# John MasonUniversity of Oxford & Open University · Education (Oxford); Maths & Stats (Open University)

John Mason

PhD Combinatorial Geometry

## About

192

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Citations since 2016

## Publications

Publications (192)

In this paper we consider implications of the current worldwide inclusion of computational thinking in relation to children's development of algebraic thinking. Little is known about how newly developed visual programming environments such as Scratch could enhance early algebra learning. The study is based on examples of programming activities used...

Is the natural number 7 rational? Is it complex? We argue that the answers to these questions relate to the ways numbers are taught. Commonly, a new kind of numbers is presented as an expansion of a previously familiar kind of numbers, which results in a nested image of the relations between number sets. In this article, we introduce an alternative...

Aspects of noticing which are often overlooked are brought to the surface and illustrated by accounting-for three accounts-of specific phenomena, two of which readers are invited to experience for themselves. These are used as a springboard for both illustrating the Discipline of Noticing as a method of sensitising oneself to notice possibilities f...

Making use of a phenomenological stance which first and foremost values the lived experience of learners, six tasks are used to illustrate what it might mean for a mathematical task to be deemed worthy of being offered to learners. These take the form of encounters with, and opportunities to develop, pervasive mathematical themes, use of mathematic...

For any convex quadrilateral, joining each vertex to the mid-point of the next-but-one edge in a clockwise direction produces an inner quadrilateral (as does doing so in a counter-clockwise direction). In many cases, a dynamic geometry measurement of the ratio of the area of the outer quadrilateral to the area of the inner one appears to be 5:1. It...

It is well known that educators such as Froebel, Dienes, and Gattegno recommend periods of free play with material objects before introducing mathematical questions designed to lead learners to encounter and articulate underlying mathematical relationships.

In this study, we identify ways in which a sample of 18 graduates with mathematics-related first degrees found the nth term for quadratic sequences from the first values of a sequence of data, presented on a computer screen in various formats: tabular, scattered data pairs and sequential. Participants’ approaches to identifying the nth term were re...

My interest here is in the way in which a single task can develop, gaining both in mathematical richness and in pedagogic purpose. This leads me to consider pedagogical affordances which can turn tasks from having rich potential into tasks used richly. The result is a domain of variations and extensions, together with possible pedagogical actions w...

In this paper, we provide an elaboration of the notion of mathematical structure – a term agreed upon as valuable but difficult to define. We pull apart the terminology surrounding the notion of structure in mathematics: relationship, generalising/generalisation and properties, and offer an architecture of structure that distinguishes and connects...

This study investigates an exploratory teaching style used in an undergraduate geometry course to help students identify an ellipse. We attempt to probe beneath the surface of exploration to understand how the actions of teachers can contribute to developing students’ competence in justifying an ellipse. We analyse the complex interactions between...

After a brief remark about methods, readers are invited to consider two related phenomena, and then to connect their experience with my comments about them. Two mathematical tasks follow, both intended to highlight particular features of attention and how it shifts. After a brief commentary about attention linked to pedagogical actions that might b...

Expressing generality, by learners, is seen as the heart and soul of school mathematics. To engage successfully requires shifts of attention from recognizing structural relationships in a particular situation to perceiving properties as being instantiated. Some examples of tasks which involve learners in encountering mathematical structure are prov...

Experience of using two different digital games is compared and contrasted in a dialogue format, because through discussion we became aware of features of which we had not previously been aware when working alone.

A fictional narrative format between a teacher and a mathematics educator is used to introduce pedagogical considerations around the theme of combining geometrical transformations. Questions about scalings from different centres, scaling from and rotating about different centres, and rotations about different centres are brought up and used to illu...

My answer is that it is never too early. In order to learn arithmetic it is necessary to think algebraically, although not necessarily using symbols. Some evidence for algebraic thinking amongst young children is given, followed by suggestions as to why such thinking has not always been promoted and developed. Specific pedagogic actions are outline...

It is well known and widely experienced that mathematical problem solving in general, and algebra in particular, form a watershed for many students. Something puts people off, whether in how it is presented (for example, when algebra is treated simply as “arithmetic with letters” and problems consist of burying arithmetic calculations in some conte...

Using a phenomenological stance, the chapter invites readers to undertake a variety of mathematical and pedagogic tasks. These are used as the basis for conjectures about the human psyche that draw upon ancient psychology and underpin both dual systems theory (Kahneman, 2012) and the notion of a bicameral brain (Jaynes, 1976; McGilchrist, 2009) as...

Consistent with a phenomenographic approach valuing lived experience as the basis for future actions, a collection of pedagogic strategies for introducing and developing algebraic thinking are exemplified and described. They are drawn from experience over many years working with students of all ages, teachers and other colleagues, and reading algeb...

The papers in this issue provide an excellent example of how pertinent distinctions which emerge from analysis of significant data sets can be communicated to colleagues and used to make sense of observations. The distinctions are vividly described, illustrated and exemplified, and used to chart maturation in student use of examples from middle sch...

Using a phenomenological stance which values lived experience, I probe beneath the surface of data presented as descriptive accounts-of incidents by focusing on attention. This includes both what is being attended to, and the form of that attention. Practical actions are proposed which can afford access into the lived experience of others, by askin...

Our overall concern is with helping students learn to construct and reconstruct proofs. Here we investigate an exploratory style which invites learners to think for themselves, with the instructor circulating among them while listening, probing, and suggesting. The objectives of this investigation are, to understand how the actions of teachers can...

This chapter is both a response to and a development from the preceding lucid and inspiring chapters in this book. It is a response because there are some clear issues arising from the examples of variation that have been provided, and I wish to elaborate on these. It is a development because my own thinking about the role of variation has been aug...

The chapters in this section, drawn together as a whole, illustrate the complexity of mathematical problem solving, both as an activity itself and as a way of teaching mathematics. Attention is paid to affect, to problem posing, to the largely absent support from textbooks for a problem solving, much less problem-posing stance, and to the effective...

Bill Brookes (1976) suggested that something is a problem only when a person experiences it as a problem. Ten years later, Christiansen and Walther (1986) suggested, following Vygotsky, that a task is what students are offered or inveigled to undertake, and activity is what happens as they attempt to carry out their interpretation of the task. Comb...

This papers comments on the contributions of ZDM Mathematics Education dedicated to the theme “Perception, interpretation and decision making: understanding the missing link between competence and performance”. The papers within this issue are brought together under the perspective of the stated aims of this issue, and then questions are raised abo...

In response to Gowers (Mathematical knowledge. Oxford University Press, Oxford, pp 33–58, 2007) who sought qualities of proofs that make them memorable, we suggest converting this into qualities that make proofs reconstructible. But Re-constructibility is a property of the person, situation and proof. This leads us to consider what values are being...

Upshot: In striving to support transition or bridging between arithmetic and algebra through software, Geraniou & Mavrikis come up against the need for learners not simply to "reflect" on what they have been doing, but to withdraw from action every so often, consider what actions have been effective, and construct their own narrative to hold togeth...

Becoming aware of something unexpected can be a form of awakening: sharpening attention, enriching noticing, opening up fresh possibilities of action, and educating awareness so as to enable a sensitive response to similar situations in the future. However it can also trigger tunnel vision (few if any actions available) or freezing (no actions avai...

This article draws on extensive experience working with secondary and tertiary teachers and educators using an applet to display rational polynomials (up to degree 7 in numerator and denominator), as support for the challenge to deduce as much as possible about the graph from the graphs of the numerator and the denominator. Pedagogical and design a...

Everyone gets stuck sometimes, and it can sometimes be frustrating, even debilitating rather than stimulating. However, being stuck is an honourable and useful state because that is when it is possible to learn about mathematics, about mathematical thinking, and about oneself. This applies especially to teachers, because the best way to sensitise y...

Some of Jeremy Kilpatrick’s contributions to mathematics education are used to contextualise the question of ‘when is [something] a problem?’ leading to different approaches to or value systems for engaging students in mathematical thinking. My own preferred approach is experiential and hence phenomenological, in which the reader is invited to comp...

Traducción al español y edición: Cecilia Agudelo Valderrama agudelo.cecilia@gmail.com El desarrollo del pensamiento algebraico – particularmente la identificación y expresión de regularidades – está al alcance de todo estudiante (todo ser humano), y es vital para su activa participación como ser social y como ciudadano productivo. Esta obra constit...

In this chapter three examples of teacher-guided use of ICT stimuli for learning mathematics (screencast, animation and applet) are critically examined using a range of distinctions derived from a complex framework. Six modes of interaction between teacher, student and mathematics are used to distinguish different affordances and constraints; five...

Gila Hanna and John Mason - for the learning of mathematics. Vol. 34 Num. 2 (2014)
Key ideas and memorability in proof
12-16
ABSTRACT:
This article discusses the concepts of "key ideas" and "memorability" and how they relate to the metric "width of a proof" put forward by the Fields medalist Timothy Gowers (2007) in a recent essay entitled "Mathem...

Consider two related tasks:
A) The diagram below displays a fragment of a pattern known to extend indefinitely and to be generated by an endlessly repeating pattern. What kind of shading will the 137th cell have? What number cell will the 137th white cell be in?
B) What is the next term in the sequence 3, 6, 11, 18, 27, …?
Tasks like the first can...

We report on our analysis of data from a dataset of 26 videotapes of university students working in groups of 2 and 3 on different proving problems. Our aim is to understand the role of example generation in the proving process, focusing on deliberate changes in representation and symbol manipulation. We suggest and illustrate four aspects of situa...

We propose that what teachers can say or write about their views of mathematics and about a range of pedagogic strategies and didactic tactics that they use is of minor importance compared to what comes to mind moment by moment when they are planning or leading a lesson. Beginning with a quote from a teacher that provoked a focus on the role and na...

Given a family of p ≥ 3 points in the plane, some three of them have the property that the smallest circle encompassing them encompasses all p points. Similarly, we show that for p ≥ 3 circles, there are three of them such that the smallest circle encompassing them encompasses all the circles, and that there are three circles for which the smallest...

Teaching “well” is seen here not as an engineering problem in which specific practices are located and demonstrated to “work,” but rather as a human endeavor calling upon the whole of the human psyche: behavior, cognition, affect and attention. The construct of different forms of attention as different ways of attending to something is used to anal...

A model describing factors affecting students’ learning will be presented. The model was based on Norman and Prichard's work which identified three major components of the teaching and learning situation, namely Entry Mastery, Student Motivation and Opportunity to Learn. I have focused on two components: Opportunity to Learn and Students’ Motivatio...

Teacher educators’ processes of establishing “mathematics for teaching” in teacher education programs have been recognized as an important area for further research. In this study, we examine how two teacher educators establish and make explicit features of mathematics for teaching within classroom interactions. The study shows how the establishmen...

Two tasks designed to encourage mathematical reasoning without any need for calculations
were presented to students with the aim of seeking evidence of different forms of attention
(Mason, 2003), and using these to learn about the tasks and about students’ power to reason ‘reasonably’ in mathematics. The first task that involves locating a secret p...

This paper elaborates the notion of a personal example space as the set of mathematical objects and construction techniques that a learner has access to as examples of a concept while working on a given task. This is different from the conventional space of examples that is represented by the worked examples and exercises in textbooks. We refer to...

John Mason examines the application of a traditional mathematical treatment of a geometry problem about the tangent to a cubic polynomial and its exploration with dynamic geometry software. He states that a more generalized view of the problem can be derived from each alternation, losing the tangents and the cubics along the way. John Mason chooses...

It is well known that the tangents at either end of a chord of a parabola meet in a point aligned vertically with the midpoint of the chord. In other words, the point of intersection of the tangents at the two ends of a chord on a parabola and the midpoint of that chord lie on a line parallel to the axis of the parabola. In this JOURNAL, Stenlund [...

During a sequence of tutorials conducted by the first author, it became evident that students were not seeing how to apply
the theorem concerning a continuous function on a closed and bounded interval attaining its extreme values in situations in
which it is necessary first to construct the closed and bounded interval by reasoning about properties...

Mathematically, most theorems can be seen as classifying those mathematical objects which satisfy certain properties, in terms
of other, usually more manageable, properties. Thus Pythagoras’ theorem classifies right-angled triangles as those triangles
for which the sum of the squares on two sides is the square on the third, while the law of cosines...

A methodology is offered for studying problem solving from the inside. This approach is first set in the context of millenia
of educational thinkers, and then in the context of providing support for the professional development of teachers of mathematics.
The approach is based on the development and strengthening of each teacher's own awareness of...

Using parts of a structural approach developed by Bennett (1966, 1970) for understanding complex situations, traditional and visionary approaches to teaching algebra are compared and contrasted,
informed by the chapters in this part. In particular, a four-term structure for activity is used to consider how a transition
from traditional to visionary...

Twenty-eight years ago The Open University initiated experientially-based PD courses for teachers at a distance. Video and mathematical tasks were used to provide experience which was commented on using constructs derived from the literature. Here we report on a study to find out what roles the constructs had played in people’s subsequent professio...

The process of example construction is explored phenomenologically as a study in self-communication. Five instances of mathematical
example construction are described. Four of these involve upper secondary or tertiary mathematical content (functions, sequences)
and one could be used as a task at almost any age. Comments and possible lessons to be l...

Two components of the human psyche, attention and intention, are used to expose aspects of teaching that can be learned from being awake to and aware of what is happening while teaching. The case is made that in order to learn from experience it is necessary to do more than engage in activity: It is necessary to withdraw from the action and reflect...

Student responses to arithmetical questions that can be solved by using arithmetical structure can serve to reveal the extent and nature of relational, as opposed to computational thinking. Student responses to probes which require them to justify-on-demand are analysed using a conceptual framework which highlights distinctions between different fo...

We takemathematical structure to mean the identification of general properties which are instantiated in particular situations as relationships between
elements or subsets of elements of a set. Because we take the view that appreciating structure is powerfully productive, attention
to structure should be an essential part of mathematical teaching a...

This book makes accessible to calculus students in high school, college and university a range of counter-examples to “conjectures” that many students erroneously make. In addition, it urges readers to construct their own examples by tinkering with the ones shown here in order to enrich the example spaces to which they have access, and to deepen th...

This paper is designed to raise issues around how we view teaching and some of the implication that has for thinking about teaching as a discipline. The paper is built around the concept of 'noticing' from some of my earlier work and aims to push ideas about teaching in ways that are intended to provoke readers into thinking more deeply about how t...

This book makes accessible to calculus students in high school, college and university a range of counter-examples to “conjectures” that many students erroneously make. In addition, it urges readers to construct their own examples by tinkering with the ones shown here in order to enrich the example spaces to which they have access, and to deepen th...

Children learn about number by talking and doing. This book offers closely observed accounts of what children in their early years and at primary school say and do. Some arise from classrooms; others are based on interviews and day to day interactions. Together they illuminate the learning, and therefore the teaching, of mathematics to young childr...

The issue addressed in this chapter is age-old: How can learners be stimulated to move from assenting (passively and silently accepting what they are told, doing what they are shown how to do) to asserting (actively taking initiative, by making, testing and modifying conjectures, and by taking responsibility for making subject pertinent choices). H...

In this reflective paper, we explore students' local and global thinking about informal statistical inference through our observations of 10- to 11-year-olds, challenged to infer the unknown configuration of a virtual die, but able to use the die to generate as much data as they felt necessary. We report how they tended to focus on local changes in...

Building on the papers in this special issue as well as on our own experience and research, we try to shed light on the construct
of example spaces and on how it can inform research and practice in the teaching and learning of mathematical concepts. Consistent with our
way of working, we delay definition until after appropriate reader experience ha...

Participants will be invited to experience the use of mathematical phenomena to stimulate interest and to encounter core-awarenesses which lie behind mathematical topics, and to develop richly interconnected example-spaces. These experiences will be related to a four-fold structure of the human psyche which can inform choices of pedagogical strateg...

This paper focuses on eight-year old students’ ways of approaching true/false number sentences. The data presented here belongs to a teaching experiment in which the use of relational thinking when solving number sentences was explicitly promoted. The study of the way of using this type of thinking and of students’ structure of attention, allow us...

Este artículo se centra en las formas en que alumnos de ocho años abordan la resolución de sentencias numéricas verdaderas y falsas. Los datos que se presentan pertenecen a un experimento de enseñanza en el cual se promovió explícitamente el uso del pensamiento relacional en la resolución de sentencias numéricas. El estudio del modo en que es usado...

The principal aim of this chapter is to provide a structure for mathematical topics as an aid to “psychologizing the subject matter’, as Dewey (1933) put it. The secondary aim is to reveal just how complex a matter preparing to teach a topic effectively can be, beyond trying to make the definitions and theorems as clear as possible. The chapter dev...

Bartering problems in arithmetic books appear on the surface to be merely an exercise in the rule of three. However there is a type of problem that contains a surprise for modern readers. This article traces some of the appearances of barter problems in arithmetic textbooks over a period of more than 400 years, and from Italy to England, and examin...

We believe there is a need for an effective mathematical pedagogy in the learning of advanced mathematics. The move from elementary to advanced mathematical thinking generally involves a significant cognitive reconstruction for the students. Previous research has shown that many students will struggle as they encounter the new mathematical ideas an...