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-

Copyright © 2012 SciRe s . 401