# Models of Doing Mathematics Proof by Students

## Description

The models of doing for school mathematics varies and mainly based on etymological sense of meaning according to words or phrases in a sentence. The students positioned their thinking based on word understanding of a proof problem. They also relate between presented information and formula in doing proof. When facing visual representations, the students tend to prove by algebra. The visual representations to be imagined merely as a tool for arranging a formal proof, i.e. when so many algebra symbols in proving, including formula and the procedures. For algebraic or analytic or symbolic-illustrative representation, the mathematics proof is just symbolic manipulation than the meaning. In doing proof, a meaning of word or the combination construct a pattern of proof changed to mathematics statement. For example, a word 'to test' has multi-example pattern. While, 'to judge' shows logical pattern. The etymological sense is a learning understanding of doing proof. There s an intersection of the etymological and the understanding. Data presented in Etymological-Understanding axis. The intersections moved from information (or data) toward pattern in proof problem. There are two other intersection, i.e. meaning and experience. That is an easy way to prove much of mathematics problems. When facing to the difficult proof problem, a set of the intersection have an action role in doing proof to an intervention of the etymological sense of meaning. That is a real learning activities, specifically in doing proof. < Fig. 1 >. There is also an intersection of doing proof between word meaning and logic. That is a new culture in doing mathematics proof, a belief of the learning. The data embedded in a four quadrant of Monophonic-Context axis and Velocity-Viscosity axes (Fig. 2). The students' experience also appear in thinking, drawing (a step to a proof), testing, and developing their performances. The etymological sense of word or phrase meaning brings the students to a broader or more rational (not common sense) ways to prove. The proof trajectories growth cross mathematics content knowledge. That is psychological context addressed by students' belief and determined using the empirical proof. The fitted of the empirical data mapping its model as an entropy value indicated 'creative proof', i.e. in illustrating, describing mathematics representations, making a rational (or consistence) relation, and generating a proof. The models of proof show the differences representation. When the students ask to 'try', then they elaborate a proof by cases. < Table 1 >. But, to 'determine' made more algebra thinking. Finally, there are performance levels in doing proof etymologically. Nine performances < Table 2 > aroused as variables of etymological sense of meaning. That is a measure of teaching and learning of doing proof. That is an exploring of the etymological toward more proof representation. That is way to assess doing proof through meaning of word. < Table 3 > .