Creative Education
2012. Vol.3, No.4, 400-405
Published Online August 2012 in SciRes (
Copyright © 2012 SciRe s .
A Cognitive Analysis When the Students Solve Problems
Elena Fabiola Ruiz Ledesma
Posgraduated College of Computational Studies, Institute Polytechnic National,
Mexico City, Mexico
Received May 21st, 2012; revised June 25th, 2012; accepted July 5th, 2012
The research reported in this paper shows an analysis of the cognitive process of students from the senior
technical program on food technology, whom are asked to solve a contextualized event on systems of lin-
ear algebraic equations within the context of balance of matter in situations of chemical mixtures. The
cognitive analysis is founded on the theories of Conceptual Fields and vents. For the analysis attention is
focused on the representations carried out by students regarding the invariants in the schemes that they
build when they face an event of contextualized mathematics. During the acting process of students
emerge different types of representation which are appropriate to the context in which the research de-
velops, with which a proposal for the classification of them.
Keywords: Mathematics; Conceptual Fields; Contextualize Events; Problem Solving
Te qualitative research allows the analysis of the cognitive
processes of students for the construction of knowledge of
various sciences who are linked. Addressing science events
such as a chemical or physical system or a natural phenomena,
where mathematics should be applied for their solution, is a
complex task; because on the one hand it is necessary to inte-
grate the knowledge and on the other hand carrying out activi-
ties of analysis for knowledge, which is both explicit and im-
plicit in the event to be addressed, playing an important role in
the representations which can be formulated about the events to
be dealt with, in order to promote the construction of knowl-
edge. Moreira (2002) assumes that we do not apprehend the
world directly, but we do it from the representations of the
world that we build in our minds.
Learning mathematics, among others, implies that the learner
is able to use them to solve specific events in their area of pro-
fessional and labor training (Ruiz, 2009). Thus, in the construc-
tion of knowledge of various related sciences, it is important to
know from the cognitive point of view what happens to stu-
dents when they work with contextualized mathematics.
Research Problem
There are many actions that favor the construction of knowl-
edge in students, among them stands the learning and teaching
activities with contextualized mathematics. The research that is
carried out regarding the cognitive process of students provides
a guideline for the design and redesign of these didactic activi-
ties in students at upper-level and technical programs. The re-
search reported makes use of the conceptual fields by Vergnaud,
Thus, the research problem deals with the issues of how the
cognitive processes of students carried out when they are faced
with contextualized events, from the perspective conceptual
fields bay Vergnaud.
The objective of the research seeks to analyze the cognitive
process of a group of students by means of analyzing the repre-
sentations they make about the invariants in the schemes they
build when they address the learning activities associated with
contextualized events where systems of algebraic equations are
linked to balance of matter; that is to say the events being ad-
dressed deal with mixing chemical solutions.
Theoretical Framework
Conceptual Fields
The theory of conceptual Fields deals with the formation of
mathematical concepts from a psychological and didactic ap-
proach, which leads us to consider the learning of a concept as
the set of “problem situations”, which constitute the reference
of their different properties and the set of schemes brought into
action by the subjects involved in those problem situations
(Vergnaud, 1991). The sense of the mathematical concept is
acquired through the schemes evocated by the individual sub-
ject to solve a problem situation.
Vergnaud (1996) defines Conceptual Fields as “a set of prob-
lem situations, concepts, invariants, schemes and operations of
thought that are related to each other for a specific area of
knowledge”. The theory of Conceptual Fields allows the cogni-
tive analysis in the problem situations proposed to the students
through the analysis of the conceptual difficulties, the repertoire
of available procedures and possible forms of representation.
From the perspective of the theory of Conceptual Fields un-
derstanding of the student’s action relative to the concepts in-
volved and the structure of systems of linear algebraic equa-
tions in the context of a phenomenon of balance of matter, fo-
cuses on studying some aspects of the operations of thought
which make the invariants in the schemes constructed by stu-
dents, which directly or indirectly impact on knowledge about
this mathematical structure being immersed in a problem situa-
tion where the linking of two contexts take place: mathematics
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).
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-
Copyright © 2012 SciRe s . 401
cepts such as management of concentrations, visualization of
the phenomenon as a system with inputs and outputs, mixing
substances to obtain a desired concentration and concentration
measurements. The schemes and operations of the invariants
mediate the action over reality, as well as the forms of organi-
zation and structuring of the different concepts of interest and
criteria for acquisition of meanings.
From the contextualization of the first methodological block
derive the activities to be determined in the second block,
which are shown in Table 1. Learning activities have the objec-
tive that the student analyzes the linear behavior by means of
the obtaining of combinations of concentrations and volumes,
each one separately, to later on confront him with the idea of
managing the two variables simultaneously, and the pursuit of
mathematical model the event in context, see Table 1.
Results and Discussion
With the above it can be established conceptual field of sys-
tems of linear algebraic equations in the context of balance of
matter. As it has been mentioned in the theoretical framework
section, the conceptual fields are constituted by the problem
situations (contextualized event and learning activities), the
concepts, the invariants, the schemes and the operations of
Results and Discussion
For the third block, which determines the results and discus-
sion, the behavior of students is analyzed through the opera-
tions of thought that they make of the invariants in the schemes
they build. It must be pointed out that during activities that
emerge from the contextualization of systems of linear alge-
braic equations with balance of matter, it was detected in the
actions of the students the recurrent use of different operations
of thought, which were considered proper of the linking of two
specific areas of knowledge, that is to say, under other circum-
stances they may or may not arise. These operations of thought
relate specifically to representations, for it was necessary to
define a classification of the types of representations found,
which is expressed below.
Table 1.
Contextualized event and its learning activities.
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 solu-
tion to 50%.
Activity 1. To mix the two chemical
solutions in order to obtain as a r esult 100 ml
of solution, do not consider the concentration
of the solution only the volume.
To mix chemical
solutions, from their
Activity 2. To mix the two chemical solutions
in order to ob t ain as a r esult a solution with a
sugar concentration to 50% (do not consider
the volume of the soluti on only the
To mix chemical
solutions, from their
Activity 3. With the given solutions, 100 ml to
35% and 100 ml to 60% carry out a mixture in
order to obtain 100 ml of a new solution with a
concentration to 50%.
To mix solutions
considering both volume
and concentration.
Propositional representation type: it is considered as the de-
scription of the event the made by the student, in the first in-
stance, in his own language (natural language), from the cogni-
tive point of view it is of great use to the student to understand
the activity to be performed. He can use some isolated concepts
of mathematics, the chemical science or other sciences, without
articulating them. The overall purpose is to communicate and
try to understand the event to be solved.
Non-operating figurative type representations: it is when the
student perceives the information of the event but if asked to
represent the situation in written form, he draws pictures,
wanting to see the link between the two areas of knowledge;
however, it does not allow him for a procedure of solution. For
the particular event that has been addressed in this research,
once the group of students understands the activities in their
language they proceed to make some drawings (figures) of the
area in context in which it is perceived that there is understand-
ing of what is asked. However, do not shift from these figures
to a written expression that enables the solution of the given
event, that is to say, they still fail in linking the two areas of
Figurative operating type representations: the student con-
tinues to make drawings but he already takes into account the
numerical information, the drawing can therefore, serve as a
support for the solution of the event without algorithms. Again,
the experiment presented in this document, the students recur-
rently makes schemes (figures) representing the balance of
matter of the proposed activities; which allow them on the one
hand to understand the variables and constants involved, and on
the other hand they serve to give way to the symbolic represen-
tation that allows the resolution of the event. Then, at this point
begins the comprehension of the link between the two areas of
Representations of analogous type: the student retains the
relevant information to solve the event and simplifies the in-
formation by using symbols, points, crosses, use of images. He
uses the experience of having solved previous events to solve
the new one, at least partially, by a very primitive process and
of very local scope, that is to say, the processes of similar
events are considered to verify if they are helpful in solving the
new event. He leans over understanding to resolve the event.
The analogous representation can be used after the proposi-
tional and even after the non-operating figurative one and it is
when the student refers to processes that appear similar to the
one he faces and that can be useful for understanding and reso-
lution of the event. His search for similar events allows him to
support the emerging understanding of the previous representa-
Symbolic-type representations: it has been considered that
these representations constitute a means to identify more clearly
the mathematical objects for the conceptualization of them and
the conceptualization of context. The student translates the
sentence of an event to a mathematical representation, arithme-
tic, algebraic, analytic or graphic, this representation is consid-
ered as an expert and allows providing the required response. It
is considered that knowledge is obtained that can be used in
different contexts. The group of students who have gone
through the previous representations, makes use of a symbolic
representation (it may be graphic, arithmetic or algebraic), the
same which allows them to integrate both mathematical
knowledge of the context to provide an appropriate solution to
the event posed. This representation, in the case of research, has
Copyright © 2012 SciRe s .
been considered the one that allows the proficiency of the con-
cepts of interest.
In general terms, it is observed that the different types of
representations of the invariants play an important role in the
resolution of the contextualized event and the learning activities
that have been posed to the students. The approach to learning
activities was characterized by because they commuted by dif-
ferent types of representation to until reaching a symbolic rep-
resentation and obtaining a satisfactory result. In the acting of
the students it was detected that when the group addresses the
learning activities and builds knowledge, the types of represen-
tation, found during the research and considered as being
proper to the linking of two sciences, are developed gradually
at the same pace as the conceptualization of systems of linear
algebraic equations in the context of mixing chemical sub-
stances. Additionally it enables the student to develop skills to
solve new contextualized events that they may be posed.
As it has been mentioned, the group of students moves
through the different types of representations until reaching the
symbolic one, which enables the students to construct their own
knowledge about the phenomenon of study that links mathe-
matical concepts and sciences. That is why, the generation of
symbolic representation was discussed in great depth, identify-
ing three related processes, which other types of representation
are present. Interpretation and selection process, Structuring
process and Operationalization process.
Interpretation and selection process, following the contextu-
alized event, where it is considered as a fundamental part in the
context, the group of students performs a selection of informa-
tion that seems relevant, in it the available prior knowledge is
considered and students translate the information into a type of
representation, such as algebraic (linear equations), arithmetic
(simple rule of three), tabular (tables of data) or graphics, even
in a figurative operating representation. See Figures 1 and 2.
Structuring process, in this one intervene again the knowl-
edge, we resort to analogies with events which have been pre-
viously solved and which appear to be similar to the pose done;
it seems that the previously solved events are stored in the
Figure 1.
Figurative operating representation (drawing).
Figure 2.
Symbolic representation (arithmetic).
memory and constitute a part of the knowledge brought into
play by the group of students, similarly, the procedures and
strategies which were followed undergo a restructuration and
advance progressively as attempts to solve the event are made.
See Table 2 and Figures 3.
Operationalization process, it takes place when representa-
tions are manipulated and the group of students attains the solu-
tion of the posed contextualized event. It is identified that dur-
ing this process the group of students applies the operating
knowledge that come from their experience and with that the
group formulates procedures or strategies. In this process the
previous knowledge of the group of students plays an important
role as they allow modeling the event and working with the
mathematical model by symbolic modes of representation that
are more operational than the propositional.
Following is shown the procedures used by the students in
the case showed in this paper.
Student 1: “It is required to make a solution of 100 milliliters
having a 50% of sugar starting mixing 2 solutions each one of
100 milliliters too, one of this solutions has a 60% of sugar and
the other a 35% of sugar.
Student 2: “There are several ways to obtain the 100 millili-
ters of the new solution as shown in the Figure 4”.
Student 1: If we take 10 milliliters of the first solution and
90 of the second one, how much sugar has the new solution?”
The student made arithmetic procedures as shown in Figure 5.
Table 2.
Tabular representation.
Milliliters of the solution at 35% Milliliters of the solution at 65%
50 50
90 10
10 90
40 60
60 40
30 70
70 30
20 80
80 20
Figure 3.
Structuring process.
Copyright © 2012 SciRe s . 403
Student 1: “Using arithmetic I can get the percentage of
sugar in the milliliters that each solution uses and adding I get
the total percentage of the third solution. I can fill a table.
See Figure 6.
Student 2: A 50% of sugar is obtained in the third solution
when we take 60 milliliters of the first solution and 40 millili-
ters in the second one.
Student 2: “It can also be solved using a equations system.
See Figure 7.
By way of conclusion, it can be observed from the research
that it has been sought to contribute to the understanding of the
construction of knowledge of the student in a conceptual con-
tent derived from the link between two different contexts,
mathematics and technical area of the working field of a Senior
Technical in Food Technology. The shown analysis has been
addressed paying attention to the action of the student in the
solution of contextualized events and teaching activities with
systems of linear algebraic equations in the context of balance
of matter, which constitute a means of analysis upon which the
construction of knowledge is described. This is a study of cog-
nitive type which uses as a theoretical framework the Concep-
tual Fields by Vergnaud 1996, initially developed to carry out
research in elementary education and that has been retaken to
explain a contextualized phenomenon at Senior Technical level.
During the analysis of the activities by the group of students
it was necessary to define the types of representations of the
invariants in the schemes, which emerged in a spontaneous way
during the performance of students. If students, at the end of the
sessions, have acquired appropriate schemes to face contextu-
alized events that require systems of linear algebraic situations,
it is necessary to further explore into it, above all in different
Another important element to be mentioned is that the more
Figure 4.
Student’s table.
Figure 5.
Arithmetic procedures of the student 1.
Figure 6.
Student’s table.
Figure 7.
Equations system made by student.
diverse contexts are used and the more contextualized events
are addressed by the student, he could be constructing his
knowledge in a more everlasting way and will be able to carry
out the transference of mathematical knowledge to other sci-
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