Lyndon C. Martin

The Equation, the Whole Equation and Nothing But the Equation! One Approach to the Teaching of Linear Equations

There exists an extensive range of research looking at the teaching and learning of linear equations, resulting in many papers highlighting a range of teaching approaches and illustrating a variety of significant cognitive problems and stumbling blocks to the learning of linear equations with understanding. Building on this literature, this paper presents some of the results of a case study which looked at the mathematics classroom of one particular teacher, Alwyn, trying to teach mathematics with meaning to less able pupils at secondary school level. Our interest here is those lessons which dealt specifically with the learning of linear equations, in which firstly a different approach was utilised and secondly many of the problems referred to in the literature were not present. We contrast this method with the teaching of linear equations to a variety of ability levels in several other classrooms that we have studied and we attempt through use of the Pirie-Kieren model, to analyse and account for the successful growth of understanding of the lower ability, year eight pupils in one particular classroom.

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.

The role of collecting in the growth of mathematical understanding

Folding back is one of the key components of the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding. This paper looks at one aspect of folding back, that of collecting. Collecting occurs when students know what is needed to solve a problem, and yet their understanding is not sufficient for the automatic recall of useable knowledge. They need to recollect some inner layer understanding and consolidate it through use at an outer layer in the light of their now more sophisticated understanding of the concept in question. The collecting phenomenon is described and distinguished through exemplars of classroom discourse, and implications for teachers and learners are discussed.

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.

Making Images And Noticing Properties: The Role Of Graphing Software In Mathematical Generalisation

This paper discusses the growth of mathematical understanding of two students, Graham and Don, as they use a computer graphing program to explore the properties of quadratic equations. Through analysing extracts of video data using the Pirie-Kieren theory, we discuss the ways in which the mathematical understanding of the students grows and how their interactions occasion, facilitate, and restrict this. We consider four ‘clips’ of their mathematical working, highlighting different aspects of their developing understanding, and use of the graphing software. Although we are talking about a computer based graphing package, our conclusions are equally relevant to the use of graphing calculators.

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.

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.

Images and the Growth of Understanding of Mathematics-for-Working

This article considers the nature and growth of mathematical understanding within the context of the workplace-training classroom. In doing this, it draws on elements of the Pirie-Kieren Theory for the Dynamical Growth of Mathematical Understanding, in particular the notions of “Image Making” and “Image Having.” These theoretical ideas are elaborated to allow the more appropriate description of how the understanding of construction industry apprentices is observed to grow in their training as they engage with tasks with significant mathematical components. Through doing this we illustrate how the Pirie-Kieren Theory can be used to describe the growth of understanding of “mathematics-for-working” (influenced by the work of Ball and Bass (2003) on “mathematics-forteaching”), and identify some of the key elements in this growth.RésuméCet article se penche sur la nature et l’évolution de la compréhension des mathématiques dans le cadre de la formation pratique en classe visant l’intégration dans le marché du travail. Ainsi, nous nous inspirons de certains éléments de la théorie de Pirie-Kieren sur la croissance dynamique de la compréhension mathématique, en particulier en ce qui concerne les notions de “création d’images” et de “possession d’images”. Ces notions théoriques sont construites pour qu’on puisse formuler une description plus adéquate des fac¸ons dont la compréhension, chez les apprentis dans le domaine de la construction, évolue au cours de leur formation au fur et à mesure qu’ils s’engagent dans des tâches comprenant des composantes mathématiques significatives. Ainsi, nous sommes en mesure d’illustrer comment la théorie de Pirie-Kieren peut servir à décrire l’évolution de la compréhension des “mathématiques pour le travail” (point de vue influencé par les travaux de Ball et Bass (2003) sur les “mathématiques pour l’apprentissage”), et à déterminer certains éléments clés de cette évolution.

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.

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.