Jo Towers

Collective Mathematical Understanding as Improvisation

This article explores the phenomenon of mathematical understanding, and offers a response to the question raised by Martin (2001) at the Annual Meeting of the Psychology of Mathematics Education Group (North American Chapter) about the possibility for and nature of collective mathematical understanding. In referring to collective mathematical understanding, we point to the kinds of learning and understanding we may see occurring when a group of learners work together on a piece of mathematics. We characterize the growth of collective mathematical understanding as a creative and emergent improvisational process and illustrate how this can be observed in action. In doing this, we demonstrate how a collective perspective on mathematical understanding can more fully explain its growth. We also discuss how considering the growth of mathematical understanding as a collective process has implications for classroom practice and in particular for the setting of mathematical tasks.

Improvisational coactions and the growth of collective mathematical understanding

In this paper we consider the phenomenon of the growth of collective mathematical understanding and explore its dependence on the particular way that a group of learners work together collaboratively. We label this group process as improvisational coaction. In an earlier paper (Martin, Towers and Pirie, 2006) we drew on the theoretical work of Becker (2000), Sawyer (2001, 2003, 2004), and Berliner (1994) in improvisational jazz and theatre, to characterise the growth of collective mathematical understanding as a creative and emergent improvisational process. Here, we extend that conceptual analysis to a yet-finer grain to explore one element of that framework, improvisational coaction, and its relationship to the growth of mathematical understanding at the level of the group. In particular we identify improvisational coaction as a particular form of interaction, and through using data extracts we derive four characteristics of the phenomenon and consider how these occasion the growth of collective mathematical understanding.

Using Video in Teacher Education.

This paper draws on a research study of elementary- and secondary-route preservice teachers in a two-year, after-degree teacher preparation programme. The paper includes excerpts of classroom data, taken from the author’s own university classroom, demonstrating preservice teachers’ responses to carefully selected video extracts of children learning mathematics in a high-school class also taught by the author. The paper includes commentary on some of the advantages and limitations of video as a teaching tool, develops an argument for the increased use, in both preservice teacher education and inservice teacher professional development, of videotaped episodes that focus on the learners rather than on the classroom teacher, and explores the value of having the teacher whose classroom is featured on the videos present for the discussion of the episodes. The paper explores the potential offered by video material to foster the belief that teaching is a learning activity by (i) refocusing attention on the learner rather than the teacher in the analysis of classroom practices, (ii) raising awareness of the importance of reflective practice, and (iii) providing a prompt for the imaginative rehearsal of action. Resume : Le present article se fonde sur une etude technique portant sur des stagiaires des niveaux primaire et secondaire dans un programme de preparation a l’enseignement de deux ans apres l’obtention du diplome. L’article comprend des extraits de donnees en salle de classe qui proviennent de la salle de classe de l’universite de l’auteur meme, illustrant les reponses des stagiaires a des extraits video choisis avec soins, extraits portant su des enfants apprenant les mathematiques dans une classe du secondaire dont l’enseignant est l’auteur. L’article comporte des commentaires sur certains des avantages et limites du video comme outil d’enseignement, il presente un argument pour l’augmentation accrue, a la fois pour l’education du stagiaire et le perfectionnement professionnel de l’enseignant qualifie, des episodes sur cassette video qui mettent l’accent sur les apprenants plutot que sur les enseignants en salle de classe et examine s’il est interessant que soit present au moment de la discussion sur les episodes l’enseignant dont la salle de classe figure sur le video. L’article analyse le potentiel du materiel video appuyant la croyance selon laquelle l’enseignement est une activite d’apprentissage en (i) mettant l’accent sur l’apprenant plutot que sur l’enseignant dans l’analyse des pratiques d’une salle de classe, (ii) en effectuant une sensibilisation relativement a l’importance de la pratique reflexive et (iii) en guidant la repetition novatrice de la mesure.

Deepening Understanding of Inquiry Teaching and Learning with E-Portfolios in a Teacher Preparation Program

This paper describes the first stages of a project focusing on the use of preservice-teacher-generated e-Portfolios as a means of documenting and assessing inquiry-based teaching and learning. The project is designed to explore ways in which preservice teacher-created e-Portfolios can be used to (1) document how inquiry lives in practice, and (2) help university instructors and practitioners in the field assess the knowledge, skills, and attributes of preservice teachers who are participating in an inquiry based teacher preparation program.

Building Mathematical Understanding through Collective Property Noticing.

In this article we explore the mechanisms through which one group of preservice teachers engage in Collective Property Noticing—a phenomenon in which group members integrate individual contributions such that the group, as a unit, notices mathematical properties of their collective image. Drawing on improvisational theory to help to illuminate these collaborative processes, we claim that Collective Property Noticing is a capacity that is vital for mathematical sense-making in collaborative groups and we propose several conditions under which it is appropriate for a teacher to intervene in students’ learning in a problem-solving setting in order to provoke Collective Property Noticing.FrRésumé: Dans cet article, nous nous penchons sur les mécanismes grâce auxquels un groupe de futurs enseignants participe à des activités d’observation des propriétés collectives, durant lesquelles les membres du groupe intègrent les contributions individuelles de chacun de façon à ce que le groupe en tant qu’équipe puisse observer les propriétés de son image collective. Sur la base d’une théorie de l’improvisation servant à éclairer ces processus de collaboration, nous postulons que l’observation des propriétés collectives constitue une habileté vitale pour la construction du sens mathématique dans les groupes de collaboration, et nous formulons plusieurs conditions dans lesquelles il est approprié que les enseignants interviennent dans l’apprentissage des étudiants, dans un contexte de résolution de problèmes, de façon à stimuler l’observation des propriétés collectives.

Blocking the growth of mathematical understanding: A challenge for teaching

This paper presents and discusses some of the findings of a research project that focused on teaching and learning in two high-school mathematics classrooms. The focus of the study was to consider the ways in which teachers’ classroom interventions promote the growth of students’ mathematical understanding. Analysis of the data resulted in the generation of a number of themes describing the teachers’ interventions. One of these themes, that I callblocking, is the subject of this paper. The paper discusses the implications of this intervention strategy for teaching, learning, research, and teacher education.

Structure and Improvisation in Creative Teaching: Improvisational Understanding in the Mathematics Classroom

In many mathematics classrooms, students learn largely by memorization; they memorize procedures, such as how to multiply two fractions, and they are then assessed by being presented with similar problems, which they can solve if they have memorized the procedure. The problem with this approach is that all too often, students fail to acquire any deeper understanding of mathematical ideas and concepts – for example, what does a fraction represent? How is it similar to a decimal, a ratio, or a percentage? What does it mean to multiply two fractions? Almost all experts in mathematics education agree that understanding mathematical ideas and concepts is a critical and desirable component of the mathematics classroom, yet teachers continue to struggle with meaningful ways to teach for mathematical understanding. Almost all teachers agree that understanding involves more than procedural knowledge and that it includes the ability to reason with and to make sense of what is learned, but the translation of this into concrete teaching strategies that can be implemented in the classroom remains a challenge.

Students’ Emotional Experiences Learning Mathematics in Canadian Schools

In this chapter, we draw on Canadian Kindergarten to Grade 9 students’ autobiographical accounts of learning mathematics in schools and their drawings of their feelings about doing mathematics in order to explore students’ relationships with mathematics and the emotions associated with doing mathematics. Drawing on enactivist thought, we offer insight into the complex relationship between emotion and learning. Our analysis reveals a nuanced emotional landscape associated with learning mathematics, including positive, negative, and highly topic-dependent relationships with mathematics among this population, together with narratives of changing relationships that shed light on the kinds of pedagogies that support and detract from learning. Drawings of students’ heads feature widely in the data, prompting us to raise questions about the disembodied nature of mathematics learning in schools.

The teacher's responsibility in whole‐class debriefing of mathematical activity

Prompted by recent moves in the United Kingdom to guide teachers’ practices in whole-class, direct interactive teaching, in this article, we offer an opportunity for North American mathematics educators to reflect on possibilities for whole-class teaching of mathematics. We focus particularly on the plenary aspect of lessons—what might be considered the debriefing of mathematical activity—and specifically on the teacher’s responsibility during those sessions, both to his or her students and to the authenticity of the discipline of mathematics. Drawing on data from a Grade-3 classroom and invoking complexity science as a theoretical lens to explore the classroom as a complex learning system, we present implications for teaching in whole-class debriefings of mathematical activity.RésuméDans cet article, en réaction à de récents développements en Grande Bretagne visant à guider la pratique de l’enseignement direct et interactif dans des classes nombreuses, nous souhaitons offrir aux enseignants des mathématiques en Amérique du Nord ‘l’occasion de réfléchir sur les possibilités que représente l’enseignement des mathématiques à des classes pleines. Nous centrons surtout notre attention sur les aspects ‘pléniers’ des leçons — ce que nous pourrions qualifier de débriefing des activités mathématiques — et surtout sur la responsabilité des enseignants au cours de ces sessions, en particulier envers les étudiants et envers l’authenticité qui caractérise la discipline des mathématiques. Notre argumentation se fonde sur des données provenant d’une classe de troisième année, et nous nous servons de la complexité des sciences comme lunette théorique permettant d’explorer la salle de classe comme système complexe d’apprentissage. Nous présentons ensuite certaines implications pour l’enseignement dans le cadre de débriefings de classe pour ce qui est des activités mathématiques.

Folding back and growing mathematical understanding: a longitudinal study of learning

Purpose The purpose of this paper is to summarize some of the key findings and approaches used in documenting the authors’ longitudinal studies of mathematical learning and understanding. In particular, it focuses on “folding back,” a theoretical construct originally developed by Susan Pirie and Tom Kieren, to show how, over the last two decades, the authors have taken up, built-upon, and elaborated this construct in relation to Pirie and Kieren’s wider theorizing and in relation to classroom practice. Design/methodology/approach The paper documents the various methodologies and methods the authors have used to elaborate theory and contribute to extending teaching practice in a number of related research studies. Findings This paper describes the role of folding back in the growth of students’ mathematical understanding, initially at the level of the individual, more recently at that of the collective – and currently with a specific consideration of the role of the teacher. It notes that the longitudinal nature of the work has allowed it to respond to shifting perspectives in the field of mathematics education and to become a more nuanced and powerful analytic and teaching tool. Originality/value The paper discusses the significance of a longitudinal, shared program of research, deeply rooted in mathematics classrooms, that builds theory systematically and over an extended period of time.