JURNAL PENDIDIKAN SAINS & MATEMATIK MALAYSIA
VOL.3 NO.2 ISSN 2232-0393
METACOGNITIVE STRATEGIES IN QUADRATIC EQUATION
WORD PROBLEM
Mariam Bt Ahmad Maulana, 2Nor’ashiqin Mohd Idrus
1,2
Faculty of Science and Mathematics
Universiti Pendidikan Sultan Idris
35900 Tanjong Malim, Perak Darul Ridzuan
1
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 Questionnaire
Self- Monitoring
Monitoring/Regulation
Questionnaire Items
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Before you started
I read the problem more than once.
I made sure that I understood what the problem
was asking me.
I tried to put the problems into my own words.
I tried to remember whether I had worked on a
problem like this before.
I identified the information that was given in a
problem.
I thought about different approaches I could try
for solving the problem.
As you worked
I checked my work step by step as I went through
the problem.
I made a mistake and had to redo some
working.
I re-read the problem and to check that I was still
on track.
I asked myself whether I was getting any closer
to a solution.
I had to rethink my solution method and try a
different approach.
After you finished
I checked my calculations to make sure they
were correct.
I looked back over my solution method to checked
that I had done what the problem asked.
I asked myself whether my answer made sense.
I thought about different ways I could have
solved the problem.
6
Assess knowledge
Assess understanding
Assess understanding
Assess knowledge and
understanding
Assess knowledge and
understanding
Assess strategy appropriateness
“Red Flag”: Error detection
Assess strategy execution
Correct error
“Red Flag”: Lack of progress
Assess understanding
Assess progress
Assess strategy appropriateness
Change strategy
“Red Flag”: Anomalous result
Assess result for accuracy
Assess strategy appropriateness
and execution
Assess result for sense
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
Percentage (%) of
Red Flag
Questionnaire Statement
Students Responding Yes
Lack of Progress
I asked myself whether I was getting
40
closer to a solution
Error Detection
I made a mistake and had to redo some
60
working
Anomalous Result I asked myself whether my answer
60
made sense
Table 3 Questionnaire responses to Assess Strategy Appropriateness
Group 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 group
Group 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.
References
Ana Kuzie (2012). Patterns of Metacognitive Behavior During Mathematics ProblemSolving in a Dynamic Geometry Environment. International Electronic Journal of
Mathematics Education, .8(1), 20 - 37.
Anca Moga (2012). Metacognitive Training Effects on Students Mathematical
Performance from Inclusive Classrooms. Unpublished doctoral thesis. BabeșBolyai University, Cluj-Napoca.
Artz, A.F. & Armour-Thomas, E. 1992. Development of a cognitive-metacognitive
framework for protocol analysis of mathematical problem solving in small groups.
Cognition and Instruction, 9(2), 137-175.
Carlson, M. P. & Bloom, I. (2005). The cycle nature of problem solving: An
emergent multidimensional problem-solving framework. Educational Studies in
Mathematics, 58, 45–75.
Chi, M.T.H., Bassok, M., Lewis, M.W., Reimann, P., & Glaser, R. (1989). Selfexplanations: How students study and use examples in learning to solve probeelms. Cognitive Science, 13, 145–182.
Chih Ting Yang & Shin Yi Lee (2013). The Effect of Instruction in Cognitive and
Metacognitive Strategies on Ninth-Grade Students’ Metacognitives Abilities.
Unpublished paper. Taipei Municipal University of Education, Taiwan.
Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. B. Resnick (Ed.),
The Nature of Intelligence. Hillsdale NJ: Lawrence Erlbaum Associates.
11
JURNAL PENDIDIKAN SAINS & MATEMATIK MALAYSIA
VOL.3 NO.2 ISSN 2232-0393
Garofalo, J. & Lester, F. K. 1985. Metacognition, Cognitive Monitoring and
Mathematical Performance. Journal for Research in Mathematics Education,
16( 3), pp. 163 - 176.
Goos, M. (2000). Metacognition in context: A study of collaborative metacognitive
activity in a classroom community of mathematical inquiry, Unpublished doctoral
thesis. The University of Queensland, Brisbane.
Hartman, H. J. (2001). Developing students’ metacognitive knowledge and strategies. In
H. J. Hartman (ed.), Metacognition in Learning and Instruction: Theory, Research,
and Practice, chapter 3, pp. 33–68. Kluwer Academic Publishers, Dordrecht, The
Netherlands.
Mevarech, Z. & Kramarski, B. 1997. IMPROVE: A multidimentional method for
teaching mathematics in heterogeneous classrooms. American Educational
Research Journal, 34, 365-394.
Mohini, M. & Tan Ten Nai (2005). The use of metacognitive process in learning
mathematics, Reform, Revolution and Paradigm Shifts in Mathematics Education,
159-162, Johor Bahru, Malaysia, Nov 25th – Dec 1st.
Nelson, T. O., & Narens, L. (1990). Metamemory: A theoretical framework and new
findings. In G. H. Bower (Ed.), The Psychology of Learning and Motivation:
Advances in Research and Theory (Vol. 26, pp. 125–173). San Diego, CA:
Academic Press.
Polya G. (1973). How to Solve It. Princeton: Princeton University Press.
Schoenfeld A. H. (1992). Learning to Think Mathematically: Problem Solving,
Metacognition and Sense Making in Mathematics. In D. Grouws (Ed.), Handbook
of Research on Mathematics Teaching and Learning, pp. 334 - 370. New York:
Macmillan.
Von Glasersfeld, E. (1995). Radical Constructivism: A Way of Knowing and Learning.
London: Falmer Press.
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