metacognitive strategies in quadratic equation word problem

JURNAL PENDIDIKAN SAINS & MATEMATIK MALAYSIAVOL.3 NO.2 ISSN 2232-0393

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METACOGNITIVE STRATEGIES IN QUADRATIC EQUATION WORD PROBLEM

1Mariam Bt Ahmad Maulana, 2Nor’ashiqin Mohd Idrus1,2Faculty of Science and Mathematics

Universiti Pendidikan Sultan Idris35900 Tanjong Malim, Perak Darul Ridzuan

Abstract

This research was done to determine which group of students used metacognitive strategies frequently when answering word problems; and to observe metacognitive skills behavior among the groups by using time line graphs. This quantitative research was carried out in one secondary school in Batang Padang involving form four students. Students were divided into three groups, which are higher achievers; middle achievers and lower achievers based on their pretest scores and teacher reference. The instrument used in this research is a test that consists of word problems in quadratic equation. The metacognitive strategies used were assessed from the administered test while the researchers identified metacognitive skills behavior portrayed by students while answering the test by observation. Data was collected from an answer sheet and metacognitive strategies assessment. Students’ responses to the metacognitive red flag questions also had become indicator of the presence of metacognitive strategies used by the students while solving mathematical word problems. The outcomes of the research show that higher achiever group is the group that use metacognitive skills frequently when answering word problems.

Keywords Metacognitive strategies, metacognitive skills, word problems

Abstrak

Kajian ini dijalankan adalah untuk menentukan kumpulan pelajar yang sering menggunakan strategi metakognitif dalam menyelesaikan soalan matematik berayat; melihat kemahiran tingkah laku metakognitif dalam kalangan pelajar dengan menggunakan graf garis masa. Penyelidikan kuantitatif ini telah dijalankan di sebuah sekolah menengah dalam daerah Batang Padang, melibatkan pelajar tingkatan empat. Pelajar dibahagikan kepada tiga kumpulan, iaitu berpencapaian tinggi; pencapaian sederhana dan pencapaian rendah berdasarkan kepada skor ujian pra dan penilaian guru. Instrumen yang digunakan dalam kajian ini adalah soalan matematik berayat bagi tajuk persamaan kuadratik. Strategi metakognitif yang digunakan oleh pelajar dikenalpasti oleh penyelidik melalui tingkah laku digambarkan oleh pelajar semasa menjawab ujian. Data telah diperolehi daripada kertas jawapan ujian dan ujian strategi metakognitif. Jawapan pelajar kepada soalan ‘red flag’ juga telah menjadi petunjuk kewujudan strategi metakognitif yang digunakan oleh pelajar semasa menyelesaikan masalah berayat. Hasil kajian


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ini menunjukkan bahawa kumpulan berpencapaian tinggi adalah kumpulan yang kerap menggunakan kemahiran metakognitif apabila menjawab soalan matematik berayat.

Kata kunci Strategi metakognitif, Kemahiran metakognitif, soalan matematik berayat

INTRODUCTION

The purpose of this research is to determine whether metacognitive skills can be used as one of the reliable ways to solve mathematics word problems. It aims is to assess the usage of metacognitive skills among three groups of form four students, which are higher achievers, middle achievers and lower achievers, on quadratic equation word problem. f you have a bunch of keys to open a door, you should know that only one right key would open the door. What should you do to get a right one? Should you try it one by one or should you identify the characteristic of the door lock in order to open it? If you ask me, I would rather try to identify the characteristic or suitability of the key and the door lock, instead of trying it one by one, as it is time consuming. This ability also will help me during emergency time, as I do not want to waste my time in front of door for a long time. Let us apply this situation into mathematics with the bunch of keys represent the cognitive knowledge that you have while the door represents metacognitive skill which is the ability to identify the characteristic of the key to decide which will fit in and the answer to the problem. Malaysian students who have gone through what Streefland (1991) calls a mechanistic education, that is a rule advertisement algorithm oriented, often do not see the connections between mathematics and real life. It is therefore not surprising that their mathematical experiences lack meaning and purpose. This may also explains why students who are successful mathematical problem solvers in school fail to use mathematical insight when making decision in real life. Word problems enable mathematics to be related into real life as students could see the dynamic of mathematics and how close we are with mathematics in our real life. Word problem in mathematics have becoming something that students wish to avoid, as they do not see any number inside it. What they see are only words that contain relational statements as the sentences express a numerical relation between two variables. They find it hard for them to translate the statement into numbers and decide what approach to be used. Word problem solving requires more thinking process and analyses beyond the key word. Students with average disabilities are unable to distinguish between relevant and irrelevant information, having difficulty in paraphrasing and imaging problem situation. There are a lot of methods introduce to overcome students inabilities to answer word problem. Bar model method, direct translation, concrete represential abstract (CRA) methods are the examples of the methods that widely used to solve word problem in mathematics. Concrete represential abstract (CRA) was one of the frequently used methods to solve word problem. It was made up of three levels, which are concrete level, representational level and abstract level. Concrete level is the level that involves manipulation of an object for example block, by students to model the


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problem. Representational level or visualization level is the level where the students represent the concrete manipulative with picture or diagram. The last level, which is abstract, refers to symbolic representation such as numbers and letters to demonstrate understanding of the task. Direct translation has become one of the methods that usually being used in problem solving. Students tend to translate directly the key words without understanding the problem first. Students have used this method widely during middle school. Students start to memorize the key word such as total and sum for addition process, difference and less for subtraction and other key words. However, this approach is suitable for simple problem only which exist during middle school only. Student’s start having problems during secondary school as the students do not able to identify the key word and the relevant information that they need in order to solve the problems. Some problems require analysis of the unknown while others provide extraneous, too little or incorrect data. Some can be solved in more than one way or have more than one correct answers and some require multiple steps to attain a solution (Baroody, 1987). In addition, problems can be presented in written or oral form (Carraher, Carraher & Schliemann, 1987), and very rarely present themselves in a nicely formulated textbook manner. Schoenfield (1992) mentions that metacognitive skill as essential elements that determine one’s success or failure in problem solving. It is because through metacognition it enables the students to become more flexible when solving the problem as the students with metacognition abilities have the ability to change their strategies when it do not lead them to the answer. This type of ability will lead them to become the successful problem solver. Most of the unsuccessful problem solver was not flexible with their strategies as they kept to the same strategies even do it did not lead them to the answer. Research (Cardella-Ellawar, 1995, Oladunni, 1998) employing metacognitive training had also demonstrated that students who were trained to monitor and control their own cognitive process for solving mathematics problems did better than untrained students. Metacognitive features of expert problem solver (Glaser &Chi, 1988). Through metacognitive strategies for example plan, it enables the experts to adapt to changing condition, eliminate unnecessary step and apply alternative in order to solve the problem. According to Schoenfield (1987, 1992) metacognition is thinking about our thinking and it comprises of three important aspects which is the knowledge about our own thought process, control on self regulation and lastly belief and intuition. Metacognitive knowledge is about reading and memory improves strategy used by providing the children with knowledge about when, where and why they should use different strategies, information about their own capacity and limitation and knowledge about task (Schneider and Pressley, 1989). Control of self regulation is how one use that knowledge to regulate cognition, for example individual modify their thinking process when they realize that the strategy they used is not fruitful to solve the problems. Metacognition is thinking about thinking. It enables awareness and control over how teachers think about teaching. It enables them to self-regulate teaching activities with respect to students, goals and situation. Some metacognition is domain-specific and some is domain-general Two general types of metacognition are: executive


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management strategies that help to plan, monitor and evaluate or revise thinking processes and products, and strategic knowledge about information/strategies/ skills you have, when, why and how to use them. On the other hand, Anderson’s (2001) model of metacognition consists of four aspects: Preparing and planning for effective learning, Evaluating strategy use and learning various strategies, Monitoring strategy use, Selecting and using particular strategies. The importance of problem solving is to valuing the processes of mathematization and abstraction and having the preference to apply them. Cobb et al. (1991, p.187) suggested that the purpose for engaging in problem solving is not just to solve specific problems, but also to “encourage the interiorization and reorganization of the involved schemes as a result of the activity”. Not only does this approach develop student’s confidence in their own ability to think mathematically (Schifter and Fosnot, 1993), it is a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others (NCTM, 1989). Because it has become so predominant a requirement of teaching, it is important to consider the processes themselves in more detail. Ability to solve problem solving is important to develop critical thinking among students as our industrial sector and our country need a lot of worker that poses critical thinking to make the right decision. Study on metacognition had been done widely in our neighbor country, which is Singapore. One of their objectives in curriculum is to develop an ability to solve problem concerning the physical world or the world of imagination. The curriculum consists of constructing mathematical models of situation, events of thought, solving problem in their mathematical form and then translating the solution into ordinary language. Singapore is one of the top three countries in TIMSS. So, should we sit back and close our eyes towards our neighbor success or should we start implement “Dasar Pandang Ke Timur” which is one of Tun Dr. Mahathir Mohammad vision by start looking into the same thing as our neighbor does which is metacognition. Students from all over the world have the same cognitive knowledge but what differentiates them is their metacognitive skill which had made students from Asia become top in TIMSS.TIMSS is the series of international assessments of students’ achievement dedicated to improving teaching and learning in mathematics and science. This research is important because through this research it could be the guide lines for teachers when they teach their students on word problem solving as metacognitive skill is one of the skills that their student must poses in order for them to apply all the cognitive approach effectively. Metacognitive has not been widely adapted in our education as we spend most of the time for drill exercise and finishing our syllabus. Metacognition is important as the fulfillment of our KBKK, Kemahiran Berfikir secara Kreatif dan Kritis as through metacognitive development it prepares our students to be able to use the source or strategy wisely instead of memorizing how to use it. Our country needs the decision maker not memorizer. This research is also important for the students as through the development of metacognitive strategies it will reduce their time to decide on what strategies to be used as time management in exam is crucial.


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This research is also significant in order to increase the awareness of Ministry of Education and educator towards metacognitive skill that lead our neighbor country into top three in TIMSS. Previous researches in metacognitive involve metacognitive intervention in problem solving and word problem which shown significant effect of this intervention.

METHODOLOGY

Subject

Thirty form four students from one of the secondary schools in Batang Padang District participated in this study. Students are chosen based on their monthly test mark and teacher reference in order to provide rich information to the researcher. These students are divided into three groups, which are higher achiever, middle achiever and lower achiever based on their monthly test mark.

Questionnaire

The Self Monitoring questionnaires produce student’s self reports on metacognitive strategies that they employe while answering quadratic equation word problems. To make questionnaire more appropriate for the students, several modification has been made by limiting the respond from three option which are Yes, No and Unsure into two options only which is Yes and No box of respond only. Table 1 below explains the relationships between the questionnaire and metacognitive framework in Figure 1. Questionnaire statements that target the “red flags” which are error detection, lack of progress and anomalous result are identified.


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Table 1 Metacognitive strategies examined by Self – Monitoring QuestionnaireSelf- MonitoringQuestionnaire Items

Monitoring/Regulation

Before you started1. I read the problem more than once. Assess knowledge2. I made sure that I understood what the problem

was asking me. Assess understanding

3. I tried to put the problems into my own words. Assess understanding4. I tried to remember whether I had worked on a

problem like this before.Assess knowledge and understanding

5. I identified the information that was given in a problem.

Assess knowledge and understanding

6. I thought about different approaches I could try for solving the problem.

Assess strategy appropriateness

As you worked “Red Flag”: Error detection7. I checked my work step by step as I went through

the problem.Assess strategy execution

8. I made a mistake and had to redo some working.

Correct error“Red Flag”: Lack of progress

9. I re-read the problem and to check that I was still on track.

Assess understanding

10. I asked myself whether I was getting any closer to a solution.

Assess progress

11. I had to rethink my solution method and try a different approach.

Assess strategy appropriatenessChange strategy

After you finished “Red Flag”: Anomalous result12. I checked my calculations to make sure they

were correct.Assess result for accuracy

13. I looked back over my solution method to checked that I had done what the problem asked.

Assess strategy appropriatenessand execution

14. I asked myself whether my answer made sense. Assess result for sense15. I thought about different ways I could have

solved the problem.Assess strategy appropriateness


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Figure 1 An episode - based model of metacognitive activity during problem solving

Task

Questions for test had been taken from form four mathematics textbooks that align with the syllabus at school. Pilot test has been done in order to determine the suitability of the questions with the students. Based on the pilot test, several modifications on the test have been made. Test was administered to the students and questionnaire was given after the students complete the test.


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Procedural framework

The research contains two phases. The first phase is to determine which group of students that are made up by higher achievers, middle achievers and lowers achievers groups uses metacognitive frequently while answering word problems. Pretest was carried out and the students were assigned to the group according to their scores. Answer sheet, problem solving behavior time line graph, recorded interviews and questionnaire papers were analyzed in order to determine which groups used metacognitive frequently. Figure 2 illustrates the the procedure of the first phase.

Figure 2 Procedural framework of Phase One

Once, the group was identified, the second phase begins. The second phase aims to study the effect of metacognitive training towards lower achievers in quadratic equation word problems. Second phase consists of post test, retrospective interview, problem solving behavior time line graph and instruction on metacognitive skills by the researchers.

Analyzing metacognitive processes in quadratic equation word problem solving

When faced with obstacle and uncertainties, good problem solver displays skills in choosing and testing alternatives strategies and they will to maintain their engagement with the problem (Good, Mulryan& McCaslin, 1992). Effective mathematical thinking in solving problems includes not only cognitive activity, but also metacognitive activity such as metacognitive monitoring that regulates problem-solving activity and allows decision to be made with regards to the allocation of cognitive resources. Schoenfeld (1985) developed a procedure for a parsing verbal protocol into 5 types of episode, which is Reading, Analysis, Exploration, Planning or Implementation and Verification. Artztand Armour (1992) separate Schoenfeld’s Planning and Implementation into two distinct categories. Characteristic of each episode defined by Artzt and Armour are described below:


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According to Goos (2000), deficiency in both frameworks is the lack of detail in describing the types of monitoring and regulating activities that would be appropriate and expected in each episode of problem solving activities. When students meet difficulties during problem solving, it will trigger controlled monitoring and regulatory process. These triggers are labeled as metacognition “red flags” which signal the need for a backtracking while corrective action is taken. There are three types of red flags, which are lack of progress, error detection and anomalous result. Recognizing lack of progress during fruitless exploration will lead students back track to analysis of the problem or reread the problem. Error detection, prompts checking and correction of calculation are done during implementation episode. While anomalous result takes place when they attempt to verify the result that does not make sense leads students back to implementation episode.

Result

The analysis of student’s written solution and questionnaire responds were presented in these section. These section summaries the responses to questionnaire items that corresponds to metacognitive “red flag “ that was identified in Figure 1. Students written solution provide insight either their responses to the questionnaire match their solution work.

Questionnaire Response

Response rate for the four statements that prompt initial recognition of metacognitive “red flags” are shown in Table 2. Table 2 represents the percentage of students respond yes to metacognitive “red flag” questionnaire statements. It is shown that lack of progress gives the least respond of yes that indicate why most of the students spend most of their time on exploration episode. The students didn’t realize that they were lack of progress while solving the word problems. Error detection and anomalous


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result show same percentage of respond, which is 60%. Most of the students avoided thinking different approach that they could try for solving problem. They sticked to the method that had been taught by their mathematics teacher or they only being exposed one-way method. As in Table 3, only 20% of students respond yes to the statement that asking whether they would think about different approaches to solve the problems. This shows that students are lacking of flexibility in procedural and strategy.

Table 2 Questionnaire responses to Metacognitive “Red Flags” statements

Red Flag Questionnaire Statement Percentage (%) of Students Responding Yes

Lack of Progress I asked myself whether I was getting closer to a solution

40

Error Detection I made a mistake and had to redo some working

60

Anomalous Result I asked myself whether my answer made sense

60

Table 3 Questionnaire responses to Assess Strategy AppropriatenessGroup of Achiever Percentage (%)of Yes

Higher 20%

Middle 0

Lower 0 %

This research focuses on determining the group of achiever that used metacognitive frequently. Table 4 shows that higher group achiever used metacognitive frequently as their “yes” respond were high compare to two other groups. The written work and questionnaire responses match high achiever but not really for middle achiever and lower achiever as their responses and written work show a little bit of mismatch. According to the literatures, student that evaluates himself or herself correctly poses high metacognitive skill. Lower achiever often overestimates their ability that lead to poor performance.

Table 4 Percentage of metacognitive used frequently according to the groupGroup of Achiever Percentage (%) of Yes

Higher 73.3

Middle 66.7

Lower 53.3

This result is aligning with the literature review that successful problem solver and expert problem solver used metacognitive skill frequently. According to Chi (1988), metacognitive features of expert problem solver as it enables them to modify their strategy when faced with nonroutine questions and challenging questions.


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DISCUSSION AND CONCLUSION

The study was conduct to determine which group of achiever used metacognitive frequently while answering word problem, which had become a great burden to students. Most students find word problem hard, as it requires them to develop the solution by analyzing and transforming of the keyword from the questions. Quadratic equation word problems require students to have ability to do factorization and develop quadratic equation from the word problem given. The difficulties lie in the word itself, as students do not see any number in word problem. They find it hard to implement the strategy that they already memorize through drill practice. Metacognitive enables the students to choose appropriate strategy to use as there is no single way of solving word problem, as each method has it own pro and cons and each student has it own weakness and strength. Metacognitive had been claim as the skills that are needed to be successful learner and problem solver in math. It generates procedural flexibility that is lacking in our students as most of our students are the one way students. They copy and follow what their teacher writes on the white board and master by doing a lot of practices. This research shows that higher achievers use metacognitive frequently and that represents the relationship between metacognitive and successful problem solver. This also has become the reason why Singapore made metacognitive as one their education that leads their students to the top spot in TIMSS. So, we should implement metacognitive skill in our education system and further studies on how metacognitive should be used in mathematics classroom should be done thoroughly.

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metacognitive strategies in quadratic equation word problem

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