Cognitive development in advanced mathematics using technology

@article{Tall2000CognitiveDI,
  title={Cognitive development in advanced mathematics using technology},
  author={David Tall},
  journal={Mathematics Education Research Journal},
  year={2000},
  volume={12},
  pages={196-218}
}
  • D. Tall
  • Published 1 December 2000
  • Psychology
  • Mathematics Education Research Journal
This paper considers cognitive development in mathematics and its relationship with computer technology, with special emphasis on the use of visual imagery and symbols and the later shift to formal axiomatic theories. At each stage, empirical evidence is presented to show how these forms of thinking are enhanced, changed, or impeded by the use of technology. 

Technology and Versatile Thinking in Mathematics.

TLDR
This presentation considers a theoretical cognitive development in arithmetic, algebra and the calculus and reflects on empirical research to show how the computer can be used both well and badly in supporting mathematical learning.

THEORETICAL-COMPUTATIONAL CONFLICTS AND THE CONCEPT IMAGE OF DERIVATIVE

Recent literature has pointed out pedagogical obstacles associated with the use of computational environments on the learning of mathematics. In this paper, we focus on the pedagogical role of

What Mathematics is Needed by Teachers of Young Children ?

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

USING THEORETICAL-COMPUTATIONAL CONFLICTS TO ENRICH THE CONCEPT IMAGE OF DERIVATIVE

TLDR
The pedagogical role of the computer's inherent limitations in the development of learners' concept images of derivative is focused on, and how the approach to this concept can be designed to prompt a positive conversion of those limitations for the enrichment of concept images.

Construction of mathematical knowledge using graphic calculators ( TI-84 plus & CAS ) in the mathematics classroom

Mathematics education researchers are asking themselves about why technology has impacted heavily on the social environment and not in the mathematics classroom. The use of technology in the

Construction of mathematical knowledge using graphic calculators (CAS) in the mathematics classroom

Mathematics education researchers are asking themselves about why technology has impacted heavily on the social environment and not in the mathematics classroom. The use of technology in the

Representational Fluency and Symbolisation of Derivative

The nature of mathematical concepts has been the subject of some scrutiny in the mathematics education literature. One of the key ideas described in the literature is the distinction between the

A framework for examining characteristics of computer-based mathematical tasks

TLDR
A tool which allows teachers, researchers and task designers to isolate, a priori, the possibilities and limitations of computer-based tasks is developed in the form of a framework that supports the notion that the use of technology in a task may afford the design of high-order thinking activities.

USING TECHNOLOGY TO INTEGRATE CONSTRUCTIVISM AND VISUALISATION IN MATHEMATICS EDUCATION

This paper provides a discussion of the pros and cons of instructivism and constructivism in the mathematics classroom, and endeavours to show why the latter is a preferable methodology to the former
...

References

SHOWING 1-10 OF 44 REFERENCES

Knowledge construction and diverging thinking in elementary & advanced mathematics

TLDR
This paper considers the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum and shows how students cope with the transition to advanced mathematical thinking in different ways leading once more to a diverging spectrum of success.

Interrelationships Between Mind and Computer: Processes, Images, Symbols

TLDR
The article considers the relationship between processes carried out by the computer, images generated by thecomputer and in the mind of the individual, and the resultant relationship with mathematical symbols and their mental manipulation.

The psychology of learning mathematics

Contents: Part A:Introduction and Overview.The Formation of Mathematical Concepts. The Idea of a Schema. Intuitive and Reflective Intelligence. Symbols. Different Kinds of Imagery. Interpersonal and

Duality, Ambiguity and Flexibility in Successful Mathematical Thinking

In this paper we consider the duality between process and concept in mathematics, in particular using the same symbolism to represent both a process (such as the addition of two numbers 3+2) and the

The School Mathematics Project

The School Mathematics Project is one of several organizations for the improvement and renewal of parts of the school curriculum in Britain. It is now concerned with problems of application as well

Duality, Ambiguity and Flexibility: A Proceptual View of Simple Arithmetic

In this paper we consider the duality between process and concept in mathematics, in particular using the same symbolism to represent both a process (such as the addition of two numbers 3+2) and the

Information Technology and Mathematics Education: Enthusiasms, Possibilities and Realities.

Abstract This presentation addresses critical issues in the use of information technology in Mathematics Education. By reflecting on human thinking processes, it will consider developments of

Advanced Mathematical Thinking

Preface. Introduction. 1. The Psychology of Advanced Mathematical Thinking D. Tall. I: The Nature of Advanced Mathematical Thinking. 2. Advanced Mathematical Thinking Processes T. Dreyfus. 3.

Encouraging versatile thinking in algebra using the computer

TLDR
Empirical evidence from several related studies shows that such an approach significantly improves the understanding of higher order concepts in algebra, and that any initial loss in manipulative facility through lack of practice is more than made up at a later stage.