ON CREATIVITY THROUGH MATHEMATIZATION IN SOLVING NON-ROUTINE PROBLEMS

This study aimed to describe and compare the students’ fluency, flexibility, and originality in solving non-routine problems in the Palembang context. They were depicted from the student’s fluency, flexibility, and originality of solving the horizontal and vertical mathematization forms. This qualitative study employed. The subjects of this study were 30 students of grade nine of junior high schools in Palembang. The instruments used were tests and interviews. The tests were employed to investigate the written horizontal and vertical mathematizations forms. Meanwhile, the interviews were to explore the students’ ideas with inadequately detailed answers. Then, the test and interview data were reduced and grouped based on the indicators of creativity. The reduced data were presented in a descriptive form for conclusions. The results of the data analysis showed that the high-ability students were the most fluent and flexible in solving the problems. Still, the provided solutions were less original and tended to use formal mathematics in the forms of formulas, symbols, and operations. Meanwhile, the moderate-ability students tended to start to solve problems by simplifying them, then presenting them in visual images. The answer sheets of the moderate-ability students revealed their fluency in understanding the problems and solutions, flexibility, and originality of thinking. This study obtained different results from the low-ability students who tended to have difficulties understanding the problems and made many errors in solving them. Such a condition showed their inability to write the known data and relate the data to other facts they had already learned. As a result, their answers did not represent fluency, flexibility, and originality.

. The skill plays important roles in the mathematics classroom and a real-world situation.Moreover, to solve mathematical problems, students are frequently demanded to be creative and use various strategies.Therefore, creativity is an important skill to develop (Puccio, 2017).Problem-solving skill is one reason why these two competencies become the focus of mathematics curriculum in Indonesia.The focus of mathematics class in Indonesia is to develop competencies based on non-routine, open, and real-world problems (Cai & Ding, 2015;Maulana & Yuniawati, 2018;Chong, Shahrill, Putri, & Zulkardi, 2018;Minarni, Napitupulu, & Husein, 2016).
The problem-solving ability is related to solving non-routine problems (Celebioglu, Yazgan, & Ezentas, 2010).Non-routine problems refer to things challenging and encouraging students to use different heuristic approaches in their solution (Dendane, 2009, Heffernan & Teufel, 2018;Sudia & Lambertus, 2017).Therefore, the solution process needs mental and intellectual processes to find solutions based on accurate data and information and drawing precise and accurate conclusions.These complex situations often cause students to solve non-routine problems difficultly (Murdiyani, 2018;Hartono, 2014).
Problem-solving has six principles.First, successful problem-solving can be achieved if the idea of the problem is recognized (Carson, 2007).Second, problem-solving uses existing data or information (Csapó & Funke, 2017).Third, the starting point of problem-solving is to discover possible solutions (Fischer, Greiff, & Funke, 2012).Fourth, realizing the core of the problem comes before trying to solve it.Fifth, ideas that create innovation should be separated from the process of evaluating ideas because the evaluation process will inhibit the idea creation (Madzík, 2019).Sixth, selected situations should be converted into a problem situation that is sometimes necessarily changed to a choice situation.When facing a difficult or complex problem, the first step to solve it is to analyze and describe it in simpler ways, such as a sketch or a more detailed paragraph; thus, the problem is more easily solved (Miller & Ranum, 2013).Furthermore, the problem-solving process is continued by looking for some possible ways to finally find the best, most appropriate, and easiest solution (Fischer, Greiff, & Funke, 2012).This process allows students to design a problem-solving strategy that they will utilize during the calculation.The process of problem-solving lets students freely use their ideas or create new ideas without being bound or associated with old ideas (AlMutairi, 2015).
There are four factors influencing problem-solving: motivation, beliefs, habits, and emotions (Ozturk & Guven, 2016).Therefore, the ability to solve problems is measured not only by students' ability to find solutions, but also by the problem-solving process.Students who can solve problems will have understood what they solve and why the solution is chosen.The problem-solving ability is measured and focuses not only on the truth of substantial mathematical solutions and procedures performed but also on the coherence and wrinkling of ideas or mathematical procedures to support these solutions.Related to this, problem-solving is a process of communicating ideas or mathematical thoughts coherently and clearly.Chamberlin (2010) points out that one of the keys to successfully solve problems is representing the problem correctly.For example, representing all mathematical ideas related to the problem in concisely can more simplify the process, operation, and discovery of the solutions.These steps can be represented in models, schemes, and symbols.In a realistic mathematical view, the process of conveying ideas in models, schemes, and symbolizations is called the mathematization process.
Mathematization is divided into two: horizontal and vertical mathematization.The activities of the horizontal mathematization include (1) identifying specific mathematics in general contexts, (2) scheming, (3) formulating and visualizing problems in different ways, (4) finding relationships, (5) finding regularity, (6) introducing isomorphic aspects in different problems, (7) turning daily problems into mathematical problems, and (8) turning daily problems into a familiar mathematical model.
The most important aspect of a problem-solving ability needs not only the mastery of factual and procedural knowledge relevant to the problem but also high creativity by noticing the problem from various points of view (Cropley & Cropley, 2009).Creativity will emerge if someone can know the relationship of the existing elements and provide new ideas to create innovation (Diyanni, 2016).On the other hand, creativity is defined as the ability to offer fresh ideas and apply them in problem-solving (Siswono, 2010).Moreover, it can be defined as the ability to combine, solve, or answer problems and reflect creative children's operational abilities in various possible answers or problem-solving based on the provided information, including triggering many ideas for a problem (DeHaan, 2009).Creative thinking and creativity come from sharply thinking with intuition, moving imagination, and uncovering all amazing and inspiring possibilities and new ideas (Barnard & Herbst, 2018).In addition, creativity usually arises because of habits, such as curiosity, enjoyment of asking questions, and constant search for new experiences (Diyanni, 2016).
Three of four Indicators reflecting someone's creativity are fluency, flexibility, and originality (Torrance, 1972;Siswono, 2010).The fluency in thinking is reflected in generating many relevant ideas or answers.Therefore, fluency in thinking is emphasized more on quantity, not quality.Fluency is defined as the ability to produce several ideas and various answers or questions, investigate a problem from different points of view, find alternatives or different directions, and successfully use various approaches or ways of thinking in formulating, demonstrating, and communicating strong mathematical ideas (Tjoe, 2019).The flexibility in thinking is reflected from likely different results of ideas and the ability to change a way or approach of problem-solving quickly.The originality is usually reflected in answers or unusual solutions and the tendency to differ from other students' answers.A student with creativity and the ability to think in high divergence does not have much difficulty in solving problems.
As defined by experts, creativity is always related to thinking and behaving abilities (Starko, 2013).Therefore, to develop their creativity, students need internal and external impulses.The external impulses include tasks and teaching materials that can facilitate students to develop their creativity.The teaching materials with problem-solving questions about everyday life situations or phenomena can develop students' creativity (Novita & Putra, 2016).
Non-routine tasks and problems that are not well-structured refer to activities that potentially develop student's creativity (Novita & Putra, 2016).Thus, creativity in solving mathematical problems can be defined as students' ability to formulate mathematical problems freely, inventively, and currently (Saragih & Habeahan, 2014).The raising ideas result from compiling information and producing divergent answers following the flexibility and fluency concepts that exist in creativity (Benedek, Könen, & Neubauer, 2012).Creativity always involves imagination, intuition, and invention by developing divergent, original, and curious thoughts making predictions and guesses, and likely employing trial and error strategies (Gilhooly, 2016).Many studies in mathematics education (Celebioglu, Yazgan, & Ezentas, 2010;Mabilangan, Limjap, & Belecina, 2011;Villareal, 2014;Yazgan, 2015) have shown that non-routine problems most effectively improve mathematical problemsolving skills.
Moreover, Pitta-Pantazi and Christou ( 2009) believe that the use of non-routine problems most effectively improves students' mathematical creativity.Yazgan (2015) has analyzed the role of strategy in solving non-routine problems and finds that high ability students' success in solving problems is different from low ability students' success.Another investigation shows that students find solutions when given free will to solve; they employed seven of the eight problem-solving strategies to solve non-routine problems (Mabilangan, Limjap, & Belecina;2011).
Research related to creative thinking shows that non-routine problems positively influence students' mathematical creativity.The use of non-routine contextual problems potentially affects students' thinking skill improvement in real-life situations.Therefore, this study employed contextual non-routine problems.Based on the description above, it was assumed that the investigation on using non-routine problems in the Palembang context would discover many strategies and representations of solutions to produce positive effects on students' mathematical creativity.Thus, this study aimed to describe and compare the fluency, flexibility, and originality in solving non-routine problems in the Palembang context.This study illustrated the students' creativity from their fluency, flexibility, and originality in making horizontal and vertical mathematization.

Design
This study employed a descriptive design because the study aimed to describe the existing phenomena taking place at this time or in the past.Moreover, the study aimed to describe and compare the fluency, flexibility, and originality in solving non-routine problems in the Palembang context.The creativity was reflected in the students' fluency, flexibility, and originality in making horizontal and vertical mathematization.
The research procedure consisted of three stages: the preparation, implementation, and data analysis stages.The preparation stage covered three activities.They were (1) assessing theories and components of creativity to measure students' creativity in solving mathematical problems, (2) arranging non-routine problems in the Palembang context, and (3) preparing interview instruments.This stage was implemented through (1) distributing test questions containing non-routine problems in the Palembang context to students to explore the emergence and forms of horizontal and vertical mathematization, and (2) interviewing the research subjects.Meanwhile, the data analysis stage included (1) data reduction, (2) data presentation, and (3) conclusion.

Subject
The subjects of this study were 30 grade 9.b students of SMP Negeri 3 Palembang.The students' characteristics were categorized as heterogeneous because they consisted of high-ability, moderateability, and low-ability students.

Instruments
The data-collecting instruments of this study were tests and interviews.The test consisted of two questions of non-routine problems in the Palembang context.The tests were employed to describe students' creativity and referred to three components: fluency, flexibility, and originality in making horizontal and vertical mathematization in solving non-routine problems in the Palembang context.This study employed a semi-structured interview because the questions could be developed as needed.The interviews aimed to gain more explicit information on the students' creativity in solving mathematical problems, particularly to explore deeper flexibility aspects.

Data Analysis
The data analysis techniques were divided into two parts: data analysis of written test results and data analysis of interview results.The data of written test result were reduced by grouping and separating the boundaries between horizontal and vertical mathematizations and calculating how many students were categorized in each aspect.These aspects are presented in Table 1 and then in tabular forms.The results of the reduced data were employed to describe the students' creativity components, including fluency, flexibility, and originality in horizontal and vertical mathematization.
Meanwhile, the data analysis of the interview began with transcribing the conversations between teachers and students.Next, the transcript was reduced, and each information categorized as important data was selected.The results of this reduction were presented descriptively to juxtapose with the test result data.The conclusion phase was the process of compiling the information obtained from the results of the tests and interviews.Then, all of the data were compared with the theories that form the basis of this study.

Creativity Aspects Description
Fluency Fluency emphasized the ability to produce several ideas and multiple answers or questions, investigate a problem from different points of view, find alternatives or different directions, and successfully apply various approaches or ways of thinking.

Flexibility
The flexibility in thinking was reflected from the likely different ideas and ability to change the way or approach of problem-solving quickly.

Originality
The originality was usually reflected in the unusual answers or solutions and the tendency to have different answers from other students.A student with creativity and the ability to think in high divergence did not have much difficulty in solving the problems.

RESULTS AND DISCUSSION
Creativity in mathematization means the fluency, flexibility, and authenticity in displaying horizontal and vertical mathematization forms when solving non-routine problems in the Palembang context.Figure 1 shows the problems employed in this study.

Pagoda Roof (Problem 1)
There is a pagoda on Kemaro Island.When viewed from above, it appears that the roof of the pagoda is an   2.

Horizontal Mathematization on the First Problem
Table 2 shows that seven students are categorized as fluent in performing the horizontal mathematization.They were fluent in identifying aspects of mathematics in context, making schematics, formulating problems to other forms, visualizing problems, seeking connection of information, finding regularity, turning everyday problems into mathematical symbols, or changing everyday problems into a known mathematical model.Some of the students' answers are presented in Figure 2.
Moreover, Table 2 presents that the students' horizontal mathematization forms vary.The first type was visualizing form.The second type was written in numbers 1, 2, up to 9 to indicate that there were nine floors in the building.The written numbers 13 m, 12.5 m, etc., were to mark the diameters.
The third type was sketch and numbers.
The interview results showed that the students confidently wrote their answers.Besides answering the question correctly, they did not do any streaks or other forms of correction.The test results showed that the other students' answers were also relatively similar to the first type.The results of the interviews showed that only high-ability students could display horizontal mathematization in other forms.Another alternative strategy demonstrated by SA showed one of the forms as presented in Figure 2. The students' answer sheets revealed that only five students were categorized as flexible.
The originality in horizontal mathematization was depicted from the students' unique representation form.Producing unique answers considered as an original form is indeed quite complex.
The originality of the students' written answers was seen from their distinctive answer forms (once the student's answer was totally or nearly similar to that of other students).This phenomenon indicated that there was little difference in the students' answers.

Vertical Mathematization in the First Problem
Fluency in vertical mathematization forms were characterized by the students' ability to produce several ideas and various answers or questions, investigate a problem from different points of view, find alternatives or different directions, and successfully apply various approaches or ways of thinking.The students' fluency was discovered from their ability to write mathematical symbols and the mastery of applying mathematical concepts and procedures for changing one symbol into another more abstract mathematical symbol.Specifically, fluency in a verbal-mathematical form was illustrated by the students' ability to express a relationship in a formula, show the regularity of using different models, combine and integrate models, and generalize to formal form.Moreover, the students' fluency was also discovered from the detailed strategy described in each step.
This study revealed that there were three levels of vertical mathematization.The first level showed fluency and proficiency in expressing the concept of diameter, radius, and the application of an octagon as a sum of eight triangles.The second level showed using incompleteness and wrong concepts caused wrong assumptions.The third level showed the absence of sufficient concept knowledge caused a fatal mistake.During the interview, the students were asked to think of other strategies possibly applied to answer the first problem.The interview results found that only three students could display another vertical mathematization form while the others did not have another idea to solve this problem.
These results are presented in Figure 3.

Horizontal Mathematization on the Second Problem
The students applied the same method as that in the first problem to solve the second problem.
This study revealed three types of horizontal mathematization.The first type was visualizing a problem using a rectangle.The second type is applying visualization sightly different from the three images made.This type utilized the visualizing shape as five floors.The third type did not apply shape as horizontal mathematization.The curved lines represented the movement from one floor to another.The horizontal mathematization is shown in Figure 4.The flexibility was explored during the interview.In addition to telling the solution written on the answer sheets, the students delivered other strategies possibly used to answer the second problem.
Researcher : Can you explain this form?(pointing at student's graphic, type 1).
Because I think this is really interesting.JM : well, for example, they want to move the weapons from the 1 st floor.
Then it is up to them to move the weapons either to the 2 nd , 3 rd , 4 th , or The interviews and answer sheets revealed that 14 students could display horizontal mathematization form in other forms.Although they could make other alternative strategies, they still led to draw one of the forms as shown in Figure 4.
The originality in horizontal mathematization forms was drawn from the students' unique form of representation.Producing unique answers considered as the original form is quite difficult.The students' written answers discovered that the originality was derived from the not common form.This phenomenon indicated that there was a little difference in the students' answers.The students' answer sheets discovered only four students were categorized as original.

Vertical Mathematization on the Second Problem
The fluency was discovered from the ability to write mathematical symbols and mastery of applying mathematical concepts and procedures for changing one symbol into another more abstract mathematical symbol.The students only applied two nearly similar strategies to solve the second problem (see Figure 5).However, only a few students applied mathematical symbols and operations to solve the second problem.The analysis revealed 14 students were categorized as fluent in operating vertical mathematization forms.Meanwhile, the interviews discovered that ten students were categorized as flexible.They generally stated that they applied the form of '4 + 4 + 4 + 4 + 4' for two reasons.First, one floor had four-item possibilities to fill.Second, because the Monpera building has five floors to reorganize, the students used the '4 + 4 + 4 + 4 + 4' strategy (see Figure 5).The second problem was the most difficult one to determine the originality in vertical mathematization forms because almost all of the students likely applied identical strategy to answer the problem.During the interview, the students stated that the second problem was not too complicated even though some deception elements were found.
The main objective of this study was to describe the students' creativity in solving the non-routine problems.The results of this study implied that the non-routine problems in the Palembang context could be solved not only by high-ability students but also by medium and low-ability students.Highability students likely applied short strategies and directly used formal mathematical symbols.This finding agrees with Minarni, Napitupulu, and Husein (2016), who suggest that students with formal understanding likely employed symbols to represent mathematics.Students who were fluent in vertical mathematization were automatically fluent in the horizontal mathematization form.This finding agrees with Siswono (2010), stating that people's ability to think creatively is higher if they can show many possible answers for a problem.Moreover, they can process knowledge better than others, combine their ideas, and create ideas from the knowledge they have learned.This study discovered that highability students tend to be more flexible than moderate-ability students.However, they likely displayed not original answers.This finding agrees with Siswono (2010), who opines that students' capability to math subject can solve problems clearly but are unable to use more than one alternative solution and do not provide an element of novelty.
The students employed their strategies and likely started with informal forms, such as describing the situation of the problem in the horizontal mathematization form.This finding agrees with Arcavi (2005) pointing out that students used informal forms, such as pictures and schemes, as an effort to understand a problem.The students were fluent in horizontal mathematization forms but not in vertical mathematization one.They could understand the problem but are confused in choosing and using the procedures.This finding agrees with Yimer and Ellerton (2009) assert that many students can understand a problem but lack the skills to create procedures that will guide them to the right direction.
The moderate-ability students in this study displayed fluency, flexibility, and originality.This phenomenon agrees with Siswono (2010), stating that creative thinking has two assumptions.First, everyone can be creative to a certain degree in a certain way.Second, the ability to think creatively is a learnable skill.In other words, each individual has a different creativity degree and distinctive way to realize their creativity.When someone can have the ability (higher or lower degree) to produce new work according to their fields, he is considered as creative.
Meanwhile, the moderate-ability students could understand problems but did not know what procedures they had to apply to solve the problems.This finding agrees with Tambychik and Meerah (2010) pointing out that students who lack heuristic knowledge in problem-solving will have difficulty solving mathematical problems and providing incorrect answers, conclusions, and recommendations.
Moreover, some of them cannot get points because they fail to find the correct answers for some problems and difficultly estimate the solution to the problems.İncebacak and Ersoy (2016) state that such failures result from the fact that the students do not have enough knowledge to solve problems.
Moreover, their solutions seem incomplete and unclear.Dendane (2009) emphasizes that students should learn mathematical contents and the use of mathematical contents to develop thinking skills and solve mathematical problems.

CONCLUSIONS
The high-ability students had the most fluent and flexible solving-problem skills.However, they provided less genuine-solutions and likely used formal mathematics in formulas, symbols, and mathematical operations.Meanwhile, moderate-ability students tended to start their work by simplifying problems and displaying them in visual images.Their answer sheets presented their originality of thinking, flexibility, and fluency understanding the problems and solutions.Meanwhile, the low-ability students had difficulties understanding problems.Moreover, they created many errors when solving the problems because they could not write the familiar data and relate them to other facts they had already learned.Consequently, their answers did not represent the aspects of fluency, flexibility, and originality.
octagonal shape.If the diameter of the lowest floor of the pagoda (on the ground) is 13 m, and the distance of each adjacent octagon is 50 cm, predict the diameter of the pagoda's roof and the area of the pagoda's roof.Explain the strategy you use! Monpera (Problem 2) Monpera consists of eight floors.The five lower floors are filled and turned into a museum.The museum has various collections serving as the witness to the fiveday-and-five-night war in Palembang.On the 1st floor, there is a collection of weapons.On the 2nd floor, there are various documents and photos of the war period.On the 3rd floor, there is a collection of old money.Meanwhile, on the 4th and 5th floors, there are statues and clothes of the heroes.If the Monpera officers want to rearrange all the existing collections by placing them on different floors, determine how many possible ways the officers can organize the collections.

Figure 1 .
Figure 1.Non-Routine Problems in the Palembang Context

Figure 3 .
Figure 3. Students' Answers to the Vertical Mathematization

Figure 4 .
Figure 4. Horizontal Mathematization on the Second Problem

5
th floor, right?Then these lines represent the movements.And these lines are similar in other cases.Researcher : What about the 4 th and 5 th floor?Why is there no line between them?JM : Because they contain the same thing.Researcher : I see.Besides this answer, do you think you can use another strategy to solve the problem?JM : I think I can.(started to write down another strategy).

Figure 5 .
Figure 5. Vertical Mathematization on the Second Problem