- Kenilworth, Warwickshire, United Kingdom
David Tall
University of Oxford, Education, Emeritus
Research Interests:
Research Interests:
Research Interests:
This study investigates the manner in which students build up meaning to draw the gradient of a given graph. Some researchers claim that students tend to use algebra to solve calculus questions. This research suggests this may happen when... more
This study investigates the manner in which students build up meaning to draw the gradient of a given graph. Some researchers claim that students tend to use algebra to solve calculus questions. This research suggests this may happen when students are encouraged to ...
Research Interests:
The editors of MERJ would like to express our sincere appreciation to the following people who reviewed manuscripts for Volume 20 of the Mathematics Education Research Journal. Also, we would like to thank the members of the Editorial... more
The editors of MERJ would like to express our sincere appreciation to the following people who reviewed manuscripts for Volume 20 of the Mathematics Education Research Journal. Also, we would like to thank the members of the Editorial Board (listed on the inside front cover), ...
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
... experience. (Dubinsky & Leron, 1994, p. xiv) ... learning. Dubinsky & Leron use the programming language ISETL (Interactive SET Language) to get the students to engage in programming mathematical constructs in... more
... experience. (Dubinsky & Leron, 1994, p. xiv) ... learning. Dubinsky & Leron use the programming language ISETL (Interactive SET Language) to get the students to engage in programming mathematical constructs in group theory and ring theory. ...
The major focus of this study is to trace the cognitive development of students throughout a mathematics course and to seek the qualitative differences between those of different levels of achievement. The aspect of the project described... more
The major focus of this study is to trace the cognitive development of students throughout a mathematics course and to seek the qualitative differences between those of different levels of achievement. The aspect of the project described here concerns the use of concept maps constructed by the students at intervals during the course. From these maps, schematic diagrams were constructed which strip the concept maps of detail and show only how they are successively built by keeping some old elements, reorganizing, and introducing new elements. The more successful student added new elements to old in a structure that gradually increased in complexity and richness. The less successful had little constructive growth, building new maps on each occasion. (Contains 19 references.) (Author/MM) Reproductions supplied by EDRS are the best that can be made from the original document. 1PERMISSION TO REPRODUCE AND
This presentation considers how the peculiar structure of the biological brain may be supported by the computational power of the computer to enhance mathematical thinking. It considers how we think and learn mathematics with particular... more
This presentation considers how the peculiar structure of the biological brain may be supported by the computational power of the computer to enhance mathematical thinking. It considers how we think and learn mathematics with particular reference to the use of visualisation and symbol manipulation. Visualisation occupies a major portion of the brain’s cortex and enables Homo Sapiens to ‘see ’ how ideas can be formed and related. Mathematical symbols in arithmetic, algebra, calculus particularly suit the biological brain, acting as pivots between concepts for thinking about mathematics and processes to calculate and predict. We use the term ‘procept ’ to describe this particular combination of symbol as process and concept. Analysis of procepts reveals that the development of symbols does not follow an easy cognitive path for the growing individual because they operate in significantly different ways in arithmetic, algebra and the calculus. We therefore advocate a versatile approach ...
Research Interests:
Research Interests:
In their critique of " object as a central metaphor in advanced mathematical thinking " , Confrey and Costa (1996) describes members of the Advanced Mathematical Thinking Group, including myself, as " reification theorists... more
In their critique of " object as a central metaphor in advanced mathematical thinking " , Confrey and Costa (1996) describes members of the Advanced Mathematical Thinking Group, including myself, as " reification theorists ". By selective quation they attibute theores largely developed independently by Dubinsky and Sfard as being broadly shared. Whilst it is true that many share an interest in the relationship of process and object and the mediating role of symbols, the notion of " reification " is only part of the domain of discourse. In the book " Advanced Mathematical Thinking " to which Confrey and Costa refer, only two chapters out of thirteen can be considered as " reificationst " — a chapter by Harel and Kaput which focuses on the notion of " conceptual entity " as part of a wider theory and a chapter by Dubinsky on " Reflective Abstraction ". Quite different persepectives are also presented, for instance t...
The framework of ‘three worlds of mathematics’ was designed to reveal the growth of mathematical thinking in individuals over a lifetime to include all abilities and interests (Tall, 2013). This paper outlines the framework and introduces... more
The framework of ‘three worlds of mathematics’ was designed to reveal the growth of mathematical thinking in individuals over a lifetime to include all abilities and interests (Tall, 2013). This paper outlines the framework and introduces new developments that simplify and extend it. These involve aspects that anyone can observe, such as how we say mathematical expressions, how we hear someone else speak mathematically, how we see moving objects, how we think about mathematical symbols as processes or mental objects. These new developments have direct application to mathematical thinking at all levels, in different individuals, in differing cultural settings, and also in our understanding of the historical growth of the subject. They offer new ways of making sense of ‘math wars’ in which different approaches to mathematical ideas cause debates over which is preferable or even correct. The broader framework takes account of differing approaches by different communities of practice. T...
What do you see as the most significant advances, changes, and/or gaps in the field of research in university mathematics education? These advances, changes, or gaps might relate to theory, methodology, classroom practices, curricular... more
What do you see as the most significant advances, changes, and/or gaps in the field of research in university mathematics education? These advances, changes, or gaps might relate to theory, methodology, classroom practices, curricular changes, digital environments, purposes and roles of universities, social policies, preparation of university teachers, etc. Please elaborate on just one or two advances, changes, or gaps most relevant to your experience and expertise. If possible, please include a few key references.
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... more
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 resonance imagery (MRI) gives a three-dimensional picture of brain structure and fMRI (functional MRI) reveals changes in neural activity by measuring blood flow to reveal which parts of the brain are more active over a period of time. But this blood flow can only be measured to a resolution of 1 or 2 seconds and does not reveal the full subtlety of the underlying electrochemical activity involved in human thinking which operates over much shorter periods.
Research Interests:
In the teaching and learning of mathematics, while it is important to focus on what happens at each stage of development, what matters even more is the cumulative effect of learning over the long-term. As mathematics grows in... more
In the teaching and learning of mathematics, while it is important to focus on what happens at each stage of development, what matters even more is the cumulative effect of learning over the long-term. As mathematics grows in sophistication, new contexts require new ways of thinking that can act as barriers to progress. Passing through such a barrier may be called a transgression. This presentation focuses on aspects of mathematics that remain consistent over several changes in context and contrasts them with others that cause conflict at any given stage. For instance, how we speak, and write mathematics reveal new insights into making long-term sense of increasingly sophisticated mathematical symbolism in arithmetic and algebra. How the eye tracks a moving object affects how we interpret the notion of variable in the calculus both visually on a number line and symbolically as a variable quantity. Studying successive changes in mathematics and the positive and negative emotional aff...
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... more
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 only problematic transitions for learners but also fundamental theoretical differences between specialisms. At university and college level, mathematicians may have sophisticated knowledge that they believe will offer enlightenment to their students, but these insights may not be shared either by learners in their own community, nor by experts in other communities. The proposed framework reveals simple insights related to mathematical thinking that are visible to both teachers and learners, offering new insights into long-term meaningful growth from practical to theoretical mathematics and on to formal definition and proof. It invites rea...
Research Interests:
Research Interests:
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... more
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 ...