Davydov’s concept of the concept and its dialectical materialist background

Abstract

The goal of this article is to investigate Davydov’s concept of the concept against the backdrop of its philosophical system, namely, dialectical materialism. In the first part, after briefly sketching the context of Davydov’s work, I consider some ontological and epistemological ideas on which Davydov bases his concept of the concept. I pay particular attention to Hegel’s and Marx’s contributions. Then, I discuss Davydov’s concept of the concept and the relationship between the logical and the historical—a relationship that proved to be crucial in the making of the educational curricular program he and El’konin launched in the 1960s in Russia. I argue that, in tune with the dominant epistemology of the twentieth century, Davydov’s concept of the concept is based on a scientific outlook of the world, one in which theoretical scientific thought is considered the pinnacle of human cognition. I conclude with a critique that intends to place the notion of the concept in a broader dialectical materialist perspective.

Full text

Radford, L. (2021). Davydov’s concept of the concept and its dialectical materialist
background. Educational Studies in Mathematics, 106, 327–342.
Davydov’s concept of the concept and its
dialectical materialist background
Luis Radford
Abstract
The goal of this article is to investigate Davydov’s concept of the concept against the
backdrop of its philosophical system, namely, dialectical materialism. In the first part,
after briefly sketching the context of Davydov’s work, I consider some ontological and
epistemological ideas on which Davydov bases his concept of the concept. I pay
particular attention to Hegel’s and Marx’s contributions. Then, I discuss Davydov’s
concept of the concept and the relationship between the logical and the historical—a
relationship that proved to be crucial in the making of the educational curricular program
he and El’konin launched in the 1960s in Russia. I argue that, in tune with the dominant
epistemology of the twentieth century, Davydov’s concept of the concept is based on a
scientific outlook of the world, one in which theoretical scientific thought is considered
the pinnacle of human cognition. I conclude with a critique that intends to place the
notion of the concept in a broader dialectical materialist perspective.
1
Introduction
One of the best-known works of Vasily Vasilyevich Davydov, at least in the Western world, is
Types of Generalization in Instruction (Davydov, 1990). In this book, Davydov provides a
definition of what a concept is. He says:
A concept functions here as a form of mental activity by means of which an idealized
object and the system of its connections, which reflect in their unity the generality or
essence of movement of the material object, are reproduced. (Davydov, 1990, p. 116;
emphasis in the original)

328
This article is a critical commentary on Davydov’s functionalist concept of the concept. My
critical commentary—which is part of an increasing interest in better understanding the nature
of mathematical concepts (de Freitas, Sinclair, & Coles, 2017)—is an attempt to investigate
Davydov’s concept of the concept against the backdrop of its philosophical system, namely
dialectical materialism. My contention is that it is only through an examination of the philo-
sophical system that underpins Davydov
’
s work that we can fully appreciate both his concept of
the concept and its educational implications. I start by offering a short sketch of the historical
context of Davydov
’
s work and the general vision that oriented his curricular research. Then, I
discuss some key ideas of Davydov
’
s dialectical materialist background. The rest of the article
revolves around Davydov’s concept of the concept and the problem of the historical and the
logical, a problem that proved to be crucial in the making of the educational curricular program
Davydov and Daniil El’konin launched in the 1960s in Russia. Drawing on Heidegger’s view
of Being and Adorno’s negative dialectics, I conclude with a critique that intends to place the
notion of the concept in a broader dialectical materialist perspective.
2
The context
In 1959, 2 years after having defended his doctoral thesis under the guidance of Piotor
Gal’perin, Davydov started working at the Institute of General and Pedagogical Psychology
of the Academy of Medical Sciences of the USSR (Rubtsov, 2015). In collaboration with
El
’
konin, he started developing a
“
modeling experiment,
”
which came later to be known as the
El’konin-Davydov curriculum. Davydov’s modeling experiment was embedded in a Soviet
educational reform initiated in 1958 by Nikita Khrushchev’s government. The reform sought
to overcome the poor
“
academic level of students and their [in]ability to apply their knowledge
to practical tasks and workplace-related activities” (Boyko, 2019, p. 83). In mathematics, the
reform was led by the famous mathematician Andrey Kolmogorov. It is in this context that
governmental efforts were made to rebuild the curriculum and to produce new textbooks
(Abramov, 2010). Commenting on the inadequacy of the mathematics curriculum of the time,
Davydov (1975a) notes that “Deficiencies in the traditional mathematics curricula for the
school are being discussed frequently both here and abroad” (p. 55).
Davydov cites Kolmogorov, saying that mathematics
studies the material world from a particular point of view, that its immediate subject is
the spatial forms and quantitative relationships of the real world. These forms and
relationships themselves, in their pure form, rather than specific material bodies, are the
reality which mathematics studies. (Kolmogorov, cited in Davydov, 1975a p. 68;
emphasis as in Davydov’s passage)
From this viewpoint, Davydov notes:
The curriculum should provide the child with work in which he (sic) will be able to
“move away” from concrete bodies accurately and at the proper moment, after having
distinguished their spatial forms and quantitative relations and having given them their
‘
pure form.
’
Only on the basis of this can he (sic) develop an accurate understanding of
mathematics. (Davydov, 1975a, p. 68)
Naturally, the child cannot start from the pure forms of mathematics. “What the research
mathematician has before him (sic) in its
‘
pure form
’
has to be constructed in the child
’
s head.

329
This ‘form’ is not given to him (sic) at the start” (Davydov, 1975a, p. 68). The fundamental
question was then as follows: “What organization of the course and what method of intro-
ducing concepts contributes best to the solution of this problem?” (p. 68). Davydov’s
experimental research, carried out for about 25 years and mainly conducted at the Experimen-
tal School N. 91 in Moscow, was an attempt at answering this question. Davydov argued that
the curriculum should be based on clear logical principles to structure its content. He claimed
that it is “particularly necessary to determine the most appropriate concepts with which to
begin mathematics instruction in school” (1975a, p. 56). Structuring the curriculum through
logical principles was, however, not enough. An appropriate psychological approach was also
needed. As he put it, the outdated curriculum of his time was failing to “provide for the
necessary development of children’s mathematical thought” (1975a, p. 55).
But the distinctiveness of Davydov’s approach is not to be found in his resorting to
logical and psychological principles to structure the mathematics curriculum. What
makes it really distinctive is Davydov’s understanding of the logical and psychological
principles. Drawing on dialectical materialism, he derived the overarching goal of his
whole enterprise. The goal was not about developing a curriculum that would allow the
child to simply acquire mathematical knowledge. To think so is to miss entirely
Davydov’s whole point. As Schmittau and Morris aptly put it, “Davydov’s curriculum
has as its overriding goal the development of the ability to think theoretically” (2004, p.
61; see also Ivashova, 2011, p. 59) or, in the words of Libâneo and Freitas, “the
formation of theoretical-scientific thought” (2013, p. 318). Now, for Davydov, the
attainment of theoretical-scientific thought is not the result of a speculative mind getting
hold of concepts through ruminative cogitations. It is rather a sensuous developmental
process that, in tune with Vygotsky’s school of thought, privileges the process of
generalization. Starting from material objects and “the immediate character of empirical
knowledge” (Davydov, 1990, p. 115; emphasis in the original), generalization allows one
to recognize the inner structure of scientific concepts and the system of their constitutive
connections.
Thus, when Davydov intends to understand what a number is, he purposely moves beyond
the immediacy of empirical knowledge and tries hard to find the theoretical connections that
would constitute the scientific concept of number. After a theoretical archeological excavation
(of which I will have more to say below), rather than counting aggregates of objects—rather
than cardinality as such
—
he finds relations: relations like
“
more than,
” “
less than,
”
and
“
equal
to” (Davydov, 1975a, 1975b, 2008). From here he devises a curricular organization of
arithmetic and algebra that might look puzzling when considered from traditional epistemol-
ogies like those offered by idealism and empiricism.
Davydov presented a schema of the main curricular ideas in various papers (see, e.g.,
Davydov, 1975b; see also Schmittau & Morris, 2004). I will limit myself here to merely
mention that the curriculum starts with a focus on a quantitative comparison of quantities
(length, width, volume, and area), first on instructional material, then on representations
through line segments and letters, in order to assert whether A = B, A ≠ B, A > B, or A <
B, without reference to numbers. In Davydov’s view, the use of symbols is required to
help children move away from specific objects and to pay attention to the connections at
the heart of theoretical concepts. Symbols, indeed, help the children to divorce them-
selves from using objects so that they can focus on “verbal and logical evaluations
(constructions of the type: ‘if…and…then…’)” (Davydov, 1975b, p. 140). As a result of
plunging into the theoretical realm through symbols, “What becomes central for the

330
children is the relationship itself, its type, rather than the [material] objects” (Davydov,
1975b, p. 164).1
In a similar vein, the children use letters to represent unknown parts of a whole (e.g., h + r =
t) and reflect on their mutual connections (see Schmittau & Morris, 2004, p. 70; see also
Freiman and Fellus, this Special Issue). The use of letters, however, should not be seen as
something that makes the child enter automatically into the realm of algebra. For Davydov,
algebra is not about using letters. To understand the use of letters in Davydov’s approach, we
have to bear in mind that Davydov is following the Vygotskian tradition where theoretical
thinking is mediated. And it is this mediation—a theoretical mediation—that signs are
accomplishing. Signs, in this account, allow the child to perceive, deal with, and reproduce
the essence of the object under study. This is why it is crucial, for example, that in the
reproduction of the essence of number, children shift from equality (A = B) to inequality (A +
K > B) and back (A + K = B + K) (Davydov, 1975b, p. 137).
Davydov’s logical approach to the concept of number and its “shift from ‘techniques of
calculation
’
to the study of the structural characteristics of mathematical
‘
objects
’”
(Davydov,
1975b, p. 141) is a reminiscence of Piaget’s epistemological analyses. Yet, while Piaget talks
about invariants, one-to-one correspondence, and abstraction (Piaget, 1964), Davydov (1990),
by contrast, talks about generalization—the generalization of material objects into idealized
objects, which, as it proceeds in practical, sensuous activity, discloses the generality or essence
of the object. The general that this general-ity produces should not to be confounded with the
general of empiricist generalizations. The general (or essence) that Davydov refers to has to be
understood in its dialectical materialist sense, that is, as “the genetically initial ‘cell’ of some
developing integral system which generates all its particular manifestations
”
(Davydov, 1967,
p. 50; emphasis in the original).
The goal of the two following sections is to offer an overview of Davydov’s dialectical
materialist background. Although the content is theoretical, I hope that it will be helpful in
better appreciating Davydov’s approach and in particular his concept of the concept, which I
see as the “cell” of his curricular enterprise. My overview is organized in two parts, one
dealing with ontology (which explains, for instance, Davydov’s insistence that a concept is
made up of internal connections), and one dealing with epistemology (which explains how
things are known).
3
The ontology of dialectical materialism
3.1
The principle of universal connection
Davydov
’
s dialectical materialism starts from a Spinozist, systemic, ontological premise about
the nature of the world (Spinoza, 1989). According to this premise, the world is an “aggregate
1
Here is an example:
The students are shown a picture of two balloons. The volume of one balloon is labeled L; this balloon is
completely drawn. The other balloon of volume P is only partially drawn. The problem says: If L = T and
T > P, then L
P. The students are unable to directly compare the volumes (one is only partially drawn)
so
they have to make an inference about the relationship between L and P. (Schmittau & Morris, 2004,
pp. 63–64)

331
of objects and phenomena linked with one another by extremely diverse relations and
connections” (Spirkin, 1990, p. 119).
The idea that objects are linked to other objects through systems of relations and connec-
tions is articulated in the work of Hegel, upon which dialectical materialism draws. In the
Encyclopaedia, Hegel notes that “A determinate content . . . contains a manifold connection
within itself and is the basis for connections with many other objects” (Hegel, 1991, p. 89;
emphasis in the original). As a result, in this view, material and ideational objects are not
merely substantial things. Objects are relational through and through. The essence of an
object—and in the previous section we saw this idea in Davydov’s account of number—
consists indeed in the connections that link such an object to other objects.
3.2
Motion and development
Dialectical materialism considers the world in constant motion. All is in flux, “on the path
toward something else” (Spirkin, 1990, p. 123), pushed, so to speak, in one direction or
another by connections and relations. In social life, for instance, the needs and interests of
people constitute basic connections that underlie the forms of production of a community.
Among the different types of connection in the universe, there is one that plays a significant
role in dialectical materialism: it is the law-governed connection, which serves to explain
development:
“
Development is an irreversible, definitely oriented and law-governed change of
material and ideal objects resulting in the emergence of new qualities
”
(Spirkin, 1990, p. 123;
emphasis in the original).
In general terms, law-governed connections can be conceived of in several different ways.
They can be conceived of as laws of the world—objective laws that have nothing to do with
the individuals. These laws are not human-made. The discovery of the laws that govern the
universe is indeed, in this line of thought, the task of scientific research. Dialectical materialism
adopts here a kind of Galilean view of the universe, which provides it with scientific-rationalist
ontology.
Law-governed connections can also be understood as laws created by the individuals—
subjective laws that individuals produce to make sense of their world. So, while the first
Galilean view gives the primacy to the object (truth lies in the object), the second one gives the
primacy to the subject (truth lies in the subject).
Davydov draws on a different dialectics—a Hegelian dialectics as articulated further by
Marx. This materialist dialectics rests on a view where subject and object become entangled,
so that law-governed connections always contain the imprint of human activity, which makes
them neither transcendental vis-à-vis the individuals nor merely subjective.
There is another question that we need to discuss—the meaning of the essence that, as we
saw in the Introduction, Davydov brings to the fore in his understanding of what a concept is.
3.3
Essence and phenomena
Dialectical materialism makes from the outset a distinction between essence and phenom-
ena. Essence and phenomena are part of two different ontological realms. A phenomenon
(or appearance) is what is observable, sensible. Essence, by contrast, is inaccessible to
human perception. In the words of Davydov, essences are “internal, essential relationships
[that] cannot be observed directly by the senses” (Davydov, 1990, p. 119; emphasis in the
original).

332
While in Kant’s (2003) dualistic theory of knowledge, the realm of the essence (the realm
of things-in-themselves) remains beyond human cognition, for Hegelian dialectical materialism
this is not the case. There is a clear and explicit relationship between them: within the dynamic
ontological view of the universe, essences are conceived of as continuously passing into
appearances.
Essence, Spirkin says, “is the basis of all the forms of their external manifestation . . .
Essence is in this sense something internal, a certain organizing principle of the object’s
existence in the forms of its external expression” (Spirkin, 1990, p. 155; emphasis in the
original).
The problem of theoretical thought is precisely to reveal the connections and relations of
objects, as these connections manifest externally or actualize themselves in “the realm of
objectively interconnected phenomena” (Davydov, 1990, p. 119).2
Within the dialectical materialist framework, essences lose the metaphysical meaning with
which idealism endowed them. Essences as a system of internal connections manifest them-
selves in motion, in practical activity—as when children compare lengths of visible, non-
visible or partially visible objects (as in footnote 1) or add fractions (Swanson & Williams,
2014).
4
The epistemology of dialectical materialism
The previous section offered a short account of the ontology of dialectical materialism. Now
we turn to its epistemology. The question is: What does it mean to know? The answer is in
Hegel’s work. As previously mentioned, for Hegel, an object of knowledge is a determinate
content; that is, a manifold of connections within itself and the basis for connections with other
objects. In this context, “To be cognizant,” Hegel says, “means nothing else but the knowing
of an object according to its determinate content
”
(Hegel, 1991, p. 89). In other words, I know
something if I go beyond the superficial aspect of the object and know its essence (internal
relations and connections). The question that naturally arises in this context is the question of
the kind of knowing that we, as humans, are capable of. Hegel notes that
if we inquire into the truth of knowledge, it seems that we are asking what knowledge is
in itself. Yet in this inquiry knowledge is our object, something that exists for us; and the
in-itself that would supposedly result from it would rather be the being of knowledge for
us. What we asserted to be its essence would be not so much its truth but rather just our
knowledge of it. (Hegel, 1977, p. 53; emphasis in the original)
Knowledge, hence, would not be knowledge of the object itself, but our knowledge of it.
“
Consciousness provides its own criterion from within itself, so that the investigation becomes
a comparison of consciousness with itself” (Hegel, 1977, p. 53).
The object, it is true, seems only to be for consciousness in the way that consciousness
knows it; it seems that consciousness cannot, as it were, get behind the object as it exists
for consciousness so as to examine what the object is in itself. (Hegel, 1977, p. 54;
emphasis in the original)
2 The adjective “objective” should not be understood as independent of the individual. As mentioned previously,
in dialectical materialism, the connections between things and phenomena refract the entanglement between
subject and object.

333
Imagining the object as concealed behind a curtain, Hegel says that
“
there is nothing to be seen
unless we go behind [the curtain] ourselves” (Hegel, 1977, p. 103; emphasis in the original).
However, what we would see, were we to walk behind the curtain, is ourselves at work. “The
inside of things is a construction of the mind. If we try to lift the veil that covers the real, what
we will find is only ourselves, the universalizing activity of the mind that we call understand-
ing” (Andler, 1931, p. 332).
It seems, then, that we have reached an impasse and that, as humans, we are limited to have
a subjective knowledge of things, knowledge of appearances only. This was Kant’s (2003)
conclusion. In Kant’s account, this was the limit of pure human reason. But it is precisely at
this point that Hegel went further. He reasoned that if the comparison between the object and
our knowledge about it (the object in itself and the object for us or for consciousness) do not
correspond, we still can alter our knowledge
“
to make it conform to the object
”
(Hegel, 1977,
p. 54).
However, Hegel’s crucial innovation is not to be found in the adjusting endeavor of our
knowledge to make it come closer and closer to the object, to its essence, to Truth. What Hegel
is articulating in these passages is a new theory of knowing that moves epistemology beyond
the classical objectivist and subjectivist paradigms of Western thought.
We see that consciousness now has two objects: one is the first in-itself, the second is the
being for-consciousness of this in-itself . . . the first object, in being known, is altered for
consciousness; it ceases to be the in-itself, and becomes something that is the in-itself
only for consciousness. (Hegel, 1977, p. 55; emphasis in the original)
As a result, there is an entanglement between object and subject, world and consciousness.
They co-produce each other mutually.
Hegel’s theoretical position gained great popularity during the nineteenth century in
German intellectual circles. German philosophers engaged in discussions about his work
one way or the other. And Karl Marx was certainly not the exception. He found Hegel’s
account very idealistic. For Marx, the entanglement between object and subject is not the result
of a majestic flight of consciousness in its march to perfection. In the first of the Theses on
Feuerbach (Marx, 1998), the co-creation of subjects and objects and their entanglement are
explained as occurring in something terrestrial: in praxis, or sensuous collective human
practical activity. Through praxis human affection and cognition become
“
inscribed in matter
”
(Aron, 1981, p. 91). This inscription constitutes the essential structure of reality
—
Wirklichkeit,
which for Marx signifies effective reality; that is to say,
“
everything that offers itself first of all
to consciousness as constituting reality
”
(Macherey, 2008, p. 47). In this way, effective reality
embraces much more than the products of human activity (i.e., les oeuvres humaines): it also
embraces the embodiments of social needs built culturally and historically and the ideas and
values of a given culture at a certain moment (Descombes, 1996). As a result, objects (material
or ideal) do not present to us as untouched by humans but as something already carved by
history. In this line of thought, truth is neither on the side of the object (as ahistorical
materialism suggested) nor on the side of the subject (as idealism and empiricism contended).
In articulating the epistemological question of the subject and object in this way, by going
beyond the classical dichotomy objective/subjective, Marx opened up a fresh avenue to look at
human cognition in a new anthropological way (Dupré, 1983; Henry, 1976). To know an
object amounts to putting into action “a dynamic of reflection and explanation that reproduces
the very movement of reality without . . . fixing it in the consideration of some of its isolated
results, cut off from the complex conditions of their elaboration” (Macherey, 2008, p. 74).

334
It is from this anthropological perspective on human cognition that Davydov—and before
him, Vygotsky, Luria, and Leont’ev—explored a new path to the study of human psychology.
And it is in this sense that consciousness and objects appear in Davydov
’
s work. For Davydov,
objects are objects of need. He quotes Rubinstein: “Objects of needs and actions, rather than
objects of contemplation, are given initially” (Davydov, 1990, p. 111). He also quotes G. A.
Kursanov: “For a thing to function for human consciousness, it should function as an object of
activity” (p. 190; emphasis in the original).
Bearing in mind the previous dialectical materialist background, we can now turn to
Davydov’s concept of the concept.
5
Davydov’s concept of the concept
As mentioned in the introduction, Davydov defined a concept as follows:
A concept functions here as a form of mental activity by means of which an idealized
object and the system of its connections, which reflect in their unity the generality or
essence of movement of the material object, are reproduced. A concept simultaneously
also functions as a form for reflecting the material object and as a means of mentally
reproducing, constructing it—that is, as a particular mental action. (Davydov, 1990, p.
116; emphasis in the original)
He explains that, ontogenetically speaking, this object appears first as existing independently
of the individual. There is, hence, an element of passivity where the object appears having an
objective content vis-à-vis the individual and an element of activity since
“
to have a concept of
an object means to reproduce or construct it mentally” (Davydov, 1990, p. 116). Davydov
continues: “This action of constructing and transforming a mental object is an act of under-
standing and explaining it, of discovering its essence” (p. 116).
He sympathetically cites Kant who “has astutely noted that ‘thinking’ means ‘acting’”
(Davydov, 1990, p. 116). In this context, Davydov mentions Kant
’
s concept of the schema and
reminds us that Kant’s key terms such as “drawing” or “describing” in the formation of
schemas are
“
none other than reproducing or constructing an object on an ideal level
”
(p. 117).
He ends up approvingly citing Y. M. Borodii saying that a concept “is a rule for reproducing
an object, or, expressed in Hegel’s language, a measure” (p. 190).
There are three things in Davydov’s concept of the concept:
(1)
A concept is a form of mental activity.
(2)
It reproduces or constructs on an ideal level the essence of the object.
(3)
It works normatively.
I will dwell on these points below.
(1)
A concept is a form of mental activity.
The first point indicates that a concept is a psychological entity. It is an activity carried out
by an individual. This activity is of a certain type. It is mental.
(2)
A concept reproduces or constructs the essence of the object.
The second point states that, in the concept of an object, the essence of the object can be
captured in its mental construction or reproduction. Not any reproduction will fit here. The
construction has to satisfy a condition: it is supposed to reproduce the object’s essence. We

335
can ask the question: is such a thing possible? For Davydov, it is. And, as we can see, this is a
very strong epistemological assumption.
Of course, Davydov was very well aware of the epistemological commitment he was
making. In fact, on this point, as we saw above, he drew on Kant’s concept of the schema and
the role imagination plays therein. He wrote: “Kant was right in indicating its [the imagina-
tion
’
s] role in constructing a concept
”
(Davydov, 1990, p. 118). But he disagreed with Kant on
two counts:
First, he disagreed with Kant’s assumption that schema-making and imagination are
both powers of the mind (something that is built into the architectonics of the human
mind). On this point, Davydov and Kant are far away from each other. As a good
dialectical materialist thinker, Davydov understood human cognition as something deep-
ly related to the cultural-historical context. We find the following statement at the very
beginning of his very dense chapter “Basic Propositions in the Dialectical Materialist
Theory of Thought”: “An individual person’s thought is the functioning of historically
developed forms of society’s activity which have been conferred on him” (Davydov,
1990, p. 108; emphasis in the original). And then, this other statement: “An analysis of
the origin and development of thought must begin with a clarification of the features of
human labor activity” (Davydov, 1990, p. 108). These statements, which derive from
Marx’s ideas of effective reality (Wirklichkeit) and human cognition discussed above, are
antithetic to Kant’s conception of human thought. Kant’s philosophy and dialectical
materialism parted ways here and ended up informing two radically different
pedagogies—one articulated around the work of Piaget and the other articulated around
the work of Vygotsky, one of which is the El’konin-Davydov Program.
Second, Davydov is in disagreement with Kant on the role of material culture in the
constitution of the schema. For Davydov, the question of material culture is crucial as it is
through the practical activity with concrete objects, and more specifically with their conceptual
transformation, that the essence of an object can be disclosed. Davydov wrote that through the
transformation of material things
“
the learning person comes to understand the relation between
a material
’
s external appearances and changes in appearances
”
(Davydov, 1999, p. 126).
So, there are important disagreements in the ways Davydov and Kant understood the
human mind. Both thinkers recognize the active side of the mind in concept formation, but
they end up taking different routes.
The differences may be better appreciated if we bear in mind the fact that Davydov’s
concept of the concept draws from an epistemological shift that occurred in sixteenth century
Europe. Before this shift, to know a thing was generally understood as to know the properties
of such a thing. This conception of knowing was epitomized by classical Greek geometry.
Then, with the arrival of new forms of production in the late Middle Ages and early
Renaissance, with the invention of Western capitalism, the conception of knowing changed.
As Arendt put it, within the new epistemology of early modernity, to know something became
associated with knowing the manner of its production: “I ‘know’ a thing whenever I under-
stand how it has come into being” (Arendt, 1958, p. 585). We find this conception clearly
articulated in Spinoza’s 1667 text (modern edition in Spinoza, 1989), De Intellectus
Emendatione (Improvement of the Understanding). Spinoza says that a true idea “shows
how and why something is or has been made” (1989, p. 29). Thus,
If a circle be defined as a figure, such that all straight lines drawn from the center to the
circumference are equal, every one (sic) can see that such a definition does not in the

336
least explain the essence of a circle, but solely one of its properties. (Spinoza, 1989, p.
32)
Within the new epistemological paradigm, the “how and why something is or has been made”
have to be revealed in what Spinoza calls its
“
proximate cause.
”
Spinoza then gives the proper
definition: “the figure described by any line whereof one end is fixed and the other free. This
definition clearly comprehends the proximate cause” (Spinoza, 1989, p. 32).
Spinoza’s concept of proximate cause is what appeared later thematized as a schema or a
rule, as we have seen in Davydov’s concept of the concept. In his commentary on Spinoza’s
definition of the circle, Davydov says:
Spinoza perceived the essence of a circle in the act of its emergence or construction
(“creation”). Its definition should express the reason why the given thing arose, the
method of constructing it . . . Here a method of obtaining any and infinitely varied
circles is given. (Davydov, 1990, p. 117)
Spinoza’s definition of a circle is cited again and again by dialectician thinkers—Davydov
himself, as we have seen, but also other dialecticians such as Ilyenkov (1977). Its importance
does not reside only in that it provides us with a neat and short example of a concept (i.e., as
something rule-based). The example can also be thought of in terms of an activity with a
concrete object—the compass—through which a transformation of material marks on a paper
may acquire a theoretical content.
(3)
A concept works normatively.
The third point about Davydov’s concept of the concept refers to the regulatory feature of
concepts. As Davydov put it, individuals “act and produce things according to the concepts
which exist as norms in the society beforehand” (1990, p. 118). From an ontogenetic
viewpoint, in this perspective, individuals do not create concepts; they encounter them. This
point has to do with one of the main tenets of dialectical materialism that I mentioned before,
according to which individuals find the conditions of possibility of their thinking in the
historically developed forms of activities of their society.
I shall come back to the implications of Davydov’s ontological position in the concluding
section. For the time being, I want to discuss the question of the historical and the logical
aspects of concepts.
6
The historical and the logical
Davydov did not embrace a recapitulationist idea of concept development. That is, he did not
accept the idea that ontogenesis is a recapitulation of phylogenesis, or, more simply put, that
human cognitive development follows the same path as the historical path in the development
of ideas. To accept a full repetition would mean that knowledge would repeat itself regardless
of time and culture. On the other hand, to accept a full independence between ontogenesis and
phylogenesis would amount to reducing the production of knowledge to what individuals do
right now, independent of historical influences. To accept a full independence between
ontogenesis and phylogenesis could easily lead to the kind of epistemological subjectivism
that we find in Kant’s theory of knowledge, for instance. These two extreme points (full
recapitulation and full independence) are at odds with the tenets of Hegel’s dialectics and its
ensuing dialectical materialism. Davydov offers an intermediate position. Ontogenesis repeats

337
phylogenesis in some qualified sense only. As he put it, “Ontogenesis . . . does not repeat
phylogenesis in totality
”
(Davydov, 1975a, p. 65; my emphasis). There must be some kind of
relationship between them. But what is it?
Davydov’s educational work is, in practice, articulated around this question. As mentioned
in Section 2, Davydov believed that the school curriculum should start from the basic or
essential connections that make up the essence of the conceptual objects to be learned. There is
an important ontological assumption here: according to this assumption, objects of knowledge
can be broken down into a few relations or connections. The problem, then, is to find those
small relations or connections. This ontological assumption is supplemented with a method-
ological one: history, on the one hand, and current mathematics on the other, can provide us
with the clues to find them. In other words, we need to look at the historical and the logical
dimensions of knowledge, and their relationship.
In dialectical materialism, such a relationship is often considered in the following terms.
The logical is associated with the movement of essences. The historical is associated with the
actual movement of the phenomena of effective reality. In this context, the logical is consid-
ered as the theoretical reflection of the historical (see Kopnin, 1966, p. 84). In terms of the
concepts discussed in the previous sections, history finds its material in the ontological
category of Appearance or Phenomena (the Concrete)—not in the category of essences (the
Abstract). But, as mentioned before, in dialectical materialism, these categories are deeply
interconnected. History, it turns out, is but another name for the theoretical principle of the
unity between the Abstract and the Concrete, Essence, and Appearance. This is why the logical
aspect of an object,
“
besides reflecting the history of the object itself, also reflects the history of
its knowledge
”
(Kopnin, 1966, p. 188). It does not come as a surprise, then, that the problem of
the relation between the logical and the historical constitutes
“
the most important [problem] of
dialectical logic” (Kopnin, 1966, p. 84).
From a methodological viewpoint, “To reveal the essence of an object it is necessary to
reproduce the actual historical process of its development
”
(Kopnin, 1966, p. 186; emphasis in
the original). But at the same time, the reproduction of the actual historical process of the
object’s development “is only possible if we know the essence of the given object” (Kopnin,
1966, p. 186). It seems that we run into a vicious circle. Marx (1993) agonized in front of this
problem in the Grundrisse, where he was trying to find the proper categories to understand
production and consumption. So, how do we get out of this vicious circle? The dialectical
answer is this: We can grasp the essence by studying its more mature forms as they appear in
the phenomenological realm. This is why, “The researcher must begin the study of the object
by the end, by the most mature form; for in this most mature form, [the] essential aspects [of
the object] appear in a more developed way
”
(Kopnin, 1966, p. 186). Hence, it does not come
as a surprise that Vygotsky cites Marx arguing that “the anatomy of man (sic) is the key to the
anatomy of the ape” (Vygotsky, 1978, p. 94). It does not come as a surprise either that in his
investigations Davydov turned to the most evolved mathematical forms of his time. What did
he find? Bourbaki. The Modern Mathematics of the twentieth century: Bourbaki, but also
Lebesgue, Kolmogorov, and others. Through his habitual sharp, penetrating, incisive, and
brilliant multidisciplinary approach, he embarked in a theoretical archeological excavation to
understand these most mature mathematical forms. What he found was a structural conception
of mathematics. Davydov cites Bourbaki extensively and his idea that the fundamental
mathematical notion is not the number. In mathematics—like in life, as understood by the
Marxist structuralists of Davydov
’
s time, such as Althusser (1965)
—
there are structures rather
than objects. Davydov hence searched hard to find those essential connections and relations

338
out of which mathematics as a school subject should be erected. He found that it is not
numbers we should start with, it is not counting, but other more primitive or fundamental
relations that underpinned, in the twentieth century modern conception of mathematics, the
notion of number, namely, the relations of order (less than, more than, equal to) and
equivalence. For him, the primitive notion is quantity (understood as a collection of objects
endowed with a total order relation). He wrote:
the concept of number arises within the context of measurement of a continuous quantity
so that a multiple relationship is established between that quantity and a part of it that is
used as a unit of measure. According to this it is possible to consider counting as the
measurement of a set of discrete objects. (Davydov, 1982, p. 228)
Thus, in this view, the origin of arithmetic thinking is not to be found in cardinality.
Cardinality is here a derivative concept. Given the evanescence of a Bourbakist approach to
mathematics today, we can imagine that if Davydov were to engage in the same archeological
digging, he would come up with a different result—and, likely, with a different curriculum
proposal as well.
7
A critique of Davydov’s concept of the concept
By way of conclusion, let me summarize some of the ideas that I have discussed about
Davydov’s concept of the concept and outline a brief critique. I hope that my critique may
spark some reflections on the theoretical premises, entailments, and commitments embedded
in the notions of the mathematical concept that we, as mathematics educators, use implicitly or
explicitly in our research.
I mentioned that Davydov drew on the dialectical distinction between the two chief
categories of dialectical materialism: essence and appearance, and he assumed a dialectical
Hegelian-Marxist ontology that links these two categories (as opposed to Kant’s ontology that
strives to keep them apart). From there he articulated a dialectical concept of the concept: a
concept is a form of mental activity derived from human activity, a reproductive rule that
works ontogenetically speaking in a normative way. From an epistemological viewpoint, to
know an object is to reproduce it mentally in its theoretical movement.
Davydov adopted a view of the concept that is grounded on the model of scientific concepts.
For him, scientific concepts are the basis
“
of theoretical thought
”
(Davydov, 1990, p. 116), and as
we saw in Section 2, it was precisely the child
’
s acquisition of theoretical thought that he strived to
promote through his curricular enterprise. One of the main features of Davydov
’
s epistemological
stance is the belief that the essence of objects is graspable through the revolutionary concept of
modern science, namely law (see Section 3.2. above). Davydov says: theoretical thought “is an
idealization of the basic aspect of practical activity involving objects, and of the reproduction in
that activity of the universal forms of things, their measures, and their laws
”
(1990, p. 116).
We cannot fail to see that Davydov’s concept of the concept is imbued with a general
outlook of the world that favors a certain form of knowability—the one that predominated
throughout the twentieth century and considered scientific theoretical thought as the summit of
human cognition. Scientific theoretical thought was perceived as the model par excellence by
which to understand the world. Certainly, Davydov was not the only educational psychologist
to embrace such a view. We only need to think of Vygotsky’s famous distinction between
everyday and scientific concepts.

339
One of my reviewers argues that Vygotsky’s (and Davydov’s) use of the adjective
“scientific” to designate a specific type of concept does not amount to any commitment with
a scientific outlook of the world. I would like very much to agree, perhaps more than anyone,
but I do not think that the use of the adjective
“
scientific
”
is merely circumstantial. I side here
with van der Veer (1991, 1996), who has shown that Vygotsky’s work was deeply influenced
by the modern idea of culture that was built around the model of science, a model that
transpires in Vygotsky
’
s marked interest in tools as means to master nature (and oneself). If we
remove the concept of tool from Vygotsky’s work, internalization and Vygotsky’s laws of
cultural development vanish into thin air. In Ape, Primitive Man, and Child, Vygotsky and
Luria argued that the first phase in the child’s cultural development lies precisely in the use of
tools:
We have seen how the small child, for whom the world of external objects was initially
alien, gradually comes closer to it, and begins to master those objects and make
functional use of them as tools. This is the first phase in cultural development. (Luria
& Vygotsky, 1998, p. 117)
Vygotsky’s commitment to a scientific outlook of culture also transpires in his emphasis on
abstract thinking. So, when Luria (1931, 1934) carried out the “Psychological Expeditions to
Central Asia” (that he prepared with Vygotsky), what were the questions he posed to the
Uzbekistan peasants? Aristotelian syllogisms—i.e., the embodiment of decontextualized
thinking.
Undoubtedly, there were very powerful historical reasons in the twentieth century that led
people to see the world through the lenses of science. First, there was the modern general
“belief in technology and the promise it held for cultural and social progress” (van der Veer,
1996, p. 258). Then, there were the two world wars; then the Cold War (i.e., the time in which
Davydov developed his work). The military successes of these wars were absolutely depen-
dent on scientific and technological development. The famous founding document of
Khrushchev
’
s reform,
“
Law on Strengthening the Links Between School and Life, and Further
Development of the System of Public Education in the USSR
”
conveyed in an explicit manner
the idea of scientific knowability as the true one: “The Soviet school educates the younger
generation in the spirit of the most progressive ideas, the ideas of communism, and forms in
young people a materialistic worldview—the basis of truly scientific knowledge of the world”
(Government of the USSR, 1958; cited in Boyko, 2019, pp. 81–82; my emphasis).
Within this scientific outlook, law, measure, and calculation became the key concepts to
understand the world. Commenting on this epistemological scientific outlook specific to the
modern period of the Western world, Heidegger (2002, p. 48) says, “Beings became [con-
ceived of as] transparent objects capable of being mastered by calculation.” Heidegger gives
the example of a stone. “The stone presses downwards and manifests its heaviness” (p. 24).
But can heaviness be reduced to its numerical value?
If we try to grasp the stone
’
s heaviness . . . by placing it on a pair of scales, then we bring
its heaviness into the calculable form of weight. This perhaps very precise determination
of the stone is a number. (Heidegger, 2002, p. 25)
Now, can we say that we have grasped the stone in this way? Heidegger’s answer is no. Can
we say at least that we have grasped the stone’s heaviness? Again, Heidegger’s answer is no:
“
the heaviness of the weight has escaped us
”
(Heidegger, 2002, p. 25). The stone
“
shows itself
only when it remains undisclosed and unexplained” (p. 25). We could say the same of the

340
circle and its roundness: the process of its construction does not exhaust the essence of the
object (notwithstanding Spinoza and the Spinozist dialecticians).
Davydov’s epistemology is articulated around this modern view where truth is of the order
of the logical and the scientifically conceptual. Certainly, the focus on this order of reality is
the strength of scientific thought. But it is also its weakness. The reason is that scientific
thought and its scientific laws necessarily marginalize the centrality of the singular, the
concrete, the sensible, and the complex ever-changing feature of cultural contexts. In doing
so, scientific thought marginalizes a part of the whole complex conditions of the object’s
cultural-historical elaboration (e.g., aesthetical, ethical, political, and economic conditions).
But I want to go a bit further. So far, I have been concerned with, and critical of, the totalizing
stance with which scientific thought has been endowed in the understanding of reality. My
argument has been epistemological. Now I want to articulate an ontological argument. What
Heidegger is saying in the citations above is that there is a substrate in any object that is
nonconceptual. In dialectical terms, the nonconceptual of an object is precisely the object’s
own negation
—
the non-identical-with-itself
—
that which, in the object
’
s historical movement, is
what the object is not. It is the unsettling and inexhaustible realm of the object that resists
consciousness. As a result, there is always a partial inadequacy of the concept and the real.
“
Knowledge does not wholly possess any of its objects
”
(Adorno, 2008, p. 187). In other words,
what individuals produce through practical sensible activity—through praxis—is both concep-
tual and nonconceptual, the latter being
“
the sheer heterogeneity of thought
”
(Huhn, 2004, p. 17).
In the creative historical movement of effective reality objects emerge. In their emergence,
each one of them brings to the fore a conceptual dimension, but also a nonconceptual one.
Now, it is in the nature of the nonconceptual to resist translation into the symbolic—scientific
laws, painting, poetry, music, linguistic expression, etc. It resists encapsulation:
no matter how hard we try for linguistic expression of such a history congealed in things
[the movement of the real
—
LR], the words we use will remain concepts. Their precision
substitutes for the thing itself, without quite bringing its selfhood to mind; there is a gap
between words and the thing they conjure. Hence, the residue of arbitrariness and
relativity in the choice of words as well as in the presentation as a whole. (Adorno,
1973, pp. 52–53)
Thus, “The cognitive utopia would be to use concepts to unseal the nonconceptual with
concepts” (Adorno, 1973, p. 10). It is only through the reflexive and critical praxis that
Marx (1998) talks about in the third thesis on Feuerbach (see analysis in Macherey, 2008)
that humans have the possibility to change their circumstances and their ideas. Thus, only
through such praxis, by going to the heart of contradictions, a dim light may be shed on the
nonconceptual. In the course of this movement, human consciousness and the object have been
transformed, but a new nonconceptual has emerged. It is impossible to get rid of the
nonconceptual. It is what propels consciousness forward again and again. This is why the
nonconceptual is the vitality of life. To cite Heidegger again, it is what “remains undisclosed
and unexplained
”
(2002, p. 25). In the case of numbers, to come back to Davydov
’
s work, the
nonconceptual is that which resists being encapsulated by the concept of quantity and the
structure of total order. In the case of the Euclidean circle, the nonconceptual is that which is
more than the traces left by a radius in motion. Were there a total coincidence between concept
and object, we would find ourselves in a closed universe, a bound reality. Dialectics—i.e., the
movement of life,
“
the consistent sense of nonidentity
”
(Adorno, 1973, p. 5)
—
would collapse:
life would become movement without movement in a tautological silent identity.

341
Like Davydov, I do think that “An individual person’s thought is the functioning of
historically developed forms of society’s activity which have been conferred on him” (sic)
(Davydov, 1990, p. 108; emphasis in the original). I concur with him that labor
—
what I call in
my work joint-labor (Radford, 2019)—“is the basis of all human cognition,” and that it is
“Only within historically developing modes of . . . activity [that] all forms of thought [are]
formed” (Davydov, 1990, p. 108). Yet, I prefer not to conceive of a concept as a mental rule,
but as something more poetic, something that brings together the cultural rationality and
worldview of the contexts where the object has emerged and evolved with all its historical and
political tensions. In this view a concept would be what we enact with others in joint activity
—
a cultural-historical enactive experience that is not merely conceptual and theoretical, but also
esthetic, ethical, political, and emotional; something that questions us.
Acknowledgments I would like to deeply thank Anna Sfard and two anonymous reviewers of the journal for
their precious comments.
Funding information This article is a result of a research program funded by the Social Sciences and
Humanities Research Council of Canada/Le conseil de recherches en sciences humaines du Canada (SSHRC/
CRSH)
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