E. F. R. LEDESMA
and chemistry.
For the representations that are one of the operations of
thought, Vergnaud mentions that the student transforms an
action on the mathematical object, establishing control by him-
self through the relationships and classifications in his reality,
in such as way that invariants arise in the development of
knowledge, in this way the representations for Vergnaud are all
those tools, whichever notation or sign of symbols which rep-
resent something that typically some aspect of the of our world,
that is from our imagination.
Several research results show that student cognitive charac-
teristics seem to have a great effect on the strategies of how
they learn, navigate and search learning resources (Liu & Reed,
1994; Ford & Chen; 2000; Chen & Macredie, 2002). However,
there is little evidence showing that the differences in cognitive
characteristics have a great impact on learning performance.
Some research results show there is a positive relationship be-
tween cognitive styles and learning performance in hypermedia
environments (Andris, 1996; Parkinson & Redmond, 2002).
Chen and Macreedie (2002) point out that it is necessary with
the problem of learning and teaching of mathematics at univer-
sity level programs where mathematics is not a goal in itself but
a tool to support sciences and a subject of training for students
of them. To this end, the theory conceives the process of learn-
ing and teaching as a system where the five phases of the theory
intervene: curricular, cognitive, didactic, epistemological and
educational, moreover, factors of emotional, social, economic,
political and cultural kind appear as well. All the phases are
necessary to ensure the completion of the philosophical as-
sumption posed, moreover, all stages are interrelated among
them, and none of them is unrelated to the others. As a theory,
in each of the phases a theoretical methodology is included, in
accordance with the paradigms upon which it is based, which
serve as a guide for the steps of curricular design, the didactics
to be followed is described, the cognitive functioning of stu-
dents is explained and epistemological elements are provided
on mathematical knowledge relating to the activities of profes-
sionals, among others.
The cognitive analysis addressed in this research directly af-
fects the cognitive phase of the theory, where a contextualized
mathematical concept acquires sense by means of the activities
of the context itself, because the concepts are not isolated, they
are constituted as a network and they bear a relationships
among them. Therefore, for the cognitive analysis it is impor-
tant to set the contextualized events and define from them the
learning activities that lead to the construction of knowledge,
both of the concepts of each science involved in the event and
the linkages between them.
To finish with this section it is important to mention that the
“problem situations” referred to in the theoretical framework by
Vergnaud correspond to the “contextualized events” and the
“learning activities” proposed by the Contextualize Events (Liu
& Reed, 1994; Ford & Chen, 2000; Hen, Macredi, & Andris,
1996; Parkinson & Redmond, 2002).
Methodology
The methodology used to carry out the analysis of the cogni-
tive process of students around the conceptual content of sys-
tems of linear algebraic equations in the context of balance of
matter involves the following three blocks:
Contextualization of systems of linear algebraic equations
in the balance of matter.
Determination of the learning activities to be applied to the
group of students.
Analysis of the cognitive process of students through the
operations of thought that they make upon the invariants in
the schemes that they construct , constituting, this block, the
section for results and their disc ussion.
The Sample
Being an analysis of qualitative kind, we worked with a
group of two students in the first quarter of the Technician in
Food Technology program, who are currently enrolled in a
mathematics course that includes the topic of systems of linear
algebraic equations as well as a chemistry course which ad-
dresses the topic of balance of matter by mixing chemical solu-
tions, both courses are not related in a curricular way. The ac-
tivities are carried out by the students in the chemistry labora-
tory, in different sessions, covering a total of twelve hours.
Observation Instruments
The data collection from the cognitive process is made by
obtaining written and film productions that help to refute or
confirm the analysis conducted with the written information.
The analysis is qualitative focusing on the operations of thought
that they perform on the invariant patterns that build in their
cognitive processes in the activities of contextualized event.
Development of Research
For the first block the contextualized event students are to
face is contextualized, which is a phenomenon that recurrently
appears in specific operations in the area of professional and
labor training of the technician in food. It says: We have 100 ml
of sugar solution to 60% and 100 ml of sugar solution to 35%.
From these solutions it is desired to obtain 100 ml of a sugar
solution to 50%.
The contextualization stages described in the section of the-
oretical framework, which are immersed in the didactic strategy
of Mathematics in Context, is the methodological process that
is used for the contextualization of systems of linear algebraic
equations in the balance of matter, generally in contextualized
events of mixing of solutions.
In the didactic phase of Mathematics within the Context of
Sciences, contextualization is set by the teacher prior to imple-
menting the didactic strategy of mathematics in context, be-
cause this contextualizing provides him with elements for the
design of learning activities. Similarly, to take time, see the
necessities of the cognitive infrastructure and consider the pos-
sible paths of solution. Also, it allows determining the mathe-
matical concepts and of the context (chemistry) that are present
in the event, establishing the relationship between concepts that
belong to two different areas of knowledge and observing the
close relationship between them. In terms of Vergnaud, the
concepts and invariants to which the student must converge for
the construction of knowledge are identified. In relation to
mathematics: The concept is “systems of linear algebraic equa-
tions” and the invariants are mathematical concepts such as
algebraic equation, linear algebraic equation, systems of equa-
tions and solution methods. In relation to the context: The con-
cept is balance of matter and the invariants are inherent con-
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