Importance of the demonstration in the teaching of mathematics

Teaching mathematics is a challenge when it comes to learning certain mathematical topics, but it becomes more difficult when the teacher takes the idea of ​​wanting to teach the proof of some theorem, axiom and / or corollary. The demonstration of certain aspects of mathematics such as: Euclidean geometry, certain aspects of linear algebra and concepts of differential calculus are of considerable complexity, which means that the teacher has a multifaceted role in facing adversities such as understanding mathematical demonstrations of students at a higher level. The first question that arises at the moment of wanting to teach to demonstrate in mathematics is the following:

What are the tools that a teacher must have to teach to demonstrate in Mathematics?
The role of the demonstration in the training of teachers in Mathematics at a higher level can be approached from different points of view. Although in this publication I want to analyze the aspects that lead me to believe that competences are necessary to evaluate the leading role of the teacher when demonstrating and providing meaningful learning to students in the higher education process.

This aspect of the competences in us educators leads me to think about the elements that are addressed in the demonstration, from the knowledge of the content that the teacher has both Mathematical and pedagogical, as well as how the student perceives this knowledge.

Let's try and analyze these two fundamental competences:

Knowledge of mathematical content: the knowledge of mathematical content refers to the knowledge that the teacher has about the mathematics he is teaching, this means that if the educator tries to explain the demonstration of some mathematical aspect without knowing the subject, it is most likely that his students can not understand the demonstration.
We make a practical example: Suppose I want to demonstrate the formula of the distance between two points P1 (X1, Y1) and P2 (X2, Y2) that from my point of view these are the knowledge that must be had to demonstrate the formula of the Distance between two points:

Knowing what an ordered pair is and how to graph it in the Cartesian coordinate system, in order to know the location of each of these points it is necessary to know the coordinates of the two points, P1 and P2, these points in turn are made up of ordered pairs , these ordered pairs once they are graphed in the Cartesian graphical system, makes it possible to plot the line segment to which we will finally calculate the distance between them.

Generalities about projections, it is necessary to know and know about projection of points, since this will help us to take to the background a rectangle triangle in which we will finally apply the Pythagorean theorem for the resolution of right triangle.

Knowledge about the Pythagorean theorem, basically knowing that the hypotenuse squared is equal to the sum of each leg squared, we would finally be completing the demonstration of the distance between two points through which a line segment passes.

These are only three types of knowledge among perhaps others, which I consider necessary in this case for the test you do, at the moment you want to demonstrate the expression of the formula to calculate the distance between two points of a line segment. The objective of this practical exercise is to make it understood that for us teachers it is very important to know about the subject in order to enter the world of mathematical demonstrations, and above all to assess the level of responsibility in learning for our students. of the upper level.

knowledge of pedagogical content: the knowledge of the pedagogical content refers to the knowledge that the teacher has about the teaching of Mathematical content, it is necessary to explain in this aspect the following, why now speak of knowledge about teaching? we have a knowledge in mathematical content. The answer is that we need knowledge in the teaching of that content that we previously know, since the fact that a teacher knows the subject does not guarantee that he will be able to explain it in the best way, and is where we are for example with these sayings of the students in the universities: "he knows a lot about his area, but he does not understand what he explains", and it is because it is of no use to know much about the subject but we have the necessary pedagogical and andragogical tools so that the student learn to know the important meaning of demonstrating in mathematics.

Conclusion and reflections

Talking about the importance of the aspect of demonstrating in mathematics, is to respond to simple facts of normal thinking about the nature and behavior in the human intellect, since no one likes to learn without knowing what it is that learns, how learn, and above all and most importantly, the knowledge of where the arguments of what is explained come from.

As everything is explained, and the origin and existence of things is demonstrated, to that proportion our learning is set in a significant way, making it so that the demonstrations take a leading role in the mathematical learning.

There are two fundamental competences that a teacher must have when explaining a mathematical demonstration, these are:

Knowledge of the mathematical subject.
Knowledge of the teaching of the mathematical subject.
I only hope this article will be very useful for my readers and followers, any questions or suggestions I hope you will make in the comments, and if you want to talk about the two competences mentioned above, let me know so I can develop it in the next publications.

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