Developing Instruction Materials Based on Joyful PBL to Improve Students Mathematical Representation Ability

Solving problem either within mathematics or beyond is one of the ultimate goal students learn mathematics. It is since mathematics takes role tool as well as vehicle to develop problem solving ability. One of the supporting components to problem solving is mathematical representation ability (MRA). Nowadays, many teachers and researchers find out secondary school students are so poor in MRA. Considering the problem, this research developed instruction materials based on Joyful Problem-based learning (JPBL) to endorse eighth graders of public secondary school students grasp their MRA. The study took place in the Province of North Sumatera, Indonesia. A number of 88 students engaged as subjects. The study administered MRA test before and after implementing the instruction materials. Data is analyzed descriptively and by t-test. The result showed that the learning materials developed is effective in improving the students’ MRA. It increased significantly, though categorized mediocre. The study contributed to the existing instruction materials in pursuing MRA.


Introduction
In contrast to the industrial era, the information and communication technology era requires individuals not only able to pass procedural activities but also able to reason or to think critically and creatively (higher order thinking skills, HOTs).This era needs thinker as well as inventor or creator more than labor at all.HOTs has characteristics such as non-algorithmic (the path of action is not fully specified in advance), tends to be complex, often yields multiple (ways) solutions, involves judgment and interpretation nuances, the application of multiple criteria.It often involves uncertainty, self-regulation of the thinking process, imposing meaning, finding structure in apparent disorder, and effortful (Hmelo and Ferrari, 1997;Resnick, 1987;Saragih & Napitupulu, 2015).It makes sense whilst Baker (1990) emphasizes that HOTs measures constitute tasks demanding more than information retrieval and either procedural or conceptual understanding.It is so important then to take into account this competence as early as possible in the part of the students.Since by embracing the competence aforementioned they become potential thinker and problem solver.
One of the most important competencies constitutes HOTs is mathematical representation.As Cobb et al. (1992) asserted, "Many of the current attempts to develop theories in mathematics education reflect the view that learning is a process of constructing internal mental representations".The authors proceed arguing, "learning is a process in which students modify their internal mental representations to construct mathematical relationship or structures that mirror those embodied in external instructional representations".While, Pape and Tchosanov (2001) state that representation may be thought of broadly as mental states, narrowly be thought of as mental reproduction of mental state.A structural equivalent presentation through picture, symbol, or signs.Or 'something' in place of 'something'.According to theory of recognition and brain investigation, brain works more effectively while making representation patterns for encoding (internalizing) and decoding (externalizing) information.
Teachers can and should develop representation ability at school through learning process.One of the most responsible subjects to mediate and facilitate students grasping this requirement is mathematics.Therefore, the goals of school mathematics should be to teach students to think, to make them comfortable with problem solving, to help them pose questions and formulate hypotheses, investigate, and make mathematics vehicle to think (Kulm, 1990), as well as grasping representation ability (NCTM, 2000).
The preliminary research carried out in 2015 (Minarni, Napitupulu, & Husein, 2016) showed that learning in the classroom is still apparently without attempting to encourage students to contribute their own mathematical representation.This is reasonable, because expository method still dominated learning in the classroom and very rarely gives students chances to get involved in solving mathematical problems.Teacher still acts as the center of learning process in the classroom.Fogarty (1997) mentioned that such learning could not make the students fully understand what is learned.Yet understanding is a major component that enables students solving problem (Schöenfeld, 2007).Indeed, this kind of learning is not going to foster students' ability to solve problems since the students do not have opportunity to do so (Ronis, 2008).
Furthermore, observations and interviews in this preliminary study revealed the teachers' complaints on the lower secondary students' weakness in drawing pictures, creating table and building mathematical equation to represent mathematical problems.These skills are clearly aspects of mathematical representations.As known for a long time, mathematical representation ability supports problem-solving skill.Concerning this, it is then important to note suggestion Pape and Tchosanov (2001) posed that teachers and students must develop classroom norms that facilitate explanation and justification and the use of representations in the service of supporting argument.

Developmental Research
The study is developmental in nature, which aimed at both developing innovative intervention and identifying underlying design principles or theories (Plomp, 2013).Innovative intervention includes implementing developed learning model and learning materials.Nieveen and Folmer (2013) distinguish four quality criteria that are applicable to a wide array of educational intervention (these criteria suggest a logical hierarchy):"

1) Relevance (also called content validity)
There is a need for the intervention and its design is based on state-of the art (scientific) knowledge.

2) Consistency (also called construct validity)
The intervention is 'logically' design.

3) Practicality
Expected: The intervention is expected to be usable in the settings for which it has been designed.
Actual: The intervention is usable in the settings for which it has been designed.

4) Effectiveness
Expected: Using the intervention is expected to result in desired outcome.
Actual: Using the intervention results in desired outcome".

Mathematical Representation Ability
"Representation is a configuration that can represent something else in some manner" (Goldin, 2002).People develop representations in order to interpret and remember their experiences in an effort to understand the world.Kilpatrick, Swafford, & Findell (2001) stated that representation could be used to understand mathematics.Mathematics requires representations because of its abstract nature such that people have access to mathematical ideas only through the representation of those ideas.Carpenter & Lehrer (1999) classify representations into two categories, namely internal representation and external representation.Internal representation is everything that exists in the cognitive structure of students, while the external representation is pictures, tables, graphs, sketches, symbols, equation (mathematical model), words (verbal) made by the students as a reflection of internal representation.Meira (2002) states that representations can bridge difficulties in understanding mathematics and can make mathematics more attractive and interesting (not rigid and monotonous).Representation is helpful in presenting problems more clearly and makes the students comprehend concepts or ideas of mathematics better.Representation enables students to create and use representation to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representation to solve problems; use representation to model and interpret physical, social, and mathematical phenomena (NCTM, 2000).
Based on various meaning of representation above, it can be viewed that mathematical representation presented by the students in the form of figure, graph, table, equation (mathematical model), explanation in words or sentences that constitute word problem faced by the students, are pointed out representation in the cognitive structure of the students, vice versa.

Joyful Problem-Based Learning
Students' mathematical representation ability will grow by giving them chances to solve various mathematical problems since by solving mathematical problems they will grasp various mathematical representations (NCTM, 2000).One of the best moments for teachers giving students chances to solve various mathematical problems surely in the teaching and learning processes in the classrooms.
On the other side, Arends (2004) states that learning approach which is in accordance with the purpose mentioned earlier is problem-based learning (PBL) since it is designed to promote students' ability in mathematical problem solving.Problem solving requires five others mathematical ability i.e. mathematical comprehension (understanding), reasoning, connection, communication, and representation.Therefore, through solving problems students will develop their capabilities wholly in doing mathematics.
PBL begins with presenting students authentic and meaningful problem situation that can serve as springboards for investigation and inquiry.PBL integrate learning from different sources and disciplines through synthesis and collaborative inquiry.PBL respect the situation in which students are not sure about the data, information, as well as a solution, situation that students will face in real world.All students, regardless of their abilities, can benefit from PBL.
One important feature in PBL is a problem.Problem should be able to act as trigger in early learning and should be designed to improving high order thinking.Such problem can be designed by relating it to the characteristics of high order thinking which proposed by Resnick (1987).
Meanwhile, applying learning approach in eighth grade need considering the development of students' mental.At this age, students are in the transition from concrete to formal stage.Therefore, combining PBL with a joyful learning environment will lead the students feel comfortable and happy to learn mathematics.The idea is in line with Puri's (2014).In her work, she suggests that learning in the classroom should be made as interesting as game so that students addicted to learn as they addicted to play the game.
Joyful Problem-Based Learning is problem-based learning designed such that the learning activity conducted in pleasant environment and convenient situation.Sometimes learning is conducted outside the classroom and using a variety of hands-on manipulatives around the classroom or school yard, which is interesting for students; the teacher scaffolds the students with friendly language (tone of the teachers' voice, facial expression, body posture and gesture makes the child comfortable).The objects that exist inside and outside the classroom, as well as the environment around the classroom used as devices to increase the interest of students to learn, but the focus remains on efforts to lead the students making a variety of representations of a problem so they truly understand the subject matter at hand.We presented in Table 1 learning activity follows the modified PBL in joyful learning environment.
The learning activity took place not only in the classroom, but also outside.For example, in solving the problems related to the topic of two-variable linear equation system, the students are invited to a bookstore located around the school area.At the bookstore, students learn to determine the price of goods sold such that the solution of problem-price become more meaningful.Meanwhile, for the problems related to Pythagorean rule, the students solved the problem in the classroom using hands-on manipulatives.
During conducting the lesson, the teacher should confirm that the students show their understanding on the problem, show the effort to represent it in the form that made it easier or aid them attaining the solution.Of course, the teacher, when needed, provides assistance or scaffolding to students who have difficulty in understanding the problem.
While working on problems in the students' activity sheet (SAS), teacher asked the students did it individually in time duration of 10 to 15 minutes.They at least attempted to grasp what information the problem gave and what it demanded.Then teacher asked them to work collaboratively, while teacher observed and gave scaffolding to who needed.Helps students to reflect on their investigation, using appropriate representations and the processes they used to get the solution of the problems.

Subjects, Design, and Data Sources
The subject of the research is the students from three public junior high school (PJHS) in the Province of North Sumatera, Indonesia.Of each, the researchers took one eight graders class.On the other side, observation sheet on students' activities is based on students' involvement in doing math problems, students' participation in-group discussions as well as in classroom, students' activity in responding questions, posing suggestions and critics to others, and students' response towards teacher's direction.
Last, the researchers used responses scale to reveal students and teachers' responses toward and perception to the learning materials developed.
In our developmental research, the intervention is the implementation of instruction materials developed based on Joyful Problem-Based Learning (JPBL).According to Nieveen and Folmer (2013), intervention in instruction is effective if it results in desired outcome.Cai et al. (2009) state "the quality of mathematics instruction can be judged by two criteria: desirable outcomes in students' learning and the processes that yield those desirable learning outcomes".Concerning to the first aspect Nieveen, Folmer, and Cai mention, in their work Napitupulu, Suryadi, and Kusumah (2016) find the highest average score of upper secondary students in mathematical reasoning ability, which is part of HOTs, was 58 of 100 and only achieved by 26% of the students.Considering the result and the effectiveness criteria they propose, to be judged effective, we determined the desired intervention of our developed instruction materials should fulfil the following criteria: 1) More than 65% of the students attain minimal score 65 in MRA test (from ideal score 100) and the students' MRA improved significantly.
2) Students' activities while learning, especially in solving mathematical problems, and teachers' activities in teaching process can be classified well, that is ̅ 85 ( ̅ is activity average score).
3) Both students and teachers mark positive perception to and response towards the learning materials that is the average score is greater than 75.
All data was analyzed descriptively except the improvement of students' MRA, which used t-test to draw conclusion.

MRA test, Observation, Perception and Response of Students & Teachers
Table 4a presents the students' average score on MRA-test.The table shows a relatively high increasing score from the Pre-test to the Test-1.On the contrary, from Test-1 to Test-2 the increasing was small.Albeit small, it was very important for it contributed to the more amounts of the students who passed the minimum criteria.The result of observation on the teachers' activity is presented in Table 7, meanwhile on the students' activity is presented in Table 8.From the Table 7, we saw that teachers' activity average score is higher than 85.At the same time, from Table 10, we got that the students' activity average score is also higher than 85.In overall, the students engaged actively in the learning and teaching process.In this case, the result exceeded the target has been set for the second effectiveness criteria.The scale on students and teachers' perception to and response toward the learning materials developed revealed the average score is 83.9.In consequence, the data fulfilled the third criteria of the effectiveness accordingly.To sum up, from the objective viewpoint the instruction materials developed is effective.It implies that the learning devices is eligible used at a wider scope (disseminate) (Nieveen & Folmer, 2013).

Students' Performance in Learning Process
We investigated Students' performance during the learning process by analyzing their work on the problems in the SAS.Examples of the problems showed in Table 9.An example of full SAS attached in Appendix 1.

Studen
The averag is the aspe aspect of t aspect of a to engage in self-directed and problem-solving activity 2. Organize the students to learn Helps students define and organize learn tasks related to the problem presented in SAS.3.Assist independent and groupInvestigation-Encourage students to gather appropriate information, understand the problem (make various appropriate representation as an aid to attain the solution), and search for explanations and solutions -Give appropriate scaffolding to the students who could not move or develop their thinking in attempting to solve problem.4. Develop and present artifacts and Exhibits -Assists students in planning and preparing appropriate report associated with the solution they get to the problem -Help the students displaying their work on the white/black board -Helps students share their work with others 5. Analyze and evaluate the problem-solving process Figure 1.Prob

Figure
Figure 2. Pr

Table 1 .
Syntax of modified problem-based learning

Table 2 .
Indicators of MRA Totally, there are 88 students engaged in the research.They are from PJHS in Medan, Pematang Siantar, and Percut Sei Tuan.The study is developmental in nature.Learning material consists of Lesson Plan, students' activity sheets (SAS), Student's Book, MRA-test consisting Pre-test, Post-test (Test-I and Test-II).In addition, Observation and responses sheet and Interview Guidelines are also used.Teaching material constitute of Linear Equation of Two Variables and Pythagorean Theorem.The problems posed in SAS are an important part of learning materials.Most of them as well as the MRA-test are developed by considering some thoughts of Resnick on HOTs.Observation and interview guidelines are set-up based on survey research theory.Last, teaching materials are chosen by considering Indonesian 2013-Curriculum, which is well suited for modified JPBL model.3.2Instruments,DataCollection and AnalysisMRA-test comprises of two aspects, i.e.(a) Represent word problem into picture, figure, table, graph, or mathematical model as an aid to solve problem; (b) Interpret problem presented in figure, table, graph, mathematical model verbally with supporting explanation.Indicators as the details these aspects set up for measuring the students MRA-test conveyed in Table2.There were five problems designed and posed to measure the students' MRA for each pre-test and post-test.Guideline for scoring students' performance in the MRA-test is depicted in Table3.This rubric is developed by considering Baker (1990) concerning assessing high order thinking.

Table 3 .
Rubric scoring for MRA-Test Components of Mathematical Representation Score Construct appropriate mathematical representation accurately and manipulated it perfectly to get the right solution.4 Construct appropriate mathematical representation accurately and manipulated it perfectly but there are some mistakes in numerical Observation sheet on instruction is used to record the learning and teaching processes.Using it, we analyze to what extent the degree of the learning materials developed was implemented and to analyze the way of teachers conducted the lessons in the classroom.It comprises of teachers' effort in endorsing students' willingness to learn, directing students to study hardly (keeping perseverance), forcing the students' engagement in discussion to solve problems, encouraging students to do investigation, providing opportunity to students solving problems in different ways and different mathematical representation, endorsing interaction within group-work, and giving necessary scaffolding and guiding reflection activity.

Table 4a .
Average Score on Pre-test, Test-I, and Test-II Percentage and the number of the students attained minimum score 65 is presented in Table4b.It means that part of the first criteria of the effectiveness is fulfilled.
Note.Minimum Score = 65; Ideal Score = 100 Moreover, the improvement of students' MRA is investigated through the difference score between pre-test to Test-II.Statistical t-test used to draw the significance of the differences.Result of t-test between pre-test and Test-II is significant at 5% level.It means that the students' MRA improved.Data on the difference score between Pre-test and Test-I is presented in Table5, while the difference score between Pre-test and Test-II is presented in Table6.Output of t-test for the difference of test I and pretest as well as for test I and test II is presented in Appendix 2. The test is conducted after test of normality of the data and homogeneity of variance was fulfilled.

Table 7 .
Percentage of teachers' activity in learning process

Table 8 .
Percentage of students' activity in doing SAS on MRA problems

Table 9 .
Examples of problems in SAS , a swimmer started to swim as far as 3000 straight.Then, he went on swimming to left as far as 4000 .Determine the distance (in meter) if he directly swam from the starting point to the end position.

Table 10
The following table shows the charge should be paid when using a certain taxi to travel.
1. Determine a mathematics equation which governs the data.