The history of algebra dates back to ancient Egypt and Babylon, where they were able to pose and solve problems that contained equations of the first and second degree. The ancient Babylonians solved equations, using essentially the same methods that we use today.
Problem Statement.
In the teaching and learning processes very often we find many determining factors in the construction of knowledge, one of them is the little interest or love towards studying mathematics, causing concerns in basic secondary education, more precisely in the Manuel German Cuello Gutiérrez School in the afternoon, where the observation of the learning processes shows that the students do not achieve a significant learning of algebra, specifically in the factorization of trinomials.
It is the duty of teachers to propose alternatives that enable the improvement of teaching and learning processes that lead the student to awaken interest in mathematics and appropriate knowledge, helping the development of semantic memory, which will lead to the strengthening of knowledge prior, turning them into meaningful learning.
All of the above leads to the following question:
Will it allow the development of software, as an innovative methodological strategy, to achieve a significant learning of the factorization of trinomials?
goals
Overall objective
- Design and validate innovative methodological strategies that allow a significant learning of the concept of algebraic expressions in the decomposition of trinomials.
Specific objectives
- Apply pedagogical activities related to solving problems on algebraic expressions and their decomposition into their prime factors. Improve interpersonal relationships between student-student and student-teacher. Reduce students' apathy towards the study of mathematics.
Justification
The 1991 Constitution of Colombia in article 67 states that “Education is a right of the person and a public service that has a social function: it seeks access to knowledge, science and technology and others cultural assets and values ”. In the same sense, the General Law of Education in its article 1 states that “Education is a process of permanent, cultural and social training that is based on an integral conception of the human person, their dignity, their rights and their homework"; This indicates that teachers are a permanent builder of knowledge and are obliged to establish innovative strategies that help improve the quality of education.
Learning as a natural, social, active and non-passive process can be linear or non-linear; Furthermore, it is integrated and contextualized, based on a model that must be changing; it is strengthened in contact with the student's abilities, interest and culture. This natural learning must be accompanied by the teacher, who is responsible for being an active agent in the process and not the machine that knows everything; on the contrary, he must learn with his students.
Usually, in the teaching of mathematics the transmissive-receptive model dominated, where the teacher elaborates content that the student receives passively. This didactic model, which adopts the master class as a prototype, transmits a very orthodox vision of mathematics, with already-made knowledge, where the contents are clearly rote. Some research on the vision and attitude that students acquire towards the study of mathematics, throughout their educational life at school, reveal a worrying situation.
Studies more interested in learning mathematics reflect a growing apathy of young people towards mathematics. The panorama worsened when verifying that those same young people had initiated the first contacts with science from curiosity and even enthusiasm, that is, from the direct manipulation of theoretical contents. Somehow it seems to happen that the teaching of mathematics itself alienates an important part of children from their initial interests in knowledge.
The teaching of mathematics, under the traditional model of receiving elaborated knowledge, puts all its concern in the contents, in such a way that a carefree vision of the teaching process itself stands out, understanding that teaching constitutes a simple task that does not require special preparation. This conception has weighed on the initial training required of mathematics teachers, so that the demands are reduced to the knowledge of the subjects and content to be taught, and very little or nothing to didactic questions or how to teach.
These teaching methods prospered until the end of the 20th century, when pedagogy appeared and one of its precursors: Erasmus of Rotterdam, broke with the old way of educating, whose sterilizing and repetitive aspect had been widely denounced. It is the first to highlight the value of affectivity and play in learning knowledge. With this reflection, Juan Amos Comenio presents a new methodology of education, based on the union of pedagogy with didactics, his project of a "magna didactics" or "universal instruction" inspired by religious and humanistic principles, helps the teacher to design strategies that allow students to assimilate knowledge easily. Even so, there are some difficult potholes to solve, it is here where the psychology of education comes to play a very important role,which is the application of the scientific method to the study of the behavior of individuals and social groups in educational environments.
The psychology of education is not only concerned with the behavior of teachers and students, but also applies to other groups such as teacher aides, early childhood, immigrants and the elderly. The areas of study of educational psychology inevitably overlap with other areas of psychology, including developmental (child and adolescent) psychology.
For all the above, teachers have the obligation to seek strategies that lead the student to use semantic memory to solve problems, achieving meaningful learning.
Theoretical framework
Just as arithmetic arose from the need that primitive peoples had to measure and count; the origin of algebra is much later, since many centuries had to elapse for man to arrive at the abstract concept of number, the foundation of Algebra. The great development experienced by Algebra was due mainly to Arab mathematicians. The Arabs introduced numbering and Algebra to the West, collecting the scientific heritage of the Greeks, assimilating the practical spirit of Indian mathematics, and perfecting the positional numbering system. The word algebra comes from Ilm al-jabr w 'al mugabala ("science of restoration and reduction"), the name of a book written in the 9th century by the Arab mathematician Al-Khwarizmi.Some experts define algebra as a generalization of mathematics thanks to the use of symbols or letters to represent arbitrary numbers.
The topic of solving algebraic equations has interested mathematicians of all times, including the ancient civilizations of Babylon and Egypt. There is evidence that the Egyptians solved certain quadratic equations 2,000 years BC, the Hindus and the Arabs made some important advances on this issue around 800 BC; but the first steps towards the development of the theory of equations were taken by Diophantus of Alexandria towards the third century BC. C.
Many are the contributions that countless mathematicians have made to Algebra. Newton, the greatest of English mathematicians and one of the greatest scientists in the history of mankind; made great contributions, among them is the binomial that bears his name and the method of successive approximations to find the attraction basins. The Frenchman Francois Viete, considered by many to be the founder of modern Algebra, introduced algebraic notation, making Algebra definitively free from the limitations imposed by arithmetic, and become a purely symbolic science; solved sixth grade equations, author of "Isagoge in artem analyticum", considered the first algebra treatise.
Paolo Ruffini; In addition to the rule that bears his name for dividing a polynomial in x, by x - a, he was the first to make a serious attempt to demonstrate the impossibility of solving polynomial equations of greater than fourth degree by means of radicals, known as the theorem of Abel-Ruffini; whose formulation and demonstration was completed by the Norwegian Hiels Henrik Abel.
Joseph Luís Lagrange, worked "on solving numerical equations"; Karl Friederich Gauss, proved the fundamental theorem of algebra and Fermat, worked on factoring and conjectured that the numbers of the form 22n + 1 were primes, known today as Fermat numbers, who conducted research on the properties of numbers, which never wanted to publish; He even wrote to his friend Pascal: "I don't want my name to appear in any of the works considered worthy of public exposure." He contributed to probability theory, calculus, and number theory. One of his most important contributions was finding the second pair of friendly numbers. "Two natural numbers n and m are friends if the sum of the divisors of is equal to m and the sum of the divisors of m is equal to n".The Pythagoreans discover the first pair: 220 and 284. Fermat, discover the second: 17296 and 18416.
Factoring
1. Perfect square trinomial
One quantity is a perfect square, when it is the square of another quantity; that is, when it is the product of two equal factors. An ordered trinomial with respect to a variable is a perfect square when the first and third terms are perfect squares and the second term is the double product of their square roots. To factor a perfect square trinomial, the square root of the first and third terms of the trinomial is extracted and these roots are separated by the sign of the second term. The binomial thus formed, which is the square root of the trinomial, is multiplied by itself or squared.
Example: factor x² + 2x + 1.
The root of x² is x; and the root of 1 is 1
So:
2. Square trinomial of the form: x² + bx + c
This trinomial meets the following characteristics: the first term must have an exact square root, the variable that accompanies the second term must be the square root of the first term.
To factor a trinomial in this way, the trinomial must be organized in decreasing form and written as the product of two binomials, such that the two second terms of the binomials give as a product the third term of the trinomial and its sum, the coefficient of the second; that is to say:
x² + bx + c = (x + M) (x + m), where: M + n = b; Mn = c
Example: factor x² + 5x + 6
Therefore: x² + 5x + 6 = (x + 2) (x + 3)
3. Square trinomial of the form ax² + bx + c
This trinomial must meet the following characteristics: be organized in a decreasing way, the first term has a coefficient different from 1 and the literal part must have an exact square root, the variable in the second term must be the square root of the variable of the first term
To factor the trinomial ax² + bx + c, proceed as follows: multiply and divide the trinomial by the coefficient of the first term, leaving it as follows: a (ax² + bx + c.) / A, then it is operated, resulting in: / a; the trinomial obtained is a trinomial of the form x² + bx + c.
Significant learning.
Learning is significant when the contents: Are related in a non-arbitrary and substantial way (not literally) with what the student already knows. By substantial and non-arbitrary relationship it should be understood that the ideas are related to some specifically relevant existing aspect of the student's cognitive structure, such as an image, an already significant symbol, a concept or a proposition (AUSUBEL; 1983, 18). This means that in the educational process, it is important to consider what the individual already knows (previous ideas) in such a way that it establishes a relationship with what they must learn. This process takes place if the learner has concepts in her cognitive structure, these are: ideas, propositions, stable and defined, with which the new information can interact.
Meaningful learning occurs when new information "connects" with a pre-existing relevant concept in the cognitive structure, this implies that new ideas, concepts and propositions can be learned significantly to the extent that other relevant ideas, concepts or propositions are adequately clear and available in the cognitive structure of the individual and that function as an "anchor" point to the former.
Types of meaningful learning
It is important to emphasize that meaningful learning is not the "simple connection" of new information with that which already exists in the cognitive structure of the learner; on the contrary, only machine learning is the "simple connection", arbitrary and not substantive; meaningful learning involves the modification and evolution of new information, as well as the cognitive structure involved in learning.
Ausubel distinguishes three types of meaningful learning: representations, concepts and propositions.
1. Learning Representations
It is the most elementary learning on which all other types of learning depend. It consists of the attribution of meanings to certain symbols, in this regard AUSUBEL says:
It occurs when arbitrary symbols are equated in meaning with their referents (objects, events, concepts) and signify for the student whatever meaning their referents allude to (AUSUBEL; 1983, 46).
This type of learning generally occurs in children, for example, learning the word «Ball», occurs when the meaning of that word comes to represent, or becomes equivalent to the ball that the child is perceiving at that moment consequently they mean the same thing to him; It is not a simple association between the symbol and the object, but rather the child relates them in a relatively substantive and not arbitrary way, as a representational equivalence with the relevant content existing in their cognitive structure.
2. Learning Concepts
Concepts are defined as "objects, events, situations or properties that have attributes of common criteria and are designated by means of some symbol or signs" (AUSUBEL 1983: 61), based on this we can affirm that in a certain way it is also learning of representations.
The concepts are acquired through two processes. Training and assimilation. In concept formation, the criteria attributes (characteristics) of the concept are acquired through direct experience, in successive stages of formulation and hypothesis testing, from the previous example we can say that the child acquires the generic meaning of the word « ball ", this symbol also serves as a signifier for the cultural concept" ball ", in this case an equivalence is established between the symbol and its attributes of common criteria. Hence, children learn the concept of "ball" through various encounters with their ball and those of other children.
The learning of concepts by assimilation occurs as the child expands his vocabulary, since the criteria attributes of the concepts can be defined using the combinations available in the cognitive structure, so the child will be able to distinguish different colors, sizes and affirm that they are it is about a "Ball", when you see others at any time.
3, Learning propositions.
This type of learning goes beyond the simple assimilation of what the words represent, combined or isolated, since it requires capturing the meaning of the ideas expressed in the form of propositions.
The learning of propositions involves the combination and relationship of several words, each of which constitutes a unitary referent, then these are combined in such a way that the resulting idea is more than the simple sum of the meanings of the individual component words, producing a new meaning that is assimilated to the cognitive structure. That is, a potentially significant proposition, expressed verbally, such as a statement that has denotative meaning (the characteristics evoked when hearing the concepts) and connotative (the emotional, attitudinal and ideosyncratic load caused by the concepts) of the concepts involved, interacts with the relevant ideas already established in the cognitive structure and, from that interaction, the meanings of the new proposition emerge.
Semantic memory
Semantic memory refers to our general archive of conceptual and factual knowledge, not related to any particular memory. It is an eminently declarative and explicit system, but clearly different from that of episodic memory, because in fact memory of events can be lost and memory of concepts can be maintained. Semantic memory shows our knowledge of the world, the names of people and things and their meaning.
It is more especially located in the inferior inferior temporal lobes. But in a broad sense, semantic memory can reside in the multiple and diverse areas of the cortex related to the various types of knowledge. Again, the frontal lobes are involved in its activation to retrieve information.
Methodological processes
The methodology to be used in a first stage is direct observation, through which the records of the students and the algebra teacher will be kept, in order to establish possible problems that are occurring in the teaching and learning processes of algebra.; Interviews will be carried out to establish shortcomings in the learning and teaching processes that were not observed in the previous stage, in the second stage the activities corresponding to the didactic unit for the teaching of trinomials will be launched, at the end of the activities a evaluation of the process with the different components that participated in the process.
Lines of research
The research is framed along the lines: quality in the teaching and learning of mathematics because the mathematics teacher is obliged to seek the improvement of teaching and learning methods.
Population
The population under study are the eighth grade students of the Manuel German Cuello Gutiérrez School afternoon session, their ages range between 13 and 16 years old, they belong to strata 1 and 2.
Geographic delimitation
The research was carried out at the Manuel German Cuello Gutiérrez School in the afternoon session, located in the Santa Rita neighborhood, south of the city of Valledupar.
Budget
Photocopies ……………………………….. $ 150,000
Purchase of books ……………………….. $ 250,000
Transportation ………………………………. $ 190,000
Advisors ……………………………..… $ 1,200,000
Impressions …………………………… $ 100,000
Others …………………………………….. $ 300,000
Total ……………………………….. …… $ 2,190,000
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