Cognitive and Social Development of Proof through Embodiment, Symbolism & Formalism
نویسنده
چکیده
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 genetic facilities set-before birth. It unfolds a developmental framework based on perception, action and reflection that leads to distinct ways to construct mathematical concepts through categorization, encapsulation and definition, in three distinct mental worlds of embodiment, symbolism and formalism, which provide the foundation of the historical and cognitive growth of mathematical thinking and proof.
منابع مشابه
From Embodiment to Metaphor: A Study on Social Cognitive Development and Conceptual Metaphor in Persian-Speaking Children
This study explores the metaphoric comprehension of normal Persian-speaking children, as well as theories of cognitive development and cultural and social impacts. The researchers discuss the improvement of the understanding of ontological conceptual metaphors through age growth and cognitive development, and how it helps to expand children’s thoughts and knowledge of the world. In this study, ...
متن کاملEmbodiment, Symbolism and Formalism in Undergraduate Mathematics Education
In recent years I have been working on a theoretical framework of long-term learning that presents three ways in which mathematical thinking develops that operate so differently as to present essentially three distinct ‘worlds of mathematics’—conceptual embodiment, proceptual symbolism and axiomatic formalism. Long-term human learning is seen to begin with facilities set-before birth in the gen...
متن کاملThe Long-term Cognitive Development of Different Types of Reasoning and Proof
This paper presents a long-term framework for the development of mathematical thinking from the thought processes of early childhood to the formal structures of formal mathematics and proof. It sees the development building on what the individual has met before which affects current thinking. Initially the child begins to coordinate perception and action to build thinkable concepts in at least ...
متن کاملThe Transition to Formal Thinking in Mathematics
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 a formal framework of axiomatic systems and ma...
متن کاملWritten for a special edition of ZDM entitled Transforming Mathematics Education through the use of Dynamic
This paper considers the role of dynamic aspects of mathematics specifically focusing on the calculus, including both physical human action and 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 quantities and al...
متن کامل