Pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rig our, abstraction and beauty.
One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. Generality has many different manifestations, such as proving theorems under weaker assumptions, or defining mathematical structures using fewer assumptions. Although generality is sometimes pursued or valued for its own sake, it has certain benefits, including:
·Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures: by exploring the implications of weakening the assumptions, one gains a better understanding of the role those assumptions play in the original theorems or structures.
·Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.
·One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently, or using results from other areas of mathematics.
·Generality can facilitate connections between different branches of mathematics, by emphasizing commonality of structure that may not be apparent at less general levels. Category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math.
Subfields in pure mathematics
·Analysis is concerned with the properties of functions. It deals with concepts such as continuity, limits, differentiation and integration, thus providing a rigorous foundation for the calculus of infinitesimals introduced by Newton and Leibniz in the 17th century. Real analysis studies functions of real numbers, while complex analysis extends the aforementioned concepts to functions of complex numbers. Functional analysis is a branch of analysis that studies infinite-dimensional vector spaces and views functions as points in these spaces.
·Abstract algebra is not to be confused with the manipulation of formulae that is covered in secondary education. It studies sets together with binary operations defined on them. Sets and their binary operations may be classified according to their properties: for instance, if an operation is associative on a set which contains an identity element and inverses for each member of the set, the set and operation is considered to be a group. Other structures include rings, fields and vector spaces.
·Geometry is the study of shapes and space, in particular, groups of transformations that act on spaces. For example, projective geometry is about the group of projective transformations that act on the real projective plane, whereas inversive geometry is concerned with the group of inversive transformations acting on the extended complex plane. Geometry has been extended to topology, which deals with objects known as topological spaces and continuous maps between them. Topology is concerned with the way in which a space is connected and ignores precise measurements of distance or angle.
·Number theory is the theory of the positive integers. It is based on ideas such as divisibility and congruence. Its fundamental theorem states that each positive integer has a unique prime factorization. In some ways it is the most accessible discipline in pure mathematics for the general public: for instance the Goldbach conjecture is easily stated (but is yet to be proved or disproved). In other ways it is the least accessible discipline; for example, Wiles' proof that Fermat's equation has no nontrivial solutions requires understanding automorphic forms, which though intrinsic to nature have not found a place in Physics or in public discourse.
School mathematics
1.Mathematics is a pattern
2.Mathematics is communication
3.Mathematics is investigation
4.Mathematics is problem solving
Nature of student of mathematics
1.Students need motivation and spirit
Motivation is the set of reasons that determines one to engage in a particular behavior. It is one of important thing needed by student, to make their dream come true. Motivation comes from two things: you, and other people. There is extrinsic motivation, which comes from others, and intrinsic motivation, which comes from within you. Intrinsic motivation occurs when people engage in an activity, such as a hobby, without obvious external incentives. Students are likely to be intrinsically motivated if they:
·attribute their educational results to internal factors that they can control (e.g. the amount of effort they put in),
·believe they can be effective agents in reaching desired goals (i.e. the results are not determined by luck),
·are interested in mastering a topic, rather than just rote-learning to achieve good grades.
Extrinsic motivation comes from outside of the performer. Money is the most obvious example, but coercion and threat of punishment are also common extrinsic motivations.
2.Students have certain characteristics
3.Students have competence
Competence is a standardized requirement for an individual to properly perform a specific job. It encompasses a combination of knowledge, skills and behavior utilized to improve performance. More generally, competence is the state or quality of being adequately or well qualified, having the ability to perform a specific role. Competence is also used to work with more general descriptions of the requirements of human beings in organizations and communities. Examples are educations and other organizations who want to have a general language to tell what a graduate of an education must be able to do in order to graduate or what a member of an organization is required to be able to do in order to be considered competent. An important detail of this approach is that all competences have to be action competences, which means you show in action, that you are competent. Every student must develop theircompetence, because it can make them easy to get the job.
In an education, there are three main components. The components are teachers, students, and education system. Teachers have a role as leader who organized the way of education. teachers lead their students to enter into a lesson. As participants in education, student must obey with all the teachers says, so the education process can run smoothly. Education system also has an important role in education.With a complete educational facilities, education can run smoothly. And the purpose of education is to create human progress can be achieved.
my name is Alifa Desiarini. now I am studying in Yogyakarta State University. I take Mathematics Education Program. I want to be a mathematics teacher who can teach my knowledge about mathematics well.
