HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 16: 57-75, 2023

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ETHICS IN THE MATHEMATICS CLASSROOM
Luis Radford
Laurentian University, Ontario, Canada

Abstract
Despite some recent progress, ethics remains a peripheral topic in mathematics education
research. My intention in the first part of the article is to show that ethics is omnipresent in themathematics classroom. However, ethics does not always operate in the same way. To better grasp
the nature of ethics, in the second part of the article, I discuss two ethical systems that have been
influential in Western thought (Hobbes’s and Kant’s). These systems have largely informed theunderstanding of the mathematics classroom, even if, more often than not, they remain implicit.
Then, I move to a short discussion of ethics in postmodern times and try to pinpoint what is atstake in ethics. The previous theoretical considerations pave the way to approach, in the last partof the article, ethics from an educational viewpoint. I argue that, in educational contexts, such as
the school, ethics appear framed by the way in which we understand teaching and learning. I end
the article with an outline of the communitarian oriented relational ethics articulated in the theoryof objectification—a communitarian ethics whose practice features responsibility, commitment,
and care.
Keywords: ethics, theory of objectification, Vygotsky, Spinoza, Kant, Hobbes

1. INTRODUCTION
The first question that might arise is the following: What does ethics have to do with mathematics education?
Let me present a short twofold answer.
First answer
Teaching-learning mathematics cannot avoid facing the question of the legitimation of particular forms ofknowledge and knowing that arise in the classroom. Classroom discussions usually lead to conflicting views
about what counts as mathematically valid and authentic.
Here is an example. In a Grade 5 class (10-11-year-old children) of a French-speaking school in
Sudbury, Canada, the students were invited to write a text for a student from another class explaining how tosolve linear equations. To write and solve simple equations, the students had been using an iconic semiotic
system (ISS) that comprises four signs:
⎧
⎨
⎩

,

⎫
, = , ↑⎬.
⎭In the ISS, the small rectangles represent cards; the envelope represents the unknown (as each envelope
Corresponding author: Luis Radford

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Luis Radford

contained the same unknown number of cards). The problems with which the students had been dealing
involved two individuals (e.g., Claudine and Sylvain) who each had a known number of cards and one ormore envelopes. The individuals’ total number of cards was the same.1
Some students suggested a text based on a concrete example: they used the equation=

These students divided the two sides of the equation by a vertical line (see Figure 1, left). But other studentssuggested a text without any concrete example (see Figure 1, right; a translation of the texts can be found in
the Appendix).
Which text is better? And if you were the teacher, what would you say to the students?

Figure 1. Two mathematical texts. Which one is better?

Since there are plenty of ways in which to think mathematically, taking sides or suggesting something else
involves a question of power, and since there is a question of power, there is also a question of subjecting
people to a particular image of mathematics. This is an ethical matter.
Second answer
My second answer is not about positions taken on questions of mathematical legitimacy but about relations
between people.
Teaching-learning mathematics in the school involves interaction between people. Teaching-learning
is based on relations with others, and these relations involve necessarily an ethical dimension: in classroom
interaction we have, for instance,●

relations of power and subjection,●

relations of authority and obedience, and●relations of solidarity and inclusiveness.

Here is an example. In a Grade 4 classroom (9-10-year-old children) of a French-speaking school in Ottawa,
Canada, the students were working in small groups of three or four trying to solve geometric problems. The
1 A paradigmatic example is the following: “Sylvain and Chantal have some hockey cards. Chantal has three cards andSylvain has two cards. Their mother puts some cards in three envelopes and makes sure to put the same number of cards ineach envelope. She gives one envelope to Chantal and two to Sylvain. Now the two children have the same number ofhockey cards. How many hockey cards are inside each envelope?” (Radford, 2017, p. 18). See also, Radford, Demers, andMiranda (2009).

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Ethics in the Mathematics Classroom

first problem revolved around the classical question of whether squares are rectangles. The pictures in Figure
2 provide a sample of body positions of the interaction of a group of four students. In Picture 1, Laura istalking to Sandra, the girl in front of her: “Yes, but they all have four sides.” In Picture 2, Mirna tries tocontribute to the group and says: “The squares have same umm … the same edges …” Laura turns to look at
Mirna for a short period of time; then, turns back to look at Sandra to continue their discussion. In Picture 3,
Híria (front left) tries unsuccessfully to get Sandra’s and Laura’s attention and says: “The squares haveparallel faces … because there is … Look!” In Picture 4, after recurring attempts to be heard, Mirna expresses
her frustration and utters an anguished “Ahhhhh!!!”

1

2

3

4

Figure 2. A group of Grade 4 students dealing with a geometry problem.

We are here in the presence of an ethics of exclusion. Híria and Mirna try hard to enter the conversation,
to contribute to the discussion, but they are not heard. The practice of an ethics of exclusion impedes a
genuine collective engagement in mathematics and raises an invisible, yet important, wall between us andthem.
The previous examples illustrate one of the main ideas of this article: teaching-learning is unavoidably
an ethical event—and this is so regardless of the pedagogical model that underpins it.
I would like to go a step further and contend that ethics is not only omnipresent in mathematics teaching
and learning but is also a crucial component of it for at least two reasons.
First, ethics shapes the manners in which teachers and students engage and assume (or not) certain
responsibilities in the mathematics classroom. Consequently, ethics shapes how teachers and students cometo understand mathematics and conceive of themselves as practicians of mathematics.
Second, ethics shapes the students’ and teachers’ relationships with others—for instance, in the various
manners by which the students voice (or not) their values and understandings, and how their voice is heard(or not). In this sense, ethics affects how teachers and students assert themselves as mathematical subjectivities(Radford, 2020).
Perhaps what is surprising in the claim that I am making, namely that mathematics teaching andlearning is unavoidably an ethical event, is not that ethics is omnipresent in teaching and learning but the fact
that, some exceptions put aside (e.g., Atweh, 2014; Atweh & Brady, 2009; Boylan, 2016; Dubbs, 2020;
Ernest, 2009; Maheux & Proulx, 2017; Neyland, 2004; Maheux & Roth, 2014; Roth, 2013; Silva D’Ambrosio

60Luis Radford& Espasandin Lopes, 2015; Walshaw, 2013), we have not been able to notice this presence before—or atleast not with the importance it deserves. I would like to suggest that the root of that omission is entrenchedin the long-standing conception of mathematics education as a technical process, a conception that reduces
mathematics education to a matter of acquiring knowledge, making the question of being and becoming
peripheral aspects of teaching and learning. With this same technicist stroke, we erase from view the fact thatthe mathematics classroom is not only a site of knowledge production but also a site where subjectivities areproduced every day.
The importance of ethics in an encompassing account of learning leads us to ask the question of thekind of ethics that we, implicitly or explicitly, nurture in classrooms and research. This question seems to beimportant in discussions on the diversity of theories in mathematics education; considering theories from the
ethical standpoint they convey (even if they do so only obliquely) may shed new light on the differences,similarities, and complementarities between theories.
In an attempt to better grasp the nature of ethics, in Section 2, I discuss two ethical systems that havebeen influential in Western thought (Hobbes’s and Kant’s ethical systems). These systems have largely
informed the understanding of the mathematics classroom, even if, more often than not, they have remained
implied. Then, in Sections 3 and 4, I discuss what is at stake in ethics. Intended always as a means to bettergrasp the nature of ethics, Section 5 revolves around some ideas of ethics in postmodern times. The previous
theoretical considerations pave the way to approach, in Section 6, ethics from an educational viewpoint. I
argue that, in educational contexts, such as the school, ethics appear framed by the way in which weunderstand teaching and learning. I end the article with an outline of the communitarian oriented relational
ethics articulated in the theory of objectification (Radford, 2019, 2021a)—a communitarian ethics whose
practice features responsibility, commitment, and care.

2. HOBBES AND KANTAll ethical perspectives are directly tied to conceptions of self and the social world. Try to think of what self
is and you will quickly realize that it is entangled with how we perceive ourselves and perceive others. This
perceiving of the other involves social relations, some to keep, some to discard. Our concepts of self areindeed inseparable from the landscape of ethical issues, “from how one ought to be” (Taylor, 1989, p. 112).
These ethical issues change from culture to culture and within a same culture through spans of time. Thus,
from the end of the Middle Ages and the dawn of Renaissance, when European feudal structures were being
shaken by the rise of mercantile capitalism and the ascending power of the concomitant new economic andcommercial relations, the social world appeared more and more as an aggregate of monadic individuals(Elias, 1991). On the social plane, life became increasingly regulated by agreements or contracts between
people (Le Goff, 1956; Jeannin, 1957). In the 17th century, in his Philosophical Rudiments Concerning
Government and Society, Thomas Hobbes conceives of these contracts as means to regulate human conduct
and to avoid aggression against each other. In his account, these contracts bring individuals out of the stateof nature, which “is nothing else but a mere war of all against all” (Hobbes, 1841, p. xvii). These contracts

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elevate the individuals to civil society; they embody personal aspirations and interests while trying tosafeguard each other’s security. Ethics appears here as a contractual mechanism to preserve the social orderwhile affirming the monads’ interests in a way that reflects the underpinnings of the emergent bourgeois
ideology.
One century later, when Kant articulated his ethical system, capitalism had expanded and become more
systematic. Kant’s society, of course, is still perceived as an ensemble of monadic individuals, but with anincreased complexity out of which the modern concept of self finds a more precise shape.
Kant assumed that individuals are by nature provided with a same reason—a legislative Reason. He
also assumed that the individuals’ reason moves in a world of universal laws that can define the social andmoral orders. From those assumptions Kant formulated his famous categorical imperative: “act only inaccordance with that maxim through which you can at the same time will that it become a universal law”
(Kant, 2006, p. 31, original emphasis).
As we can see, the categorical imperative answers the question: What should I do? Through it, moral
content becomes specified not in terms of social ends that the rule would help to attain or achieve. Moralcontent can only be derived from the form of the rule as expressed and appraised by the individual.
MacIntyre (1966, p. 209) notes that “The eighteenth-century individualist sees the good as theexpression of his feelings or the mandate of his individual reason.” The categorical imperative is an attempt“by the individual to supply his own morality, and at one and the same time, to claim for it a genuine
universality” (p. 208). As I argued in (Radford, 2021a), the categorical imperative makes the Other appear asa reflection of the self in a mirror through which the self can see how the actions of the Other might affectitself. It assumes a universal human subject and a universal reason that needs to be understood as a response
to “the emergence of a formal law in Europe at that time, the state homogenization that was being imposed[on society], [and] the development of an instrumental rationality” (Bohy-Bunel, 2022, pp. 23-24). This
universal human reason is nonetheless an exclusive ethnocentric reason in its most intimate nature. As BohyBunel remarks, the Enlightened modern bourgeois individual “arrogates to himself all possible humanity”(p. 24). This individual is
the Westerner whose social, classist, patriarchal and colonial project of domination becomes clearer inthe modern age . . . Its “science”, which ends up fitting perfectly into this project, postulates that it
relies on some universal “human reason,” precisely in order to better assign to non-humanity those
whose activity and being must be submitted to a total managerial control. (Bohy-Bunel, 2022, p. 25;
emphasis in the original)
It is hence under the assumption of one and only one universal reason, the Enlightened managerial
reason, that Kant can think that he solves the problem of the tension between the individual and the socialworld and the tension of human action in that world.

3. WHAT IS AT STAKE
What is hence at stake in conceptions about ethics? The previous discussion provides us with a hint: what is
at stake is the social meaning of choices about human conduct. This social meaning refracts, on our dealings

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with others, the forms of life found in the social world. In the case of Hobbes and Kant, the forms of life were
determined to a large extent by the capitalism of their time, which became the “dominant reality in thehistorical life of European societies” (Foufas, 2020, p. 24), that is capitalism in its commodity form.
Confronted with a social understood as an assemblage of monads, Hobbes found in the contractual
mechanisms of capitalists the prototype of a juridical form where individuals could negotiate and combine
their own interests. Through contractual mechanisms individuals became subject to a new entity, an assembly,
through which the multitude becomes a state or civitas. Pressured by the needs of a more sophisticated socialand economic organization and the emergence of a formal law, Kant did not need to have recourse tosomething extraneous to the individuals, as Hobbes’s did through the idea of assembly, to which theindividuals relinquished the control of their own actions. Kant did not need to do something like that, as heassumed, as we just saw in the previous section, that there was a universal reason legislating choices abouthuman behaviour.
It might not come as a surprise that the growth and implementation of ethics during the 20th century
unfolded torn by two opposite forces that we find already in play in Kant’s categorical imperative: on the one
hand, there was an emphasis on the individual (we saw that the categorical imperative comes from and isenunciated by the individual). This is the view of the existentialist ethics (Sartre’s ethics, for example) where,
standing in the debris of WWII, authentic existence was “to be found only in a self-conscious awareness ofan absolute freedom of choice” (MacIntyre, 1966, p. 269). On the other hand, there was an emphasis on theuniversal validity of the law, without assuming, however, that the law emanates from the individual itself.
The ethical self should be educated through a series of prescriptions. This is what has been called“prescriptionist” ethics; that is, the ethics practiced in particular by the legislators facing the problem ofgoverning the masses. We find modern prescriptionist ethics at work in post-revolutionary France. For
example, in 1833, French Minister of Education François Guizot explains that
The great problem of modern societies is the government of minds… It is not only for the commune
and in a purely local interest that the law wants all the French to acquire, if it is possible, the knowledge
indispensable to the social life, and without which the intelligence languishes and sometimes becomes
stultified... This is because freedom is only assured and regular among people enlightened enough tolisten in all circumstances to the voice of reason. (Guizot, cited in Mayeur, 2004, p. 335)
Thus “modern legislators and modern thinkers alike felt that morality, rather than being a ‘natural trait’of human life, is something that needs to be designed and injected into human conduct” (Bauman, 1993, p.
6). The result was that, in the footsteps of Kant, modern ethics was mainly understood as deontology; that is,
as duty and obligation driven by general rational laws to be applied regardless of the context. At the practicallevel, and with the push of modernity, this deontology was supplemented with a utilitarian outlook of theworld. In the contemporary educational context, Neyland pictures the problem as follows:
Recent modernist reforms . . . have resulted in teachers increasingly being represented as objects rather
than as subjects in policy discourse; professionalism has been replaced by accountability, andcollegiality by competition and surveillance; initiative, creativity and teacher-led innovation have been
constrained; teaching has become technicized, and learning experiences impoverished. (Neyland,
2004, p. 62)
Here, as Neyland suggests, ethics amounts to proceduralism; that is, the implementation of the dictatesEthics in the Mathematics Classroom

63of the educational apparatus as how teachers and students should behave and what they have to accomplish.
Postmodernist thinkers have quarrelled with this proceduralist conception of ethics. What bothers
many of them is not the subjective aspect involved in ethical contexts. They retain the Kantian idea that
morality does not come from outside the subject but from within. Bauman, for example, argues that “Themoral call is thoroughly personal; it appeals to my responsibility” (1993, p. 60). What bothers these
postmodernists is the deontological dimension. They have reacted suggesting a non-deontological ethics, anethics that is context-sensitive and hence impossible to frame through a priori laws or principles. As Baumanputs it, “Moral phenomena are inherently ‘non-rational’… They are not regular, repetitive, monotonous andpredictable in a way that would allow them to be represented as rule-guided (1993, p. 11; emphasis in theoriginal) . . . morality is endemically and irredeemably non-rational” (1993, p. 60; emphasis in the original).
Now, how can such an ethics possibly work if postmodernist thinkers themselves have been very
successful in showing that the individuals are subtly shaped by a multitude of social mechanisms ofgovernance that make them think and act in specified ways? How can we make sure that the ideas with which
we ponder a delicate ethical situation are not those prescribed by the system even if they do not appear in alaw form? In other words, how can we make sure that it is me and not the system through its webs of power
and knowledge that is talking through me?
Foucault, the last modern Kantian and one of the first postmodern thinkers, raises the question as
follows: How can the subject be the locus of meaning, feeling, intentionality, and ethical decisions if thissubject must talk, feel, and intend through thoughts and words that are not its own?
Can I, in fact, say that I am this language I speak, into which my thought insinuates itself to the pointof finding in it the system of all its own possibilities, yet which exists only in the weight of sedimentations[that] my thought will never be capable of actualizing altogether? (Foucault, 1966, p. 335)
Rothenberg articulates this problem asking how the individuals produced within the social modes ofpower, “subsisting as nothing other than the intersection of various discursive determinants” (2010, p. 26),
could be able to act independently of ideological operations. These individuals, Rothenberg argues, “have totranscend their ideological determinations somehow and lift the veil of misrecognition, even though they
themselves are nothing other than the expression of ideology and thoroughly blinded by it” (Rothenberg,
2010, p. 25).
What is at stake, I argued before, is the social meaning of choices about human conduct. We now see
that, going a step further, what is at stake at a deeper level, is our conceptions of human nature. Are humans
capable of transcending their ideological determinations or are they trapped in them forever?

4. SELF AS LACK AND SELF AS EXCESSHistorically speaking, feminist, multiculturalist, psychoanalytic, and dialectical materialist thinkers, andscholars in related fields have contended that human nature is such that ideological determinations can beovercome. We are not trapped. The “trapped forever” position, they argue, is based on a conception of theself as lack. It features a self that lacks wholeness or plenitude, subjected forever to the limits of the given
structures and confining discourses. They counter this conception of self with a conception of self as excess.

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In feminist theory the subject is conceived of as having the power “to reflect on the social discourse andchallenge its determinations” (Alcoff, 1988, p. 417). Following a Hegelian thread, for Butler (1999), thesubversion of the subject is possible because all acts of signification not only restrict the subject’s actions but
are, at the same time, in their enactment, always located within the possibility of a variation in the “alternative
domains of cultural intelligibility” (p. 185).
Rothenberg (2010) approaches the problem from a psychoanalytic perspective and claims that meanings
have the peculiar property of being impossible to be totally controlled. Although individuals are socially
produced, the producing causes leave “a remainder or [meaning] indeterminacy, so that every subject bears
some unspecifiable excess within the social field. Every subject is an ‘excessive’ subject [and] the excess isineradicable” (2010, p. 10).
Drawing on Lacan, Žižek refers to a fundamental characteristic of the Symbolic: its “openness.”2
The cause of this irreducible “openness” of the Symbolic is not its excessive complexity (we neverknow in what decentered context our statement will be inscribed), but the much more refined, properly
dialectical impossibility of taking into account the way our own intervention will transform the field.
(Žižek, 2010a, pp. ix-x)
Through our interventions in the world, Žižek contends, we affect our reality, which means that wechange and affect our social determinations. These are not outside an undisturbed reality, nor is our free will.
When we feel thwarted in our freedom by the constraining pressure of external reality, there must be
something in us, some desire or striving, which is thus thwarted, but where should this striving come if
not from this same reality? Our “free will” does not then in some mysterious way “disturb the natural
course of things,” it is part and parcel of this course. (Žižek , 2010a, p. xii)
Dialectical materialist thinkers (e.g., Fischbach, 2014; Macherey, 2008) have drawn on Marx’s (1998)
work, particularly on the idea of transformative praxis quickly sketched in Thesis 3 of Theses on Feuerbach.
Following Marx’s ideas, in dialectical materialism the subject is featured as one that, while being produced
by its circumstances, has, inversely, the power to reflect on these circumstances and to transform them (foran enlightening analysis see Macherey (2008) and Fischbach (2015a)). Following Marx, what contemporary
dialectical materialists add to the agentic conceptions of self mentioned above (i.e., the self as excess) is thatsubversion is accomplished in praxis, with others (Freire, 1998). Subversive praxis is the arena of theemergence of a new form of social consciousness. For consciousness is not only a refraction of reality.
Consciousness, through its varying layers of depth, is a concrete relation that, given our biography andcultural background, propels us towards the world and leads us to act on/in it and transform it (Clot, 2015).
In short, what dialectical materialists add to the agentic conception of self mentioned before is that thesubversion of self is not the result of the deeds of a solitary being: it is a political and social project strictlytied to the transformation of the individuals’ consciousness. There is still hope, then, that in our Grade 4
example, Mirna and her twin sister, Híria, will be heard. However, for this to occur, there must be atransformation of circumstances. The classroom culture must be transformed. This transformation requires a
new praxis, a classroom praxis, out of which a new form of social consciousness can emerge.
To recap, in each one of these accounts, although for different reasons, there are always possibilities to2 The Symbolic is the sphere of the Law. Without submitting ourselves to the Symbolic, we would not be able to function insociety. The Symbolic includes grammatical rules, social norms, and unconscious prohibitions (see Žižek, 2007).

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interrupt the quotidian train of our actions and thinking. It is at this point that Scott’s (1990) The Question of
Ethics as a question of interruption of habits and values acquires its whole sense. For Scott, The Question of
Ethics indicates our capacity to bring forward “an interruption in which the definitive values that govern
thought and everyday action lose their power and authority” (p. 4). This interruption makes sense precisely
because of our cultural-historical agentic nature (i.e., our power to sublate and surpass the cultural andhistorical possibilities on which we draw when we re-act to, and re-enact, the world).

5. ETHICS IN POSTMODERN TIMES
5.1 Ambiguity
As stated before, one of the main features of postmodern ethics is to consider ethics as having an intrinsicallyambiguous nature. To be ambiguous means that ethics is neither something contractual (like teachers do this,students do that), nor is it something that works on the basis of rules and abstract principles (like do yourhomework!). Valero and Jørgensen (2021) give an example of the ambiguous nature of ethics in contemporary
ethical thought. A student submits an essay that, according to a plagiarism software, shows 21.5% ofcompatibility between a student’s text and a referred text. However, the institutional maximum ofcompatibility is 20%. For the institution, plagiarism is
unacceptable and grounds for expulsion. The teacher knows the student and knows that there is a caseof terminal cancer in the immediate family that has affected the student’s academic ability … For theteacher, ensuring access to education, particularly for poor people, is a fundamental part of his
understanding of fairness in education. It is part of his social responsibility and commitment as ateacher. What does the teacher decide to do: follow institutional norms or his personal norms? (Valero
and Jørgensen, 2021, p. 271)
As the authors remark, “The issue is to enter into a deep reflection to generate an ethical rule that is
applicable in this situation” (Valero and Jørgensen, 2021, p. 272; my emphasis).
So, to say that ethics is intrinsically ambiguous means that our acts and relations to others do not haveone obvious meaning. They are context sensitive. Ethics is seen here as a context-sensitive dynamic andopen-ended relational stance that is continuously materialized and assessed as teachers and students engage
with each other.
5.2 Ethics as thoroughly personal
Another main feature of postmodern ethics, or at least the one we find articulated in Bauman’s work, is, asmentioned before, that ethics is something “thoroughly personal,” something that “appeals to myresponsibility” (1993, p. 60).
Bauman continues here the Western modern tradition that posits the individual as the origin of
intentionality (Husserl, 1982) and ethical concerns (Kant, 2006). This feature of postmodernism is anything
but new. It is easily understood once one realizes that postmodernity “one may say, is modernity without
illusions (the obverse of which is that modernity is postmodernity refusing to accept its own truth)” (Bauman,
1993, p. 32). In other words, postmodernity is modernity disenchanted; modernity removed of the illusions

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Luis Radfordof the universal abstract reason of the Enlightenment and its overarching grand narratives. Of course,
postmodernism, even if it continues the modern tradition of the “new,” the “novel,” the “creative,” is more
complex than this (Macherey, 2006). Capital (understood as the general system of production of theindividuals’ lives, the system of production that characterized modernity) has not disappeared. It continues
to be the mechanism through which human life is shaped and quotidianly produced. But capital has beenrefined and expanded: it has become more complex, more nuanced, and much more oppressive than inmodern times. Bauman, hence, is right in seeing in postmodernism the story of an unending end; it is still amodernism, but one that seeks with relentless vigour to get rid of the encumbrances of the Symbolic (the“law” in Žizek’s sense) that has continuously threatened the autonomy of the modern subject. In this sense,
postmodernist ethics is still (or again) a subjective ethics, this time pushed to its radical limits—with all thecontradictions that such a push entails (Lipovetsky, 1989; Žizek, 2010b). If postmodern ethics recognizes therole of the Other in some qualified Levinas’s sense (as Bauman does), it does so within a monadic
understanding of the social featured by modernity, that is, as an arena of “competitive struggle” whose
pinnacle is “individual choice” (Bauman, 1993, p. 46).

6. ETHICS IN THE THEORY OF OBJECTIFICATION
6.1 Reconceptualizing the social
Ethics in the theory of objectification is neither modern nor postmodern. Although ethics is understood in a
non-deontological, context-sensitive manner, ethics is not considered as coming from within. It is not
something “thoroughly personal.” The reason is not because the individual is exempt of all moral
responsibility. The reason is that it understands the social not as an agglomeration of monads that, once
constituted as humans, enter the social realm. This monadic conception of the individual, where the social isseen as a derivative of the individual’s life, has been one of the chief marks of modernity and postmodernity,from where we have inherited the “acids of individualism” (MacIntyre, 1966, p. 266).
The conception of the social that underpins the theory of objectification is such that there is no
separation between the individual and the social. The social is always appearing within the individual andbetween individuals. One cannot exist without the other. The social is neither a container, nor a flat surface
where we live our life. The social is made up of relations (rapports) between individuals. These relations are
not merely connectors; they are susceptible of modifying the individuals they put into relations.
To the extent that social relations express and manifest in their enactment political, economic, culturaland social conceptions about the world, ourselves and others, they affirm certain forms of dealing withothers. This is what we see in our discussion of Kant’s and Hobbes’s ethical systems: each one of themsuggests a social form of self-other dealings (Kant’s ethics turned to a universal reason legislating thetransactions between individuals; Hobbes’s one turned to transactions of a contractual nature). Ethics isfound precisely in these forms, which are nothing else but forms of alterity. Ethics, then, appears as the formof Self-and-Other.
If the social is understood as made up of social relations in movement, what would then be the kind of
relations that are targeted in the theory of objectification?

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6.2 Ethics and learning
To answer this question, we need first to make explicit the educational project in which this theory is
subsumed. The theory of objectification inscribes itself in a Freirean and Vygotskian educational project that
posits the goal of Mathematics Education as a political, societal, historical, and cultural endeavour aimed at
the dialectical creation of reflexive and ethical subjects who critically position themselves in historically andculturally constituted mathematical discourses and practices, and who ponder new possibilities of action andthinking.
As in other educational theories and pedagogical models, ethics in the theory of objectification appears
in the relations between students and teachers. Ethics, understood as the form of Self-and-Other that manifests
itself in our dealings with other individuals, is directly linked to the understanding of how learning occurs.
Let us consider two examples.
Think of the theory of transmissive instruction. Learning is conceived of as the assimilation, throughpractice and repetition, of knowledge that the teacher possesses. This theory positions teachers as knowers
and the students as lacking knowledge. The ethics of the transmissive instruction, manifesting itself in theform of alterity that it promotes, reflects, but also operationalizes. It manifests itself concretely in thealienating relations of power and subjection it fosters and maintains, relations that are thematized along theobedient lines of superior/inferior, potent/impotent, knower/ignorant, authority/vassal (Radford & Lasprilla
Herrera, 2020).
Now think of constructivism. In contradistinction to the theory of transmissive instruction,
constructivism is based on an ethics that stresses the freedom of the student: since knowledge is conceivedof as what results from the autonomous deeds of the student, and learning is the very process of the student’s
subjective construction of knowledge, teachers and students are positioned otherwise: the student’s freedom
and autonomy configure the constructivist ethical space (Radford, 2012). It alienates teachers and students,
but in a way that differs from how transmissive instruction does (Radford, 2021a).
Learning in the theory of objectification is neither about transmitting knowledge (as in direct ortransmissive teaching), nor is it about the students’ constructing their own knowledge (as in constructivist
approaches). In the theory of objectification, learning is conceived of as a collective and truly social embodied
and material process through which students critically encounter culturally and historically constituted waysof thinking mathematically. This encounter happens in what the theory terms joint labour (Radford, 2021a,
2021b).
6.3 Joint labour
Joint labour is a sensuous, practical, material activity—activity understood as driven by collective concerns.
The German and Russian languages have a specific term for this type of activity: Tätigkeit and deyatel’nost’,
respectively. Activity in this sense is opposed to activity as being merely busy with something (as in watchingTV). Again, the German and Russian languages have a specific term for this other type of activity: Aktivität
or aktivnost’. Unfortunately, in the translation into English (and several other languages), the distinction islost and both types of activity are rendered as activity.
In the case of the theory of transmissive instruction, classroom activity is not oriented towards thesatisfaction of collective needs. This activity corresponds hence to Aktivität or aktivnost’. In joint labour, by68Luis Radford

contrast, students and teachers work hand in hand to produce something together, what Hegel termed “acommon work,” in our case, mathematics. It is this sense of labouring together (as opposed to simply
interacting or exchanging with others) that makes joint labour a truly social activity and learning a collective
process.
6.4 Transforming ethics
Classroom research has shown us, however, that for learning to become a truly collective process, radical
changes in the classroom culture might need to occur. Often, drawing on experiences shaped by transmissive
instruction, the students conceptualize the teacher as the possessor of knowledge and power, and conceptualize
themselves as submissive to the teacher and her knowledge, even when the teacher tries to conceptualize
herself differently and encourages the students to learn collectively and organizes the classroom into small
groups. Often, the students configure small, enclosed groups and erect aggressive or exclusive antagonistic
barriers between their group and other groups—they resort to what we may term a clique ethics or gang
ethics (Radford & Lasprilla Herrera, 2020), which is also what we see in the example from the Grade 4classroom briefly mentioned in the Introduction.
Of course, moving towards a more encompassing, democratic, and inclusive ethics cannot be achievedby imposing new social forms of conduct. “It is not obedience to someone or obedience to something, but
the free adoption of those patterns of behavior which will vouchsafe the consonance of all of behavior”(Vygotsky, 1997, p. 233). The educational problem around ethics becomes the problem of the creation ofclassroom conditions for new ethical relations (new forms of alterity) to emerge and to be collectively
pondered and discussed against the always contested background of culture and history. The process of
emergence of such an ethics has to overcome the dynamic power of capital, which constantly tries toappropriate the social “in order to put it at the service of its own development.” It is hence “a question of notletting ourselves be dispossessed: it is necessary to maintain and to develop [the spring of the production ofsocial life] against capital” (Fischbach, 2015b, p. 71). This can only be done by developing a form of
democracy that would “increase our power to generate ourselves our own social life” (p. 71).
6.5 A communitarian ethics
So, what are the new ethical relations that we strive to nurture in the theory of objectification? We focus ona classroom mathematics practice featuring a communitarian ethics which is consonant with the conceptual
historical-cultural bases of the theory and its conception of learning as a collective process.
Communitarian ethics is based on what we call three ethical vectors. They define a dynamic ethicalspace. Rather than be seen as fixed terms, we see them as repères; that is, reference points against which togauge our actions. These vectors work as “primary orientation of the ethical self” and should not be seen as“a modern uniform code”; they are “shared ethical ideas, priorities and principles that are subject to ongoingand ‘antagonistic’ negotiation’” (Neyland, 2004, p. 57). These vectors (or “virtues” to use MacIntyre’s term)
are responsibility, commitment, and care. It is here where we resort to the construct of “voice”—not voice ina linguistic sense exactly; rather we resort to voice as something that brings in the notion of difference andthe primacy of the political (Giroux, 2005). Finding one’s voice or having a voice is “moving from silence
into speech,” it is “a gesture of defiance that heals, that makes new life and new growth possible” (bell hooks,

Ethics in the Mathematics Classroom

69

2015, p. 29), something that “assumes a primacy in talk, discourse, writing, and action” (p. 33).
Coming back to our Figure 1, when Mirna utters an anguished “Ahhhhh!!!” and moves her right hand
towards the two other girls who are not listening, she is moving into speech to express her frustration about
not being counted and heard. Her voice (which is much more than what she discursively says, as it also says
things in her body posture, facial expression, pitch, gesture) opens up new possibilities for action (for herself
and the other teammates). Mirna’s embodied utterance is, indeed, a call to the Other.
6.5.1 Responsibility
The call now must be responded to, and it is responded to within a certain node of social relations that tie thestudents together. Whatever path the teammates’ response takes, it is cast in a general ethical attribute that
Lévinas calls responsibility. For Lévinas, responsibility is “the essential, primary, and fundamental structureof subjectivity … [where] the very node of the subjective is knotted” (1982, p. 101). Since all educational
theories put into motion a certain ethics—for, as mentioned before, ethics is the substrate and the form thatis continuously materialized in our dealings with the Other—responsibility is a common denominator of allof them. Yet, the meaning of responsibility is not the same. In the theory of direct instruction, the students’
responsibility consists in assuming a submissive role vis-à-vis the teacher. The teacher’s responsibility
consists in subjecting the students through the power of knowledge. In the case of the theory of didacticalsituations (Brousseau, 2005), responsibility follows Hobbes’s contractual outlook and appears as a didacticalcontract; that is, a reciprocal distribution of duties. In the case of the theory of objectification, responsibility
means living and acting with and for others; it means to respond to the call of others as they are on their own
terms: in their “existence, in [their] being-for-other[s] . . . as free being[s]” (Hegel, 1978, p. 57).
6.5.2 Commitment
Commitment is both the promise and its realization of doing everything possible to work side by side withothers in the course of our joint labour (e.g., trying to understand the process being followed to solve a
problem, trying to contribute to the classroom common work).
The classical utterance “I do not understand” of classrooms operating within the transmissive instruction
model often conveys the stance of a de-responsabilization; it intimates that it is the teacher who has not done
his work properly. Through this utterance, the student invests the teacher with the role of patriarch ofknowledge and requests the teacher to deliver knowledge. However, if teachers and students understand thatknowledge is produced collectively in the classroom, the same utterance would mean something like: “I donot understand, but let’s work together so perhaps things will become clearer.” Commitment is this resolution
to engage in joint labour, to participate in the creation of the classroom common work (oeuvre commune),
which can be establishing collectively how equations can be solved or theorems proved, for example.
6.5.3 Care for others
Far from being an act of condescending or simply caring for someone, the care for others is a relational
involvement entailing the attention to, and recognition of, others and their material and spiritual needs.
Although caring for the Other opens up the possibility of seeing ourselves in the Other, of recognizing ourvulnerability in the vulnerability of the Other, the importance of caring for the Other is to go beyond ourselves

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Luis Radford

and to be dragged powerfully into the world, to position ourselves there, with-the-Other.
To understand Mirna’s “Ahhhhh!!!” within an ethical practice of responsibility, commitment, and care,
we need to broaden our conception of language and recognize (in its Hegelian sense) this painful expression
as voice; that is, as something where, as Lévinas suggests, the saying (that which I want so much to utter but
I will never be able to do) moves beyond the totalizing enclosure of the said (that which I manage to say) andbecomes rather the possibility of openness to the other (Radford, 2021a). In this conception of voice, power
does not disappear since power is not a thing, but something imbricated in our relations to others. What we
can expect in the transformative movement towards a communitarian ethics is that, through conscious,
reflective, and critical stance, power in the classroom goes beyond its own subjecting mechanisms of social
order and becomes rather something fluid, dynamic, to be exercised with responsibility for the Other.

7. CONCLUDING REMARKSIn the first part of this article, I argued that mathematics teaching and learning is unavoidably an ethicalevent. There are at least two reasons for this. The first reason has to do with the fact that teaching and learningalways involve interaction between teachers and students and this interaction is based on relations (relationsof power and subjection, relations of exclusion, or relations of solidarity, inclusivity, etc.). The second reason
has to do with what is to be learned, with knowledge. It has to do with the legitimation of particular forms ofknowledge and knowing. Teachers are confronted in their everyday practice with a variety of ways thatstudents bring about how to solve mathematical problems. Legitimation appears here as what counts as true,as mathematically right, sound, worth pursuing, etc. I presented an example in which students produced two
different mathematical texts explaining how to solve linear equations. The question revolved around theepistemological role of examples: should the text include an example? Should it be rather general? What text
is better? As soon as we move beyond Eurocentric views of mathematics (D’Ambrosio, 2006; Powell &Frankenstein, 1997), the question of legitimation acquires its full sense.
In the first part of the article, I complained that despite its omnipresent nature in the mathematics
classroom, ethics still remains largely unexplored (although it is gaining some traction, as we can see in theNo. 38 (December 2021) issue of the Philosophy of Mathematics Education Journal, edited by Paul Ernest).
To better understand conceptions of ethics, in the second part of the article, I dealt with two ethicalsystems—Hobbes’s and Kant’s—that have been influential in Western thought and have informed theunderstanding of the mathematics classroom, even if, more often than not, those ethical systems haveremained implicit (for instance, Kant’s subjectivist ethical stance has influenced constructivism; Hobbes’s
contractualism has influenced transmissive learning, and both Kant and Hobbes have influenced the theoryof didactical situations).
In the third part, I moved to a discussion of what is at stake in ethics. We saw that, on a first level, theanswer lies in the social meaning of choices about human conduct. On a second and deeper level, the answer
lies in our conception of human nature.
The discussion of ethics in postmodern times, intended always as a means to better grasp the nature of
ethics, allowed us to see that, on the one hand, the postmodern project breaks with the modern project, while,Ethics in the Mathematics Classroom

71

on the other hand, continues it. The postmodern project breaks with the modern one in refusing to resort to auniversal reason as the basis of the ethical self. At the same time, it continues the modern project informulating a subjective ethics; that is, an ethics that is at the end of the day “thoroughly personal” (Bauman,
1993, p.60). The postmodern understanding of ethics still rests on the modern conception of the social as anaggregate of individuals, monadic self-constituted agents.
The previous theoretical considerations paved the way to approach, in the last part of the article, ethics
from an educational viewpoint. I outlined the ethics in a Vygotskian theory of teaching and learning, thetheory of objectification (Radford, 2021a). To do so, drawing on the work of Fischbach (2015a, 2015b), I
started offering a reconceptualization of the social as made up of social relations, which are much more thansimple links between individuals. So, instead of being a mere background, the social appears to be a dynamic
totality in perpetual movement and transformation. The relations of which the social is made up express
conceptions about ourselves and others. They are always tinged with political and cultural valences. These
considerations led me to suggest that ethics can be considered as the form of alterity (the form of Self-andOther) that manifests in our dealings with other individuals. I argued that, in educational contexts, such as
the school, ethics appear framed by the way in which we understand teaching and learning.
Against this background, I outlined the communitarian oriented relational ethics that is articulated in
the theory of objectification—a communitarian ethics. This ethics follows Spinoza in stressing the body’s
power of acting in the world (agendi potentia; Spinoza, 1989, p. 210). The communitarian ethics also follows
Lévinas’s (1982) work which radicalizes previous ethical systems in acknowledging that our actions anddeeds are always modulated by the presence of the Other—a presence that comes to us in a sentient andfleshy manner: through the proximity of our bodies. In this proximity our conceptual epistemologicalcategories and mechanisms are put on hold, and we encounter the Other as is. “The Other is appreciatedprecisely as Other, in her radical alterity and irreducible singularity, only when thought renounces its
totalitarian hubris and learns to think of the Other on her own terms . . . ‘beyond essence’” (Min, 1998, pp.
573–574; emphasis in the original). The self appears here conceived not as the prerequisite of existence of
ethical relations. The self is rather seen as the result of those relations.
In conceiving of ethics as the form of alterity, the focus turns not to moral precepts but rather to thefluid and content-dependent relationships between subjects as they appear in the immediacy and banality of
everyday life. In this view, ethics is continuously materialized in praxis out of a myriad of possibilities, for,in this view, the individual appears as “full of unrealized possibilities every minute” (Vygotski, 2003, p. 76).
If we come back to Figure 2, we see that Mirna’s teammates could have opted for other actions. The
materialization of the students’ actions reflects their understanding of the context (consciousness) and themanner in which the context is lived through their unfolding collective affective experience (emotions), as
well as the relational stances (ethics) they (consciously or not) adopt towards each other. Through theprevious classroom examples, we see that embodied, emotional, contextual, cultural-historical action isrelated to ethical postures that students assume and show in practice.
What is specific to the communitarian ethics of the theory of objectification is that, braking with thesubjectivist stances of modern and postmodern ethics, its practice features responsibility, commitment, andcare. This communitarian ethics provides reference points for pedagogical actions in the classroom, where
teachers and students explore together new critical spaces that promote engagement, inclusiveness, debate,

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and respect (for some classroom examples, see Radford, 2021, 2021b). Underneath the communitarian ethics
lies the recognition that our historical, cultural, and material origin embeds and refracts dynamic andantagonistic visions and conceptions of the world and of what a good life can mean. Although impossible toposit and describe a priori, the good life and the common good are understood here as ideas (idéalités), more
specifically, as “visées éthiques” (aims), generated in a joint unending project created itself within the
dynamic that produces the social relations, a project that will always be paved with tensions, contradictions,and difficulties. “Let us call ‘ethical aim’ (visée éthique) the aim of the ‘good life’ with and for others in justinstitutions” (Ricœur, 1990, p. 202). It is the vitality of contradictions that gives substance to social human
life, always changing, always challenged, and that makes ethics to always unfold in a context of struggle.

ACKNOWLEDGMENTS
This article is a result of a research program funded by the Social Sciences and Humanities Research Councilof Canada and Laurentian University Research Fund. The article was presented at the TSG 57 (Diversity oftheories in mathematics education) of ICME 14. A previous version of it appeared in Radford (2021c).

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Ethics in the Mathematics Classroom

APPENDIX
Translation of the texts shown in Figure 1.
First text:
Step 1: Count all cards and envelopes in the equation.
Step 2: Take the number with fewer envelopes and remove all the envelopes. Ex.

Step 3: Same thing for the cards. Ex.
Step 4: Give an answer ex.

Second text:
To write an equation you have to:
1. Read the problem.
2. Take the smaller amount of card[s] on one side and cross it out.
3. Cross out the same amount of cards on the other side.
4. Take the smaller amount of envelopes on one board and cross it out.
5. Cross out the same amount of envelopes on the other side.
6. Circle the remaining cards and envelopes.
7. Count the remaining cards and envelopes.
8. Divide the cards into each envelope if you have more than one envelope remaining.

Luis Radford
Laurentian University, Ontario, Canada
Email: lradford@laurentian.ca
https://orcid.org/0000-0001-6062-0605

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