Algebraic Symbols Research Articles

En route from patterns to algebra: comments and reflections

This article is a commentary on the papers in this issue of ZDM on Understanding Generalization in K-12 Algebra. It tries to interpret and understand at least some of the reported phenomena within a constructivist framework. The latter locates meaning not in the external representations but in the individual’s activity on and with them. This activity on the other hand is strongly regulated (but not determined) by social contracts and belief systems. From that, and considerations of the more general mathematical context, various suggestions for further and extended research can be drawn. One special aspect is that generalization processes will have to be complemented by some kind of instruction on the conventional algebraic symbolism and its usage.

Algebraic thinking with and without algebraic representation: a three-year longitudinal study

An Algebraic Thinking Test was given to 116 students aged 12–14, at the end of each of three years. This age span crosses two levels of school in New Zealand. This test assessed their ability to represent compensation in the four arithmetic operations both numerically and with letters for variables. The analyses of these results, together with the results from separate interviews designed to report individual progress of students in the New Zealand Numeracy Project, showed that students who had developed advanced mental strategies for dealing with additive, multiplicative and proportional operations, were the students who were capable of making full use of the alphanumeric symbols of algebra. These results, taken together with earlier studies by the authors, led to a proposal for a “pathway for algebraic thinking” accessible to all students.

Profiles of algebraic competence

The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map algebraic symbols onto word problems was evaluated. Two weeks later, unassisted algebraic problem-solving tests were completed. Cluster analyses revealed four meaningful, relatively well-ordered, arithmetic/algebraic competence profiles reflecting partial knowledge states associated with the acquisition of algebraic understanding. The findings are discussed in terms of the conceptual changes associated with the acquisition of algebraic competence.

Algebra in a man with severe aphasia

We report a dissociation between higher order mathematical ability and language in the case of a man (SO) with severe aphasia. Despite severely impaired abilities in the language domain and difficulties with processing both phonological and orthographic number words, he was able to judge the equivalence of and to transform and simplify mathematical expressions in algebraic notation. SO was sensitive to structure-dependent properties of algebraic expressions and displayed considerable capacity to retrieve algebraic facts, rules and principles, and to apply them to novel problems. He demonstrated similar capacity in solving expressions containing either solely numeric or abstract algebraic symbols (e.g., 8−(3−5)+3 versus b−(a−c)+a). The results show the retention of elementary algebra despite severe aphasia and provide evidence for the preservation of symbolic capacity in one modality and hence against the notion of aphasia as asymbolia.

Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking

Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and other symbols (e.g. “+”, “=”) made it possible to achieve what is considered one of the greatest cultural accomplishments in human history, namely, the constitution of a symbolic algebraic language and the concomitant rise of symbolic thinking. Because of their profound historical ties with natural language, the emerging syntax and meanings of symbolic algebraic language were marked in a definite way by the syntax and meanings of the former. However, at a certain point, these ties were loosened and algebraic symbolism became a language in its own right. Without alluding to the theory of recapitulation, in this paper, we travel back and forth from history to the present to explore key passages in the constitution of the syntax and meanings of symbolic algebraic language. A contextual semiotic analysis of the use of algebraic terms in 9th century Arabic as well as in contemporary students' mathematical activity, sheds some light on the conceptual challenges posed by the learning of algebra.

Taylor Expansion Diagrams: A Canonical Representation for Verification of Data Flow Designs

A Taylor expansion diagram (TED) is a compact, word-level, canonical representation for data flow computations that can be expressed as multivariate polynomials. TEDs are based on a decomposition scheme using Taylor series expansion that allows one to model word-level signals as algebraic symbols. This power of abstraction, combined with the canonicity and compactness of TED, makes it applicable to equivalence verification of dataflow designs. The paper describes the theory of TEDs and proves their canonicity. It shows how to construct a TED from an HDL design specification and discusses the application of TEDs in proving the equivalence of such designs. Experiments were performed with a variety of designs to observe the potential and limitations of TEDs for dataflow design verification. Application of TEDs to algorithmic and behavioral verification is demonstrated

Developing Algebraic Thinking: An Academy Model for Professional Development

To think algebraically, one must be able to understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; and analyze change in various contexts, according to Navigating through Algebra (NCTM 2001). Many teachers, however, often view algebra as a set of isolated skills and procedures.

Toeplitz Algebras and Extensions of Irrational Rotation Algebras

AbstractFor a given irrational number θ, we define Toeplitz operators with symbols in the irrational rotation algebra , and we show that the C*-algebra generated by these Toeplitz operators is an extension of by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group KK1(, ℂ). We also prove an index theorem for that generalizes the standard index theorem for Toeplitz operators on the circle.

Human Symbol Manipulation Within an Integrated Cognitive Architecture

This article describes the Adaptive Control of Thought-Rational (ACT-R) cognitive architecture (Anderson et al., 2004; Anderson & Lebiere, 1998) and its detailed application to the learning of algebraic symbol manipulation. The theory is applied to modeling the data from a study by Qin, Anderson, Silk, Stenger, & Carter (2004) in which children learn to solve linear equations and perfect their skills over a 6-day period. Functional MRI data show that: (a) a motor region tracks the output of equation solutions, (b) a prefrontal region tracks the retrieval of declarative information, (c) a parietal region tracks the transformation of mental representations of the equation, (d) an anterior cingulate region tracks the setting of goal information to control the information flow, and (e) a caudate region tracks the firing of productions in the ACT-R model. The article concludes with an architectural comparison of the competence children display in this task and the competence that monkeys have shown in tasks that require manipulations of sequences of elements.

Sorting Symbol Strings

Atheme Throughout All Grade Bands of the Algebra Standard of the NCTM's Principles and Standards for School Mathematics is the ability of students to “represent and analyze mathematical situations and structures using algebraic symbols” (p. 222). In the band for grades 6–8, this theme is further articulated as asking students to “develop an initial conceptual understanding of different uses of variables” (p. 222). Although variables sometimes occur alone, more often they occur in expressions, equations, and inequalities. We will refer to letters and numbers combined in equations, inequalities, and expressions as “symbol strings” as do Chazan and Yerushalmy (2003). The role of the variable is often determined by the symbol string in which it occurs; therefore, if students understand the different kinds of symbol strings, they will understand the roles that variables play. We have developed activities that ask students to identify, describe, compare, and classify symbol strings—in other words, to develop a feel for symbol strings. Chazan and Yerushalmy (2003) discuss it in this way: “Skilled performance [in school algebra] involves developing a feel for symbol strings … that indicates what sorts of creatures they are and what should be done with them.”

Teaching Linear Functions in Context with Graphics Calculators: Students? Responses and the Impact of the Approach on Their Use of Algebraic Symbols

This study analyses some of the consequences of adopting a functional/modelling approach to the teaching of algebra. The teaching of one class of 17 students was observed over five weeks, with 15 students undertaking both pre- and post-tests and 6 students and the teacher being interviewed individually. Use of graphics calculators made the predominantly graphical approach feasible. Students made considerable progress in describing linear relationships algebraically. They commented favourably on several aspects of learning concepts through problems in real contexts and were able to set up equations to solve contextualised problems. Three features of the program exerted a ‘triple influence’ on students’ use and understanding of algebraic symbols. Students’ concern to express features of the context was evident in some responses, as was the influence of particular contexts selected. Use of graphics calculators affected some students’ choice of letters. The functional approach was evident in the meanings ascribed to letters and rules. Students were very positively disposed to the calculators, and interesting differences were observed between the confidence with which they asked questions about the technology and the diffidence with which they asked mathematical questions.

Developing Algebraic Reasoning Through Generalization

NCTM's (2000) recommendations for algebra in the middle grades strive to assist students' transition to formal algebra by developing meaning for the algebraic symbols that students use. Further, students are expected to have opportunities to develop understanding of patterns and functions, represent and analyze mathematical situations, develop mathematical models, and analyze change. By helping students move from specific numeric situations to develop general rules that model all situations of that type, teachers in fact begin to address the NCTM's recommendations for algebra. Generalizing numeric situations can create strong connections between the mathematical content strands of number and operation and algebra (as well as with other content strands). In addition, these generalizing activities build on what students already know about number and operation and can help students develop a deeper understanding of formal algebraic symbols.

Generalization of Patterns: The Tension between Algebraic Thinking and Algebraic Notation.

This study explores the attempts of a group of preservice elementary school teachers to generalize a repeating visual number pattern. We discuss students' emergent algebraic thinking and the variety of ways in which they generalize and symbolize their generalizations. Our results indicate that students' ability to express generality verbally was not accompanied by, and did not depend on, algebraic notation. However, participants often perceived their complete and accurate solutions that did not involve algebraic symbolism as inadequate.

Descartes'o matematinės idėjos pirmuosiuose lietuviškuose vadovėliuose

Straipsnyje įvardijamos pagrindinės R. Descartes‘o matematinės idėjos ir apžvelgiama jų dėstymo istorija pirmuosiuose lietuviškuose vadovėliuose. Teigiama, kad R. Descartes suformulavo analizinės geometrijos ir algebrinės simbolikos pagrindus, uždavinių sprendimo metodiką, įvedė koordinačių sistemą, kintamojo dydžio sąvoką, priartėjo prie diferencialinio skaičiavimo, pritaikė integralinio skaičiavimo metodus. Pirmoji lietuviška matematinė knyga – aritmetikos uždavinynas (1885 m.), kuriame devynis uždavinius rekomenduojama spręsti sudarant lygtį iš sąlygos. Su algebrine simbolika lietuvių skaitytojus pirmasis 1916 m. supažindino A. Smetona. Prof. Z. Žemaičio ir kitų matematikų pastangomis funkcijos sąvoka mokant algebros nuo 1928 m. imta vartoti kaip pagrindinė. Analizinės geometrijos bei diferencialinio ir integralinio skaičiavimo pradmenys tuo metu dėstyti gimnazijų baigiamosiose klasėse, vadovėlius jiems parašė J. Stoukus, B. Ketarauskas, A. Juška.

Using Spreadsheets to Promote Algebraic Thinking

For students to find algebra conceptually meaningful, as well as useful in modeling and analyzing real-world problems, they must be able to reflect on, make sense of, and communicate about general numerical procedures (Kieran 1992). Such procedures consist of set sequences of arithmetic operations performed on numbers. Examples include computing an average and performing the standard division algorithm. Thinking about numerical procedures starts in the elementary grades and continues in successive grades until students can eventually express and reflect on the procedures using algebraic symbolism. This article outlines how such thinking can progress to algebraic reasoning and illustrates how computers used to promote this progression.

Enriching distance teaching and learning of undergraduate mathematics using videoconferencing and audiographics

Technical subjects such as mathematics do not transfer easily to the distance education mode, so teaching and learning in such environments present special challenges. Any technologies employed for this purpose must satisfy the need for visual material, algebraic symbolism, and geometric representation. There should be opportunities to explore ideas, to work on problem solving and to provide specific feedback, preferably through prompt verbal and visual explanations. Most importantly, the medium should also facilitate the development of motivation, enthusiasm for the subject, and the potential for interactivity. An interactive teaching and learning model involving Desktop Videoconferencing (DVC) and other audiographic facilities was developed and trialled for distance education in undergraduate mathematics. It appears that very little has been reported previously in this area of mathematics teaching, certainly not on the scale of this development. Weekly teaching and learning sessions of two hours duration were held with a first‐year group throughout the teaching semester. This experiment demonstrated that it was possible to integrate the systems used, together with computer applications/graphics software, to enable the difficult task of representing algebraic, geometric, and numeric concepts, all of which are essential for the development of higher level mathematical topics. Various qualitative measures were used for analysis of (i) complexity of the environment, (ii) effectiveness of the medium/methodology, (iii) improvement in skills, and (iv) the development of interactivity.

Mathematizing Verbal Descriptions of Situations: A Language to Support Modeling

In this article, I concentrate on the linguistic aspects of modeling, primarily investigating the uses of notations over tasks that consist of multievent temporal phenomena. Specifically, I describe attempts by a group of 7th graders who were algebra beginners to reformulate narratives using two lexical sets: verbal and iconic. The 2 sets are parallel representations of the characteristics of any process that can be described by a function of a single variable. A software environment provides an opportunity to manipulate visual objects and to use a verbal set that is simpler and more abstract than natural language descriptions. The use of these lexical sets offers a path toward modeling that does not require the use of algebraic symbols, typically the only first route of employing mathematical notations. Over the course of the intervention, notations were introduced, and syntactic and semantic aspects of the language were developed. Along the course of the study, I observed how natural language text turns into a script of events and processes, how the qualitative graphic description of the script is sketched using iconic notations, and how the same graph turns into a subject for qualitative analysis of rate of change.

A computational approach to George Boole's discovery of mathematical logic

This paper reports a computational model of Boole's discovery of Logic as a part of Mathematics. George Boole (1815–1864) found that the symbols of Logic behaved as algebraic symbols, and he then rebuilt the whole contemporary theory of Logic by the use of methods such as the solution of algebraic equations. Study of the different historical factors that influenced this achievement has served as background for our two main contributions: a computational representation of Boole's Logic before it was mathematized; and a production system, BOOLE2, that rediscovers Logic as a science that behaves exactly as a branch of Mathematics, and that thus validates to some extent the historical explanation. The system's discovery methods are found to be general enough to handle three other cases: two versions of a Geometry due to a contemporary of Boole, and a small subset of the Differential Calculus.

A student's construction of transformations of functions in a multiple representational environment

This paper reports on a case study of a 16-year-old student working on transformations of functions in a computer-based, multi-representational environment. The didactic approach to reflections, translations and stretches began with visualization exercises, and then was extended to investigate the implications of visual changes in data points, and subsequently, in algebraic symbolism. A detailed analysis of the student's work during the transition from the use of visualization and analysis of discrete points to the use of algebraic symbolism is presented. Two new semiotic forms are introduced as an alternative kind of algebraic symbolism and as a means to facilitate the transition to f(x) notation: covariational equations and the horseshoe display for transformations. The implications of this case for the redesign and modification of the software are discussed.

A chemical equation as a representation of a class of linear Diophantine equations and a system of homogeneous linear equations

A chemical equation, comprising nelements interspersed among mspecies (reactants + products) and where the stoichiometric coefficients are represented by algebraic symbols, is shown to be equivalent to a class of linear Diophantine equations as well as a system of khomogeneous linear equations in munknowns, where k^ n.The use of conservation conditions for the elements in the chemical equation affords a non‐conventional approach for the generation of some of the integral solution sets of the Diophantine equations and the integral solutions of the system of homogeneous linear equations.