Designing Mathematical Thinking Tools

Click here for pictures from the Symposium

A Fields Institute Symposium

  • 10-12 June 2005
  • Faculty of Education, University of Western Ontario

Keynote addresses (open to the public):


  • The goal of the Symposium is to explore mathematical thinking tools.
  • We are especially interested in investigating the role of metaphor (or 'seeing-as') in the development of mathematical understanding.
  • And we are interested in the role of technology, in this context.



Friday June 10

  • 6:00-7:00 pm - Registration
    • Faculty Lounge, Faculty of Education
  • 7:00-7:30 pm - Opening remarks
    • Room 2038, Faculty of Education
  • 7:30-9:00 pm - Richard Noss keynote+discussion
    • Room 2038, Faculty of Education
  • 9:00 pm - Reception
    • Faculty Lounge, Faculty of Education

Saturday June 11

  • 9:00-10:30 - Working meeting
  • 10:30-12:00 - Rafael E. Núñez keynote+discussion
    • Room 2038, Faculty of Education
  • 12:00-1:00 - Lunch
  • 1:00-2:30 - Anna Sfard keynote+discussion
    • Room 2534, Elborn College
  • 2:30-5:00 pm - Working meeting
  • 6:30pm - Symposium Dinner

Sunday June 12

  • 9:00-12:00 - Working meeting
  • 12:00-1:30 - Lunch
  • 1:30-3:30 - Working meeting

Organized by

  • George Gadanidis (, Faculty of Education, University of Western Ontario
  • William Higginson (, Faculty of Education, Queen's University
  • Robin Kay, Faculty of Education (, University of Ontario Institute of Technology
  • Kamran Sedig (, Department of Computer Science (Faculty of Science) and Faculty of Information and Media Studies, University of Western Ontario
  • Christine Suurtamm (, Faculty of Education, University of Ottawa

Previous Symposia


All meetings are at the Faculty of Education, University of Western Ontario, 1137 Western Road, London, ON, Canada, N6G1G7.

Accommodation for out of town participants is in our Essex Hall B&B Residence, which is opposite the Faculty of Education - please see map.

Parking is available in the lot shown on the map for $4 per entry (two $2 coins).

Richard Noss

Director, London Knowledge Lab, Institute of Education, University of London

Keynote address


I take as a starting point two assumptions: that there is a rising demand on individuals and communities to understand something of how social and technical systems operate; and that it is becoming more and more difficult to catch sight of how things work, as the mechanisms that drive them become ever more hidden in invisible computer code. Competence in constructing, interpreting and critiquing mathematical models has become a core part of social and professional life in the twenty-first century, but it is becoming more and more difficult to achieve that competence.

The implications for the design of learning systems are manifold. I will focus here on just three. First, we should try to design learning environments where people can make models of things, either physically or virtually. Second, learners need to experience how to share their models, to talk about interesting mathematical phenomena that underpin them and – if possible – rebuild and critique each others' models as they (and their thinking) evolve.

A third and critical implication is that researchers and teachers will need to find new ways to represent mathematical knowledge that are designed simultaneously to be learnable (in the way that, for instance, algebra was not) and rigorous (so that mathematical integrity is not sacrificed on the altar of simplicity). I take it as axiomatic that the apparent complexity of a mathematical idea is often located in the representational infrastructure in which it is expressed, rather than an ontological facet of the idea itself. So designing new representational infrastructures (as well as activity structures) that facilitate mathematical expression is a critical challenge for the design of mathematical thinking tools. I will illustrate the approach based on several strands of work that have emerged from recent studies with colleagues in London including the "Playground" and "Weblabs" projects.


Richard Noss is Professor of Mathematics Education at the Institute of Education. His overarching research interest is in trying to understand what kinds of knowledge people really need in the 'knowledge economies' of the twenty-first century, and in building tools that help them acquire that knowledge. Over the last ten years he has worked with bank employees, nurses, airline pilots and engineers, among others, trying to describe their professional and mathematical knowledge. He has also engaged them in 'playful' learning experiences, including (most ambitiously) teaching the programming language Logo (often mistakenly described as a 'language for kids') to a group of investment bank employees. At the other end of the age scale, the recently-completed Playground project has involved very young kids in designing and building video-games.

Richard is chief editor of the The International Journal of Computers for Mathematical Learning, which aims to realise the vision of its chair, Seymour Papert, to 'foster a new, creative and more learnable mathematics with digital technologies'. He is the co-author, with Celia Hoyles, of Windows on Mathematical Meanings: Learning Cultures and Computers. (Kluwer, 1996).

Current projects

Recently completed projects

Rafael E. Núñez

Associate Professor at the Department of Cognitive Science, University of California, San Diego, United States.

Keynote address


The term "Metaphor" is widely used in ordinary language. It is used in literature, advertising, art, and politics, to mention a few. Metaphor is often seen as a figure of speech, and as such, as a matter of words, a *linguistic* phenomenon that helps illustrating what is being said. In mathematics education, quite often is this general ordinary meaning that is used to understand mathematical understanding and mathematical thinking. In contemporary cognitive science, and especially in cognitive linguistics, "metaphor" involves various technical distinctions, such as the one between "metaphorical expressions" and "conceptual metaphors". Moreover, in these disciplines, "metaphor" is seen as one among many cases of conceptual mappings, which also involve conceptual metonymies, conceptual blends, fictive motion, and others. Together, and often working in complicated networks, they are hypothesized to form the vast family of cognitive mechanisms that make human abstraction and imagination possible. In this talk I will explore some issues regarding the study of these conceptual mappings and their inference-preserving properties. In particular I will focus on methodological and theoretical problems involving (a) the level at which the subject matter "metaphor" is defined, (b) the role of bodily experience, (c) the nature of "selective projection" in conceptual mappings, and (d) the study of "metaphor" via convergent methodologies such as real-time gesture production, priming psycho-linguistic experiments, and studies involving the neuroscience of "metaphorical" understanding. Some implications for research in mathematics education will be discussed.


I investigate cognition from the perspective of the embodied mind (see my website). I am particularly interested in high-level cognitive phenomena such as conceptual systems, abstraction, and inference mechanisms, as they manifest themselves naturally through largely unconscious bodily/mental activity (e.g., gesture production co-produced with conceptual metaphors and blends). My multidisciplinary interests bring me to address these issues from various interrelated perspectives: mathematical cognition, the empirical study of spontaneous gestures, cognitive linguistics, and field research with the Aymara culture in the Andes. My recent book, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (with UC Berkeley linguist George Lakoff) presents a new theoretical framework for understanding the human nature of Mathematics and its foundations.

I am also the director of the Embodied Cognition Laboratory at UCSD, with lab space and members dedicated to investigating how cognition is grounded on the peculiarities, experiences, and limitations of the human body.



Anna Sfard

Professor of Mathematics Education at Michigan State University (US) and the University of Haifa (Israel).

Keynote address


For many years now, researchers in mathematics education have been using the notion of metaphor as a conceptual tool for describing development of mathematical thinking. In this talk I will take a double, object-level and meta-level view while describing advantages and pitfalls of metaphorizing in two different types of discourse: (1) in mathematics itself and (2) in mathematics-education research that investigates the development of the mathematical discourse. In both these cases, I will focus on the metaphor of object, this special type of “discursive transplant” that allows us to replace talk about processes with talk about objects. Processes of objectification deserve particular attention because of their apparently crucial role in human thinking. At a closer look one realizes that the metaphor of object, while massively present in all human discourses, is also practically transparent to its users. The transparency makes it liable to misguided, unhelpful applications. Thus, while objectifying renders our discourses effective and, in particular, makes us able to plan for the future by recycling our past experience, it may also bear the main responsibility for the fact that research on human thinking in general, and research in mathematics education in particular, are quandaries-ridden and are often criticized for their being fragmented, unable to cumulate knowledge, and incapable of living up to their major commitment of improving teaching and learning.


With a formal background in mathematics and physics, and with a life-long interest in history, philosophy and language, I specialize today in mathematics education, focusing my research on the intricacies of human learning and creative thinking.

The overarching theme of my work is the constitutive role of language. More specifically, I  investigate the implications of the assumption that human thinking as a particular case of communicative activity. In my recent studies, dealing with the question of the origins of mathematical objects and with the issue of transition from operational to structural thinking (reification), I use an approach the roots of which  go to semiotics and discourse analysis. The role of metaphor in the development of mathematical discourse and the issue of building a common focus in mathematical communication are among the themes on which I have been working for the last few years. These are also the subjects which have been investigated in a series of studies on learning and teaching introductory algebra I have been conducting in Israel and in Canada.

The cognitive role of metaphors and, more generally, investigating cognition through discourse analysis, were the topics of my most recent graduate courses and are the focus of several empirical studies that I am carrying out with Ph.D. students. In addition to my research work, I have been participating for many years in developing new mathematics curricula for Israeli senior secondary schools and serving as the editor  of  the Israeli Journal for Mathematics Teachers.