Symbolic and Verbal Representation Process of Student in Solving Mathematics Problem Based Polya ’ s Stages

The purpose of this research was to reveal how the construction process of symbolic representation and verbal representation made by students in problem solving. The construction process in this study referred to the problem-solving stage by Polya covering; 1) understanding the problem, 2) devising a plan, 3) carrying outthe plan, and 4) looking back. This research was qualitative research by getting involved 4 students of Junior High School class VIII. This study obtained results that the construction process of symbolic representation made by students since in the process of understanding the problem. In understanding the problem, students were able to identify the problem well. Then students could make the symbol used as a variable that the value was not known. By using the symbol students could perform a series of calculation to obtain the value of the symbol. Symbols created by students were very helpful and facilitated students in solving problems. While in the construction process, verbal representation was done by students since the process of understanding the problem. The form of verbal representation was manifested by writing down all information known from a slightly changed problem because they used a language that they understood more. Based on the known information, students could plan and performed a series of calculations using written sentences so that the problems could be solved properly.


Introduction
National Council of Teachers of Mathematics (2000) set five standard processes that must be mastered by students such as representation.Mathematics is an abstract science.Mathematics really needs representation because one can get mathematical ideas that one of them is through representation (Kilpatrick, Swafford, & Findell, 2001).Through representation students can make mathematical ideas to be more concrete and facilitate to do reflection (NCTM, 2000).Even the representation is seen as an important part of mathematical activity and means to capture mathematical concepts (Cobb et al., 2002;Cobb, 2003;Gravemeijer et al., 2003;Meira, 2003;Kaput et al., 2008;Cai, 2005;Anwar et al., 2016).
Student success in solving the problem is not out of the representation role (Akkus & Cakiroglu, 2006;Hwang et al., 2007;Delice & Sevimli 2010;Bal, 2014;Boonen et al., 2014).Thus, the mathematical problem solving using representation is strongly recommended because it can be used as a tool to facilitate during the problem solving process (Stylianou 2008).Schoenfeld (in Stylianou, 2010) calls representation as a means to understand the information obtained in problem situations.Newell and Simon (1972) see that representations can be as a means of recording while Stylianou and Silver (2004) call representation as a tool that facilitates the exploration of concepts or problems.In addition Gibson (1998) and Izsak (2003) view representation as a means of monitoring and evaluation to assess progress in solving problems.
Representation is a form of ideas disclosure embodied using various means such as spoken language, written language, symbols, drawings, diagrams, models, graphs, or using physical thing (Goldin, 2002).Representation can arise as a result of an individual response to a problem he faces.The resulted representation is strongly influenced by one's learning experience (Hutagaol, 2013), so if some people are faced the same problem then it is possible to obtain several different representations.
According to Goldin and Steingold (2001) there are two systems of representation namely external representation system and internal representation system.External representation system includes a conventional representation which is usually symbolically while internal representation system is created in one's mind and used to define mathematical meanings.Villages et al. (2009) mention the types of external representations namely: 1) verbal representation; it is fundamentally expressed in writing or speech.2) Pictorial representation; Including images, diagrams or graphs, as well as some interrelated activities.3)Symbolic Representation; Consisting of numbers, operations and connection signs, algebraic symbols and some interconnected actions.
Several studies have been conducted to see the kinds of representations that are effective in solving the problem (Çelik & Arslan, 2012;Boonen et al., 2014;Dündar, 2015).These researches specifically describe the construction process of symbolic and verbal representations done by students in problem solving.Problem solving in this research referred to the problem solving stages by Polya (1973) which include: 1) Understanding the problem: understanding all the words in the problem.2) Devising a plan: making a strategy to solve the problem in the form of guessing, drawing, looking for patterns, modeling.3) Carrying out the plan: applying the selected strategy so the problem can be solved.4) Looking back: reconsidering the whole process of problem solving.So that it will get the picture of symbolic and verbal representation process in problem solving.

Method
This research revealed how the construction process of symbolic representation and verbal representation were done by students in problem solving.So that through this study will illustrate the construction process theory of symbolic representation and verbal representation.According to Cresswell (2012) this research was qualitative research.In this study involving 4 Junior High School students of class VIII, used as a research subject.4 students as the subject of the study were selected with the consideration of being able to make symbolic representations and verbal representations in problem solving.The process of collecting data in this study begins by giving problems to the subject.The problem used in this research was "Uncle will distribute 20 candies to Budi, Dian and Anwar.Budi gets 8 candies, Dian gets three times more candies than Anwar.How many pieces candy does Dian get?".After the subject solves the problem, the researcher conducts an interview based on the subject's work.Interviews conducted aim to reveal how the subject can make symbolic representations and verbal representations.In addition, the interview is also to clarify certain parts that have not been known by researchers.In data collection, researchers also use video recorder to record all the subject activity in solving the problem.

Result
The research data was in the form of symbolic representation and verbal representation made by students during the problem-solving.Referring to the data, it is necessary to encode the symbolic representation and verbal representation first;1) Symbolic representation is characterized by the use of mathematical symbols in solving problems, 2) Verbal representation is characterized by the use of verbal form of sentences in writing in solving the problem.
Finding 1.The Process of Symbolic Representation in Problem Solving.

Description of Subject 1:
The process of symbolic representation was made by the subject during the understanding of the problem.In understanding the problem, the subject wrote all the information that was in the problem.In addition, subject also wrote symbol x which was used as the variable of the number of candies given to Dian and Anwar whose value was not yet known.Because Dian got candy three times more than Anwar, then symbolized by 3x.In addition to writing down the information in the problem, the subject also understood and wrote down what was asked by the problem.
Based on the result to understand the problem, the subject could create a linear equation of one variable that was 4 x + 8 = 20 (4 x was obtained from summing the number of candies received by Dian and Anwar symbolized by 3 x and x).Equation 4x + 8 = 20 was made by the subject appropriately.The equation had accurately described the problem situation.Based on the equation, the subject could determine the value of x = 3.Thus, subject could determine the number of candies received by Dian that was Dian received three times more candy than Anwar (symbolized by 3x).Therefore Dian received 3 x 3 = 9 candies.
To clarify the process of symbolic representation performed by subject 1, the following is represented the process structure of symbolic representation;

Discussion
This research described the construction process of symbolic and verbal representations done by students in problem solving.The first findings show that the research data specifically indicates the symbolic representation made by the students at the stage of understanding the problem.The students with symbolic representation used the symbol x and A was used as an information variable that the value was unknown.Based on these symbol, students could make the equations 4x + 8 = 20 and 8 + 3A + A = 20.The equations generated by the student had precisely matched the problem situation that was in the source of the problem.It was in accordance with opinion of Lesh et al. (1987) that the skill of symbolic representation is characterized by the skill in transforming mathematical problems into the representation of arithmetic formulas.The symbol was used effectively by students in problem solving.The symbol simplified the problem solving process.This finding was in accordance with the opinion of Dahlan and Juandi (2016) that in high school students, representations are useful for solving problems or clarifying, or expanding mathematical ideas starting from the process of collecting facts (data), arranging tables or graphs, until the development of symbolic representation (algebra).In addition, the result of this study also supports Schoenfeld's theory (in Stylianou, 2010) that representation is as a means to understand the information obtained in problem situations.Someone uses representation as a means to combine different aspects of the problem so that all obstacles can be seen and how they interact.This is an important function of representation since not all aspects of the problem situation can be readily captured by the readers.
The second finding showed that verbal representation was made by students at the stage of understanding the problem.Based on the student worksheet it appeared that students with verbal representations wrote down all the information that was in the problem with little change according to what they understood from the source of the problem.Based on the information they obtained in understanding the problem, students could perform a subtracting operation of 20 -8 = 12.The subtracting operation described the problem situation that existed on the source of the problem.Based on the results of the subtracting operation the students could perform a series of calculations to solve the problem.The calculation performed by the students did not involve the formal operation of mathematical calculations.However, students were more dominant to use sentences in writing to solve the problem.The result was consistent with the opinion of Lesh et al. (1987) that the procedure in solving the problem, one of them can be done by explaining the problem solution by using verbal (written) or oral (speech).
Besides Kaput (1987), Lesh (1987), Zhang (1997) Johnson (1998), and Milrad (2002) explain that language representation skills (verbal) are characterized by a passion in translating the nature and relationships in mathematical problems into verbal or verbal representations.The verbal representation used by students is very effective in problem solving.This is supported by the opinion of Hwang et al. (2007) that students who are skilled in manipulating translation between representations using their language are students who have good problem-solving skills.

Conclusion
Based on the results of this study it can be concluded that the construction process of symbolic representation done by students starts since at the stage of understanding the problem.In understanding the problem, students are able to identify the problem well so that students can make the symbol used as a variable that the value has not known yet.Symbols created by students are very helpful and facilitate students in solving problems.While in the construction process of verbal representation is also done by students at the stage of understanding the problem.The form of verbal representation is manifested by writing down all known information from a slightly changed problem because they use a language that they understand more.Based on known information, students can plan and perform a series of calculations using written sentences so that problems can be solved properly.

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