*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.

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.

**Bibliography**

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.

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.

**Bibliograpghy**

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.

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.

**Bibliography**

What Is Math? By Dan Falk, Smithonianmag.com, 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.

Having defined mathematics as the activity of telling stories about mathematical objects, I ponder in this module about mathematical objects. These latter entities seem quite unlike those investigated in physics, biology or astronomy. The objects of the scientific discourses, just as those we encounter and talk about in our everyday life, can be experienced directly even before one is able to say anything about them. The case of mathematical objects is quite different. Numbers, functions and derivatives, unlike stones, stars and living creatures, do not wait for us out there to be seen, smelled, heard or felt. If so, what are these objects called mathematical (or MOs, for short), where we can find them and how we can get aware of them and their properties? This two-part module is devoted to all these questions.

More specifically, in this module I will address the questions of what mathematical objects are and why we need them.

According to the communitive answer to query 1, mathematical objects are discursive constructs, that is, a special form of speech that allows us to say more in less words. This position contrasts with the classical Platonic view, according to which the entities we talk about in mathematics are of a different ontological status than the talk itself (the latter plays the mere auxiliary role of a medium for describing the preexisting reality).

**Bibliography**

Sfard, A. (2000). Symbolizing mathematical reality into being: How mathematical discourse and mathematical objects create each other. In P. Cobb, E. Yackel, & K. McClain (Eds.) Symbolizing and communicating: perspectives on mathematical discourse, tools, and instructional design (pp. 37-98). Mahwah, NJ: Lawrence Erlbaum Associates.

Sfard, A. (2008). Chapter 6: Objects of mathematical discourse: What mathematizing is all about. In A. Sfard, Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, UK: Cambridge University Press.

Having described mathematical objects as discursive constructs, that is, as a special form of speech that allows us to say more in less words, I will ask in this module how these constructs come into being. More specifically, a close look will be given to objectification - the process of constructing mathematical objects. This process, its causes and its mechanisms, will be discussed as it occurs on the historical and ontogenetic levels. In both cases, three basic discursive transformations will be identified: those of saming, reifying and encapsulating. A distinction will be made between primary (unnamed) and discursive objects, and also between concrete and abstract objects. In the end, the question will be asked about the main challenges mathematics students face while faced with the task of constructing a new mathematical object.

**Bibliography**

Nachlieli, T., & Tabach, M. (2012). Growing mathematical objects in the classroom – the case of function. I*nternational Journal of Educational Research, 51-52*, 1-27.

Sfard, A. (2015). Metaphors in mathematical thinking and in research on mathematical thinking: a prop or a trap? In F. Caluori, H. Linneweber-Lammerskitten, & C. Streit (Eds.), *Beiträge zum Mathematikunterricht *(pp. 42-49). Münster: WTM-Verlag.

Having discussed the first two signature features of mathematical discourses, words and visual mediators – two types of signifiers that yield mathematical objects – I focus in this module on another characteristic of mathematical discoruse, the routines.

I begin with the question of what routines are. The commognitive definition of routines is followed by a close look at some of their general properties, among them recursivity, interconnectedness, generosy and situativity. A plausible explanation of how we recruit rouitnes in most life situations is derived from the state-od-the-art developments in neuroscience. In the end, a seemingly counterintuitive claim is made: routine moves, rather than being obstacles to the creative action, constitute its vital components.

**Bibliography**

Lavie, I., Steiner, A., & Sfard, A. (2019). Routines we live by: From ritual to exploration. *Educational Studies in Mathematics, 101(2),* 153-176.

Feldman-Barrett, L. (2020). *Seven And A Half Lessons About The Brain*. New York: Houghton Mifflin Harcourt.

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