vygotsky's everyday concepts/scientific concepts dialectics in school ...

VYGOTSKY'S EVERYDAY CONCEPTS/SCIENTIFIC CONCEPTS
DIALECTICS IN SCHOOL CONTEXT: A CASE STUDY
Nadia DOUEK
IUFM de Nice, France
The aim of this paper is to analyse, through a case study, how Vygotsky’s everyday
concepts/ scientific concepts dialectics, intentionally promoted by the teacher, can
result in an evolution of conceptualisation in mathematics. I will present the case at
study, then, on a theoretical level, I will characterise conceptualisation in school
context. This will result from combining Vygotsky’s elaboration about scientific
concepts with Vergnaud’s definition of concept in a dynamical perspective inspired
by the Vygotskyan dialectics. On a pragmatic level, I will specify the conditions that
can favour building up this dialectics in school. This will be backed by Boero’s
"experience fields" didactic theory, and will concern argumentation and student’s
implication in experience field activities.
VYGOTSKY'S EVERYDAY CONCEPTS / SCIENTIFIC CONCEPTS
DIALECTICS
Scientific concepts are characterised by Vygotsky (1985, Chapt. 6) through their use,
as they are used in a conscious, intentional and explicit way. They are related to other
concepts in systems. They are general, have a theoretical character and can be
presented through a definition. Everyday concepts are rich in meaning, but their
relation to systems is rather implicit. They are developed spontaneously through
subject’s experience, within culture and social context. Generally they have a local
scope and do not need to be defined. The existence and development of everyday
concepts and scientific concepts are closely related: scientific concepts evolve from
abstract to particular groundings; everyday concepts can be refined into more precise
(and eventually into scientific) concepts; but relating to a system and the conscious
use of a concept imply that some everyday concepts have already been developed,
which can back its scientific use. I wish to stress the relative and dynamical character
of these two poles of the dialectics.
THE EXPERIENCE FIELDS DIDACTICAL SETTING
Our example comes from a second grade class involved in the Genoa Project for
primary school (20 children, about 8 years old). The aim of this Project is to teach
mathematics, native language, natural sciences, history, etc… through systematic
activities concerning "fields of experience" from everyday life (Boero, 1994; Boero et
al., 1995). These activities are intended both to develop mastery of the fields of
experience themselves, and to approach and develop concepts and skills belonging to
those disciplines. For instance, in Grade I the development of numerical knowledge is
rooted in the "money" and "class history" fields of experience; moreover they allow

to promote students' argumentation as well as the use of specific symbolic
representations.
In the "fields of experience" didactical setting, a fairly common classroom routine
consists: of individual production of written hypotheses on a given task; of classroom
comparison and discussion of selected students' products, orchestrated by the teacher
(cf Bartolini Bussi, 1996); of individual written reports about the discussion; and of a
classroom summary, usually constructed under the guide of the teacher and finally
written down in the students' copybooks. In most cases, classroom summaries
represent students’ reached knowledge (with possible ambiguities and hidden
mistakes). Final institutionalisation phases (Brousseau, 1986) are attained only in few
circumstances. From the researcher's point of view, this style of slowly evolving
knowledge offers the opportunity to observe the transformation of students'
knowledge in a favourable climate. Argumentation has a strong status in this
didactical setting, and most problem situations, particularly in mathematics, are
solved through argumentation, avoiding manipulative activities (cf Douek, 2003).
Argumentation reflects fairly well each student’s level of mastery of knowledge,
his/her level of use of references, and the class' common stable references (cf Douek,
1999). Justification and explanation are essential in the didactical contract. As in all
Italian primary schools, the student group remains with the same two teachers for the
first five years of primary education. Therefore didactical contracts are very stable. I
must add that Italian curricula are rather loosely prescriptive around a core of
necessary knowledge.
Students deal with the plant cultivation experience field for about 4 months in Grade
II. Students had measured wheat plants taken in a field, then had to follow the
increase in time of the heights of plants of the classroom pot. They described them
and drew them on their copybooks. They solved various problems concerning their
growing up, which involved mathematical knowledge; our example concerns one of
those problem situations. We will consider data deriving from direct observation,
students' texts, and videos of classroom discussions concerning this problem situation
as well as previous activities.
THE CASE AT STUDY
Here is the problem posed to the class:
On Friday December 3, the plant in the pot, kept in class and in the light, was 26 cm
tall, and the one in the dark was 7 cm tall. Today, Friday December 10, the plant in the
pot kept in the light is 28 cm tall and the one in the pot in the dark is 10 cm tall. Which
plant has grown the most from December 3 till now?
Students had to deal with two kinds of conceptual difficulties. The first difficulty is
related to the comparison of transformations (increases) in the additive conceptual
field, a complex and new situation for them (cf Vergnaud’s categorisation, in
Vergnaud, 1990). The second difficulty is inherent in the students' concept of
increase, which was at the level of an everyday concept, although students were able
to solve problems as “how much such plant has increased from that date till now”,

and to compare heights of two plants. The closeness between the concepts of height
and increase added complexity. However, it will be interesting to see how the
evolution of the conceptualisation of increase depended on, and simultaneously
favoured, the mastery of the additive field and its extension to include the comparison
of transformations.
Phase 1: Individual oral production with teacher’s mediation
13 students (out of 20) engaged correctly into the problem solving, through
calculations and comparison of results. But they were confused when the teacher
asked them to explain their problem solving. Only 6 of them arrived easily at a stable
correct conclusion. Some students answered “the plant that increased the most is the
one which is the tallest”. They deduced increase from height, thus contradicting their
own numerical result. Others said “this plant increased the most, so it is the tallest”.
They deduced height from increase, disregarding the visible state of the plants.
Performing the right calculations was not enough to grasp the situation as a whole. To
solve problem numerically, most students interpreted increase as a supplementary
length in the plant. But the dynamical character of increase was not grasped with its
dependence on time and the implicit fact that it may be slow or quick between two
given moments: they were not able to use such an argument to interpret the situation.
Their everyday conceptualisation of increase was not sufficient. But through
argumentation guided (or stimulated, depending on cases) by the teacher, this concept
evolved and most students became able, at least momentarily, to handle the problem
situation through verbal activity, paying attention to the various elements of the
situation, and were able to interpret them according to a hierarchy that organised
calculations. Many students used an expression that became meaningful and
stabilised: “pezzo cresciuto” ("the bit that grew").
The other 7 students had difficulties to master and interpret data, and did not succeed
in producing a solution. The teacher encouraged them (as well as 2 students out of the
group of 13 who had produced a correct solution) to draw the two plants, mark the
drawing with signs corresponding to the different heights at different dates, and
follow the increases on the drawing. Hence for all the students the situation gained a
spatial character, either through the drawing or through the image of “the bit that
grew” (which is not correct scientifically, since growth is an evolution of the whole
plant). This spatial frame helped most students to “see” and grasp the dynamical
aspect of the plant growing as well as to fix for each plant the elements that need to
be determined and compared. Given students' mastery of the additive field, I
conjecture that those spatial representations allowed the representation of numbers on
the number line to be transferred (gradually, depending on students) to this new
additive situation. This favoured the mathematisation of this new situation while
putting its dynamical character into new evidence.
As a whole, student/teacher interactions developed argumentation concerning three
types of references: reality of plants cultivation, their “stories” with their dynamical
character and time dependence; schematisation through drawing and/or its equivalent

verbal spatial organisation, which helped to master the hierarchy between dynamical
and static elements (either as data or as parts of the sketched plants); and calculations
that helped finding and fixing intermediate elements, but which could have been
mastered abstractly and independently from the particular situation.
Confusion due to the insufficient everyday conceptualisation of increase was
overcome thanks to a sufficient mastery of the additive situation and related external
representations (numerical, verbal, schematic). This made the situation challenging
and attainable at the same time. We can conjecture that such conditions made the
scientific/everyday concept dialectics effective. Argumentation was the way by
which that dialectics enhanced students conceptualisation. It also allowed some
students to appropriate elaborate verbal expression reflecting refined understanding
of the situation. We heard sentences such as: “the plant that increased the most is the
one which was in the dark although the highest was the one in the light.”
At the end of Phase 1, all students could distinguish and calculate increases and
heights; 9 expressed the relation between them as in the excerpt mentioned above.
Phase 2: comparison activity and collective discussion
This text was distributed for individual comparison, before collective discussion:
JESSICA : the plant in the dark increased the most because it increased 3 cm, on the
contrary the plant in the warm place increased 2 cm. I say that the one in the dark
increased the most because the centimetres of the part of the plant that increased are the
most.
GIUSEPPE : the plant that increased the most since Friday, December 3 is the one in
the dark, because it increased 3 cm, but the one in the warm increased 2 cm and so 3 are
more than 2. But in fact the plant which is the tallest is the one that makes 28 cm, and it
is the one in the warm.
Questions :
1) In which way did Giuseppe and Jessica reason ?
2) In Giuseppe’s text there is a consideration we cannot find in Jessica’s. Which one is
it?
This metacognitive reflection about their problem solving led students to share the
reasoning and stabilise it within the class, as well as to deepen, for some students, and
to pursue, for the others, the effort of clarification and interpretation of the situation.
At the beginning of the collective discussion, the weak everyday conceptualisation of
increase appeared again: many students mixed it up with height in their reasoning,
even though they had been able to discriminate them during their interaction with the
teacher. But once some students referred to the bit that increased, the evoked spatial
situation helped to restore the “right” argument. However, the drawing was not
needed anymore. A new example given by a student established another criteria to
distinguish increase from height in relation to time: a child’s height is his/her
increase since birth (note that this is rigorously true for plants, not for children…);
while increase is determined in relation to two dates. All along the discussion the

arguments to interpret and compare the texts concerned “reality” (the known
development of the plants), spatial organisation, and calculations, with their
hierarchical organisation.
Phase 3: Individual synthesis
The teacher brought the students to a synthesis by the following question:
Is it the same to say “which plant increased the most” and to say “which plant is the
tallest”? Why?
The generality of this question aims a more scientific handling of the involved
concepts. 12 students out of 20 produced a general and valid synthesis, most gave
examples (and some of them were creative in doing it). 14 students referred to time. 6
out of these 14 had produced a drawing in Phase 1.
A CHARACTERISATION OF SCIENTIFIC CONCEPTUALISATION IN
SCHOOL CONTEXT
Let us now describe scientific conceptualisation as we consider it pertinent in school
context, and present a characterisation that allows to evaluate its evolution. We will
use it to analyse the effect of Vgotsky’s everyday concepts/ scientific concepts
dialectics on conceptualisation, as well as the conditions that helped this dialectic to
take place. This characterisation captures two entangled aspects: the epistemological
construction, and the development of cognitive activities. Vygotsky (1986; cf
Vergnaud, 2000) helps us to recognise the quality of the use of the concept, and its
evolution. Vergnaud (1990) helps us to recognise explicit links with experienced
“reality” through reference situations and external representation; and with subject’s
activity interpreted through operational invariants 1 . External representation, in
coherence with the experience field didactical theory, includes gesture and language
in the broadest way (cf Radford, 2003). We consider two characters of
conceptualisation and at the same time we try to describe how to detect them:
- The evolution of systemic links of concepts. Linked objects can be reference
situations, external representations, other concepts, or components of concepts
(according to Vergnaud’s definition). Links can be propositions about the objects
(e.g. propositions that express properties, characteristics, analogies or differences or
hierarchies), metaphors, examples, etc. Operational invariants or generalisations can
underlie these links.
- Students’ awareness in handling the concept, its components or its systemic links: it
consists in the mastery of the concept, and the intentional recourse to it or to some of
its components, that can be detected specifically in using and expressing operational
invariants and through using its external representations.
AN ANALYSIS OF CONCEPTUALISATION IN OUR EXAMPLE
1
Vergnaud defined a concept as composed by the set of its reference situations, the set of its
operational invariants ('schemes', theorems in actions, etc.) and the set of its symbolic
representations (see Vergnaud, 1990).

Students had to discover the crucial elements of the situation and organise their
hierarchical relations to back a procedure of calculation. This reflects the
development of an intentional and explicit use of additive relations and a mastery of
important systemic links. But they also dealt with the central notion of increase. This
everyday use of increase was crucial to interpret the problem situation in terms of
additive relations; at the same time the possibility of interpreting it with the scientific
concepts of the additive field made it necessary, and possible as well, to develop the
conceptualisation of increase (and specially to make explicit its dependence on time).
Teacher’s stimulation and direction of argumentation made this dialectics alive and
productive. As a result of setting this dialectic, the everyday concept of increase was
made more “scientific” (though remaining in an everyday use).
Evolution of the concept of increase
Development of systemic links of increase:
- distinction between height and increase: It was possible thanks to the implicit
relation to the number line, either by drawing or indirectly by mental reference
(because of student’s mastery of such elements of the additive conceptual field),
which contributed also to determine their relation to time. Increase was determined as
a variation of length through time;
- linking increase to reference situations: The real situation made it possible through
its dynamical character, thanks to the interpretation needed to solve the problem;
indeed increase it is a variation related to time that is not really visible, but has to be
elaborated. Also referring height to increase since birth as different from increase
between two dates had a positive effect: students had another reference situation and
another explanation of the role of time at their disposal.
Concerning students’ awareness:
students syntheses revealed that they assimilated this new understanding of the
concept (all of them referred to time) and were able to use it in a personal way to
answer the question put in general terms (it was not possible to just repeat the newly
established knowledge). The ability to determine and distinguish the different
components of height and increase was related to the possibility to master and
combine intentionally that set of additive situations (variation of length, reference and
dependence to time, and comparison of variations).
Extension of the scientific conceptualisation of the additive field: comparison of
additive transformations
Comparing transformations in the context of plants' increase (familiar because of the
long term work in the experience field of plant cultivation) was meaningful and
efficient: the growing of the plants confers to the additive situation a dynamical
character and a possibility of mental spatial representation. This allowed to master
and organise the various phases of calculus.
Development of systemic links of the additive conceptual field:

they can be identified through the relationship established between comparison of
transformations and a “real” dynamical situation, as well as through schematic (direct
or mental) treatments of the static visible aspects and the dynamical, only thinkable,
aspects. Hierarchic organisation between the various additive calculations, in relation
to their interpretation as static or dynamical, was an operational invariant, as well as
the arithmetic theorems in action underlying the numerical treatment.
Concerning students awareness:
at the beginning of the sequence one part of the students mastered the additive
situation only through calculations. At the end, though, all of them were able to
interpret calculations in relation to the reference situation and give examples of
comparable additive situations. They revealed then a conscious mastery of this
additive situation and an ability to relate it or compare it to other situations (as
comparing increases of children by comparing increases between two dates, and
heights as increases since “birth”).
CONDITIONS THAT ALLOWED THE VYGOSTKIAN DIALECTICS TO
WORK
The analysis of the didactical situation put into evidence two main conditions that
favoured the setting of this dialectics and its functioning.
- The important use of the “reality”, the development of various expression about it
and the development of everyday concepts in school context. This comes from the
experience fields didactical setting, which implies making sure that students are
sufficiently involved in some activities related to this “reality”, and important
everyday concepts are explicitly shared, and enriched with verbal and other external
representation through observations and discussions. In fact, everyday concepts do
not quite correspond to spontaneous ones, though they remain in the everyday stream
of conceptualisation (cf Vygotsky, 1985, Chapt. 6).
- Argumentative activity managed by the teacher allowed students to interpret the
situation, express and discuss their views, handle and compare various (verbal,
schematic, numerical) external representations. It helped students to assimilate the
new construction, because they had to express themselves in a variety of ways to
answer to a variety of detailed questions. Argumentation played a dynamical role so
that the dialectic could evolve.
CONCLUSION
Vygotsky’s scientific concepts/everyday concepts dialectics works as a powerful
model for conceptualisation in teaching and learning contexts. But through
observation of examples such as the one reported here, I found that it can turn out to
be also an efficient tool to plan teaching situations in order to achieve some important
educational aims in mathematics. The "experience fields" didactical setting does use
this dialectics as one of the levers to develop conceptualisation, with some specific
favourable conditions:

- the development of everyday concepts in school context up to a certain level of
expression and activity in the perspective of a “mastered reality”, in parallel to the
building of scientific conceptualisation of related notions;
- the explicit rooting of mathematical concepts within systems that may contain
nonmathematical concepts, and by this way can support the students' mathematical
activity by providing them with meaningful references.
We can also note how, in the "experience fields" didactical setting, a continuous
development of argumentative skills allows to nourish the backing of mathematical
reasoning on other (and easier to master) kinds of reasoning. As a result,
argumentation can effectively move the everyday concepts /scientific dialectics under
the teacher's guide.
REFERENCES
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