In this volume, the authors address the development of students’ algebraic thinking in the elementary and middle school grades from curricular, cognitive, and instructional perspectives.
Author: Jinfa Cai
Publisher: Springer Science & Business Media
In this volume, the authors address the development of students’ algebraic thinking in the elementary and middle school grades from curricular, cognitive, and instructional perspectives. The volume is also international in nature, thus promoting a global dialogue on the topic of early Algebraization.
Early algebra and algebraic reasoning . ... In J. J. Kaput , D. W. Carraher , & M. L.
Blanton ( Eds . ) , Algebra in the early grades ( pp . ... In J. Cai & E. Knuth ( Eds . )
, Early algebraization : A global dialogue from multiple perspectives ( pp .
Britt, M. S., & Irwin, K. C. (2011). Algebraic thinking with and without algebraic
representation: A pathway for learning. In J. Cai & E. Knuth (Eds.), Early algebraization (pp. 137–159). New York: Springer. Brizuela, B., & Schliemann, A.
Author: Carolyn Kieran
This survey of the state of the art on research in early algebra traces the evolution of a relatively new field of research and teaching practice. With its focus on the younger student, aged from about 6 years up to 12 years, this volume reveals the nature of the research that has been carried out in early algebra and how it has shaped the growth of the field. The survey, in presenting examples drawn from the steadily growing research base, highlights both the nature of algebraic thinking and the ways in which this thinking is being developed in the primary and early middle school student. Mathematical relations, patterns, and arithmetical structures lie at the heart of early algebraic activity, with processes such as noticing, conjecturing, generalizing, representing, justifying, and communicating being central to students’ engagement.
In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple
perspectives (pp. 5–23). ... The development of children's algebraic thinking: The
impact of a comprehensive early algebra intervention in third grade. Journal for ...
Author: Sinan Kanbir
In this well-illustrated book the authors, Sinan Kanbir, Ken Clements, and Nerida Ellerton, tackle a persistent, and universal, problem in school mathematics—why do so many middle-school and secondary-school students find it difficult to learn algebra well? What makes the book important are the unique features which comprise the design-research approach that the authors adopted in seeking a solution to the problem. The first unique feature is that the authors offer an overview of the history of school algebra. Despite the fact that algebra has been an important component of secondary-school mathematics for more than three centuries, there has never been a comprehensive historical analysis of factors influencing the teaching and learning of that component. The authors identify, through historical analysis, six purposes of school algebra: (a) algebra as a body of knowledge essential to higher mathematical and scientific studies, (b) algebra as generalized arithmetic, (c) algebra as a prerequisite for entry to higher studies, (d) algebra as offering a language and set of procedures for modeling real-life problems, (e) algebra as an aid to describing structural properties in elementary mathematics, and (f) algebra as a study of variables. They also raise the question whether school algebra represents a unidimensional trait. Kanbir, Clements and Ellerton offer an unusual hybrid theoretical framework for their intervention study (by which seventh-grade students significantly improved their elementary algebra knowledge and skills). Their theoretical frame combined Charles Sanders Peirce’s triadic signifier-interpretant-signified theory, which is in the realm of semiotics, with Johann Friedrich Herbart’s theory of apperception, and Ken Clements’ and Gina Del Campo’s theory relating to the need to expand modes of communications in mathematics classrooms so that students engage in receptive and expressive modes. Practicing classroom teachers formed part of the research team. This book appears in Springer’s series on the “History of Mathematics Education.” Not only does it include an important analysis of the history of school algebra, but it also adopts a theoretical frame which relies more on “theories from the past,” than on contemporary theories in the field of mathematics education. The results of the well-designed classroom intervention are sufficiently impressive that the study might havecreated and illuminated a pathway for future researchers to take.
In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple
perspectives (pp. 137–159). New York, NY: Springer. Brown, G. (2006, January
14). The future of Britishness. Speech to the Fabian Society New Year
Author: M.A. (Ken) Clements
Publisher: Springer Science & Business Media
The four sections in this Third International Handbook are concerned with: (a) social, political and cultural dimensions in mathematics education; (b) mathematics education as a field of study; (c) technology in the mathematics curriculum; and (d) international perspectives on mathematics education. These themes are taken up by 84 internationally-recognized scholars, based in 26 different nations. Each of section is structured on the basis of past, present and future aspects. The first chapter in a section provides historical perspectives (“How did we get to where we are now?”); the middle chapters in a section analyze present-day key issues and themes (“Where are we now, and what recent events have been especially significant?”); and the final chapter in a section reflects on policy matters (“Where are we going, and what should we do?”). Readership: Teachers, mathematics educators, ed.policy makers, mathematicians, graduate students, undergraduate students. Large set of authoritative, international authors.
An approach to geometric and numeric patterning that fosters second grade
students' reasoning and generalizing about functions and co-variation. In J. Cai &
E. Knuth (Eds.), Early algebraization: A global dialogue from multiple
Author: Carolyn Kieran
This book highlights new developments in the teaching and learning of algebraic thinking with 5- to 12-year-olds. Based on empirical findings gathered in several countries on five continents, it provides a wealth of best practices for teaching early algebra. Building on the work of the ICME-13 (International Congress on Mathematical Education) Topic Study Group 10 on Early Algebra, well-known authors such as Luis Radford, John Mason, Maria Blanton, Deborah Schifter, and Max Stephens, as well as younger scholars from Asia, Europe, South Africa, the Americas, Australia and New Zealand, present novel theoretical perspectives and their latest findings. The book is divided into three parts that focus on (i) epistemological/mathematical aspects of algebraic thinking, (ii) learning, and (iii) teaching and teacher development. Some of the main threads running through the book are the various ways in which structures can express themselves in children’s developing algebraic thinking, the roles of generalization and natural language, and the emergence of symbolism. Presenting vital new data from international contexts, the book provides additional support for the position that essential ways of thinking algebraically need to be intentionally fostered in instruction from the earliest grades.
Reston, VA: Lawrence Erlbaum and NCTM. Kieran, C. (2011). Overall
commentary on early algebraization: Perspectives for research and teaching. In J
. Cai & E. Knuth (Eds.), Early algebraization (pp. 579–593). New York, NY:
Author: Monica Neagoy
Publisher: Corwin Press
Give your students a foundation of algebra for math success – now and in the future! Students and teachers must become friendly with algebraic foundations, as they have increasingly become the gateway to careers in the STEM fields. Monica Neagoy empowers teachers to embrace algebra and connect it to higher math concepts, tuning you and your students to algebraic thinking, reasoning, and doing. You’ll discover: ?Four explorations to help you weave key algebraic ideas into everyday mathematics Step-by-step lessons from real classrooms that will guide you in teaching concepts and in establishing their relevance and applicability New methods that break down difficult algebraic concepts and build a critical foundation for higher math
In the above-mentioned papers the Lamé equation was successfully algebraized
only for integer values of £, corresponding to the Lamé polynomials. A first
indirect algebraization of the Lamé equation for half-integer 6 was studied by
Ward in ...
algebra is constructed which is equivalent to the first one in terms of its natural
complexity . R . Sh . Grigolia ( GE - AOS - A ; Tbilisi ) 2001e : 03108 03F55 03C25
03F30 Polacik , Tomasz ( PL - SILS - IM ; Katowice ) Induction schemata valid in ...
It can be said that this is the very beginning of all algebraization , although we
understand by it a deeper aspect than ... in reasonings of continuity which has
been an early source of " algebraization " : only finitary operations can be trusted
The intended title was Algebraic Analysis of NonClassical First Order Logics . 2 .
Algebraic logic . ... In the following , we shall briefly outline the basic elements of
her method for an algebraization of logics . Rasiowa ' s first monograph The ...
Algebraization of analysis started early — during the latter half of the 19th century
- - and the habit of looking at whole classes of functions , instead of at the
behavior of individual functions , led naturally to borrowing from topology for the ...
Author: Raymond Louis Wilder
Category: Social Science
Mathematics as a Cultural System discusses the relationship between mathematics and culture. The book is comprised of eight chapters discussing topics that support the concept of mathematics as a cultural system. Chapter I deals with the nature of culture and cultural systems, while Chapter 2 provides examples of cultural patterns observable in the evolution of mechanics. Chapter III treats historical episodes as a laboratory for the illustration of patterns and forces that have been operative in cultural change. Chapter IV covers hereditary stress, and Chapter V discusses consolidation as a f ...
A version of this theorem was known early on in the logic community , e.g. , [ 73 ,
274 ] . It is used to show Tarski's Algebraization Theorem , see [ 150 ] . Tarski's Algebraization Theorem provides a more general “ algebraization ” of the f.o. ...
Author: Bozzano G Luisa
Publisher: Elsevier Science
The second part of this Handbook presents a choice of material on the theory of automata and rewriting systems, the foundations of modern programming languages, logics for program specification and verification, and some chapters on the theoretic modelling of advanced information processing.
quite a number of adherents in the early and midseventies ( when
Chomskyanism went through a crisis ) and in the process ... Linguistic
structuralism , culminating in Harris ( 1951 ) , had accomplished , for the first time
, the algebraization of ...
Destined to become as indispensable to historians, librarians, teachers, students, and browsers as the acclaimed American-based classic, the International Edition lists thousands of inventions, discoveries, and first happenings from a global perspective.
This modern idea on the logical status of geometry occurred a few years earlier to
the Italian mathematician Gino Fano ... But in algebraization all other geometrical
magnitudes , such as areas and volumes , are comprehended , and no new ...
The algebraization of this graph will mean here , roughly speaking , the process
of generation of the superblock algorithmic ... In early versions of the D2M
package some PROLOG procedures were implemented for the symbolic formula
Author: H. A. Barker
Publisher: Pergamon Press
Category: Technology & Engineering
Hardbound. The tone of the Proceedings is set by the three Plenary papers, and the remaining papers are arranged under the coherent themes of environment, computational methods, modelling and simulation, design methods and applications. The papers in the Proceedings represent the state-of-the-art in the rapidly changing technology of computer aided design in control systems. They clearly show how that technology is absorbing the most recent developments in computer science and adapting them to its requirements. The reader will find that the emphasis in the technology is shifting towards open environments with object-oriented databases and modern graphical user interfaces supporting a whole range of tools for modelling, analysis and design.
The first step from sets to types is essentially the familiar idea of set theoretic
semantics for a syntactic system , i.e. ... The disjunction property is important in
the algebraization of a logic as well as in defining a relational semantics for a
Peirce's earlier work on relation - theory – until about 1883 or thereabouts - was
mainly in terms of relatives rather than relations . ... In a somewhat similar vein
the algebraization suggested as the seventh phase may be seen as a
Author: Richard Milton Martin
Publisher: Philosophia Verlag Gmbh
Category: Analysis (Philosophy)
The philosophical papers collected together in this volume cover a variety of topics centering around the three items of the title. Mereology, the theory of part-whole, appears and reappears throughout as a kind of basso ostinato for much that is said. For its full effect, however, mereology must be combined with various items treated in metalogic or logical semiotics, the modern trivium of syntax, semantics, and pragmatics. When pressed for their total philosophic richness, all of these subjects flow over into topics of perennial interest in metaphysics, including even metaphysical theology. It is thought that the treatment here brings these various subjects together in a new light and in an exact way. As a result, they are seen to gain in richness, scope, and depth, and a basis provided for the study of how intimately they "interanimate" each other. Each paper here is a critical and/or constructive adventure of ideas, not necessarily agreeing in all details with every other. Even though they are concerned with a considerable variety of philosophical topics, there is nonetheless a common methodology throughout, namely, the logica utens of first order quantification theory - or its algebraic surrogate - together with the first-order metalogic based on it, which are thought to provide the bedrock of sound philosophical method. This view has been spelled out in considerable detail in the author's previous publications and is further develop here in important ways. "Richard M. Martin's work display a wealth of ideas, proposals, and formal analyses, always on top of the ideal of precision and rigour which were so important to him" Lingua e Stile, 1988 Of interest to: Philosophers, linguists, logicians
Moreover, algebra began to penetrate other mathematical fields (e.g. geometry,
analysis, logic, topology, number theory) to such an extent that in the early
decades of the 20th century one began to speak of the algebraization of