Obstacles in Constructing Geometrical Proofs of Mathematics-Teacher-Students Based on Boero’s Proving Model

This study aimed at identifying the obstacles of mathematics-teacher-students based on Boero’s proving model. This study was conducted using a mix method by applying sequential explanatory strategy. The stages of the research were carried out by taking the quantitative data and revealing the qualitative data using semi-structured interviews. In the result of this study, it was found that most of mathematics-teacher-students had difficulties in constructing geometrical proofs of each Boero’s proving model. Even in the phase of writing formal proof, there were only 6.67% of students could write fully in the cases of indirect proving. There were 13.33% of students in the cases of direct proving. This study concluded several obstacles which students faced in constructing the geometrical proofs formally in each phase of Boero’ proving model. The obstacles included: the difficulty in making a diagrammatic sketch of conjecture which was completely made with the correct geometrical notation; the difficulty in knowing of cause-effect of geometrical problems to be proved, if it involved some conditional sentences; inability to write a conjecture made in the form of geometrical symbols, formulas and axiomatic deduction; the difficulty in selecting a valid statement of the conjecture made and the difficulty in writing formal proof.


Introduction
When we talk about mathematics, it is implied in mind that it is a science which requires us to use the logic of mathematical rules in the form of postulates and theorems to analyze and solve all the problems of algebra and geometry to form a concept or new mathematical knowledge. Since mathematics is the science of logic on the form, composition, scale, and concepts related to each other which is divided into three areas, namely algebra, analysis and geometry [1], it requires a treatment so that the construction of a concept can be understood by students.
Studying mathematics is to study the branches of mathematics which are one of them is the geometry. Everything in this universe is a geometrical case. Therefore, mathematics through the geometry is to be learned about the concepts embodied in the objects which exist in nature through the concepts of geometry. Geometry is very important to be developed in learning process of mathematics in classroom. The objectives of learning geometry are as follows [2]:  3 series of statements which are considered true. Deductive reasoning uses the laws of logic to link all the correct statements to a correct conclusion. A definition is a true statement to use in geometric proof.
In relation to the ability to verify proof in general mathematics in which there is the ability of geometrical proof, Sumarmo [15] divided it into two, as follows: the ability to read the proofs and the ability to construct the proof. The ability to construct the proof is to formulate a mathematical statement of proof based on definitions, principles and theorems, and write it in the form of proof completely.
To construct geometrical proving, Ridgway [17] stated there are five steps, namely: 1. Identifying what information given and what to be proven. It is the easiest step to determine the conjecture of a causal link that contains the statement "If ..... then .....". The word "if" shows what information is given. Meanwhile, the word "then" indicates the part which must be proven. If the conjecture is not suitable with what is given, we can restate in the form of "if .... then ...". 2. Making a diagram of information given to clarify what to be proved. 3. Presenting back the information on the diagram created. 4. Making a plan of verifying proof by reasoning. 5. Using the plan of verifying proof created to further writing a proof.
Boero (in Reis and Reinkl) [18] explained a model of the verifying a proof consists of six phases; determine the conjecture (conjecturing), formulate a statement (formulating statement), exploration (exploring), selecting and combines coherent argument (the selection and combination of coherent arguments), test the result (testing the result), write a formal proof (writing a formal proof). In the first stage, it is to determine the allegation or conjecture to explore issues which aims to establish and identify the arguments to support the proof. The second phase is to formulate a statement which aims to provide the exact formula of the allegations made, which is the basic for further process. The third phase is to explore allegations and identify the correct arguments to be validated. The fourth phase is to select and combine the arguments determined which are coherent in deductive chain. The fifth phase is to test the result of the proof whether or not suitable with the rules of mathematics which can then be written into formal proof in the sixth phase.
In Indonesia, geometry in curriculum of college is divided into several subjects in each semester. At the beginning of the semester (second semester), student learn geometry through a flat geometry subject (basic geometry). The subject of flat geometry contains of Euclidean geometry material involving axiomatic systems and geometrical proving. The ability to construct geometrical proving in this subject needs to be developed.
Considered by the importance of geometrical proving for mathematics-teaching-students, it is important to conduct a study to identify the obstacles faced by mathematics-teaching-students in Indonesia. It becomes a question whether the characteristics of the obstacles which have been revealed would be the same as the characteristics of the mathematics-teaching-students in Indonesia or there were other findings. Therefore, it is important for assessing the ability of constructing a geometric proof of mathematics-teaching-students in order to obtain information about obstacles.
With reference to the steps of verifying geometrical proving disclosed by Ridgway and the Boero's proving model, to identify obstacles of mathematics-teaching-students in constructing proofs based on Boero's proving model includes the following: knowing information given and what is to be proved; determining the cause-effect of the problem to be proven; describing a diagrammatic sketch completely with geometrical symbols and notations; making a statement based on known information and diagrammatic sketch made; writing a conjecture that has been made in the form of geometric symbols, formulas and deductive axioms; exploring valid and invalid statements of conjecture which have been made; choose a valid statement of conjectures made; creating a link between the valid statements using deductive axiom rules; testing beforehand.

Method
This study was conducted using a mixture (mix method) by applying explanatory sequential strategy [19]. In the first stage of quantitative data collection, the samples consisted of 30 mathematics- Geometrical proving test 2 (direct) Prove if the triangle of ABC was determined the center point P at side of AC, then line // AB through point P will cut BC in point Q exactly at the center.
Furthermore, students' answers were given 1 of score for a correct answer and 0 for an incorrect answer on each indicator. Students' answers were analyzed with descriptive statistics referring to the indicator of the ability to construct geometrical proofs as can be seen in Table 1.

Testing Result
Testing the first geometrical proof before writing it down formally 6 Writing Formal Proof Using the rules of method of proving (direct and indirect) correctly Writing down formal geometrical proof completely The next stage was to explore the data qualitatively by conducting interviews to four students. The qualitative data were taken from semi-structured interviews in order to obtain the obstacles in constructing the geometrical proofs. The questions were asked related to indicators of the ability to construct geometrical proof. Furthermore, the data from interviews were analyzed to determine the obstacles on the ability to construct geometrical proofs of mathematics-teaching-students.

Result and Discussion
The result of quantitative data was analyzed with descriptive statistics. After that, it was conducted an interview to several students based on the answers written to get the qualitative data about the ability to construct geometrical proof.

Phase of Conjecturing
This phase involved four indicators, namely: knowing the information given and what is to be proved; knowing the cause-effect of the problem to be proved; Describing diagrammatic sketch with geometrical symbols and notation completely; making statements based on provided information and diagrammatic sketch made. Table 2 is the result of the students' answers at each geometrical proof test:  Table 2 above shows that 80% and 57. There were 14% of students answered correctly for the indicator of knowing the information given and what is to be proved. Overall, students could determine what information is known and could understand the problem to be proven. On indicators of knowing the cause-effect of the problem to be proved, Geometrical Proof Test 1 by 74, 29% of students answered correctly. That was caused by Geometrical Proof Test 1 which only involved two simple statements with a triangle of ABC with a cause and a side of AC = BC as an effect to be proved. However, in the Geometrical Proof Test 2, there were only 54.29% of the students knew the cause-effect of the problem to be proved. That was because the Geometrical Proof Test 2 was more complex.  Figure 1, it can be seen that Endah had a wrong conjecture. She considered the problem to be proved is AB // OP and Endah did not write down the information known of the problem to be proved. To obtain the data deeper from Endah, the researcher conducted an interview. Here is the result of an interview with Endah. Endah seemed confused about the meaning of the phrase "then the line //AB through the point P will cut BC at the point Q right in the middle point". Therefore, Endah merely perceived the problem as "If the ABC  was determined the center point P on the side of AC, then the line // AB is through the point P ". Thus, Endah assumed that the center point Q of BC as a result of BQ // AB. Therefore, the problem proven is PQ // AB. In other word, Endah did not understand about the meaning of a sentence or phrase on the problem to be proved. In addition, Endah also failed to specify a conditional statement in the form of cause-effect of the problem. Therefore, she had a difficulty in identifying the information known and the problem to be proved.
On the indicator of describing diagrammatic sketch with geometrical symbols and notation completely, most students faced a difficulty that there were only 34.29% and 50% of students answered correctly. The students had a difficulty in describing diagrammatic sketch which was caused them unable to determine the conjecture solution of proof. The difficulty which the students had was that they described a wrong diagram which was not suitable with what was to be proved and wrong in geometrical notation. In addition, students were also unable to make statements about a theorem which would be used to solve the problem. For the last indicator which was making a statement using geometrical axiomatic system related to the solution of proof, there were 31.43% for geometrical proof test 1 and 34.29% for geometrical proof test 2 answered correctly.
Here is presented the results of the answers to student named Dewi in Figure 2. To explore the answer of Dewi, the writer interviewed her. Here is the result of an interview with her.
Researcher : What kind of conjecture do you think to prove this problem? Dewi : If in a triangle, the two angles are equal, then two sides are also the same anyway. The sides are pulled from a common point, namely the point C, to each corner. Therefore, it forms an isosceles triangle of ABC. Researcher : Please try to give any reasons for your answer.

Dewi : In the geometrical proof test 1, it is clear that if the two corners of a triangle are equal, the two sides are the same because it is the nature of a triangle. There is no need to prove it.
It can be seen that Dewi considered that the statement did not need to prove. Dewi assumed that triangle in the statement was an isosceles triangle with its nature of the same feet or two sides. Therefore, according to Dewi it needed obviously no prove again. Table 3 shows the result of students' answers of phase of formulating statement, as follows:  Table 3 presents that there were only 28.57% of students answered correctly in Geometrical Proof Test 1 and 31.43% in Geometrical Proof Test 2. The difficulty in describing diagrammatic sketch was the first cause which in writing the conjecture made using geometrical symbols, formulas, and axiomatic deduction. The incorrect diagrammatic sketch made students unable to write a proof. Table 4 shows the result of students' answers of phase of exploration, as follows: In this phase, it can be seen that almost all the students could not identify statements which are valid or invalid from conjecture which was made. It can be assumed by that there were only 28.57% of students answered correctly in each geometrical proof test given. The result was in line with the second phase in which can be indicated as if the second phase could not be passed, the further phases would be the same.

Phase of Selection and Combination of Coherent Arguments with Deductive Chain
Here is the result of students' answers of phase o selection and combination of coherent arguments with deductive chain, as follows:  Table 5 presents only 31.43% of students answered correctly in the Geometrical Proof Test 1 and 28.57% in the Geometrical Proof Test 2. It indicated that the students had a difficulty in selecting a valid statement of conjectures made. The problem appeared because of the inability of students to make conjectures in the form of axiomatic deduction and the lack of students' ability in sketching graphs of conjectures made. Thus, the students also had a difficulty in making a link among the valid statements by using rules of axiomatic deduction. There were only 25.71% of students answered correctly in each geometrical proof test. The mistake which students  Figure 3, it can be seen that Ratna was wrong in describing the diagrammatic sketch. The conjecture of problems to be proven was also incorrect. As the result, she was unable to make a valid statement and use the geometrical theorem to justify the statement which was made. Thus, the statement linked no connection with each other at all, even somewhat it seemed forced and made up. To obtain the further data about Ratna's answer, the writer conducted an interview. Here is how the interview with Ratna conducted.
From the interview with Ratna, she was actually not sure with her conjecture. However, Ratna kept pushing with statements according to what she knew. She did not know that the statements she made were valid or not. Ratna tried to relate all the statements which she made although there was no connection to link invalid statements to valid ones. In addition, Ratna also kept trying to relate all which she did not master with the problem to be proved. It can be seen that Ratna tried to pull line of BP so that AP = CP and AB = BC. In this case, the problem to be proved by her was not an isosceles triangle but it was a scalene triangle.

Phase of Testing Result
Here is the result of students' answers of phase of testing result, as follows:  Table 6 shows that there were only 20.00% of students tested geometrical proof test in geometrical proof test 1 and there were 25.00% in geometrical proof test 2. Most of the students wrote the formal proof without tested it first after they got confident with the conjecture they made was right.

Phase of Writing Formal Proof
Here is the result of students' answers of phase of writing formal proof, as follows:  Table 7 presents that there were 20.00% of students knew the proof used was indirect proof (contradictory proof), but there were only 6.67% of students wrote the formal proof completely. Meanwhile, in the geometrical proof test 2, there were only 25.71% of students knew the proof used is direct proof and there were 13.33% of students wrote in formal proof completely. In the first problem, students thought that all the problems of geometrical proof could only be solved by direct proof. Nonetheless, most students seemed did not understand to verify the geometrical proof using direct proof yet. As shown below Risky's answer which uses contradictory proof to solve the problem 1, but the Risky's answer was yet incomplete and found a few flaws.
Here is the answer of Risky. The writer tried interviewing Risky based on answer shown in Figure 4, focusing on the incompleteness of Risky's answer to the conclusion of the proof. Here is the interview with Risky.
Researcher : What method of proof did you use? Risky : Contradictory proof sir. Researcher : Was the proof which you wrote complete? Risky : I think it is complete sir. Researcher : I could not find the final conclusion from the proof you wrote. Can you explain? Risky : Oh, yes sir, actually statements (1) and (2) it means the assumption of BC AC  was wrong. What is correct is AC = BC, which means it was proved, sir. Researcher : I also found an error in the statement you previously wrote that AT = BC, but on the next statement AD = BC. What did it mean? Risky : That's correct sir AT = BC, I was wrong to write the statement below. I am sorry I was insecure sir. Researcher : Did you first check your answer before writing it down formally? Risky : No sir, I wrote my idea and the proof sir without checking my answer first. From the interview conducted to Risky, it can be seen that he was careless in writing the formal a proof. It can be obviously seen from the writing from the mistakes in each statement. In addition, Risky considered that the proof which he wrote was complete. Meanwhile, it was important in writing a formal proof to write it to the conclusion that the problem was already verified. Risky felt confident of the answer. Risky's carelessness was that he did not re-check the proof which he wrote. He was quite sure with his answer so that he did not have to recheck.
This study tried to identify obstacles in constructing geometrical proof based on Boero's proving model. From the result of data analysis both qualitative and quantitative, it showed that phase of conjecturing determined further subsequent phases. It can be seen that most of students had a difficulty in determining the correct conjecture with the problem to be proved. In phase conjecturing, understanding the information and what to be