Tools for Thought: The Case of Mathematics

The idea of representation is continuous with mathematics itself. Any mathematical concept must be represented in some way if it is to be present in the learner's mind. We distinguish between external representation (environment) and internal representation (mind). External representation refers to all external media that have as their objective the representation of a certain mathematical idea. We mainly use the term external representation for tangible material, graphical representation and mathematical symbols. External representation always needs an interpreter, a learner who gives it meaning. The fact is that teaching and learning mathematics is more effective in terms of understanding mathematical ideas if it focuses on investigating different representations of a particular mathematical concept and encourages pupils to find links between these representations. Representations are predicated neither in terms of the adequacy of the relationship between ideas and their representations , nor as heuristic devices in meaning-making processes; they are rather an integral part of the activity of knowledge presentation. Representing mathematical ideas has the following main roles in the process of teaching and learning: interpretation of what is represented (internal presentations), recording , representing ideas (ways of thinking, knowledge presentation externally), and communicating (e.g., discussion about representations). The last two roles are the focus of this focus issue of the CEPS Journal: we aim to bring together different issues concerning representing learners' ways of thinking, knowledge presentation, and the role of external representations in the process of teaching and learning mathematics. On the one hand, we are interested in how students explain and share their ways of thinking in order to better understand their progress in learning; on the other, we would like to rethink the role of external representations. Stated more generally, our concern is how knowledge presentations can help the learner to develop competences; not only mathematical competences, but also those that empower her/him to make well-grounded decisions and use mathematics in ways that fulfil her/his needs as a constructive and thoughtful person. In this focus issue of the CEPS Journal, we contribute to the area of research on representations of mathematical ideas with four contributions. Each of them deals with a specific issue regarding the topic, while also covering different age groups of students, from preschool children to primary teacher students.

Educational Studies in Mathematics, 1997

In this paper, we explore the relationship between learners' actions, visualisations and the means by which these are articulated. We describe a microworld, Mathsticks, designed to help students construct mathematical meanings by forging links between the rhythms of their actions and the visual and corresponding symbolic representations they developed. Through a case study of two students interacting with Mathsticks, we illustrate a view of mathematics learning which places at its core the medium of expression, and the building of connections between different mathematisations rather than ascending to hierarchies of decontextualisation.

Croatian Journal of Philosophy, 2020

This article focuses on particular ways in which visual representations contribute to the development of mathematical knowledge. I give examples of diagrammatic representations that enable one to observe new properties and cases where representations contribute to classifi cation. I propose that fruitful representations in mathematics are iconic representations that involve conventional or symbolic elements, that is, iconic metaphors. In the last part of the article, I explain what these are and how they apply in the considered examples.

Diagrams, Graphs, and Visual Imagination in Mathematics, 2023

This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. As we know from Kant, vision (as a part of human sensibility) and responsiveness to reasons (as supported by our overall conceptual capacities) are related with one another through the imagination. Mathematics is an expression of this relation based on our most fundamental intuitions about space and time. Peirce went a long way to develop Kant’s take on the nature of mathematics, and central to his interpretation of it was the idea of diagrammatic reasoning. According to Peirce, in practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of the very process of the reasoning itself. As thinking, in this case, is actually performed by means of manipulating images, seeing and understanding become one. Defining diagrammatic reasoning as a fusion of vision and thought helped Peirce find some intriguing answers to questions concerning the nature of mathematical knowledge, many of which could not even be as much as formulated by Kant. What is the role of observation in mathematics? How can we explain the fact that mathematical reasoning is deductive and, at the same time, capable of the discovery of new truths? How is mathematical necessity reconciled with the essential incompleteness and indeterminacy of our ordinary visual experience? What exactly is the relationship between the particularity of a mathematical diagram and the generality of the meaning it conveysand what is the difference (if any) between mathematics and natural languages in this respect? Etc. Peirce’s life-long, if unsystematic, work on the issues that are associated with the questions above created an intricate conceptual puzzle. The driving motivation of the research this book represents is to show that tackling this puzzle requires something more than sifting through the wealth of available historical and philosophical material. While the histories of science and philosophy do provide separate bits of the puzzle, Peirce’s theoretical interests, by his own admission, appear to be closely intertwined with certain facts of his personal history. In light of this, without considering relevant biographical data, in Peirce’s case, there is no way to understand how the pieces of the puzzle actually fit together. Due to the plurality of data impelled by the task, this book is addressed both to those specializing in philosophy, mathematics, and intellectual history, and to a wider audience that might be interested in what all those areas have in common in Peirce’s case. Last but not least, this book could not have been written without the support of family, friends, and colleagues. I am especially grateful to Eric Bredo, Marcel Danesi, Paul Forster, Nathan Houser, Henry Jackman, Steven Levine, Mark Migotti, and James O’Shea. Of great importance for the book were my conversations with Kathleen Hull and Thomas L. Short. As to the biographical part of the study, I am indebted to Joseph Brent, the author of the most comprehensive biography of Charles Peirce to date. Finally, I owe much more than I can tell to my constant companions and interlocutors, Zina Uzdenskaya and Gleb Kiryushchenko.

2006

In this text we will focus on some "geometric" judgements, which ground proofs and concepts of mathematics in cognitive experiences. They are "images", in the broad sense of mental constructions of a figurative nature: we will largely refer to the well ordering of integer numbers (they appear to our constructed imagination as spaced and ordered, one after the other) and to the shared image of the width less continuous line, an abstracted trajectory, as practice of action in space (and time).

The explicit recognition of the objects and processes implied in mathematical activity is a competence that teachers should develop. This competence allows teachers to understand, design, and manage the processes of mathematical learning, and assess them with suitability standards previously set. Consequently, formative processes to develop this competence should be designed. This paper describes a training design aimed at developing the teachers' competence of epistemic and cognitive analysis of mathematical tasks. The focus is on the teachers' recognition of the dialectical relationships between visual and analytic languages, as well as their interactions with mathematical objects. It follows that, the tasks are a challenge for the prospective teachers, but in addition, they allow unveiling naive conceptions about the use of visualisation in teaching mathematics.

Proceedings of the Twentieth Annual Meeting of the …, 1998

ZDM, 2005

In this paper we include topics which we consider are relevant building blocks to design a theory of mathematics education. In doing so, we introduce a pretheory consisting of a set of interdisciplinary ideas which lead to an understanding of what occurs in the "central nervous system"-our metaphor for the classroom and eventually in more global settings. In particular we highlight the crucial role of representations, symbols viewed from an evolutionary perspective and mathematics as symbolic technology in which representations are embedded and executable.

Researchers have emphasized the important role of visualization, and visual thinking, in mathematics, both for mathematicians and for learners, especially in the context of problem solving (see . In this paper, we examine the role that motion and time-which engage similarly sensory modes of thinking-play in mathematicians' conceptions of mathematical ideas.