mathematician sensitive to the needs of a general audience. In addition to introducing fundamental ideas in Linear Algebra through a wide variety of interesting examples, the book also discusses topics not usually covered in an elementary text (e.g. the cost of operations, generalized inverses, approximate solutions). The challenge is to show why the everyone in the title can find Linear Algebra useful and easy to learn. The translation has been prepared by a native English speaking mathematician, Professor Anthony V. Geramita.

STUDENTS’ WAYS OF REASONING ABOUT NONLINEAR FUNCTIONS IN GUESS-MY-RULE GAMES

There is a growing movement toward the introduction of algebra in early grades. This is supported by an increasing number of research studies that have reported success in getting young students to “do” mathematics considered beyond their reach. Yet, the consensus is that more research is needed to provide insights into the processes by which students make sense of algebraic ideas. In particular, more studies are needed on how nonlinear functions can be introduced in early grades. This article reports an algebra research strand that introduced seventh grade students to quadratic functions using Guess-My-Rule games. The article describes several instances of the students engaging successfully with ideas and forms of reasoning involving quadratic functions. The purpose is to contribute to the debate on the introduction of algebra in early grades by providing further evidence of young students’ ability to engage with algebraic ideas usually considered to be beyond their reach.

An algebraic method for classifying S-integrable discrete models

We discuss a method for classifying integrable equations on quad-graphs based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators of the ring generators. This function grows exponentially in the general case. Examples show that it grows more slowly for integrable equations. We propose a classification scheme based on this observation.

Subspace in linear algebra: investigating students’ concept images and interactions with the formal definition

This paper reports on a study investigating students’ ways of conceptualizing key ideas in linear algebra, with the particular results presented here focusing on student interactions with the notion of subspace. In interviews conducted with eight undergraduates, we found students’ initial descriptions of subspace often varied substantially from the language of the concept’s formal definition, which is very algebraic in nature. This is consistent with literature in other mathematical content domains that indicates that a learner’s primary understanding of a concept is not necessarily informed by that concept’s formal definition. We used the analytical tools of concept image and concept definition of Tall and Vinner (Educational Studies in Mathematics, 12(2):151–169, 1981) in order to highlight this distinction in student responses. Through grounded analysis, we identified recurring concept imagery that students provided for subspace, namely, geometric object, part of whole, and algebraic object. We also present results regarding the coordination between students’ concept image and how they interpret the formal definition, situations in which students recognized a need for the formal definition, and qualities of subspace that students noted were consequences of the formal definition. Furthermore, we found that all students interviewed expressed, to some extent, the technically inaccurate “nested subspace” conception that Rk is a subspace of Rn for k < n. We conclude with a discussion of this and how it may be leveraged to inform teaching in a productive, student-centered manner.

Quick Reads: Let's Cut a Square

Geometry is one of the oldest branches of mathematics. Many important mathematical ideas and theorems are rooted in geometry and can be visualized using geometric interpretations. Students learn about shapes, their properties, and the methods of calculating perimeters and areas of basic shapes early in their school years. Geometry can and should be used to help students visualize and better understand a broad range of mathematical concepts. In the following activity, students use the concepts of area and length to help them understand an important algebraic idea.

Connecting Geometry and Algebra in the Middle Grades

Welcome to the 2011 Focus Issue, which highlights connections between geometry and algebra that teachers can leverage in the middle grades. NCTM's Principles and Standards for School Mathematics (2000) recommends that students in the middle grades experience both the geometric representation of algebraic ideas and the algebraic representation of geometric ideas. By making these connections, students see that mathematical topics are related. They are not just a collection of isolated facts in seemingly disjoint fields but facts that often have many extensive connections.

Computational Diversions: Turtle Really and Truly Escapes the Plane

Just about anyone, when asked, can respond to the question ‘‘What is your all-time favorite book?’’ I’m no exception–in fact, for me, there’s no contest. The book is Turtle Geometry, by Hal Abelson and Andy diSessa (Abelson and diSessa 1980). Actually, Turtle Geometry is more than just a ‘‘favorite’’ book: it actually changed my life, a story I’ve never written about. In 1980, while working as a fledgling computer programmer at the Rockefeller University (my machine there was a DEC PDP-8), I happened into the University Library and plucked the book from the ‘‘new arrivals’’ shelf, intrigued by the mysterious title. A month later, I was certifiably insane, telling everyone who would listen that there was this really, really important book that you just had to read, that introduced mathematical ideas in a completely wonderful, experimental way, that made everything clear... and so forth. I had no idea who the authors were, but the book jacket said that they worked at MIT, so I applied to graduate school to study with them. My application form must have appeared sufficiently obsessive, and to my shock I was accepted. It sounds arrogant to say that ‘‘the rest is history’’, but in any event, the rest is my history. I still love Turtle Geometry, and have probably re-read the book in its entirety a half dozen times in the three decades since it was published. Abelson and diSessa use a Logolike language to present a remarkably accessible, procedural approach to geometry. In the course of about 500 pages, they introduce ideas of recursion, artificial life, Euclidean geometry, vector algebra, topology, and general relativity, to name a few. The book has lost absolutely none of its joy and freshness for me; by the way, it’s still in print, and if the interested reader wishes to consider this a plug, he or she can be my guest. The purpose of this column is to quench an ambition that has festered within me these past 30 years, since first picking up Turtle Geometry from the new-arrivals shelf. In the

Vedas and the Development of Arithmetic and Algebra

Problem statement: Algebra developed in three stages: rhetorical or pr ose algebra, syncopated or abbreviated algebra and symbolic alge bra-known as school School algebra developed rather early in India and the literature now suggests that the first civilization to develop symbolic algebra was the Vedic Indians. Approach: Philosophical ideas of the time influenced the development of the decimal system and arithmetic and that in turn led to algebra. Indeed, symbolic algebraic ideas are deep rooted in Vedic philosophy . The Vedic arithmetic and mathematics were of a high level at an early period and the Hindus used a lgebraic ideas to generate formulas simplifying calculations. Results: In the main, they developed formulas to understand the physical world satisfying the needs of religion (apara and para vidya). While geometrical focus, logic and proof type are featur es of Greek mathematics, boldness of conception, abst raction, symbolism are essentially in Indian mathematics. From such a history study, a number o f implications can be drawn regarding the learning of algebra. Real life, imaginative and creative pr oblems that encourage risk should be the focus in student learning; allowing students freely move bet ween numbers, magnitudes and symbols rather than taking separate static or unchanging view. A move f rom concrete to pictorial to symbolic modes was present in ancient learning. Real life practical n eeds motivated the progress to symbolic algebra. T he use of rich context based problems that stimulate a nd motivate students to raise levels higher to transfer knowledge should be the focus of learning. Conclusion/Recommendations: The progress from arithmetic to algebra in India was achieved th rough different modes of learning, risk taking, problem solving and higher order thinking all in li ne with current emphasis in mathematics education but at rather early stage in human history.

Education

Take a random polygon, find the the midpoints of its sides, and construct a new polygon from them. Repeat ad infinitum. What happens? This simple geometric iteration is at the heart of this issue's Education section offering, “From Random Polygon to Ellipse: An Eigenanalysis” by Adam N. Elmachtoub and Charles F. Van Loan. A cursory numerical experiment (although the traditionalist may wish to try this with a compass and straightedge) shows that the polygon shrinks to a point which the astute observer will realize is the centroid of the original vertices. If we now zoom in on the sequence of shrinking polygons, most easily accomplished by rescaling at each iteration, we see even more order. The rescaled vertices approach a binary oscillation between two polygons, both of which can be inscribed in the same ellipse. While students posed with these questions may think they are pursuing a geometry problem, the not-too-secret agenda is to introduce them to some fundamental ideas of numerical linear algebra, namely iteration, rescaling, and convergence. The authors guide us to write the problem as an iterative algorithm whose properties can be analyzed via eigenvalues and eigenvectors and whose convergence is akin to the power method for finding the eigenvalue of largest magnitude of a matrix. One might consider threading this example through a course in numerical analysis or numerical linear algebra, having the students explore the behavior at the beginning of the course (a great programming exercise), and eventually leading them through the analysis after introducing iterative methods for computing eigenvalues and eigenvectors. The problem opens the door to many generalizations, including exploring what happens for different geometric constructions and asking for geometric interpretations for other iterative linear algebra algorithms. The midpoint and rescaling construction is also simple enough to show to freshmen or even high school students; who knows, perhaps a few of them may be drawn to the study of mathematics by the allure of binary blinking ellipses?

Weighted limits in simplicial homotopy theory

By combining ideas of homotopical algebra and of enriched category theory, we explain how two classical formulas for homotopy colimits, one arising from the work of Quillen and one arising from the work of Bousfield and Kan, are instances of general formulas for the derived functor of the weighted colimit functor.