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Anna Sfard Unit

Hans Freundenthal Award 2009

This Unit is under construction, material will be uploaded gradually.

Credits : all modules, except module 0, are presented and have been filmed by Anna Sfard herself. Module 0 has been prepared in collaboration with Jean-Luc Dorier, who presents it and has been film at the university of Geneva.

Module 0

Module 1

Introduction: Why different ways of talking about learning and mathematics?

Great many stories of mathematics learning have been told, and they may sometimes be seen as competing with each other. While each one of us may favor one of these stories and call it “the best”, there is no single version that can count as the story of mathematics learning – as the one that is superior to all the rest in some absolute way.

The version to be presented in this unit is called commognitive. Explaining this approach will require engaging with the fundamental questions of what is learning, what is mathematics, and what we mean when we put these two words together and speak about learning mathematics. All this will be done in the modules that follow. This brief introductory talk evolves around the preliminary query: Why should we spend our time trying to define things as basic and obvious as learning or mathematics? In response, it is claimed that the way we speak impacts the way we act.

Sfard, A. (2008 ). Puzzling about (mathematical) thinking (Chapter 1). In Thinking as communicating: Human development, the growth of discourses, and mathematizing (pp. 3 -33). Cambridge, UK: Cambridge University Press.

Ben-Yehuda, Linchevski, L., Lavy, I., & Sfard, A. (2015). Lifting ‎the labels: A cautionary story about stories we tell about mathematics ‎students. In E. Silver & P. A. Kenney (Eds.), More Lessons Learned from Research (pp. 135 - 146). Reston, VA: NCTM.

Module 2


This module introduces the commognitive definition of learning. To clarify it fully and to explain why it was chosen from among many existing alternatives, the presentation of the definition is preceded by a brief historical account of research on human development.

Ever since its beginnings, the study of human learning has been fueled by the relentless tension between two desires: the researchers’ wish to capture human learning in all its uniqueness and their wish to do it scientifically, whatever this word meant for them at that time. Because of the conflicting nature of these two desires, it was difficult to attend to both of them at the same time. Whenever a story of learning was told that seemed to satisfy one of them, the resulting research would eventually be criticized for the neglect of the other one. The history of these zigzagging attempts is organized in this talk around the widely differing answers given by various schools of thought to the question “If learning means change, what is it that changes when people learn?” The response adopted by the commognitive researcher seems to constitute an effective tool for capturing the uniqueness of human learning, the goal that can now be pursued without compromising the scientific quality of the endeavor.

Sfard, A. (1998). Two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4-13.

Sfard, A. (2015). Learning, commognition and mathematics. In D. Scott & E. ‎Hargreaves (Eds.), The Sage Handbook of Learning (pp. 129-138). London: Sage‎.

Module 3

Mathematics as discourse

In the last module (Module 2) I defined learning as a process of changing an activity and claimed that school learning aims at those activities that have been developed throughout history and are now prevalent in our society.

In this module I focus on the question of “What is the activity that we change when learning mathematics?” The commognitive answer to this query echoes those given by postmodern thinkers such as Foucault, Lyotard, and Rorty. According to this definition, mathematics is a form of discourse, in which we create theories of mathematical objects, MOs.

Here, the term discourse signifies a form of communication made distinct, among others, by its unique keywords, visual mediators, and routines. Theories of MOs are potentially useful stories about MOs that we gradually construct as we apply these discursive tools. After explaining what is meant by story, I specify the characteristics according to which a story can be considered as useful. Having done all this, we are left  with the task of defining what is meant by mathematical object. The clarification of this latter notion will be the aim of Module 4.

For the summary of our recursive process of defining mathematics, see the diagram below.

What Is Math? By Dan Falk,, September 23, 2020
What is Mathematics? By Elaine J. Hom LiveScience. com, August 16, 2013 
Sfard, A. (2018). On the Need for Theory of Mathematics Learning and the Promise of ‘Commognition’. In P. Ernest (Ed.), The Philosophy of Mathematics Education Today (pp. 219-228). Cham, Switzerland: Springer.

Module 4

Mathematical objectsas discursive constructs

Module 5

Mathematical routines

Module 6

Historical development of mathematical discourse

Module 7

Ontogenetic development of mathematical discourse

Module 8

Cultural embeddedness of mathematics leanrning

Module 9

Identity of mathematics learner

Module 10

Teaching mathematics

Module 11

Commognitive research and its methods