the identification of the difficulties in solving mathematical problems

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The identifi cation of the diffi culties in solving mathematical problemsof junior high school mathematics teachersin Nusa Tenggara Timur and Maluku Utara

Heri Retnawati, Dhoriva Urwatul Wutsqo, Endang Listyani, Kartiko Rachman YPYogyakarta State University

e-mail: [email protected]

Abstract: This study aims to determine the diffi culties in solving mathematical problemsof junior high school mathematics teachers in Nusa Tenggara Timur and Maluku Utara,two of the 33 provinces in Indonesia. The method used in this study was descriptiveexploratory. The data analysis was based on 114 junior high schools mathematicsteachers’ responses to the fi ve items of the National Examination questions. The itemswere the most diffi cult items according to the students, particularly those of the juniorhigh schools in Nusa Tenggara Timur and Maluku Utara, whose graduation has notreached 100%. The result of the analysis indicated that the order of teachers’ diffi cultiesin solving mathematical problems is executing the problem solving plan, understandingthe problems, interpreting the results and designing the problem solving plan. Basedon this study, 46.491% teachers had diffi culty in executing the problem solving plan,45.846% in understanding the problem, 43.129% in interpreting the results and 33.063%in designing the problem solving plan.

Keywords: diffi culties, mathematical problems, problem solving

1. Introduction

Various factors infl uence the success ofeducation (Bridge, Judge & Mock, 1979).Based on the review of these various factors,the teacher factor has 15% contribution to thestudent success (Sallis, 2002). In order to be ateacher who has a great role in determining thesuccess of the students, teachers should ideallyhave personal competence, social competence,academic competence, and professionalcompetence. Supporting the success relatedto the academic professional competence,teachers must be able to master learningmaterials. That is the major component in

professional competence. These competenciescan be determined by evaluating the teacher,by administering a test, and by analyzing thetest.

The results of the study conducted byMardapi, Soenarto, and Retnawati, (2011)show that the ability of the teachers whoteach at junior high schools, whose NationalExamination score is low, is also low inmastering learning materials. In this study,teachers’ ability data in mastering learningmaterials in 100 districts/cities in Indonesiawas necessary. The data source was teachers’responses to the essay test for the subjectstested in the National Examination at junior

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high schools, senior high schools andvocational schools. The overall informationobtained from all over Indonesia was groupedinto four regions, covering Java (Region 1),Sumatra (Region 2), Sulawesi, Kalimantan,Nusa Tenggara Barat (Region 3), and NusaTenggara Timur, Maluku Utara, Papua (Region4).

However, the teachers’ responses to theessay tests have not been fully exploited andexplored. One thing that can be done in orderto maximize the utilization of the data was toidentify the teachers’ diffi culties in solvingmathematical problems. Problem solving inmathematics includes several things, namelyconcept understanding, language interpretation,algorithms, and computation. The mistakemade by the teachers which causes their lack ofability to solve the mathematical problem canbe explored using their responses to the essaytest. The results of the data exploration can bepresented as a problem solving error profi leperformed by mathematics teachers. Basedon that profi le, there are various improvementefforts that can be planned, namely policy,training, community service, and other actionsthat can improve teachers’ mastery of thesubject matter. Those improvement effortsare also useful to improve teachers’ abilitiesin solving problems in order to improvetheir’ academic professional quality. Basedon this background, this study revealed thejunior high school teachers’ diffi culties insolving mathematical problem according toPolya. Teacher’s ability to solve mathematicalproblems is the refl ection of their ability inmastering the learning material. Masteringthe learning material is very important in thelearning activities.

An educational evaluation is conductedto obtain information related to education.According to Gronlund (1976), educationalevaluation has many purposes, namely a) toprovide clarifi cation of the learning outcomes

that have been implemented, b) to provideinformation about the achievement of short-term goals, c) to provide feedback for thelearning progress, d) to provide informationabout the diffi culties in learning activitiesand the selection of learning experience forthe future. The information generated by anevaluation activity can be used to determine:a) the suitability and sustainability of thelearning goals, b) the usability of learningmaterials, and c) the level of effi ciency andeffectiveness from the teaching strategiesused in the classroom (methods and learningtechniques).

One interesting problem related to theevaluation and its results is problem relatedto mathematics. There are various opinionsexpressed by experts about the defi nitionof Smathematical terms. According to Gold(2008), mathematics has many interpretations.Mathematics can be defi ned based on itscontents (Gold, 2008), based on the objectsstudied by mathematics learners (Avigad,2008), and it can also be defi ned as a processof thinking (Lewis). Reys, et al. (1998)defi ne mathematics as a lesson about thepatterns and relationships, ways of thinking,art characterized by rules and consistency,language that uses terms and certain symbols,as well as a useful tool in everyday life andalso tools that assist the development of otherexisting knowledge.

Mathematics is also known as a structureof relationships that associates symbols.This opinion is based on the idea about theformation of mathematics. Related to thatstatement, Ruseffendi in Ismail (1998) arguesthat mathematics is formed as a result ofhuman thinking associated with the ideas,processes and reasoning. Related to the processof formation, mathematics is also known as aknowledge which belongs to human being.This knowledge arises because of the need ofhuman being to comprehend the nature around

JOURNAL OF EDUCATION, Volume 7, Number 1, November 2014

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them. Nature is the source of idea to obtainmathematical concepts through abstractionand idealization.

After the model is created, defi nitionsand axioms are created based on that model.Defi nition is an agreement used to replacesomething else, usually an expression orreplacement to replace something that is toodiffi cult to write (James & James, 1976). Anaxiom is a statement that is accepted withoutproof. A theorem is obtained through theprocess of thinking called deductive logic(Allendoerfer, 1969). The theorem resultedfrom the thinking activity is a generalconclusion that can be proven (James & James,1976). The defi nition, axioms and theorems area unitary system that constructs a mathematicalconcept.

Mathematical objects are abstract. Theycorrelate with each other and form a new morecomplex concept (Skem, 1971). They arearranged in a hierarchy, so one concept is thebasis for another concept (Herman Hudoyo,1988:3). Mathematical concepts that are

Table 1.Stages in Solving a Problem Designed by Polya

Stages in Problem Solving Ideal CriteriaUnderstanding the problem 1. Able to write down the core problem

2. Write down the obtained data3. Able to model the problem question and select the appropriate notation4. Make a sketch related to the problem solving if necessary

Designing the problem solvingplan

1. Find the pattern used for solving the problem2. Know the formula related to the problem solving3. Know the prerequisite conditions for solving the problem

Executing the problem solvingplan

Perform calculation based on the designed plan

Interpreting the results 1. Check the steps that have been done2. Interpret the results as a conclusion3. Look back and fi nd out if the core of the problem has been aswered in the conclusion.

founded are applied to the nature. People usethem to fulfi ll their needs in their life.

Mathematics is knowledge which is veryuseful in human life. This knowledge is usedto resolve problems (Polya, 1973). There arefour stages in solving a problem accordingto Polya, namely understanding the problem,designing the problem solving plan, executingthe problem solving plan, and interpreting theresults. These stages are presented in Table 1.

The variable associated with the successin mathematics learning is the teacher’scompetence. According to the Republic ofIndonesia Government Regulation No. 19 Year2005, Article 28, Section 3 and Law No.14 Year2005, Article 10, Section 1 “The competenceof educator as a teaching agent in primaryand secondary education as well as in earlychildhood education includes (a) pedagogicalcompetence, (b) personal competence, (c)professional competence and (d) socialcompetence. The teachers’ professionalcompetence according to Law No. 14 Year2005 is mastering the materials, structures,

Heri R. et al.: The Identifi cation of the diffi culties…(page 1-13)

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concepts, and scientifi c mindset that support theteaching subjects, mastering the standard andbasic competencies of the teaching subjects,developing the learning materials creatively,developing the teacher’s professionalism bydoing a refl ective action sustainably, utilizinginformation and communication technology inorder to develop their professionalism.

Polya (1984) created “Ten Command-ments For Teachers” containing ten things thatshould be noted by teachers to improve theirprofessional competency. These 10 principlesare: 1) Be interested in your subject line; 2)Know your subject; 3) Try to read the faces ofyour students; try to see the reviews of theirexpectations and diffi culties; put yourselfin the reviews of their place; 4) Realize thatthe best way to learn anything is to discoverit by yourself; 5) Give your students not onlyinformation, but also know-how, mentalattitudes, the habits of methodological work;6) Let them learn to guess; 7) Let them learnto prove; 8) Look out for the reviews of suchfeatures of the problem at hand as may beuseful in solving the problems to come - tryto disclose the general pattern that lies behindthe present concrete situation; 9) Do notgive away your whole secret at once - let thestudents guess before you tell it - let them fi ndout by themselves as much as is feasible; 10)Suggest it; do not force it down to reviewtheir throats.

2. Method

This study used the descriptive exploratorymethod with the quantitative approach. Theteachers’ diffi culties in solving matematicalproblems are identifi ed based on the stagesin the problem solving according Polya. Theobject used in this research was the responsesof the junior high school teachers to theNational Examination questions. Mathematicsteachers who participated in this study consist

of 111 junior high school mathematics teachersin Nusa Tenggara Timur and three junior highschool mathematics teachers in Maluku Utara.They teach at the junior high schools whichgraduated less than 80% of the students in2010 and 2011. The National Examinationquestions (multiple-choice type) are modifi edto an essay test. The rubric from this essaytest was designed based on Polya’s problemsolving stages, including understanding theproblem, designing the problem solving plan,executing the problem solving plan, andinterpreting the results.

The data used in this study were collectedusing the documentation method. The datawere obtained through research activitiesof Puspendik Balitbang Kemendikbud(Education Research Center, Research andDevelopment Bureau, Ministry of Educationand Culture) in 2011. The data were inthe form of the teachers’ responses to theNational Examination questions (essay testfor the 9th and 12th mathematics teachers whoteach at junior high schools and senior highschools, especially in science and socialscience department). The data have not beenanalyzed, especially in relation to the stepsof the utilization of mathematics as a tool toresolve problems. A test was used to determinethe teachers’ mastery of the teaching materialaccording to the SK/KD (standard and basiccompetencies). The test was compiled based onthe material considered diffi cult by students,based on students’ absorptive capacity inthe last fi ve or six years (2006-2011). TheNational Examination mathematics score foreach school was obtained from PuspendikBalitbang Kemendiknas.

The data were analyzed quantitatively.The quantitative descriptive analysis was usedto identify the profi le of the diffi culties facedby the mathematics teachers at junior highschools in Nusa Tenggara Timor and MalukuUtara, whose graduation is less than 80%. The


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teachers’ diffi culties included the diffi cultiesin understanding concepts, languageinterpretation, algorithms, and computationalcapabilities. The achievement of the teacherswas correlated with the average of the schoolscore in mathematics. The recommended

suggestion based on the results of this analysiswas the suggestion that can be used to improvethe education quality, especially to increasethe teachers’ ability in solving mathematicalproblems related to their professional abilities.

Table 2.Distribution of Research Sample Junior High School Mathematics TeachersWho Participated in Competency Tests Administered by Balitbang (Research andDevelopment Bureau)

Regional Province Frequency

Region I

DIY 39Jakarta 18Banten 17Jawa Tengah (Central Java) 45Jawa Timur (East Java) 31

Number of Teachers in Region I 150

Region II

Aceh 33Bangka Belitung 45Sumatra Barat (West Sumatra) 115Kepulauan Riau (Riau Islands) 10Lampung 8

Number of Teachers in Region II 211Kalimantan Barat (West Kalimantan) 65Kalimantan Tengah (Central Kalimantan) 28Kalimantan Timur (East Kalimantan) 57Nusa Tenggara Barat 23Gorontalo 4Sulawesi Barat (West Sulawesi) 9Sulawesi Selatan (South Sulawesi) 51Sulawesi Tenggara (Southeast Sulawesi) 10

Number of Teachers in Region III 247Nusa Tenggara Timur 111Maluku Utara (North Maluku) 3

Number of Teachers in Region IV 114Total 722


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Table 3.Graduate Competency Standards thatwere Diffi cult to Achieve and Used as TestInstruments by Balitbang

NumberCompetency Standards Used

as a Problem1 Determining the surface area of

the curved side2 Determining gradients, equations,

and graphics3 Solving problems with the concept

of congruency4 Determining the volume of the

curved side5 Determining the central tendency

and its use to solve daily problems

3. Findings and Discussion

Determining the Surface Area of a ConeAs many as 37.093% teachers from the

total of 114 mathematics teachers in Region4 did not understand the problem of the 1st

question well. Understanding the problem ofthe 1st question was divided into four steps orsubstages based on Polya’s Problem SolvingModel, namely write down the core of theproblem, write down the obtained data, modelthe question and select the appropriate notationand also make a sketch related to the problemsolving. The teachers’ diffi culties in answeringthe 1st question measured by mistakes that theymade are presented in Figure 1.

Mathematics teachers in Region 4 couldunderstand the core of the mathematical


Figure 1: Teachers’ Diffi culties in Solving the Problem of the 1st Question

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Table 4.Summary of the Teachers’ Diffi culty in Solving the Problem of the Test Item Number 1

Polya’s Stage Polya’s Substage PercentageUnderstand the problems Write down the core of the problem 42.982%

Write down the obtained data 29.240%Model the question and select the appropriate notation 21.930%Make a sketch related to the problem solving 85.088%

Designing the problemsolving plan

Know the prerequisite conditions for solving the pro-blem

23.684%

Find the pattern used for solving the problem or knowthe formula related to the problem solving

49.123%

Implementation Of theProblem Solving Plan

Find the prerequisite conditions for solving problem 24.123%Find a fi nal solution 52.193%

Interpret the results 62.281%

problem presented in the 1st question. It wasknown from the percentage of the teachers whomade mistakes on the fi rst stage, understandingthe problem: write down the core problem.There were just 42.982% teachers who made amistake at the fi rst stage. As many as 29.240%teachers in Indonesia did not write down theobtained data. At the sketching stage, 85.088%mathematics teachers could not use a sketch asa tool for solving the problem. In the next stage,designing the problem solving plan, there were23.684% teachers who made mistakes. Theyhad diffi culty in fi nding and using the formulato calculate the surface area of a cone. Thedetailed percentages of the mistakes made bythe teachers at each substage of Polya’s Pro-blem Solving model are presented in Table 4.

Determining Gradients, Equations, and GraphicsThe 2nd question is related to the straight

line function that was perpendicular to theother line and through a point. There was anadditional problem. The teachers also mustdraw two lines in a Cartesian fi eld. Theteachers’ diffi culties in answering the 2nd

question are presented in Figure 2.

Mathematics teachers in the Region ofNusa Tenggara Timur and Maluku Utara didnot understand the problem well. As many as38.158% teachers did not understand the coreof the problem and they were not be able towrite down the obtained data. In designingthe problem solving plan stage, there were70.833% teachers that did it well. The mostimportant thing in solving problem of the2nd question based on Polya’s model was theinterpreting of the result stage. The detailedpercentages of teachers’ mistakes in solvingthe 2nd question are presented in Table 5.

Solving Problems with the Concept of CongruencyThe 3rd question related to geometry is

focused on congruency. This question wascombined with the triangles and the PythagoreanTheorem. The percentages of the teachers’diffi culties in answering the 3rd question arepresented in Figure 3.

The teachers’ diffi culties on the 3rd

question were relatively high when comparedto the teachers’ diffi culties on the 1st and 2nd

questions.


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This can be seen from the percentage ofthe mistakes made by the teachers. The teachershad diffi culties in understanding the problem,

designing the problem solving plan, executingthe problem solving plan, and interpreting theresults. Most of the teachers had diffi culties

Table 5.Summary of the Teachers’ Diffi culty in Solving the Problem of the Item Test Number 2




Know the prerequisite conditions for solving theproblem

32.456%


56.725%


Find the prerequisite conditions for solving problem 74.123%Find a fi nal solution 35.965%



Figure 2: Teachers’ Diffi culties in Solving the Problem of the 2nd Question

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in understanding the problem and designingthe problem solving plan. The percentages ofthe teachers’ diffi culties at those stages werehigher than those at the two other stages.

As many as 72.807% teachers did not drawa sketch to help in solving a problem. There are54.386% of the teachers who had the idea touse the concept of congruence or PythagoreanTheorem in solving the problem in the 3rd

question. Most of the mathematics teachersin Region 4 only assumed without proven theexistence of a pair of congruent triangles. Afterthey assumed that there were two congruenttriangles, 40.643% of them made a mistakein the calculation. The detailed percentage ofteachers’ mistakes in solving the 3rd question ispresented in Table 6.

Determining the Volume of The Curved SideThe teachers’ diffi culties in solving the

problem of the 4th question about the volume

of the cone with unknown length radius arepresented in Figure 4.

As many as 32.602% teachers in NusaTenggara Timur dan Maluku Utara haddiffi culties in understanding the problem ofthe 4th question. Most teachers understand thecore problem but only 23.684% made sketchrelated to the problem in order to fi nd the radiuslength of a cone. Based on the percentagesof the teachers’ diffi culties in Figure 4,mathematics teachers in Region 4 did nothave diffi culties in answering the 4th question.There were common mistakes made by theteachers, namely the teachers did not write theconclusion and they did not provide the unitso they did not answer the questions correctly.The detailed percentages of mistakes madeby the teachers at each substage of Polya’s insolving the 4th question are presented in Table7.

Figure 3: Teachers Diffi culties in Solving the Problem of the 3rd Question

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Table 6.Summary of the Teachers’ Diffi culty in Solving the Problem of the Test Item Number 3





39.474%


96.491%


Find the prerequisite conditions for solving problem 40.643%Find a nal solution 48.246%


Determining The Central Tendency and UsingIt to Solve Daily Problems

The problem of the 5th question wasabout the statistical problems. The teacherswere asked to calculate the average based on

two different averages. The percentages ofthe teachers’ diffi culties in answering the 5th

question are presented in Figure 5.The percentage of the teachers’ diffi culties

in understanding the problem is 52.632%.

Figure 4: The Percentages of Teachers’ Diffi culties of the 4th Question

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This indicates that there were mistakes madeby the teachers. Most of the teachers knowthe core of the problem but they did not writedown the obtained data and they also did notmodel the question and select the appropriatenotation. There are 37.343% teachers who had

diffi culties in designing the problem solvingplan and 37.343% teachers had diffi cultiesin executing the problem solving plan. Thedetailed percentage of the mistakes made by theteachers at each substage of Polya’s in solvingthe 5th question are presented in Table 8.

Table 7. Summary of the Teachers’ Diffi culty in Solving the Problem of the Test Item Number 4Polya’s Stage Polya’s Substage Percentage

Understand the problems Write down the core of the problem 16.667%Write down the obtained data 23.684%Model the question and select the appropriate notation 23.684%Make a sketch related to the problem solving 84.211%



21.930%


19.298%




Figure 5: Teachers’ Diffi culties in Solving the Problem of the 5th Question

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Table 8. Summary of the Teachers’ Diffi culties in Solving the Problem of the Test Item Number 5Polya’s Stage Polya’s Substage Percentage

Understand the problems Write down the core of the problem 31.579%Write down the obtained data 53.509%Model the question and select the appropriate notation 56.140%Make a sketch related to the problem solving 39.474%



35.088%


34.430%




4. Conclusion

The results of this study indicate that theorder of diffi culties from mathematics teachersof junior high schools in Region 3 in solvingmathematical problems are in executingthe problem solving plan, understandingthe problems, interpreting the results, anddesigning the problem solving plan. Based onthis study, 46.491% teachers have diffi culties inexecuting the problem solving plan, 45.846%in understanding the problem, 43.129% ininterpreting results and 33 063% in designingthe problem solving plan.

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