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The proving ability in mathematics is one of the skills that must be possessed by a mathematician. It can be simplify as the process of inferential argument using only definitions and previously established results include axioms and theorems. The process can be found in all different branches of mathematics such as analysis, algebra, topology, geometry, and so on. However, the standard of geometry proof in several level of education is very different from that of high level education class, which in turn is different than that of postgraduate mathematics classes.

 

In the development of geometry, known as advanced geometry, there are two major groups of researcher who interest in euclidean geometry and non-euclidean geometry. Some theorems in euclidean geometry are easier to prove. It is because students have been introduced to this theory since primary school with basic shape of euclidean geometry which exists around. Students have known the sum of all angles of a triangle is equal to 180 degrees. They have learning experience to measure the angle using a protractor. Meanwhile, in non-euclidean geometry, students have never been taught before. Therefore, students have difficulties about the concept of the sum of all angles of a triangle which is greater than or equal to 180 degrees.

However, a formal procept which students have in the euclidean geometry can be used to understand some new concepts in non-euclidean geometry. It has been known that geometry proof is deduction process which starts from undefined term. Therefore, if one does not clearly understand one step then the next step becomes difficult and the one after that is impossible. At the first step of learning process, some students only have the brief notion of the theorem in their mind. They do not really know the essence of the theorem and have more flexible thinking until they perceive the notion of proof of the theorem. Therefore, the ambiguity of process and product represented by the notion of formal procept also provides a more natural cognitive development at the university level which gives the students enormous power to develop more flexible mathematical thinking.

 

This paper aims to address the following questions: 1) how is the ability of postgraduate students in mathematical proofing, 2) how postgraduate student use a formal procept in advanced geometry. In addition, this paper also examines the process of thinking among postgraduate students.

 

The subject in this study was a master student in mathematics postgraduate school at Universitas Negeri Surabaya whom were in the first year of their study. We were analysing the postgraduate student ability in mathematical proofing. One theorem in euclidean geometry and non-euclidean geometry had been chosen to explore the ability. The first theorem says that the sum of all angles of a triangle is equal to 180 degrees, where as another theorem says that the sum of all angles of a triangle is equal or greater than 180 degrees. Students were ordered to prove the theorem with two columns proof strategy which contains given statements and reason. Furthermore, the student's answer has been explored to figure out how student use formal procept in advanced geometry. The process of student thinking had been examined to determine what kind of thinking had been used to learn new concept in non-euclidean geometry.

 

Regard the result of student's answer analysis, it is found that the students have difficulty in proving the sum of all angles of a triangle is greater than or equal to 180. It is because the new concept is totally different from the scheme which has been owned by the students. Furthermore, the existence of the new concept leads to cognitive conflict. It is a perceptual state in which one notices the discrepancy between one’s cognitive structure and the environment (external information), or among the different components (e.g., the conceptions, beliefs, substructures, etc.) of one’s cognitive structure.

 

The use of formal procept on euclidean geometry is required to embed the concept of non-euclid geometry. By using a formal procept and cognitive conflict, students are able to assimilate and accommodate the concept of euclidean geometry so that they can understand the concept of non euclid geometry more easily. Mediation is done to facilitate the students in conducting the process of accommodation and assimilation on the concept of the sum of all angles of a triangle. Assimilation refers to the process by which a subject incorporates a perceived stimulus into the existing scheme. This process will assist students in understanding about the sum of all angles of a triangle is greater than or equal to 180 degrees. Regard to the results of examining the process of thinking students in understanding the new concepts which exist in non-euclid geometry found that students use the process of analogical thinking.

advanced geometry, geometry proof, procept, cognitive conflict, assimilation, accommodation