David Tall

The University of Warwick · Institute of Education

As mathematicians reflect on their teaching of students, they have their own personal experience of mathematics that they seek to teach. This chapter offers an overall framework to consider how different mathematical specialisms may require different approaches depending on the nature of the specialism and the needs of the students. It involves not...

Function is a multifaceted and essential concept in mathematics that includes various complex ideas such as mapping, relations between variables, algebra, input-output, and rule. Pre-service mathematics teachers’ (PMTs) should have a rich understanding of these ideas for their future teaching practices. However, research showed that PMTs had a limi...

In this chapter we consider how research into the operation of the brain can give practical advice to teachers and learners to assist them in their long-term development of mathematical thinking. At one level, there is extensive research in neurophysiology that gives some insights into the structure and operation of the brain; for example, magnetic...

This is a chapter of the book Complex Analysis by Stewart and Tall, on sale from Cambridge University Press. (c) CUP.

Cambridge Core - Real and Complex Analysis - Complex Analysis - by Ian Stewart

This chapter offers a framework for the long-term development of sense making and anxiety for mathematics in general and algebra in particular. While many may see the development of algebra building from the basic ideas of arithmetic and generalizing to algebraic techniques for formulating and solving problems, over the long-term increasingly subtl...

Lesson Study is a format to build and analyse classroom teaching where teachers and researchers combine to design lessons, predict how the lessons might be expected to develop, then carry out the lessons with a group of observers bringing multiple perspectives on what actually happened during the lesson. This article considers how a lesson, or grou...

First published in 1979 and written by two distinguished mathematicians with a special gift for exposition, this book is now available in a completely revised third edition. It reflects the exciting developments in number theory during the past two decades that culminated in the proof of Fermat’s Last Theorem. Intended as a upper level textbook, it...

A teaching experiment—using Mathematica to investigate the convergence of sequence of functions visually as a sequence of objects (graphs) converging onto a fixed object (the graph of the limit function)—is here used to analyze how the approach can support the dynamic blending of visual and symbolic representations that has the potential to lead to...

In this paper we consider data from a study in which students shift from linear to quadratic equations in ways that do not conform to established theoretical frameworks. In solving linear equations, the students did not exhibit the ‘didactic cut’ of Filloy and Rojano (1989) or the subtleties arising from conceiving an equation as a balance (Vlassis...

A teaching experiment—using Mathematica to investigate the convergence of sequence of functions visually as a sequence of objects (graphs) converging onto a fixed object (the graph of the limit function)—is here used to analyze how the approach can support the dynamic blending of visual and symbolic representations that has the potential to lead to...

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In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy's ideas of function, continuity, limit and infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and...

This chapter charts the growth of proof from early childhood through practical generic proof based on examples, theoretical proof based on definitions of observed phenomena, and on to formal proof based on set theoretic definitions. It grows from human foundations of perception, operation and reason, based on human embodiment and symbolism that may...

This study combines the Japanese lesson study approach and mathematics teachers’ professional development. The first year of a 4-year project in which 3 Dutch secondary school teachers worked cooperatively on introducing making sense of the calculus is reported. The analysis focusses on instrumental and relational student understanding of mathemati...

How Humans Learn to Think Mathematically describes the development of mathematical thinking from the young child to the sophisticated adult. Professor David Tall reveals the reasons why mathematical concepts that make sense in one context may become problematic in another. For example, a child's experience of whole number arithmetic successively af...

In this paper we study the difficulties resulting from changes in meaning of the minus sign, from an operation on numbers, to a sign designating a negative number, to the additive inverse of an algebraic symbol on students in two-year colleges and universities. Analysis of the development of algebra reveals that these successive meanings that the s...

This chapter reflects on the evolution of the mathematics of change and variation as technology affords the possibility of conceptualising and communi-cating ideas for a wider range of learners than the few who traditionally study the higher levels of the calculus. It considers the overall program of development con-ceived by Jim Kaput, instantiate...

In recent years, reform calculus has used the computer to show dynamic visual graphics and to offer previously unimaginable power of numeric and symbolic computation. Yet the available technology has far greater potential to allow students (and mathematicians) to make sense of the ideas. A sensible approach to the calculus builds on the evidence of...

This paper evolves a framework for making sense of mathematics through perception, operation and reason and uses it in a specific case of trigonometry. It is based on a fundamental theory of how humans learn to think mathematically from early childhood to the frontiers of mathematical research. It is intended to be of theoretical value in mathemati...

This is a very successful textbook for undergraduate students of pure mathematics. Students often find the subject of complex analysis very difficult. Here the authors, who are experienced and well-known expositors, avoid many of such difficulties by using two principles: (1) generalising concepts familiar from real analysis; (2) adopting an approa...

The Dirac delta function has solid roots in 19th century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his infinitesimal definition of delta.

were particularly interested in the distinction between objects formed in geometry (such as points, lines, circles, polyhedra) and concepts studied in arithmetic, algebra and symbolic calculus (numbers, algebraic expressions, limits). We concluded that the development of geometric concepts followed a natural growth of sophistication ably described...

We discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of mathematical thinking and imaginative visual interpretations of axiomatic proof.

This study combines elements of the Japanese Lesson Study approach and teachers’ professional development. An explorative research design is conducted with three upper level high school teachers in the light of educational design research, whereby design activities will be cyclically evaluated. The Lesson Study team observed and evaluated two diffe...

This chapter traces the long-term cognitive development of mathematical proof from the young child to the frontiers of research. It uses a framework building from perception and action, through proof by embodied actions and classifications, geometric proof and operational proof in arithmetic and algebra, to the formal set-theoretic definition and f...

In this paper, the development of mathematical concepts over time is considered. Particular reference is given to the shifting of attention from step-by-step procedures that are performed in time, to symbolism that can be manipulated as mental entities on paper and in the mind. The development is analysed using different theoretical perspectives, i...

question: "Is 0.999... (nought point nine recurring) equal to one, or just less than one?". Many answers contained infinitesimal concepts: "The same, because the difference between them is infinitely small." " The same, for at infinity it comes so close to one it can be considered the same." "Just less than one, but it is the nearest you can get to...

While the general notion of ‘metaphor’ may offer a thoughtful analysis of the nature of mathematical thinking, this paper suggests that it is even more important to take into account the particular mental structures available to the individual that have been built from experience that the individual has ‘met-before.’ The notion of ‘met-before’ offe...

This paper uses the framework of “three worlds of mathematics” (Tall 2004a, b) to chart the development of mathematical thinking from the thought processes of early childhood to the formal structures of set-theoretic definition and formal proof. It sees the development of mathematical thinking building on experiences that the individual has met bef...

This paper considers the role of dynamic aspects of mathematics specifically focusing on the calculus, including computer software that responds to physical action to produce dynamic visual effects. The development builds from dynamic human embodiment, uses arithmetic calculations in computer software to calculate ‘good enough’ values of required q...

What is the nature of mathematical thinking, problem-solving and proof? In the book Thinking Mathematically that John Mason wrote with Leone Burton and Kaye Stacey, the term 'proof' never appears. On enquiring the reason for this, John expressed the deep fear that the word 'proof' engendered in his Summer School students. In this paper I will refle...

Proof is a construct of mathematical communities over many generations and is introduced to new generations as they develop cognitively in a social context. Here I present a practical framework for this development in simple terms that nevertheless has deep origins. The framework builds on an analysis of the growth of mathematical ideas based on ge...

This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts towards...

Jim Kaput lived a full life in mathematics education and we have many reasons to be grateful to him, not only for his vision of the use of technology in mathematics, but also for his fundamental humanity. This paper considers the origins of his ‘big ideas’ as he lived through the most amazing innovations in technology that have changed our lives mo...

How do students think about algebra? Here we consider a theoretical framework which builds from natural human functioning in terms of embodiment – perceiving the world, acting on it and reflecting on the effect of the actions – to shift to the use of symbolism to solve linear equations. In the main, the students involved in this study do not encaps...

Recent literature has pointed out pedagogical obstacles associated with the use of computational environments in the learning of mathematics. In this paper, we focus on the pedagogical role of the computer's inherent limitations in the development of learners' concept images of derivative. In particular, we intend to discuss how the approach to thi...

At this conference we are considering the use of new technologies in mathematics. Many presenters will show the use of a wide variety of technological environments, symbol manipulators, geometry environments, graphical facilities, spreadsheets, statistical packages, interactive productivity tools, electronic books, multi-media, and so on. These ena...

This paper considers mathematical abstraction as arising through a natural mechanism of the biological brain in which complicated phenomena are compressed into thinkable concepts. The neurons in the brain continually fire in parallel and the brain copes with the saturation of information by the simple expedient of suppressing irrelevant data and fo...

In this paper I formulate a basic theoretical framework for the ways in which mathematical thinking grows as the child develops and matures into an adult. There is an essential need to focus on important phenomena, to name them and reflect on them to build rich concepts that are both powerful in use and yet simple to connect to other concepts. The...

This paper is written in the context of the conference on Mathematical Learning from Early Childhood to Adulthood, which focuses on the overall growth of mathematical thinking in individuals. It is in two parts, the first presents a framework of long-term cognitive growth and the second uses this theory to address questions posed for discussion at...

This article considers two students who both pass a preliminary course in college algebra at the same level; one proceeds satisfactorily in the next course whilst the other finds it impossibly difficult. We analyse the responses of the students in semi-structured interviews on topics from the first course to seek reasons for this difference in late...

This paper presents an analysis of the function concept. Although it is seen by many as a fundamental building block of the mathematical curriculum (NCTM, 1989), in practice, few students grasp the full extent of the notion of function linking across its various different representations. The data presented here comes from a research project at War...

Mathematics Education Research Centre University of Warwick, CV4 7AL, United Kingdom In reacting to this forum on'Algebraic Equalities and Inequalities', I take a problem-solving approach, first, asking'what is the problem?'then looking at the five presentations to see what can be synthesized from their various positions (acknowledging that they ar...

It is probably difficult for those of us looking at the huge range of current mathematics education research to imagine the state of the theory when the International Group for the Psychology of Mathematics Education was first conceived. Psychology was still in the grips of behaviorism, with the teaching of mathematics largely in the hands of mathe...

Using Technology to Support an Embodied Approach to Learning Concepts in Mathematics. In L.M. Carvalho and L.C. Guimarães História e Tecnologia no Ensino da Matemática, vol. 1, pp. 1-28, Rio de Janeiro, Brasil.This paper consists of two parts: the first is a developing theory of three modes of thinking: embodied/proceptual/formal. The second part b...

In this paper the notion of "procept" (in the sense of Gray & Tall, 1994) is extended to advanced mathematics by considering mathematical proof as "formal procept". The statement of a theorem or question as a symbol may evoke the proof deduction as a process that may contain sequential procedures and require the synthesis of distinct cognitive unit...

In this paper, we contrast the mathematical simplicity of the function concept that is appreciated by some students and the spectrum of cognitive complications that most students have in coping with the function definition in its many representations. Our data is based on interviews with nine (17-year old) students selected as a cross-section from...

In this paper we focus on students' understanding of the core concept of function, which is mathematically simple yet carries within it a rich complexity of mathematical ideas. We investigate the linguistic complexity that reveals itself through the mathematically simple notion of a constant function and the representational complexity involved in...

This presentation will consider different modes of proof that may be appropriate in mathematics and the kinds of belief that underpin them. In particular, it will distinguish between three distinct worlds of mathematical thought: the embodied, the proceptual (using symbols as process and concept in arithmetic and algebra) and the formal. Embodied p...

The transition to formal mathematical thinking involves the use of quantified statements as definitions from which further properties are constructed by formal deduction. Our quest in this paper is to consider how students construct meaning for these quantified statements. Dubinsky and his colleagues (1988) suggest that the process occurs through r...

The notion of "learning by reason" rather than "learning by rote" has long been a focus of creative teaching. In writing the oft-quoted paper on "instrumental understanding" and "relational understanding", Richard Skemp (1976) is a significant link in the chain of those who developed notions of "meaningful learning". Skemp, however, had wider goals...

Without Abstract

Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conception as natural infinities.Research has shown that individuals' natural conceptions of infinity are `labile and self-contradictory' (Fischbein et al.,1979, p. 31). The formal approach to ma...

In this paper we propose a theory of cognitive construction in mathematics that gives a unified explanation of the power and difficulty of cognitive development in a wide range of contexts. It is based on an analysis of how operations on embodied objects may be seen in two distinct ways: as embodied configurations given by the operations, and as re...

This research focuses on students using an experimental approach with computer software to give visual meaning to symbolic ideas and to provide a basis for further generalisation. They use computer software that draws orbits of x=f(x) iteration and are encouraged to investigate the iterations of fλ(x)=λx(1-x) as λ increases. The iterations pass thr...

This paper investigates the development of university students' understanding on 'equivalence relations & partitions' over a period of time. Although these ideas are taught in the same topic, they have quite different cognitive properties. We find that, although the concept of 'relation' can be visualised, an 'equivalence relation' is more subtle....

This paper presents a framework to follow students' development through a formal mathematics course. Built on learning episodes of two groups of three students attending a traditional university course, it highlights different cognitive demands learners will have according to their personal learning strategy.

There has long been a dichotomy between the mathematics that is learned by trainee teachers 'for their own personal development' and the mathematics they will need to teach to young children. My viewpoint is that a student should be encouraged to study whatever it is that interests them, but when it comes to pre-requisites for teaching mathematics...

This paper discusses the long-term cognitive development of the meaning of symbols in algebra, starting with symbols in arithmetic as procedure, process and concept, on to generalised arithmetic, evaluation algebra (treating expressions as evaluation processes), manipulation algebra and then axiomatic algebra. We discuss cognitive changes required...

This paper considers the case of students who attain the same level of performance at the end of one course and yet reveal very different levels of success on the course which follows. A comparison is made between two students attaining a grade B in college algebra who perform differently in the succeeding Pre-Calculus Course. Interviews reveal qui...