The Failure to Construct Proof Based on Assimilation and Accommodation Framework from Piaget

The purpose of this article is to describe the process of a proof construction. It is more specific on the failure of the process. Piaget’s frameworks, assimilation and accommodation, were used to analyze it. Method of this research was qualitative method. Data were collected by asking five students working on problems of proof using think aloud strategy. Student’s works were compared with the problem structures. Based on the data analysis, it shows that there were three causes of failure of proof construction. The failure occurs on constructing proof because (1) Incomplete schemes on assimilation, (2) Incomplete schemes on accomodation process, (3) Complete schemes but unrelated on assimilation and accomodation process


Introduction
Proof has a significantly important role in mathematics or mathematics education.It is known as the core of mathematics and deductive thought (Hanna et al., 2009).Proof is a mathematical tool.If someone would like to learn mathematics, he/she has to learn how to construct proof or at least he understands it (Wu, 1996).It can be inferred that if students do not have sufficient competence in mathematical proof, they will have difficulty in learning mathematics.
In fact, some researchers (Dreyfus, 1999;Cadwallder, 2009;Lee, 2009;Azrov, 2013) have found out that mostly student meet difficulties to construct proof.Therefore, there were some studies concerning with this.J. Selden and A. Selden (1987) had categorized kinds of errors resulted by students when they tried to prove theorems.Pinto (1999) found that a different path when the student failed to apply concept of definition in constructing formal proof, i.e. giving meaning unsuccessfully and extracting meaning unsuccessfully.Dreyfus (1999) concluded that students met difficulties to work on proof problem, it was due to their inability to use proper term to deliver their ideas.Gholamazad, Liljedahl, and Zazkis (2003) analyzed why a preservice teacher for elementary school could fail to construct proof.Furinghetti and Morselli (2010) analyzed the invalid proof resulted by the student was caused by affective and cognitive factors.
Study about the processes that undergraduates use when they attempt to construct proofs isnot much.Alcock and Weber (2010, p. 6) said while there has been considerable research on undergraduates' difficulties with proof, some researchers have noted that there has been comparatively little work about the processes that undergraduates use when they attempt to construct proofs, and that more research of this type is needed.To find out what made the students failed to constructproofwould of course be useful as guide to design the most effective way to teach proof.This article examines examples of wrong proof or invalid proof produced by the students.However, the study focuses more on identifying the failure of students in constructing the proof by analyzing the mistakes of the child's thinking process based on Piaget's framework of assimilation and accommodation

Assimilation and Accommodation Framework
Piaget (Kaasila, Pehkonen &Hellinen, 2009) had confidence that individuals have to adapt with environments.For this Piaget discribed two processes of adaptation that represents the ability of organisms to adjust with their environment.They are assimilation and accommodation.Based on the Subject's says, it could be concluded that she had concept scheme of subset as relation operator and union as binary operator partially.Consequently, she applied assimilation process by using thinking scheme incompletely Subject is thinking structure is not compatible with problem structure resulted by her incomplete thinking schemes when doing the process of assimilation and accommodation,.So that she was stuck on that phase and could not progress to proof process.Therefore it can be said that Subject failed to construct proof because of incompleteness thinking structure during the process of assimilation and accommodation on the first premis.This failure will then couse for E4 error to exist or the error on applying theorems (J.Selden & A. Selden, 1987) called this case an inability to read the problem as well a tendency to have a quick problem solution.
The failureto construct proof on the second premis can described as follows,the second premis is A∩B = ∅, Subject also has the knowledge scheme about intersection concept as in the statement on the premis.When she could read it well, that "there is no set intersection between A and B. However, what she say is not same by what she drew.She drew two circles intersected, although drew the symbol of empety set (∅ at the intersection area.
As we know, the diagram pict should it have been intersected.It showed that Subjecthave the knowledge of intersection but unrelated.
That sort of knowledge level disabled her to solve the proof problem to construct proof, so that she failed to construct proof.Based on the type of error made by Subject to proof according to J. Selden and A. Selden (1987) Subject had an error E5 that is circularity, an error to the statement and error of E6 that is the proof can not be understood locally.It has been seen here, the incomplete and unrelated thinking scheme made her failed to construct proof.It occure on both given premis.Consequently, she failed to relate the statements on premis to the conclusion.Herewith, Subject could be said as a whole failure to construct proof.

2) Second task proof problem
The answer sheet of Subject for the second TPP, as in the following Figure 4 berikut.Explanation of incomplete schemes on accommodation proces thinking is quite similar with the problem of incomplete thinking schemes on assimilation process, it is about the incompleteness scheme.The difference is the process of assimilation occurs just when they face the problem, while the accommodation process takes place after they face the problem and had done several thinking processes such as remembering, recalling long-term memory or associating with matters related to the problems encountered.Such efforts will emerge the scheme needed but it can not be.If it is not, then the condition is called as Incomplete thinking schemes on accomodation proces.As experienced by the subject when they try to understand the statement A ⊆B ∪C.Various attempts have been made by the subject to understand the statement like reading the questions repeatedly, make a Venn diagram.But the subject remains understand the statement as (A ⊆B) ∪C.c.Complete schemes but unrelated on assimilation and accomodation process.Condition of complete thinking schemes butun related to assimilation process also appears when they faced with statement A ⊆ B ∪ C. It was initially expected that the subject did not have a schematic of the formal definition of a subset.In fact, he/she has a union set definition.It was known when the interviews were conducted after the completion of work on the TPP.Researcher : you write the definition of subsets and it can be used for proof of this.
Subject : oo yes.This differs with the incomplete scheme.In incomplete scheme in the assimilation and accommodation, the students really do not have scheme which has been given an intervention.
In constructing proof.If one of the four condition above occurs or if all those conditions occurs it may cause impasse that student does not know how to continue.Weber (2001, p. 2) Said "Students often fail to construct a proof because they reach an impasse where they simply do not know what to do".
There are some effects of incompleteness and unrelated knowledge schemes.Incompleteness or unrelated of knowledge schemes will make students relate difficult to manipulate a mathematical expression that is equivalent to other forms.They are unable to write a concept image held in the form of formal symbols and otherwise unable to make a concept image of a known object.Students also do not have warrant to make conclusion and it was often found out that students have verbal knowledge but they were unable to write in the formal symbol.Such conditions cause the impasse or incompatibility in proving that leads to failure to construct proof.
The study related to schemes constructed by students in problem solving has been done by Subanji (2007Subanji ( , 2011)).The study found that there were three factors causing pseudo thinking processes, namely (1) Incomplete substructures in the assimilation process, (2) Incomplete substructures in the accommodation, (3) mismatching on the use of substructures in the process of assimilation and accommodation.Thinking pseudo is a result of a problem solving process and not the output of real mental activity.Remembering proof is a special form of problem solving (Weber, 2001;Furinghetti & Morselli, 2010) then pseudo thinking process can appear when students construct proofs.It can be further investigated in subsequent research .
Discovering how the construction process of proof done by the students is very essential, because it can give an idea of how success and failure proof construction going on.It is also useful to get guidance on how proof should be taught and in general how a concept should be constructed in the student's mind.Incompleteness scheme does not mean the students have not studied the scheme, they might have learned but the concept constructed were not meaningful so that the concepts were difficult to remember.Subanji (2015) said that another cause is wrong construction of scheme.He found that there were many mathematical concepts constructed by junior high school students mistakenly.It would affect to the outcome of next assimilation and accommodation in solving problems and/or constructing a new scheme.Scheme has two main functions, namely (1) to integrat existing knowledge and (2) as a mental tool for the reception of new knowledge (Skemp, 1982).
The conclusion that can be drawn from this research is that the failure of the proof construction not only as presented by Selden, Benkhalti, and Selden (2014), but also occur due to the incompleteness of the scheme knowledge of students when assimilation and accommodation and or as a complete scheme but not connected in the process asimolasi and accommodation.This failure factors could be more than what has been found in this study.What if student has a complete scheme but he/she can not use these schemes in constructing proofs.This will be further investigated in subsequent research.
scheme, it ha pear in intervi appear in thin Figure Based on Her answer, Subject drawn representation of ⊆ , then she turned the page around for several times to read the information/premis and also the conclusion.After that she drawn outer circle as represantation of set .At that time, observer confirmed to Subject about her thinking process.Consider the following transcript of conversation between observer and Subject:Researcher: Please, explain the meaning of your draw (pointed to the Subject's draw) ies.ccsenet.Subject:Because there is no parenthesis in here ( added parenthesis on the text of problem)Researcher : Ok, would you please five is equal to two plus three.Subject : (wrote down directly 5 = 2 + 3) Researcher : then, wil you read it as 5 2 and then plus 3? Subject : eh...yes I will (smile) As the following dialogue: Subject: I think definition is only a statement that it is not like this.