Thinking Process of Pseudo Construction in Mathematics Concepts

This article aims at studying pseudo construction of student thinking in mathematical concepts, integer number operation, algebraic forms, area concepts, and triangle concepts. 391 junior high school students from four districts of East Java Province Indonesia were taken as the subjects. Data were collected by means of distributing the main instrument and tracer instrument to the subjects respectively. Both instruments were deployed for the purpose of digging up the construction process. The construction was clustered on the basis of the pseudo contruction cases and followed with in-depth interviews to three subjects of each case. The findings show that pseudo construction was identified in four cases. The first case was associated with integer operations, operations of algebraic form, the concept of area, and the concept of triangle. They used analogy of “in debt” to construct concepts of negative number operation. In the second case, they used objects (book, pencil, and thing) to describe variables in the algebraic form operation. For the third case, students deciphered unit area (m) as multiplication m x m. In the fourth case, students did not pay attention to the requirements of the triangle. Although they gave a right answer to their work, their construction concept was completely false.


Introduction
Indonesian junior high school students have a lower achievement in mathematics for International Studies TIMMS 2011. Mulis et al. (2012) reported that Indonesian 8th grade students' achievement in mathematics of international studies TIMSS2011 was at a low level with a score of 386 even lower than the score in 2007, namely 397.In 2011, Indonesian students ranked 38 th out of 45 participating countries, while in 2007 ranked 36 th out of 49 countries.The students' low achievement could be reflected in terms of content domain and cognitive domain, particularly in Numbers, Algebra, Geometry, and Data and Chance.This fact has been a serious concern in the context of education in Indonesia, and the government has tried to undertake two important policies: reforming the primary and secondary education curriculum, including curriculum 2007 and Curriculum 2013 and certifying teachers so that they become more professional.Both policies were undergone to improve students' quality of learning.However, a study conducted by Subanji and Nusantara (2013) indicates that middle school students in East Java Indonesia made mistakes when dealing with mathematics.These mistakes were closely related with three typical characteristics: pseudo thinking, mistakes in the use of analogies, and misconceptions.
Pseudo thinking is a thinking process that results in an answer to a problem or construction to a concept "that is not true".The construction concept does not represent the actual thinking.Subanji (2007) explains that the pseudo thinking can be classified into two forms: true pseudo and false pseudo.In the context of problem solving, true pseudo happens when a student answers a question correctly but the process of thinking is wrong.False Pseudo occurs when a student answers a question incorrectly, but he/she is able to reason correctly.In the context of the construction of concepts, true pseudo happens when the concept a student writes seems to be correct, but his/her understanding about the concept is wrong.False Pseudo happens when a student writes the concept wrongly, yet his/her understanding about the concept is correct.This study examines the students' thinking processes in constructing a true pseudo concept of integer operations and algebraic forms, hereinafter called pseudo construction non-routine problem solving.Pape (2004) used the term Direct Translation Approach (DTA) versus Meaning Based Approach (MBA) in the context of solving word problems.Leron and Hazzan (2009) applied the Dual Process Theory of Kahneman (process S1 versus S2 processes) in the context of solving algebra problems.Studies conducted suggested pseudo thinking, but not discuss in detail the process of formation of pseudo construction.Nevertheless, a study of pseudo thinking in students' problem solving is still limited.Subanji (2007Subanji ( , 2013)), Subanji and Supratman (2015) explored the process of pseudo thinking more deeply by using the framework of Piaget's assimilation and accommodation.In this study the thinking process about the construction of the mathematical concept of "pseudo" is discussed in detail.It emphasizes pseudo thinking in the process of concept formation of integer operations, algebraic form, area concept, and triangle concept.
According to Subanji and Nusantara (2013), the mistake made by students in doing mathematics requires attention; the mistake would seriously affect the subsequent understanding of their mathematical concepts.In order to minimize the impact of the mistake in building the next concept, it is important to track the sources and causes of the mistake.The sources can possibly be found in the formation of student's thinking scheme called the construction process of student's concept.The construction can be seen in detail by using Piaget's framework, namely assimilation and accommodation, which is figured out in the form of cognitive map.Piaget (in Huitt & Hummel, 2003) stated that when someone gets a new stimulus, there are two processes used by the individual in its attempt to adapt: assimilation and accommodation.Both of these processes are used as the person increasingly adapts to the environment in a more complex manner.Assimilation is the process of using or transforming the environment so that it can be placed in preexisting cognitive structures.Accommodation is the process of changing cognitive structures in order to accept something from the environment.Both processes are used simultaneously and alternately throughout life.Piaget (in Subanji, 2007) explained that, in the learning process, someone constantly adapts and involves the process of assimilation and accommodation.The assimilation is a process of integrating a new stimulus into formed schemata.Assimilation occurs when there is a match between a given stimulus and schemata that already exists in the minds of students.Accommodation is the integration of a new stimulus to the schemes owned by changing the existing schemes.Accommodation occurs when the schema owned by the student is not in accordance with the stimulus.Someone needs to adjust the scheme by changing the old scheme or form a new scheme so that the scheme is formed according to the stimulus.Subanji ( 2007) illustrated Piaget's process i.e. assimilation and accommodation as shown in Figure 1.

Figure 1. Assimilation and accommodation process
In Figure 1 (a), assimilation occurs when the problem structure is in accordance with the owned scheme, hence directly interprets in the correct way to form a new structure scheme.In Figure 1 (b), the scheme of structure thinking is not in accordance with the structure of the problem.In order to interpret in the correct way, there should be a conversion of the old scheme or a new scheme formation is made such that the structure of thinking can be aligned with the structure of the problem.Hence, the structure of the problem can be properly integrated into the new formed scheme.
The use of cognitive map to depict scheme formation has been addressed in several studies.For example, Jacobs (2003) revealed that the cognitive map indicates the direction of thinking.It can be used as guide for the next step of thinking.Pena et al. (2007) asserted that the cognitive map illustrates the causal relationship of the various phenomena and concepts, and can be modeled.Perdikaris (2012) described the cognitive style of students in solving geometry problems in term of Van Hielle Theory.Subanji (2007Subanji ( , 2011) ) used a cognitive map to assess pseudo thinking of students in solving mathematical problem.In this study, the schematic picture of students' thinking in constructing mathematical concepts using cognitive maps was depicted to capture the occurrence of mistakes in constructing the concepts.

Method
The objective of study is to analyse the formation scheme of students' thinking by using cognitive maps.This study was conducted in early June 2014 at junior high schools.391 students from four districts of East Java Province, i.e.Malang, Blitar, Tulungagung, and Jombang were taken as the subjects.In each district, two until four schools were selected carefully, depending on whether or not they were willing to participate in the research.The subjects in Malang and Blitar were selected from "good" schools.In contrast, the subjects in the regions of Tulungagung and Jombang were selected from the mainstream schools.The subjects were taken from IX grade junior high schools, after they had joined the National Examination.There were several reasons for it: (1) students had learned all of the materials required in the curriculum; (2) the research did not disturb the school activities; and (3) students were not burdened by the school exams; they could freely express their ideas.
Two types of instruments were used in this study: main instrument and tracer instrument.The main instrument was used to explore the student's thinking process in constructing concepts of integer operations, operation of algebraic forms, the concepts of area, and the concepts of triangle.The main instrument contained statements about the mathematical concepts with which the students could justify whether or not the concept is correct or wrong as shown in Table 1.They also gave reasons to strengthen their justification.
There is a triangle with sides length 6 cm, 7 cm, and 14 cm 4 A rectangle with a size of 6 m x 5 m.The area of the rectangle is 30 m 2 .A m 2 unit derived from m x m) The tracer instrument was constructed to provide alternative variety of construction.The subjects were given an opportunity to provide reasons of their answers in order to reinforce, change, or alter their opinions.The tracer instrument was also used to justify the concept of construction process.The following tracer instrument was used in this study.This instrument was done by students after they had completed the first instrument, of course, after a given pause of two hours.True, because it has a debt 4 and debt again 3 so that its debts to 7 False, because it has a debt 4 then pay by 3 so that its debts to 1 False, because negative 4 meets negative 3, it should be positive 7 True, because there are negative 4 and negative 3, then it becomes positive 7 True, because by using a number line, from negative 4 to move backward in three steps and becomes -7 True, because by using the pattern, -4 True, because negative meets negative is positive; -4 +3 = -1 True, because negative times negative is positive; -4 + 3 = -1 True, because it has a debt 4 and is paid 3, the result is -1 False, because it has a debt 4 "minus".It means it has another debt 3, so that its debts becomes 7 3 There is a triangle with sides length 6 cm, 7 cm, and 14 cm True, because there are 3 (three) sides so it can be made a triangle False, because when side of 6 cm is added by 7 cm, the result is 13 cm, shorter than 14 cm.So, it should be made longer.
False, because it does not meet Pythagorean Theorem's; 6 2 + 7 2 is not equal to 14 2 True, because number times number (6 x 5) and unit times unit ( m x m) False, because m 2 in unit area (not mxm) True, because, let x = book, then two books plus three books, the result is five books True, because 2 books plus 3 pencils is 5 books pencils False, because the variable is different so quantity isdifferent, let x = book and y = pencil, then book and pencil could be added True, because 2 plus 3 is 5 False, because there is no property of summation (commutative, associative, and distributive) that guaranties The study was conducted in several steps.First, the subjects finished the main instrument that is related to statements of essential mathematical concepts.Here, the students' pseudo construction of mathematical concepts was obtained.Second, students did the tracer instrument the result of which was used to explore the construction process based on pseudo construction cases.Third, three students representing each pseudo construction case were interviewed in relation to integer operations, operation of algebraic forms, the concepts of area, and the concepts of triangle.The interviews were to triangulate the data and map the construction process of mathematical concepts.The construction process was then described through a cognitive map.The data analysis was carried out by grouping students' answers from the main and tracer instruments based on the types of pseudo constructions.The interviews were qualitatively analyzed and the construction process was explained by using a cognitive map.The students' pseudo construction of mathematical concepts could be explored.

Results and Discussion
In the following paragraphs, we will describe the types of mathematical construction that have been done by the subjects.The description is based on all of the subjects' answers in the instruments.All possible answers and reasons given by the subjects were identified and recorded.The coding was then done on the subjects with similar answers and reasons.Subjects with the same codes were used as one group.Next, the groups were selected for further interviews and think aloud.

Pseudo Construction Concept of Integer Number Operation
The following table presents the students' answers categorization based on the main instrument and tracer instrument.Table 4 shows that students who were pseudo thinking reached 44% (i.e.141 out of 319 students gave true answers).The result of mathematical construction by students often looks different from what they wrote on the answers.When we look up and down their work, their answers gave the effect of true, but when we trace it back by interviews and think aloud, what the thought was different form the essential concepts.Their construction is impressively true but actually is wrong.This kind of construction is named by pseudo construction.
The fault of students thinking scheme formation was captured when they constructed integer number operation concept.They were faced with problem of giving judgement and reason of the statement "True or false the statement of -4 -3 = -7 and -4 -(-3) = -1."Most of students gave true answers of the statement.The students evaluation seemed true, but when their reasons were traced, almost all students did a assimilation process by interpreting negative number (-) and minus operation by "debt" or "obligation".They mean the number -4 by 4 debt."Minus 3" was also assimilated by in debt 3.They could not distinguish between negative number symbol and minus operation, both of them were assimilated by "in debt".They did not understand that both symbols of (-) in case of -4 -3 were different.The symbol (-) of -4 constitutes a negative number symbol, but the symbol (-) of -3 means a number operation.
Students start to have dis-equilibration when they confront to the statement of -4 -(-3) = -1.Their representation of debt on the negative number and minus operation could be used to the problem -4 -(-3).Minus of negative 3 could not be represented as in debt (in debt 3), because there is no concept of in debt (in debt).Dis-equilibration causes the subjects do accommodation by changing their thinking structure.The following text is the conversation between the researcher (R) and the subjects (P2, P3, P4) related to justification of their answers.

R: Is the statement true? And what is your reason? P2:
It is true, because at the beginning, it has debt 4 then is paid 3 hence still has debt 1. Minus meets minus resulting in plus, Mam.The problem says, minus 4 minus negative 3 it will be minus 4 plus 3.

P3:
It is true, this negative meets negative it will be positive, consequently negative 4 plus 3 is the same as negative 1. Yes, this negative times negative, bracket symbol means multiplication, so negative times negative the same as positive, the same with the previous reason.
The accommodation process was not based on the true mathematical concepts.The justification of -(-3) to be 3 by saying negative meets negative to be positive or negative times negative resulting in positive was forced by the subjects to perform their next step.In the concept of number operation, multiplication can only be applied to number.There is no concept of multiplication of negative number with minus operation.Their given results seem to be true, but the construction concept of student is still pseudo, we name it as pseudo construction.Fault in the construction of concepts by students is a fundamental mistake (Bingobali et al., 2011;Brodie, 2010;Gal & Linchevski, 2010).Brodie (2010) explained that fault made by students in mathematics learning occurred when they were building mathematical reasoning, which includes: basic mistakes, appropriate mistakes, missing information, and partial insight.Mistakes in the construction of the concept of integer operations that have been performed by the students can be referred as a basic mistake.Gal and Linchevski (2010) found that there had been a mistake students made in learning mathematics, especially in a geometry representation process that includes: (1) the perceptual organization: Gestalt principles, (2) recognition: bottom-up and top-down processing; and (3) representation of perception-based knowledge: verbal vs. pictorial representation, mental images and hierarchical structure of images.The representation of perception-based knowledge can be extended to a representation of the symbol.Symbol of negative numbers and symbols from operations minus expressed by students as "in depth" is a fundamental mistake.According to Bingobali et al. ( 2011), students' mistakes in the concept construction of integer operations are included in the concept abstraction fault.Pseudo construction conducted by the subjects can be described in the cognitive map below.In constructing the concepts of integer operations, the subjects did not use the number line as a basis for work.They preferred to use an analogy of "debt" as a representation of negative numbers.The subjects also represented the epitome of operation and number as the same thing i.e. "debt".As a result, the subjects could not give reasons when there was a statement "subtract by a negative number".They made justification that negative meets negative result to positive or negative times negative yields a positive.As we know, in integer operations, only numbers can be multiplied.There is no concept of multiplication of numbers with the operation.
The construction concept of "minus a negative number" should be used by a number line or with a pattern.In terms of pattern, students are expected to find the concept of "subtract by the same negative number which means the opponent is added with negative numbers" as shown in the following pattern.

Pseudo Construction Concept of Algebraic Form
When students were faced with the problem 2x + 3x = 5x, students answered correctly but when explored further, the students' constructions were apparent (i.e.Pseudo construction).Students constructed the variables x and y is not as numbers, but as "objects" (i.e.books or apple or other).Summation 2x + 3x = 5x is assimilated with 2 books plus 3 books equals 5 books.2x + 3y is assimilated with 3 plates + 2 spoons.Because the objects were different object, they could not be summed.Their construction was not based on the mathematical concepts.
Table 5 shows that students who were pseudo thinking reached 63% (i.e.167 out of 264 students gave true answers).The following text represented the argument of students related to clarification of their answers.

P2:
True.Because both contain x, so x can be summed.If both the form of the book entirely into two books plus 3 books that is equal to 5 books P3: True.Because both variables are equal.So it can be summed.If for example x that book means 2 books plus 3 books so there are 5 books P4: Right, so we suppose that the object x, such that the 2 apples plus 3 equals 5 apples In the algebraic form, variable x is not declared as an object but expressed as a number, so that the representation 2x + 3x can be operated as x expressed as numbers.If x numbers, then there is a distributive properties which ensure that 2x + 3x = (2 + 3) x = 5x.If x represents the object, then no one can guarantee the nature of the operation can be performed.Because the context variable in the algebraic form is a number, the variable x will be true if replaced with the price of the book or price of apple not as an object of books or apple, because the price is a number.
The subjects also experienced pseudo construction when faced with the statement 2x + 3y = 5xy.The subject stated that the statement 2x + 3y = 5xy is wrong, but when explored further, their reasons were not appropriate.

P2:
Not true, because 2x and 3y have different variables.Suppose we have two dishes plus 3 tablespoons, it is true that the number of objects is five but the plate and spoon cannot be summed together P3: I am not sure the answer is correct Mom, because I think that's variable x equals to y so results remain xy and 2 plus 3 equals 5, so it would 5xy Subject P2 constructed the variables x and y is not as numbers, but as objects.P2 gave a reason "cannot be summed of 2x and 3y" not because of the properties of numbers in mathematical operations, but because they are different objects.In this case, the apparent construction occurred in P2 when building knowledge of algebra operations.The process of pseudo construction of students in algebraic form operations can be described in the following cognitive maps.The properties of the algebra operations derived from the properties of number operations, such as addition, subtraction, multiplication, division, and root or exponent.Statement 2x + 3y actually states the representation of a number to the value of x and y, and 2x + 3y representation as a number is not owned by the student.So, they constructed 2x + 3y as a set of objects, then a mistake occurred when two books and three pencils plus, as they collected two books and three pencils and there were five objects such as pencils and books.
Fault of representation of algebraic form may inhibit the transition from arithmetic to algebraic thinking (Elizabeth, 2003;Trygve and Barbro, 2006).Elizabeth (2003) emphasized the importance of the mathematical properties: associative, commutative, and distributive in the transition from arithmetic to algebraic thinking.When students represent the variables x and y as an object, it cannot use the mathematical properties and this could hamper subsequent learning algebra.Therefore, the representation 2x + 3y as numbers are very essential in learning algebra.Trygve and Barbro (2006) asserted that in the last decade there was a shift from behavioristic perspective to a thorough analysis of the cognitive competencies that are involved in learning algebra.Tracing the thinking of students in constructing algebraic concepts is very important in order to know the mistakes and make efforts to repair.The importance of student math error/mistake correction was also confirmed by Shein (2012).In learning mathematics, the construction process of the math concepts students need to be constantly monitored in order to know immediately if there is a mistake.Assessing pseudo thinking of students in constructing mathematical concepts as steps can be used to explore students' thinking mistakes.

Pseudo Construction Concept of Area Concept
Pseudo construction also occurs when students constructed concept of an area.Students could calculate area and could write unit extent by m 2 , but the process of constructing was pseudo.The concept of area has not been constructed, yet only the procedures were successfully constructed.It was characterized by a statement of the student m 2 unit area resulting from m x m instead of the unit square with sides of 1 m.The research instrument asked students to "assess the following statement" "A rectangle with a size of 6 m x 5 m.The area of the rectangle is 30 m 2 .A unit m 2 is derived from m x m" Table 6 shows that students who were pseudo thinking reached 81% (i.e.286 out of 352 students gave true answers).When the researchers examined them further, it turned out that the students thought that multiplication m x m was assimilated with multiplication in the algebra i.e. .Students did not think about the concept of area but thinking about algebra as shown in the following statement.

S3:
True.Because the formula of rectangle area is length times width.Yes, then we multiplied the meter times meter, becoming meter squared.Yes like that.This is the same as in algebra.
If a multiplied by a will produce a squared, multiplied by the same thing m times m to be m squared The students constructed the area by assimilating the multiplication of two numbers and unit area obtained from the multiplication between the units.They constructed the concept of a rectangular area by a pseudo manner.From the responses, the students were able to calculate the area and could write unit by m 2 , but the process of constructing was still a pseudo.Their constructed concept of area was not true area concept.Pseudo thinking processes of students in constructing the concept of the area can be described through a cognitive map in Figure 5.The pseudo process experienced by students in constructing the area concept is classified as a true pseudo thinking (Vinner, 1997;Subanji, 2011Subanji, , 2015)).Students "seem properly construct" the concept of the area, but when explored in depth, they constructed the wrong thing.

Pseudo Construction Concept of Triangle Concept
The following are details of the reasons given by students when assessing the statements of the triangle.Table 7 shows that students who were pseudo thinking reached 65% (i.e.162 out of 251 students gave true answers).The construction pseudo was observed in students from the absence of triangle requirement.In the concept of triangle, there is a requirement that sum of any two sides length should be greater than length of another side.When students were faced with the statement "there is a triangle by sides measure 6 cm, 7 cm, and 14 cm", the students considered that the statement was "true".Students did not pay attention to that 6 + 7 = 13 < 14, this condition was certainly not compatible with the requirements of triangle.Students did not know or did not pay attention to these conditions and immediately concluded that the triangle could be made, because there were three sides.
Translated version of the reason: Because, there is no information about triangle in the statement; right triangle or arbitrary The student correctly answered the given statement.But the reason given was not right.He gave a reason that it was not a triangle, not because they did not meet the requirements of the triangle, but because it did not meet the Pythagorean triple.Errors construction process of students in the concept of a triangle can be described in a cognitive map in Figure 9.The construction process of students in understanding the triangle is affected by the grouping procedure of right triangle.Students used to check the triangle by checking the size of the three sides meeting the size of the right triangle.The habit of checking the right triangle directly used in this process of assimilation is "checking" with the Pythagorean concept.

Conclusion
The process of students' pseudo thinking in constructing mathematical concepts is necessary to get attention.Teachers need to be aware whether or not the students really understand the concepts being taught.Teachers also need to think about how the concept that was presented to the students was strengthened with a variety of exercises, as well as reconsidering the material prerequisites prior to new concepts.As is known, the operation on integers and operations in the algebra is a basic concept in mathematics in secondary schools, which underlies the "almost" all the advanced mathematical concepts.Pseudo construction experienced by students on the concept of integer operations and algebraic forms would impede the process of studying advanced mathematics concepts.This pseudo construction will certainly bring difficulties to hinder the process of learning mathematics and subsequent construction of mathematical concepts.Students' mistakes in constructing the concept of subtraction operation integer that is dominated by the process of assimilation "in debt" as well as for negative numbers would complicate the learning of addition and subtraction of negative numbers.The mistakes in constructing the variables x and y as objects (books, apple, and so on) will complicate the learning about multiplication and rank of the algebra.2x + 3x assimilated with 2 books + 3 books still make sense, but when faced with the problem of 2x2 + 3x2 will not fit anymore, because there is no concept of "book squared".Mistakes in constructing the concept of area and triangle lead to learning mathematics meaningless

Figure 3 .
Figure 3. Cognitive map of pseudo construction of negative number subtraction

Figure 4 .
Figure 4. Cognitive map of pseudo construction of algebraic form operation Figure 5. Subjects jugdement and reason of area problem

Figure 6 .
Figure 6.Cognitive map of pseudo construction of area concept

Figure 7 .
Figure 7. Subjects jugdement and reason of triangle problem

Figure 9 .
Figure 9. Cognitive map of pseudo construction of triangle

Table 1 .
Main instrument

Table 6 .
Summary of students' reasons about the area problem

Table 7 .
Summary of students' reasons about the triangle problem