Carolyn Kieran

• PhD
• Professor Emeritus at University of Quebec in Montreal

Early algebraic thinking is the reasoning engaged in by 5- to 12-year-olds as they build meaning for the objects and ways of thinking to be encountered within the later study of secondary school algebra. Ever since the 1990s when interest in developing algebraic thinking in the earlier grades began to emerge, there has been a steady growth in the r...

"They are the same" is a phrase that teachers often hear from their students in arithmetic and algebra. But what do students mean when they say this? The present paper researches the notion of sameness within algebraic thinking in the context of generating equivalent numerical equalities. A group of Grade 6 Mexican students (10- to 12-year-olds) wa...

This paper provides evidence on how elementary school students from a Mexican public school move from using an operational sense, expressed in computational strategies, to a kind of relational thinking based on structure sense ideas and expressed by number decomposition. Even though the results are somewhat preliminary, they illustrate how students...

Theorizing about task design is a fairly recent area of attention within the educational research community, emerging in the late 1960s and continuing with growing interest to the present day. To reflect the evolution in task design theorization, this chapter focuses first on historical aspects and highlights the main theorizing initiatives of the...

This paper presents strategies identified in Mexican rural school students in seeking structure on equivalence tasks that involve the equal sign and tasks that do not. The results arise from the pilot study of a research project on the structure of numbers and numerical operations – a key aspect of early algebraic thinking.

The dominant focus on generalizing in the development of algebraic thinking has to a large extent obscured the process of seeing structure. While generalization-oriented activity remains highly important in algebra and early algebra, and in fact includes a structural component, equal attention needs to be paid to the complementary process of lookin...

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) Topi...

Part V of this volume offers the reader six chapters dedicated to the theme of “Planning and assessment.” While five of the six chapter titles suggest a greater emphasis on the assessment aspect, consideration is also given to planning in all chapters. Current perspectives on assessment view the planning for effective learning and for the kind of a...

Recent studies using neuroimaging technology with tasks touching on various areas of mathematics are raising a great deal of excitement with their findings. This chapter presents some key work related to higher level mathematical reasoning and a few insights arising from these studies with respect to our current understanding of algebra learning. A...

The development of students’ mathematical reasoning (MR) is a goal of several curricula and an essential element of the culture of the mathematics education research community. But what mathematical reasoning consists of is not always clear; it is generally assumed that everyone has a sense of what it is. Wanting to clarify the elements of MR, this...

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 document reveals the nature of the research that has been carried out in early algebra and how it has shaped the growth...

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 document reveals the nature of the research that has been carried out in early algebra and how it has shaped the growth...

In this chapter, we revisit Michèle Artigue’s classic 2002 IJCML article and draw out what we consider to be the core theoretical ideas and key dimensions of the body of work on tools and tool use that Michèle not only elaborated but also inspired others to further develop. We trace the evolutionary path of these core ideas, noting the ways in whic...

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 o...

This chapter gives an overview of the current state of the art related to frameworks and principles for task design so as to provide a better understanding of the design process and the various interfaces between teaching, researching, and designing. In so doing, it aims at developing new insights and identifying areas related to task design that a...

A wonderful part of the opening session of the ICME congresses is the ICMI Awards ceremony. The 2012 ceremony, which was presided over by Prof. Carolyn Kieran, the chair of the ICMI Awards Committee, was no exception. Congress participants shared in congratulating the recipients of the 2009 and 2011 competitions for the Klein and Freudenthal awards...

Resumen: En el presente artículo discutimos algunos ejemplos de resultados obtenidos en el proyecto de investigación doctoral en torno al desarrollo del conocimiento algebraico de estudiantes de bachillerato en un ambiente CAS. De acuerdo con dichos resultados, se observa el mismo tipo de comportamiento: un conflicto cognitivo; a partir del cual el...

While documentation work is at the core of teachers’ professional activity and professional development, this article argues for and illustrates the way in which documentation work is also central to the professional activity of design researchers. It has the double aim of contributing both to the documentational approach of didactics as well as to...

Our study concerns the conceptual mathematical knowledge that emerges during the resolution of tasks on the equivalence of polynomial and rational algebraic expressions, by using CAS and paper-and-pencil techniques. The theoretical framework we adopt is the Anthropological Theory of Didactics (Chevallard 19:221–266, 1999), in combination with semio...

The history of mathematics education provides ample evidence of the dichotomous distinction that has been made over the years between concepts and procedures, between concepts and skills, and between “knowing that” and “knowing how.” In no field of school mathematics learning has this dichotomy been so damaging as in algebra. While reform efforts o...

Teachers are regarded as having a major role in the development of mathematics teaching and students’ learning. Nevertheless, in much mathematics education research, teachers are viewed as recipients, and sometimes even as means to generate or disseminate knowledge, thus conserving a distinctive gap between research and practice. The theme of this...

The study featured in this article, with its central focus on resources-in-use, draws upon salient aspects of the documentational approach of didactics. It includes an a priori analysis of the curricular resources being used by a teacher for the first time, followed by detailed in situ observations of the unfolding of her teaching practice involvin...

In this paper we analyze and discuss students’ performance in a CAS environment related to the simplification of rational expressions. Results indicate that if students have more initial paper-and-pencil techniques, the CAS environment spurs them to deeper theoretical reflections than for students who have fewer techniques.

Our study concerns the conceptual mathematical knowledge that emerges during the resolution of tasks on the equivalence of polynomial and rational algebraic expressions, by using CAS and paper-and-pencil techniques. Our findings highlight the mathematical knowledge (technological discourse) constructed in the process of confronting, differentiating...

Little is known about the ways in which teachers take on research-based resources and adapt them to their own needs. This study focused on the manner in which three teachers who were participating in a research project on the learning of algebra with Computer Algebra System (CAS) technology spontaneously adapted the resources that were designed spe...

In this paper we analyze and discuss the influence of CAS technology and an Activity designed with a Technical-Theoretical approach on two 10th grade students’ thinking on a Task related to simplifying rational algebraic expressions. The theoretical elements adopted in this study are based on the instrumental approach. Results indicate that CAS and...

Teachers participating in a research project on the learning of algebra with CAS technology spontaneously adapted the resources designed specifically by the researchers for the project. Analysis of the classroom-based observations of teaching practice showed that adaptive shaping occurred with respect to all three key features (the mathematics, the...

The twenty-nine chapters of this volume on early algebraization, which include an introduction and commentary for each of the three main parts, reveal the rich diversity that characterizes the rapidly evolving field of early algebra. The volume articulates the ways in which researchers are currently conceptualizing early algebraization from curricu...

This article focuses on the teaching practice of a 10th grade teacher, who participated in a research project involving the use of novel tasks and CAS technology. The classroom lessons that are analyzed centered on a proving problem that was embedded within an extended task situation on factoring polynomials. A twofold analysis is presented, the fi...

The central question at stake in this chapter is: What theoretical frames are used in technology-related research in the domain of mathematics education and what do these theoretical perspectives offer? An historical overview of the development of theoretical frameworks that are considered to be relevant to the issue of integrating technological to...

Our research project aimed at understanding the complexity of the construction of knowledge in a CAS environment. Basing our work on the French instrumental approach, in particular the Task–Technique–Theory (T–T–T) theoretical frame as adapted from Chevallard’s Anthropological Theory of Didactics, we were mindful that a careful task design process...

This article is divided into four parts. The first part presents some introductory remarks on the use of Computer Algebra System (CAS) technology in relation to the long-standing dichotomy in algebra between procedures and concepts. The second part explores the technical-conceptual interface in algebraic activity and discusses what is meant by conc...

In this chapter we describe a classroom instructional activity that integrates the use of CAS in a manner aimed at engaging high school students in substantive mathematical reasoning. The activity draws on students' prior knowledge of factoring and pushes them to use and develop this knowledge beyond the usual for high school-level algebra. In part...

This paper, written within a discursive perspective, explores the co-shaping of public and private discourse, and some of the circumstances under which one occasions the other, in the evolution of mathematical thinking by pairs of 13-year-olds. The discourse of six pairs of students, engaged in interpreting and graphing problem situations involving...

In this reaction to the paper presented by Werner Peschek, I take issue with the notion put forward that, in general mathematics education, we ought to assign the field of competence for operative knowledge and skills primarily to the experts and, consequently, that this competence could be delegated almost completely to the electronic mathematical...

As is suggested by the title, this chapter presents research on algebra learning and teaching that has been carried out in various technological environments, more specifically those where the focus has been either multiple representations in computer and graphics calculator environments, or dynamic control, or structured symbolic calculation. Disc...

This paper describes some of the main contributions of research to the growing knowledge of the cognitive processes which the learning of algebra in secondary schools involves. The continuous attempts of researchers to develop a theory on the teaching/learning of algebra are also discussed. Finally, some future trends in the teaching/learning of al...

This paper addresses the dialectical relation between theoretical thinking and technique, as they co-emerge in a combined computer algebra (CAS) and paper-and-pencil environment. The theoretical framework in this ongoing study consists of the instrumental approach to tool use and an adaptation of Chevallard’s anthropological theory. The main aim is...

De nombreuses études font ressortir l'importance des explications orales des enseignants et enseignantes de mathématiques pour l'apprentissage des élèves. Cette compétence à expliquer les mathématiques en classe apparaît donc centrale à la formation des enseignants et enseignantes. Afin de mieux rendre compte de la construction de cette compétence...

This chapter is Carolyn Kieran’s Plenary Lecture that was presented at the 12th ICMI Study Conference. It presents a model for conceptualising algebraic activity that is a synthesis of three principal activities of school algebra: generational activity, transformational activity, and global/meta-level activity. The model is used as a basis for refl...

n during a CAS-based activity to develop a general factorization for 1 − n x . We illustrate how he attributes a meaning to the ellipsis symbol related to his experience of "terms cancelling out" rather than the taken-for-granted meaning of an undefined continuing process. We show his associated difficulties in making sense of the sequence of decre...

The study presented in this report is part of a larger project on the co-emergence of technique and theory within a CAS-based task environment for learning algebra, which also includes paper-and-pencil activity. In this paper, an example is presented in which 10th graders in Canada use handheld computer algebra to factor expressions of the form xn...

Equivalence of algebraic expressions is at the heart of transformational work in algebra. However, we know very little about students' understanding of equivalence. This study is part of a larger project that explores the use of CAS as a didactical tool for promoting both technical and conceptual growth in high school algebra with tasks specially d...

This research report, part of a larger study (Proulx, 2003), elaborates on a particular framework construed to analyze mathematical oral explanations given in the mathematics classroom. Originating from a lack of information on the characteristics of future mathematics teachers’ oral explanations, this study aimed to understand and make better sens...

This reaction paper is written from the perspective of a researcher whose work in the learning of algebra has primarily been with 12-to 16-year-olds. To provide some context for the comments and questions I will be raising with respect to the set of papers contained in this collection, I begin with remarks of a theoretical nature, first focusing on...

Investigated recent developments in the teching and learning of algebra and provided participants with a forum for sharing and discussing therir research endeavors, development projects, ans experiences. The teaching and learning of algebra is a difficult area for study because across different countries and even within countries, what is done in c...

La triade, tâche, technique et théorie, élaborée par des chercheurs français a servi de cadre conceptuel pour notre recherche sur l'utilisation de la calculatrice à affichage graphique par des élèves de Secondaire 1, 2 et 3 (12 - 15 ans), avec des tâches portant sur la théorie élémentaire des nombres. Dans cette recherche, nous avons d'abord analys...

Traditional approaches to research into mathematical thinking, such as the study of misconceptions and tacit models, have brought significant insight into the teaching and learning of mathematics, but have also left many important problems unresolved. In this paper, after taking a close look at two episodes that give rise to a number of difficult q...

The authors of this volume claim that mathematics can be usefully re-conceptualized as a special form of communication. As a result, the familiar discussion of mental schemes, misconceptions, and cognitive conflict is transformed into a consideration of activity, patterns of interaction, and communication failure. By equating thinking with communic...

In designing mathematics learning with the mediation of computerized tools, one of the crucial questions to be considered is how much and in what way we would like the tool «to do the work » for the students. In problem situations where the solution is achieved mostly via graphical representatives, and algebraic models are mostly used as «keys » fo...

Without Abstract

In this article we take a close look at the now popular claim that many school subjects, and mathematics among them, are best learned in an interactive way, through conversation with others. Two types of specially devised analytical tools are used to analyze the data coming from a two-month-long series of interactions between two 13-year-old boys l...

Bridging the individual and the social: Discursive approaches to research in mathematics education Guest editorial While looking at the papers collected in this volume one feels that in spite of their diverse themes, these seven studies have quite a lot in common and, as a collection, seem to be signaling the existence of a distinct, relatively new...

The main thesis of this chapter is that to prepare teachers to their new role in the reformed mathematics classroom, we must first understand better the mechanisms of the interactions supposed to take place in these classrooms. Of particular interest are the instances of students’ collective problem-solving. The student-to-student classroom interac...

A reflection on the relationship between theory and practice in the context of innovative approaches to the teaching and learning of algebra leads to take into account both the actual classroom practices and curriculum contents and the research results from new proposals.

Presents a teaching experiment to turn students from external observers into active participants in a game of algebra learning where students use graphs to build meaning for equivalence of algebraic expressions. Concludes that the graphic-functional approach seems to make the introduction to algebra much more meaningful for the learner. (ASK)

Members of the English-speaking mathematics education research community have long been awaiting the day when they could become better acquainted with Guy Brousseau's Theory of Didactical Situations. Brousseau, who is a professor of didactique des mathématiques at the Institut de Formation des Maîtres d'Aquitaine of the University of Bordeaux in Fr...

It may seem strange to some to imagine models as research results. How are research results defined? Narrowly, in the sense of empirical findings, or more broadly as the products of research endeavors -that are in some way linked to empirical work. At international conferences of mathematics education researchers, for example, the International Gro...

It may seem strange to some to imagine models as research results. How are research results defined? Narrowly, in the sense of empirical findings, or more broadly as the products of research endeavors that are in some way linked to empirical work. At international conferences of mathematics education researchers, for example, the International Grou...

The introduction of school algebra can take many different directions: the rules for transforming and solving equations (to which current teaching often reduces algebra), the solving of specific problems or classes of problems (which has played an important role historically in the development of algebra and its teaching), the generalization of law...

This chapter describes the main highlights of a seven-year program1 that was devoted to researching various aspects of introducing algebra to different groups of students by means of a technology-supported, functional approach. The software that was developed as part of the research program was CARAPACE, a functional problem-solving environment tha...

The articles in this issue arose from the keynote presentations and discussions of the Algebra Working Group of the Seventh International Congress on Mathematical Education (ICME-7) held in Québec City, Canada, in August 1992 (official title of the Working Group: The Place of Algebra in Secondary and Tertiary Education). Before presenting an overvi...

When we look back over the mathematics education research of the last 25 years as reflected in the pages of this journal and compare the per pectives of the past with those of the present, it is tempting to project onto the activities and intentions of the earlier researchers all of the notions that one finds in today's work. In fact. some current...

what compels many students to resort to memorizing the rules of algebra / what makes the comprehension of school algebra a difficult task for the majority / is the content of algebra the source of the problem / or is it the way it is taught that causes students not to be able to make sense of the subject / or do students approach algebraic tasks in...

Instructional strategies are presented that illustrate ways of developing students' understanding of nonnumerical notation that are compatible with a constructivist stance. The concepts of using letters to represent a range of values and those used to represent unknowns are discussed. (KR)

Describes a year-long geometry project involving 26 children (aged 12–13 yrs) using a Logo-based computer environment involving rectangles, circles, and isosceles triangles, with an emphasis on the latter. Aims of the project included increased awareness of (1) different quantifiable attributes of geometric figures, (2) attributes that are the inva...

Esta es una de las cinco publicaciones sobre investigación en Matemática Educativa resultado del trabajo académico, sobre investigación, realizado por la red de unidades Académicas del Programa Nacional de Formación y Actualización de Profesores de Matemáticas a lo largo del año 1990. En cada una de ellas se recogen los trabajos y los proyectos de...

This book, which was first published in 1990, is aimed at teachers, mathematics educators and general readers who are interested in mathematics education from a psychological point of view. The book describes research findings that shed light on the learning of mathematics from early arithmetic to high levels of algebra and geometry. The book is th...

This paper describes some of the main contributions of research to the growing knowledge of the cognitive processes which the learning of algebra in secondary schools involves.

This paper describes a year-long study of the way a group of six 12-year-old children went about solving a special class of geometric problems, using a computer and a limited set of Logo programming tools. We examined children's solution strategies from the perspective of whether the strategies led to insights about the underlying mathematical rela...