Concept mapping: important elements for intervention.

Flores, Rafael Perez

Abstract

The aim of this paper is to present the results of an experiment carried out in the Mexican Autonomous Metropolitan University (UAM) based on recent theories of educational research on building up knowledge. The research group was studying "Calculus 1" following the Programme of Intervention, "SAM" (Mediated Learning System), an innovative teaching project in mathematics. The Programme of Intervention, "SAM", is based on the theories of the cognitive paradigm, and an important element of such Programme is concept mapping. The Programme considers that through the use of concept mapping in the study of mathematics, cognitive abilities can be acquired leading to an improvement in the intelligence coefficient of the students. Through the Programme the teacher is the agent between the curriculum contents and the students. He designs the classroom presentation and the concept mapping according to how the student learns. The teacher considers that concept mapping is an important element to develop skills.

Introduction

In the education context, the cognitive paradigm considers that education must be oriented towards the achievement of meaningful learning (it must make sense) and towards the development of general and specific strategic abilities of learning. Teaching, concretely in the classroom, from the point of view of this paradigm, must allow for the learning of the contents of the curriculum in the most meaningful way. This has the implication that planning and organization of the didactic processes are vital for the creation of the minimum conditions for meaningful learning.

In addition, with regard to teaching within the framework of the cognitive paradigm, the teacher must start from the idea that the student is active and can learn meaningfully, that he can learn to learn and to think. The teacher must concentrate on the task of the combination and organization of didactic experiences in order to achieve learning.

In order to bring information that will contribute in some way to aspects related to the teaching and learning of mathematics, the initiative arose of undertaking research concentrating on a Programme of Intervention designed "SAM" (Mediated Learning System). Such a programme-a special way of acting in the classroom-would consider mathematic contents as a vehicle to develop awareness in the student, understanding cognitive development as the development of a collection of cognitive skills. One of the important elements of the Programme "SAM" is concept mapping.

Programme "SAM"

The Programme "SAM" takes into consideration the thought processes both of the teacher and the student; processes proper to the analysis of the cognitive paradigm. The program is based on Ausubel's theory of meaningful learning (Ausubel, Novak & Hanesian, 1988). It is a Programme that follows the model Learning-Teaching (how the one who is learning learns, based on which, it is possible to design the teaching) and serves as a tool to achieve student learning where the teacher acts as a mediator for the learning.

The Programme, on one hand, considers the abilities and cognitive skills as objectives, and on the other, mathematical contents and method are considered as a means to achieving these objectives. From this point of view, the Programme orients the teaching towards cognitive development, which is why it is considered as a Programme of cognitive intervention. The Programme has objectives by skills: Develop skills of induction and deduction considered as a part of a capacity for logical reasoning and develop the skills of situating, locating and expressing graphically as part of the capacity for spatial orientation.

The Programme "SAM" achieves its objectives when the teacher uses concept mapping as a support material during action in the classroom. The teacher has to arrange and organize the contents of the subject in order to facilitate learning. Concept mapping represents a new construction of material.

How the Program Works

The Programme is a particular way of performing in the classroom at university level. The ideas about this performance may be adapted for other levels and other subjects. What kind of work does the professor need to perform before initiating the course? The professor needs to analyze the material (the content of the material) to arrange it in conceptual maps. The conceptual maps that the professor makes are a guide for his performance with the students in the classroom. The above doesn't mean the activity in the classroom is meant to present the maps that the professor makes. Particular information, images and concepts are found in the conceptual maps. Figures 1 and 2 present examples of conceptual maps that a professor has made for a calculus course for future engineers. For a class session, the professor has prepared his conceptual map titled Real Numbers and the information within the map will guide him.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The professor always has to prompt the first stage of the learning: the perception. Thus, information attainable to the students' intellect has to be presented. It is at this moment that the professor must consider the particular information in his map. He may begin by presenting information that the students already know. This can be the numbers 1, 2, 3, 4, etc. Immediately after that a drawing, a scheme or a graphic must be prepared to help the mental representation. He may mention that a line is commonly used to locate the numbers. Afterwards the numbers such as 1, 2, 3, 4, etc. can be presented and located on the line as is shown in Figure 3.

[FIGURE 3 OMITTED]

Once certain information has been perceived and has contributed to the mental representations through drawings, two concepts may be presented. The concept of natural numbers N = {1,2,3,..., n, n +1,...} may be introduced and the concept of integer numbers as well Z = {..., -(n+l), -n,..., -3,-2,-1,0,1,2,3,...,n,n+1}. In this way one can reach the third stage of learning: conceptualization.

Subsequently, it is important to present and to locate on the line numbers such as 5/2, 1/2, -5/3, 1/3, etc. An explanation can now be given so that the student realizes that natural numbers are a part of integers and that integers are a part of those numbers which have just been located on the line as in Figure 3b.

Once the above has been analyzed the concept of rational numbers

Q = {[p/q]|p [member of] Z & q [member of] N} may be given which is supposed as in Figure 3c.

Before reaching the concept of real numbers, information about other types of numbers such as [square root of 2] or [pi] must be provided. It has to be pointed out that these numbers are not rational numbers; therefore, they are neither integers nor natural numbers. Afterwards they must be located on the line to help develop their mental representation. For the case of numbers such as [square root of 2] one must not forget that its location on the line can be obtained through geometrical procedures, by building a rectangular triangle as it is shown in Figure 3d.

Once the characteristics of these numbers have been explained the term

or the concept of irrational number may be introduced. In this way the soil has been prepared for a concept with a degree of larger generality: the concept of real number. By proceeding in this way we get to the following R = Q [union] I.

This way of proceeding from the particular to the general (natural to real numbers) initiates the starting point of a mental inductive process within the students, and the concepts are grounded in an appropriate way within the cognitive structure.

Unlike the other didactic strategies, one particular characteristic of the Programme is respect for the order of the basic stages of learning: starting from perception of particular information attainable to the student's intellect, and continuing with representations which aid the mental representation, and finally getting to the concept.

It is very common not to consider the above and to initiate the session by giving a concept. It is usually simply stated the real numbers are comprised of rational and irrational numbers. It is very common to start from such a generality. The student is then compelled to perform a deductive process without a previously existing inductive process, which encourages the memorization of the information and results in weak grounding of the concept within the cognitive structure.

Another characteristic of the Programme is the role of the conceptual maps the teacher has made. The elaboration of conceptual maps guides the performance in the classroom that respects the basic stages of learning and thus develops the cognitive skills of the students. That is why the maps are important tools for the cognitive intervention. The Programme of Intervention, supports the role of the teacher as mediator (Feuerstein, 1980; Nickerson, Perkins & Smith, 1994) and has the following characteristics. The teacher of the Programme analyzes in detail the contents of the subject. He selects, orders, frames and situates it in space and time. He groups certain information. He gives specific meaning to certain information. He directs the class dynamics so that certain information appears in context on various occasions. He doles out to the student methods of selecting, focusing and grouping, first, the information perceived, and later, the concepts elaborated.

Another example is the introduction of the concept of the line (Figure 4). In the upper part of the map, the most general concept, that is to say, the line is found. This map tells the teacher that he has to initiate by showing lines within the Cartesian plane which are located in different ways as is shown in Figure 5a.

[FIGURE 4 OMITTED]

Note that within the Line map (Figure 4) in the lower part, images similar to those in Figure 5a are found. This illustrates that the professor may if it is considered appropriate, include in the classes more information related to the information appearing within the map. Subsequently images similar to those in Figure 5b may be presented, but they will then include points, and represent the joining of two or more points to form a line.

By proceeding in this way, the instructor starts with the perception of particular information and continues with mental representation supported by drawings until reaching the concept of interest. The teacher has to consider that images and mental representations or graphics possess a high potential for conversion into concepts by means of the student's understanding of them (De Guzman, 1996). Before attempting the line concept, the concept of slope must be introduced in a simple way, and with the clear algebraical movements that the professor introduces in the classroom the concept of equation y=mx+b can be reached. In this way the concept was obtained in an inductive way allowing comprehension and avoiding memorization. Further learning of the concept of slope can be guided by the concept map in Figure 6.

[FIGURE 5 OMITTED]

If we do a simple reading of the conceptual map, the Line (Figure 4), we will probably do it downwards; in this way we are starting from the general to the particular. But this progression from the general to the particular is not the appropriate way to teach the concept to students. That is why the conceptual map that the professor elaborates guides him to know where he must start (from the information which is found below) to where he has to get (all the way to concepts which are found above). There must be a lot of insistence on the following: the class sessions are not meant to show the student the map the professor has elaborated. The role of the map is to guide the professor's classroom practice along the lines of the inductive process. Not all calculus classes are devoted to the learning of concepts. The exercises and problems are also part of the content that has to be dealt with in class, and the student has to develop skills to work out exercises and problems. Let us consider when an expression such as 5x - 4 > - 7 + 6x has to be worked out. This task can be worked out in a mechanical way without taking awareness that the left part of the mathematical expression is a line and that the right part is also a line. The Programme takes into consideration at all times that for each exercise or mathematical expression a graphic representation exists which facilitates the mental representation and aids the imagination. The professor of the Programme performs in the classroom showing that the mathematical expressions may be reduced to obtain the particular information which represents the solution. An expression such as 5x - 4 > - 7 + 6x, for which a mental representation is not obtained in a simple way, can be reduced (with an adequate algebraic development) to the following expression -x + 3 >.0. In many cases these algebraic simplification strategies are memorized as a set of stages to find the solution and they are not always used successfully. The professor of the Programme insists on the graphic representation of mathematical expressions to aid the mental representation. One feature of the Programme is that the professor insists on going from the generalities (exercises and problems) to the peculiarities (solutions), supporting himself with graphic representations to set the starting point of deductive thought processes. In this way the student develops skills to solve problems and is able to confront complex tasks. In general terms, the professor aids the students in carrying out inductive and deductive mental activities for the development of cognition.

[FIGURE 6 OMITTED]

The professor considers in each session the basic stages of learning: perception, representation and concept visualization (Roman & Diez, 1988). During the classes the deductive process is always carried out after the inductive process. The professor works with the student to practice inductive thinking by progressing in an orderly manner from perception when he begins with the general and advances towards the particular, when he follows concepts with facts, or when he relates a system of symbols to particular cases.

By analyzing the exercises and the problems that the student works, the professor is able to figure out if the inductive and deductive processes are carried out correctly, that is to say, if the student has developed the requisite logical reasoning. By observing the work of the students, the professor can determine whether they are able to follow facts with concepts and concepts with facts appropriately, and whether the mental representation supports the students during the accomplishment of the tasks.

Research Methodology

A design factor 2 x 2 was used taking into consideration a factor of independent measurements with two values or levels and a factor of repeated measurements also with two values. The independent measurement is the treatment whereby the levels are experimental groups with treatment and control group without treatment. The repeated measurement is formed by phases of application with two levels: pretest (grading before beginning training) and posttest (grading once the training is completed).

The following is the description of the methods employed. A group of students was selected who were given the pretest examinations (Cattell and Raven Test) which measures general intelligence. When the information was analyzed two groups were formed: the control group and the experimental group. Before starting the training, analyses were carried out (applying the tests Student's and Wilcoxon's Ranking) which checked that the experimental and control groups were homogenous. There were no statistically significant differences in general intelligence at the time of the pretest.

An initial knowledge examination was given to the experimental group before beginning the course. The control group began an ordinary "Calculus I" course and the experimental group began the same subject but with the Programme of Intervention.

The Programme "SAM" was designed to be applied taking into consideration the characteristics of the programming of the Universidad Autonoma Metropolitana (UAM) (Autonomous Metropolitan University) of Mexico City. The Programme lasted three months during which there were a determined number of sessions that were obligatory. In addition to these sessions, the Programme (training) posttest examinations were given to all students, in both the experimental and control groups.

Later an analysis was made of the data obtained. The following tests were applied: Student's t (Parametric test) and Wilcoxon's Ranking (Non parametric test). The aim of these analyses was to evaluate the differences in the results obtained in the pretest phase and the posttest phase, between the experimental and control group.

The following hypotheses were applied. If an experimental group of university students were given a course of "Calculus I" applying the Programme of Intervention "SAM" and the results are compared with those from another control group with homogeneous characteristics given the same course but without the Programme of Intervention:

(1) a significantly greater increase in the general intelligence-measured by the tests "Raven's Progressive Matrices"--will be observed in the subjects of the experimental group than in those of the control group.

(2) a significantly greater increase in general intelligence-measured by the "g" Factor Test- will be observed in the subjects of the experimental group than in those of the control group.

Results

The instructions used for qualitative records were a diary of each of the students, a diary of the mediator (professor) and partial evaluations (periodic classroom assessments). The information gathered from the students revealed qualitative aspects of the academic advances. These pertained to both the content of the subject and the development of abilities and skills. An analysis of all the information obtained in this way concluded: The students mentioned the emphasis the Programme had had on the basic stages of learning and the activities had helped the learning process. The students recognized the importance of perception and representation for the start of the inductive process. They appreciated the importance of the inductive process for the study of mathematics. The students spoke of one of the fundamental elements of the Programme: they referred to the fact that they had understood the theory and they had not simply memorized it. The students perceived the conduct of the teacher as mediator as favorable to learning. The favorable commentaries expressed by the students were reflected in the results of the partial evaluations (periodic class assessments). At the end of the Programme of Intervention 80% of the students passed the course.

For the quantitative register the following instruments were used: Raven's Progressive Matrices Test and Cattell's "g" Factor Test. The results obtained verified the hypothesis of the research. After processing the information from Raven's Test for the control and experimental groups with statistical models, significant differences could be observed only in the experimental group at a level of reliability of 99% between the pretest and posttest. A significant progression could be discerned in the general intelligence of the experimental group. This increase is mainly explained by the effect of the Programme of Intervention "SAM" applied to this group.

In the same way, the processing of the information of Cattell's "g" Factor Test revealed differences statistically at the level of reliability of 99% between the pretest and posttest. A significant progression could be discerned in the general intelligence of the experimental group. This increase is mainly explained by the effect of the Programme of Intervention applied to this group. Figure 7 shows the interaction of the factors.

[FIGURE 7 OMITTED]

The conclusion reached shows that a group of university students (experimental group) were given a course of "Calculus I" applying the Programme of Intervention "SAM" and the results were compared with those obtained by another group (control group) with homogenous characteristics who studied the same subject but without applying the Programme of Intervention. A significantly greater increase in general intelligence as measured by Raven's Progressive Matrices Test and by the "g" Factor Test was found in the subjects of the experimental group versus those of the control group.

Conclusions of the Research

From the information presented for the fundamental hypothesis of this research it may be concluded that the Programme of Intervention achieved its objectives: it developed the students' intelligence understood as a collection of abilities that in its turn is formed by a set of skills.

With the application of the Programme of Intervention it is possible to develop "logical reasoning" understood as a capacity with the skills of "induction" and "deduction." It is also possible to develop "spatial orientation" understood as a capacity formed by the skills "situate and locate" and "express graphically."

For the Programme "SAM," the learning of mathematics is understood as a development of abilities and skills. The statistical information leads to the conclusion that the Programme "SAM" offers a significant increase in the intelligence of the students. On the other hand the contributions of the students about what they have learnt offers evidence of the development of awareness.

To sum up: the Programme of Intervention "SAM" applied to the subject "Calculus I" developed abilities and skills as evidenced by the application of the tests. It is clear the Programme "SAM" improved learning of the mathematical content.

Finally, continuing research is needed in the universities in regard to the learning-teaching processes that combine technical elements (contents) and human elements (students and teachers) to contribute to the enrichment of pedagogical knowledge. Future research should also address aspects such as motivation, values and attitudes. It is also important to include in future research the analysis of the transfer to daily life of the intellectual improvements in the students.

References

Ausubel, D. P., Novak, J. D. & Hanesian, H. (1988). Psicologia de la educacion. Mexico: Trillas.

De Guzman, M. (1996). El rincon de la pizarra. Madrid: Piramide.

Feuerstein, R. (1980). Instrument e'nrichment. An intervention program for the cognitive modifiability. Baltimore: Univ. Press.

Nickerson, R. S., Perkins, D. N., & Smith, E. (1994). Ensenar a pensar. Barcelona: Paidos, 2a ed.

Roman, M., & Diez, E. (1988). Inteligencia y potencial de aprendizaje. Madrid: Cincel.

Rafael Perez Flores

Universidad Autonoma Metropoliatana--Mexico