Understanding on Strategies of Teaching Mathematical Proof for Undergraduate Students

Many researches revealed that many students have difficulties in constructing proofs. Based on our empirical data, we develop a quadrant model to describe students' classification of proof result. The quadrant model classifies a students' proof construction based on the result of mathematical thinking. The aim of this article is to describe a students' comprehension of proof based on the quadrant model in order to give appropriate suggested learning. The research is an explorative research and was conducted on 26 students majored in mathematics education in public university in Banten province, Indonesia. The main instrument in explorative research was researcher itself. The support instruments are proving-task and interview guides. These instruments were validated from two lecturers in order to guarantee the quality of instruments.Based on the results, some appropriate learning activities should be designed to support the students' characteristics from each quadrant, i.e: a hermeneutics approach, using the two-column form method, learning using worked-example, or using structural method.


INTRODUCTION
Proof and proving in mathematics education are an important part of mathematics, as a pillar of the mathematics building. Therefore, mathematics education especially in mathematics learning in university emphasizes the constructing of mathematical proof. Students who enter college level should develop a formal mathematical knowledge.
Students are introduced to formal proof in the study of mathematics at university. Formal SURRI DV D SURFHVV EHJLQV IURP H[SOLFLW TXDQWL¿HG GH¿QLWLRQV DQG GHGXFHV WKDW RWKHU SURSHUWLHV KROG DV D FRQVHTXHQFH 7DOO HW DO Learning how to construct formal proof is given WR PDNH VHQVH D IRUPDO GH¿QLWLRQ WKDW FDQ EH used in building the basis of deduction theorem. It is indicated that undergraduate students must develop their mathematics knowledge formally. A formal proof is based on rigorous logic and can be communicated like a discussion of PDWKHPDWLFLDQV RQ VFLHQWL¿F IRUXP 7KHUHIRUH undergraduate students need proving exercises so that they will able to understand a formal mathematics structure.
The process of proving a mathematical proposition is a sequence of mental and physical actions, such as writing, thinking tobegin the SURRI GUDZ GLDJUDPV WR UHÀHFW RQ SUHYLRXV actions or trying to remember the example. The process of proof formation of a theorem of or statement is more complex than the proof itself 6HOGHQ 6HOGHQ 7KHUHIRUH WHDFKHU LV needed for facilitating in learning mathematics thus facilitate students in mathematical proofs. In mathematics, one of the teachers'aid to help the students in order to make it easy to perform mathematical proofs is to make it into a tangible SURRI 6RZGHU +DUHO 6R WHDFKLQJ assistance on mathematical proof to the students can be done gradually and trying to create proof into something tangible.
Many researches revealed that many VWXGHQWV KDYH GLI¿FXOWLHV  can help teachers or lecturers in order to give an appropriate learning assistance. Some suitable learning activities should be designed to support the construction of this thinking process. In addition, if students' thinking process is incorrect, WKHQ UH¿QHPHQW WKLQNLQJ SURFHVV FDQ EH HDV\ LQ order to it does not occur in next learning.
In this article, we will describe a students' comprehension of proof based on quadrant model in Figure 1.

RESEARCH METHOD Participants
7KH UHVHDUFK ZDV FRQGXFWHG RQ VWXGHQWV majored in mathematics education in public university at Banten province, Indonesia. The consideration of that was because the students were able to think a formal proof in mathematics. And also, the students are more easily managed by the researcher to follow the procedures empirically planned, so that the data obtained is D VWXGHQWV ¶ DFWXDO RI UHÀHFWLRQ WKLQNLQJ *LYLQJ D SURYLQJ TXHVWLRQ LQ WKH EHJLQQLQJ DLP WR VHOHFW students who will be the subjects, which further deepened with the interview. The diagram of VHOHFWLQJ UHVHDUFK VXEMHFW LV )LJXUH

Instruments
The main instrument in explorative research was researcher itself. The support LQVWUXPHQWV DUH SURYLQJ WDVN DQG LQWHUYLHZ guides. These instruments were evaluated and validated from two lecturers in order to guarantee the quality of instruments.

RESULTS AND ANALYSIS Results
The following are the description of the data of the failures experienced by students. The tool used in the evaluation of the proofs was the (UURU 3URRI (YDOXDWLRQ 7RROV 3((7 GHYHORSHG E\ $QGUHZ The most frequent errors were an error in claiming a statement, the error in making a statement, and one in choosing a proof type. Errors in claims occurred at the beginning of proof by stating n a , with a is an integer. The error in choosing a proof type was by choosing an incorrect method.
Based on PEET results, students' response FDQ EH IXUWKHU FODVVL¿HG LQWR WKUHH FDWHJRULHV First, there are 6 students who are wrong in extracting a concept so become wrong in Students' proof construction depends on their understanding about the proving task. A different proof construction among students indicated that the understanding of proof is different too. Subject S1 knew how to begin the proof. And also, he could connect to appropriate PDWKHPDWLFDO FRQFHSWV ZHOO 6XEMHFW 6 NQHZ how to begin the proof, but he fails in connecting PDWKHPDWLFDO FRQFHSW 6XEMHFW 6 GLGQ ¶W NQRZ how to begin the proof, and so she solves the proving task using inductive reasoning. Subject 6 GLGQ ¶W NQRZ KRZ WR EHJLQ WKH SURRI DQG VR VKH fails in beginning a proof. Therefore, we suggest that the proof comprehension of these students  7KLV OHYHO encouraged student to construct a proof using GH¿QLWLRQ DQG ULJRURXV ORJLFV :HEHU DQG also using deductive reasoning and symbolic. Therefore, according to the three worlds, the WKLQNLQJ SURFHVV RI 6 LV LQ D[LRPDWLF V\PEROLF GHYHORSPHQW 7DOO According to Table 1 above, characteristics of proof comprehension is a complete comprehension. It indicated that both a local SURRI FRPSUHKHQVLRQ FRPSRQHQW DQG D KROLVWLF SURRI FRPSUHKHQVLRQ FRPSRQHQW are complete. Therefore, Subject S1 was able to construct proof correctly. According to Polya 0HHO WKLV VWXGHQW ¶V XQGHUVWDQGLQJ LV DW intuitive level. It showed that in his mind, there are three worlds of knowledge in mathematical XQGHUVWDQGLQJ L H DSSOLFDWLRQV PHDQLQJV DQG ORJLFDO UHODWLRQVKLS /HKPDQ 0HDQZKLOH following Skemp's understanding, Subject S1 has a relational understanding. And so, type of understanding of the subject is inventing OD\HU 3LULH .LHUHQ 7KHUHIRUH SURRI comprehension in this subject is called completed comprehension.
According to students' characteristics in this quadrant, we suggest that all learning strategy is appropriate for students in the quadrant. Mathematics is not only a subject to be learned and taught, but it is to be produced. However, with hermeneutics, it would be easy to develop our own idea and produce mathematics. Thus, some appropriate learning strategies for students in this quadrant are hermeneutics approach. This was the best opportunity for students to learn from the pioneers how to develop a new idea and create something new. Only with hermeneutics, teaching and learning mathematics and also UHVHDUFK LQ PDWKHPDWLFV FRXOG EH ÀRXULVKLQJ DQG IUXLWIXO 'MDXKDUL We suggest that one of appropriate leaning IRU WKLV TXDGUDQW LV XVLQJ WKH WZR FROXPQ IRUP method. This method can play a dual role, so student can connect some mathematical concept easily. While in many cases teachers do use the form in a way that limits students ability to WKLQN ÀH[LEO\ ZKHQ IRUPXODWLQJ DQ DUJXPHQW LQ other cases teachers can use the form in a way WKDW HQDEOHV JUHDWHU ÀH[LELOLW\ LQ UHDVRQLQJ DQG SURYLQJ :HLVV +HUEVW &KHQ Besides, students' characteristic is weak validation skill. It indicated that proof comprehension component of "transferring the general ideas or methods to another context" is a wrong validation. Finally, the students may not be able to distinguish proofs from supplementary or explanatory and so may not be able to distinguish proofs from supplementary or explanatory comments. It might be good to present material in a way that makes these comments. For example, explanations of GH¿QLWLRQV LOOXVWUDWLRQV RI UHOHYDQW FRQFHSWV cautionary remarks, and even remarks of proofs, except those needed to organize their linear presentation and help readers with validation, might best be treated as annotations. This might simultaneously provide prototypical examples for enhancing students' conceptions of proof and also encourage them to validate proofs carefully 6HOGHQ 6HOGHQ )ROORZLQJ *LEVRQ ¶V VXJJHVWLRQ 6XEMHFW 6 KDV GLI¿FXOWLHV in conceptual understanding and proof techniques DQG VWUDWHJLHV $FFRUGLQJ WR :HEHU :HEHU ¶V RSLQLRQ 6 ¶ GLI¿FXOWLHV LQFOXGHG WKHLU misunderstanding of a theorem or a concept and misapplying it and his inadequate conception knowledge about mathematical proof. Therefore, 6XEMHFW 6 IDLOHG LQ EHJLQQLQJ D SURRI According to Table 1 above, characteristics of proof comprehension is uncompleted comprehension. It indicated that both a local SURRI FRPSUHKHQVLRQ FRPSRQHQW DQG a holistic proof comprehension (component DUH LQFRPSOHWH $FFRUGLQJ WR 3RO\D 0HHO VWXGHQW ¶V XQGHUVWDQGLQJ LV DW UDWLRQDO level. Meanwhile, following Skemp's term, type of understanding of Subject S1 is relational understanding. Whereas, following other theory about understanding stated that type of understanding of the subject is formalizing OD\HU 3LULH

Characteristics of Proof Comprehension and
.LHUHQ 7KHUHIRUH proof comprehension in this subject is called uncompleted comprehension.
7KH VWXGHQWV ¶ FKDUDFWHULVWLFV KDYH QR SURRI structure, so that she/he failed in beginning a proof. One of learning strategies for this quadrant is structural method. Among attempts to improve students' proof comprehension in constructing a SURRI RQH FDQ GLVWLQJXLVK WZR EURDG DSSURDFKHV D FKDQJLQJ WKH SUHVHQWDWLRQ RI WKH SURRI DQG E FKDQJLQJ WKH ZD\ D VWXGHQW HQJDJHV ZLWK LW /HURQ 7KH DLP RI FKDQJLQJ WKH presentation is to make it easier in understanding SURRI VWUXFWXUH $ VWUXFWXUHG SURRI LV DUUDQJHG LQ levels, with the main ideas and approach given at the top level and subsequent levels giving GHWDLOV DQG MXVWL¿FDWLRQV RI HDFK RI WKH VWHSV LQ the preceding levels.
When the presentation is changed, explanations must be provided by the instructor. The explanations can help students who construct LQFRUUHFW SURRI WR XQGHUVWDQG KRZ FRQVWUXFWLQJ FRUUHFW SURRI ,Q WHUPV RI 0HMtD 5DPRV HW DO ¶V framework, a structured proof is designed to IDFLOLWDWH XQGHUVWDQGLQJ KLJKHU OHYHO LGHDV DQG identifying modular structure, though it does so at the expense of separating some claims from their supporting data and warrant. These changes DUH UHÀHFWHG LQ HPSLULFDO UHVHDUFK RQ WKH HI¿FDF\ of structured proofs. Fuller et al. found that, compared to those who read a traditional proof, students who read structured proofs were more successful at summarizing the key ideas of the SURRI )XOOHU HW DO 7KHVH ZD\ FDQ KHOS VWXGHQWV ZKR DUH LQ 4XDGUDQW ,9

CONCLUSION
We develop a quadrant model to describe VWXGHQWV ¶ FODVVL¿FDWLRQ RI SURRI SURGXFWLRQ Characteristics of students' proof comprehension LQ ¿UVW TXDGUDQW DUH FRPSOHWHG FRPSUHKHQVLRQ According to students' characteristics in this quadrant, we suggest that all learning strategies are appropriate for students in the quadrant. Nevertheless, we suggest that appropriate learning strategy for students in this quadrant is a hermeneutics approach. Characteristics of students' proof comprehension in second quadrant are uncompleted comprehension, especially in connecting mathematical concepts and validation skill. We suggest that one appropriate leaning for second quadrant is using WKH WZR FROXPQ IRUP PHWKRG 7KLV PHWKRG FDQ play a dual role, so a student can enable greater ÀH[LELOLW\ LQ UHDVRQLQJ DQG SURYLQJ Characteristics of students' proof comprehension in third quadrant are uncompleted FRPSUHKHQVLRQ HVSHFLDOO\ WKH GLI¿FXOW\ WR generate deductive reasoning. The students' FKDUDFWHULVWLFV KDYH QR SURRI VWUXFWXUH DQG also have a little conceptual understanding. One of learning strategies for this quadrant is asking to generate example and learning XVLQJ ZRUNHG H[DPSOH 0DQ\ PDWKHPDWLFV education researchers have suggested that asking learners to generate examples of mathematical concepts is an effective way of learning about novel concept. Characteristics of students' proof comprehension in fourth quadrant are uncompleted comprehension, especially in beginning a proof. The students' characteristics KDYH QR SURRI VWUXFWXUH VR WKDW VKH KH IDLOHG LQ beginning a proof. One of learning strategies for this quadrant is structural method.