Summary
The teaching of Mathematics takes on a new importance every day for what it means in all spheres of economic and social development, it must contribute to the development of the student with a vision of the world that favors the formation of creative, productive and scientist and achieve skills development that allows him to successfully face his future working life. In the present work, the study of one of the fundamental mathematical skills is performed: calculating, determining those elements that affect its development and proposing an alternative solution from a methodological perspective.
Summary
The teaching of the Mathematical cobra every day a new importance for what means in all the spheres of the economic and social development, should contribute to that the student is developed with a vision of the world that favors him the formation of a creative, productive and scientific thought and that I / you / he / she achieves a development of abilities that you / they allow him to face with success her future labor life. Presently work is carried out the study of one of the fundamental mathematical abilities: to calculate, determining those elements that impact in their development and proposing a solution alternative from a methodological perspective.
Introduction
The training of a broad profile professional requires that the graduate master the bases of scientific knowledge and develop skills to face their future working life. To achieve this goal, mathematical training in the student plays a central role, since, among other things, “(…) mathematics represents the most general and efficient gnoseological and methodological instrument in the investigation of the phenomena of any science, including the social sciences. Mathematical thinking, modeling, creative, heuristic thinking, is spreading more and more, becoming the characteristic thought of the man of science in general ”(Hernández, 2008).
For the Technical Sciences careers, in which Mathematics is an essential work tool, the graduate must acquire a culture that implies the understanding of this science from the point of view of its development and historicity, its method, the relationship of this with Computing and the ability to creatively apply knowledge to technical problems and technological processes of his specialty.
The University of Medical Sciences, of Ciego de Ávila, has, among others, the high responsibility of training graduates in health technology, a career that requires putting the achievements of science and technology at the service of the construction branch, which is why demand for the preparation of future graduates with high theoretical knowledge, habits and professional and investigative skills and highly committed to the Cuban revolutionary process. This career requires, to a great extent, a correct mathematical training, which is why the improvement of its teaching becomes of singular importance,For this, it must be restructured in such a way that it becomes the means through which representations are formed to solve their scientific tasks, being a trained professional in correspondence with the current demands of the present and future times.
Applied Mathematics is a fundamental discipline within the basic cycle of the career, in which it is necessary to work based on the students mastering, among others, the main methods of Differential Calculus, Integral Calculus and Analytical Geometry. Their study constitutes the basis of logical and algorithmic thinking. Applying them creates the habit of expanding their knowledge by themselves so that they can carry out the mathematical analysis of practical tasks specific to their specialty.
However, regardless of the intentions that have been present at the time of carrying out the improvement of the programs of
Mathematics in the different study plans, there are still a series of difficulties in the teaching process of this science, so that its learning in most cases is rote, reproductive, which does not allow reaching productive and creative levels of assimilation desired.
From the authors' point of view, there are still difficulties in the curriculum, including:
- The application of the director programs and the integration of the academic, labor and research components are not yet desired. The work carried out in the application of the study programs does not guarantee adequate quality interdisciplinary work based on skills development..The level of knowledge and development of basic skills at the end of a subject or discipline is not adequate.
In the program of the Differential and Integral Calculus I course for this career there is a group of skills of great importance for the future professional, but in an investigation such as the one presented, not all of them can be addressed, hence, it was decided to investigate about the ability to calculate because it has the highest incidence within the program.
In the Differential and Integral Calculus I course for the first year of Health Technologies, it is placed in relation to what was previously expressed, that of “calculating derivatives of scalar and vector functions of a real variable”. For its achievement it is necessary that the students who enter the race, from the previous level, have developed a group of related skills that are a necessary condition for the achievement of what is expressed in the content of the program, taking into account the sequential nature of Mathematics. Despite the fact that they work with them from primary education, even upon their arrival at the higher level, difficulties have continued to be detected in terms of their development, implying that their teaching is resumed at the university but in a new dimension, not as a repetition of procedures or content,but from a different perspective taking advantage of the potential offered by the content and the profile of the race.
It was logical to ask what difficulties in developing the ability to “calculate derivatives of scalar and vector functions of a real variable” present the students of the Health Technology degree program at the University of Medical Sciences of Ciego de Ávila, at finish the Differential and Integral Calculus I course, which does not allow you to fulfill the referred objective of the program?
In this work we propose to defend the following ideas:
- In the development and at the end of the Differential and Integral Calculus I course, students have difficulties when calculating derivatives of scalar and vector functions of a real variable, which hinders the achievement of the objective related to the generalization of calculus according to the demands of the program. Exercise does not take advantage of the potential of the content to achieve its development.
The objective was framed in determining the difficulties presented by students of Health Technologies, of the University of Medical Sciences, of Ciego de Ávila in relation to the ability to “calculate derivatives of scalar and vector functions of a real variable”, taking as referring to those who are receiving the subject (first year students) and those who have already completed it and are in the second year.
To comply with it, the following tasks were proposed:
- Characterization and diagnosis of the students who are receiving the subject and those who have already completed it, based on the ability to “calculate derivatives of scalar and vector functions of a real variable” Bibliographic review of procedures and development of intellectual and professional skills. normative documents, discipline program and subject used in the career. Analysis of adaptations to current programs to achieve the development of basic skills through their methods, procedures and tasks. Assessment of the current exercise system for the subject and adaptation according to to your goal system.
The object of the investigation is the teaching-learning process of the Differential and Integral Calculus I subject in the Health Technology career and its field the exercise of Mathematics for the development of the ability to “calculate derivatives of scalar and vector functions of functions of a real variable ”.
The following methods and techniques were used:
- Observation (mainly to classes to analyze the techniques and procedures used by teachers of the subject attending to the development of the ability "to calculate derivatives of scalar and vector functions of a real variable") (See annex 5). Test students who are receiving the subject and those who have already taken it and are in the second year of their degree, to learn about the development of the ability to “calculate derivatives of scalar and vector functions of a real variable” (See Annex 1). Interview with Mathematics teachers of the higher education with the aim of inquiring about the problems that arise in the development of this skill, as well as students to assess their criteria on these deficiencies (See Annex 3).Questionnaire for teachers to locate students at different levels of skill development (See Annex 4). For data processing, elements of descriptive statistics were used. The entire population of first and second year students was taken of the career.
The scientific novelty is given in that a study is carried out that allows to perfect the teaching-learning process of the Differential and Integral Calculus I subject through exercise, to achieve the development of a specific skill; system that can be re-created and generalized to the development of other career skills. In addition, a characterization of the problems presented by first-year students of health technologies is carried out, taking into account the ability to calculate.
Practical utility, as a perspective of later stages of research, will allow the development of exercise based on the particularities of the development of a specific skill in the conditions of a university career such as health technologies.
The research that is presented is of the descriptive-causal type, foreseeing for future stages the development of the formative experiment with the aim of validating in practice the theoretical development here based.
Development
When carrying out the analysis of the state planning documents for said process; as well as others of a more general nature that have an impact on this, (Curriculum for the Health Technology career, Program of the Applied Mathematics discipline and Program of the subject Differential and Integral Calculus I) were obtained as regularities that despite Paying attention to matters related to mathematical training based on social assignment, skills development and learning evaluation, among others, do not adequately explain the scope and extent to which they should materialize.
In particular, the training and development of skills in man to a maximum of possibilities is a problem that is currently the focus of attention, as a consequence of the accelerated development of science and technology, and especially, a great challenge for workers. of Health Education, for its responsibility in the training of new generations. The concept has had various interpretations and treatments by Psychology and Didactics. From the first, it refers primarily to provisions favorable to action. Physiological, social and hereditary factors enter into its determination (Universal Illustrated Encyclopedia. 1925) and from Didactics it is a component of the content that has to do with the way of acting.
Many authors have treated it exposing their criteria, reflections, points of view, etc. Among them, Carlos M. Álvarez de Zayas states “The ability as an action that is can be decomposed into operation. While the ability is linked to the intention, the operation does so with the conditions, in such a way that in each ability the links of the same or operations whose integration allows the dominance by man of a mode of action can be determined "(Álvarez, C. 1989). Héctor Brito defines it as: “that particular executing psychological formation constituted by the system of dominated operations that guarantees the execution of the subject under conscious control” (Brito, H., 1999).
As a summary, the following generalizations can be reached:
- It is a broad and complex psycho-pedagogical concept. It is only formed and developed in the process of carrying out the activity. It contains a system of operations that must be dominated by the subject. It contains a mode of action that is general, depending on the level of systematicity in question.
H. Brito's definition clearly indicates the logical sequence of skill formation, while mental activity is put into practice through a totally individual route, where a system of operations is developed with full awareness of the individual. These criteria are compatible with the treatment given to this concept in the teaching of mathematics, which is why the position adopted in this work corresponds to its general criteria.
Among the mathematical skills to develop in this engineer, calculation stands out, which belongs to the system of specific skills for this science formulated by Hernández, H. 2006, pp. 24 to 26. From the mathematical point of view there are different definitions, for example, "… calculate is the transformation of a set of real numbers, related by operations, into a number, by applying supposedly known algorithms" (Espinosa G., José R. 2007).
The previous definition could be accepted if the calculation were not restricted only to operations with real numbers, since it is known that it can be extended. This is how it is understood by calculating: “an existential form of an algorithm that can be carried out manually, mentally, orally, in writing and by means of tables or computational means. This always presupposes explicitly or implicitly the ability to algorithmize; that is, to propose a strict sequence of mathematical operations that describe a procedure that leads to the solution of a certain exercise or problem.
For the Health Technologies career, Applied Mathematics is an essential work tool and the knowledge system of the subjects of the career offers the possibility of working with the skill from different perspectives in which it can be maximized of possibilities.
For example, in the topic “Differential Calculus of real functions of a real variable”, which covers approximately 80% of the subject Mathematics I, different contents are studied that are used not only in it, but also in others that they receive. The following table shows specific examples of certain contents of some subjects of the degree program in which you use Differential Calculus to carry out demonstrations, formula deductions and / or as a calculation tool:
| Subject | Cycle | Content |
| Physical | Basic | Mechanics (Newton's laws, kinematics, work and energy, wave motion). |
| Biostatistics | Of the exercise of the Profession | Measurement calculation for quantitative and qualitative data. |
| Invest Methodology And Statistics | Of the exercise of the Profession | Calculate the Mean, the Mode, the Median, the variance, standard and standard deviation. |
The mastery of the skill is recognized when a degree of systematization has been achieved in the execution of the action that leads to the level of mastery of the system of essential, necessary and sufficient operations called functional invariants of the action.
Based on the reviewed literature, in particular the theoretical analysis developed by M. Rodríguez and R. Bermúdez in their book "The Personality of the Adolescent" and the authors' experience, they are proposed as functional invariants of the ability "to calculate derivatives of functions scalar or vector real variables "the following:
- Identify the solution path of the exercise taking into account the algebraic structure of the function. Select the necessary calculation rules. Apply the calculation rules.
For example, to calculate the derivative of a function at a point, the student must:
- Identify the type of function to be derived: scalar or vector, single or multiple variables, explicit or implicit, simple or compound.Select the appropriate rule to calculate the derivative: immediate drifts, derivation rules.Calculate the derivative using the rule. selected and perform algebraic operations to simplify the result as much as possible.
In order to evaluate the executing aspects of the action with a view to diagnosing the presence of the skill, it can be specified in a technique of evaluative scale, this must be as objective as possible and for its preparation, the phenomenon under study must be concretely modeled (in this case the skill), formulate the objectives that are pursued accurately and determine the indicators of the existence of the phenomenon through external manifestations of the student during the performance of the action through the functional invariants of the skill. The scale to use depends on the objectives of the researcher and the knowledge they have about the instrumentation they are studying.
Based on the previous foundation, the indicators and the scale (composed of four levels of depth) were defined:
Indicators to evaluate the execution of the ability to calculate derivatives:
- Select the rule (s) according to the algebraic structure of the function. Apply the rule (s). Operate.
Scale by levels according to the indicators:
| INDICATOR: | DEPTH LEVELS | ||||
| FIRST | SECOND | THIRD | FOURTH | ||
| one | The function is simple and presents an algebraic operation. | The function is simple and features more than one algebraic operation. | Simple and compound functions appear with an algebraic operation. | Simple and compound functions appear with more than one algebraic operation. | |
| two | Make up to two mistakes. | Make a mistake. | Make a mistake. | Make no mistakes. | |
| 3 | Make up to two mistakes. | Make a mistake. | Make no mistakes. | Make no mistakes. |
To fulfill the objective, various instruments were developed and applied that made it possible to scientifically demonstrate the ideas to be defended in this work, the results of which are detailed below:
Test applied to students (Annex 1)
They were made taking into account the scale by levels according to the indicators previously expressed. (It is appropriate to clarify that the exercises proposed at each level of development of this skill are considered within the average in terms of the degree of difficulty).
Analysis of the results:
Based on the results obtained in the different years (See Annex 2) it could be specified that:
Among the students who are in the first year, most of them are categorized in the first level, although 15.79% do not even reach this level. Most of the second-year students reach the third and fourth levels, although 38.9% are between the first and second levels and 13.9% have not reached the first level.
In general, it can be seen that in each year the pass rates decrease considerably from the first to the fourth level, and how there are students who have not even reached the first level, an aspect that makes the situation even more critical. There are no significant differences between the two years, although there are more favorable results in the second year, a logical matter because they have more systematized this type of calculation.
To inquire about the causes that cause this anomaly, an interview was conducted with teachers (Annex 3) from which it was found that these calculations are systematized throughout the course, but are not integrated with the different levels of depth, and although they allege Having a real diagnosis of your students, this does not correspond to reality, which could be verified in the applied survey. (See annex 4).
On the other hand, eight classes were observed; in five of them the potentialities of the content are not used in the projection of the strategy to solve the difficulties presented in these calculations, while in the others it is not used to the maximum. (See annex 5).
In addition, student notebooks were observed in which it was found that there are no varied exercises where the different levels of depth of these calculations can be exercised and that they work with functions that present little complexity in their algebraic structure for periods of time. excessively long, which violates the principles of the systematization of knowledge.
With the aforementioned, it can be argued that in essence the skill is not mastered because:
- A correct diagnosis is not made in the different moments that require it. The strategy that is drawn does not take into account the sequential nature of Mathematics. When treating the different contents, the potentialities that they provide are not fully exploited., methodological and content in the exercise is not adequate.
The exercise class for the development of the ability to calculate
The exercise class is of vital importance to achieve the development of skills in the teaching of Mathematics in which: the potentialities of the knowledge system must be exploited based on the development of skills, taking into account the different levels of assimilation by which the student must go through the execution of the action, the exercises must be generalizing to be able to face various problems of the professional reality, must contribute to the motivation of the students, there is a logical derivation that allows linking the student with specific problems of their field of action and spheres of action, evaluation and control should be carried out from a perspective that makes the student aware of the processes that take place in the execution of the action.
The elements previously expressed allowed to elaborate a system of standard exercises that are being used in the exercise in classes and extra classes, with their respective indications and methodological recommendations that contribute to the development of the ability to “calculate derivatives of scalar and vector functions of real functions of a real variable. "
When making a partial cut in the investigation, an adequate quantitative and qualitative advance was verified in the students regarding the mastery of the skill, in which the amount they have promoted at a level according to the value scale created is considerable. (See Annex 6)
Conclusions
- The ability to calculate occupies an important place in the program of the course Differential and Integral Calculus I for the health technologist, due to the impact that it has on the professional sphere of the future graduate. Functional invariants were established, indicators were established and created a scale that allowed to evaluate the ability to calculate. Difficulties in the development of the ability to calculate were evident in the first-year students who are taking the subject and in the second-year students who already took it. The exercise class becomes a conducive framework for skill development.
Bibliographic references
- Álvarez de Zayas, Carlos M. 1989. Theoretical foundations of the management of the Teaching Process in Cuban Higher Education. Havana. Cuba. P. 72 Álvarez, C. and others. 1983. On the system of skills in a university specialty. Cuban Physics Magazine. Vol. III. No. 1.Bermúdez, R. and Rodríguez, M. 1996. learning theory and methodology. Editorial Pueblo and Education. Havana City.Brito, H. 1988. Habits and Skills and Capacities. Varona Magazine. Havana: 13: VI, Jul.-Dec. Universal Illustrated Encyclopedia. 1925. Barcelona. Editors Sons of Espasa. Volume XXVII, pp. 447-448) González, José R. 2007. Thesis of degree. Hernández, H. The trace of mathematics in thought. 2008. MONTH, p. 2 (Material on magnetic support).
Annexes:
Appendix 1
Student test:
Calculate the indicated derivative in each case:
Student test
Note: Exercises a) to d) correspond to the first to the fourth level, respectively.
Appendix 2
Behavior of the “calculate derivatives of scalar and vector functions of real functions of a real variable” ability:
| LEVEL | First year | Second year | ||
| Quantity | % | Quantity | % | |
| Less than the first level | 6 | 15.79 | 5 | 13.9 |
| First level | 13 | 34.21 | 6 | 16.7 |
| Second level | 8 | 21.05 | 8 | 22.2 |
| Third level | 5 | 13.16 | 8 | 22.2 |
| Fourth level | 6 | 15.79 | 9 | 25 |
| Total | 38 | 100 | 36 | 100 |
Annex 3
Interview with Mathematics teachers:
- What do you think are the greatest difficulties that your students present in calculating derivatives of real functions of a real variable? Are the different depth levels of these calculations systematized through the subject? Are these levels integrated into new contents? Are students given differentiated attention to intensify the development of skills in these calculations? Are methodological activities carried out to find solutions to eradicate these difficulties?
Annex 4
Survey of Mathematics teachers of different profiles.
Next, they are shown different levels of development of the ability “to calculate derivatives of real functions of a real variable”. Place your students as accurately as possible in one of the categories: majority, medium, minority, or none. (To see the different levels, see the table on pages 6 and 7)
Teach classes on the profile of: __________________________________
Annex 5
Class observation guide:
Objective: To determine the relationships in the didactic-methodological work of the teachers and the result of the students' learning.
Aspects to consider:
- Main difficulties of students in calculating derivatives of scalar and vector functions of a real variable. Taking advantage of the potentialities of the content to achieve the development of the skill. Exercise system based on that skill.
Annex 6
Behavior of the “calculate derivatives of scalar and vector functions of real functions of a real variable” ability:
| LEVEL | First year | Second year | ||
| Quantity | % | Quantity | % | |
| Less than the first level | 3 | 7.89 | two | 5.55 |
| First level | 6 | 15.78 | 3 | 8.33 |
| Second level | 7 | 18.42 | 5 | 13.88 |
| Third level | 12 | 31.57 | 16 | 44.44 |
| Fourth level | 10 | 26.31 | 18 | fifty |
| Total | 38 | 100 | 36 | 100 |
