2013. Vol.4, No.7, 585-591
Published Online July 2013 in SciRes (
Copyright © 2013 SciRes. 585
Students’ Metacognitive Strategies in the Mathematics Classroom
Using Open Approach
Ariya Suriyon1, Maitree Inprasitha2, Kiat Sangaroon3
1Doctor of Philosophy Prog ram in Mathematics Education, Khon Kaen University, Khon Kaen, Thailand
2Center for Research in Mathematics Education, Khon Kaen University, Khon Kaen, Thailand
3Department of Mathematics, Faculty of Scie n ce, K hon Ka en Un iversity, Khon Kaen, Thailand
Received March 24th, 2013; revised April 25th, 2013; accepted May 22nd, 2013
Copyright © 2013 Ariya Suriyon et al. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
This paper describes a study investigating students’ metacognitive behavior and abilities in the math-
ematic class using the open approach. Four 1st grade students, ages six to seven years, served as a target
group from the primary school having participated since 2006 in the Teacher Professional Development
Project with innovation of lesson study and open approach. The research was based on Begle’s conceptual
framework (1969), focusing on observing the nature of occurrences in order to describe emerging facts in
the class. In addition, the data were examined by triangulation among three sources: video recording, field
notes, and students’ written works. Data analysis rested upon 4 open approach-based teaching steps (In-
prasitha, 2010). The study results showed that the open approach-based mathematic class helped students
exhibit metacognitive behavior and abilities relevant to the four teaching steps: 1) posing open-ended
problem, 2) students’ self learning, 3) whole class discussion and comparison, and 4) summarization
through connecting students’ mathematical ideas emerging in the classroom.
Keywords: Problem Solving; Metacognitive Strategies; Lesson Study; Open Approach
As for significance of problem solving, National Council of
Teachers of Mathematics (NCTM) (1989) mentioned in the
curriculum standard on the item 1, “mathematics as problem
solving”. Also, the NCTM Curriculum and Evaluation Stan-
dards obviously demonstrated that mathematics might truly be
the best teaching through problem solving situations (Kroll &
Miller, 1993), and in problem solving, it is necessary to empha-
size mathematics at school levels (NCTM, 1980) in relevance
with the study of Inprasitha (1997), concluding in his research
that problem solving was fundamental teaching reform. The
problem solving approach supports education reform as a bot-
tom-up process. “Bottom” means “class”, and “up” means “so-
ciety” as a whole”. In addition, NCTM (2000) referred to the
importance of problem solving as integration of all mathemati-
cal learning by determining a main issue of teaching and learn-
ing programs from the elementary level to grade 12 that stu-
dents should be able to investigate and reflect on mathematical
problem solving, which serves as a basic provision regarding
metacognitive traits. The provision should begin to be used
with students at the lowest school grade in mathematical prob-
lem solving. In the research on mathematical problem solving,
based on the fundamental concept of Flavell (1976: p. 232), the
metacognitive aspect is significant for many researchers. Ac-
cording to Flavell’s definition of metacognition, it can be con-
cluded that “In any kind of cognitive transaction with the hu-
man or non-human environment, a variety of information proc-
essing activities may go on. Metacognition refers, among other
things, to the active monitoring and consequent regulation and
orchestration of these processes in relation to the cognitive
objects or data on which they bear, usually in service of some
concrete goal or objective.”
There is increasing interest concerning the study on roles of
metacognition in mathematical problem solving. However, little
is known about the nature of elementary students’ use of meta-
cognitive strategies, and how these strategies are applied when
students solve problems. Goos and Galbraith (1996) conducted
a study on the nature of using metacognitive strategies by two
secondary students and studied how the students applied those
strategies when they took part in problem solving. In Inpra-
sitha’s study (2003) of Thai students’ metacognition, he found
that when they read mathematical problems, they knew what
were given in the questions, but they could solve problems only
to a certain extent. As for metacognitive strategies, students
conducted observation and investigation in advance of problem
solving by developing plans, monitoring, and evaluating their
own learning or thinking; this approach improved students’
efficiency in open-ended problem solving. These strategies
were still used among students at low levels, although there
were still some pairs of students who did not employ metacog-
nitive strategies during open-ended problem solving. From
review of literature regarding metacognitive strategies, the
study of Pressley, Veenman et al. (2004) shows that to develop
metacognition among student groups, teachers need to have
tools to apply metacognition within classes, beneficial to those
activities. In general, metacognitive learning and teaching is not
only essential to each teacher but also in systematic school
Context of the Study
The objective of this study was to investigate students’
metacognitive behavior and abilities in the open approach based
mathematic class of the school which has participated in the
Teacher Professional Development Project with the innovation
of lesson study and open approach. Therefore, in order to un-
derstand the context as a source of data collection, in this re-
search, the researcher had a role as a participating observer.
Also, details of observations leading to research issues are ex-
plained hereinafter. This study was conducted at Koo Kham
Pittayasan School located in Khon Kaen Province in the
Northeast of Thailand. It is a school with extensive educational
opportunities and teaching management from kindergarten level
to secondary level in grade 9. The innovation of lesson study
and open approach has been introduced through the Teacher
Professional Development Project with collaboration and su-
pervision and monitored by the Center for Research in Mathe-
matics Education, Khon Kaen University since April, 2006.
The use of the applicable innovation as an important method
was emphasized to develop mathematical thinking in the inte-
gration of the lesson study process and open approach. For the
lesson study process, it was adapted from the Japanese practice
by integrating open approach and focusing on participation in
all steps in the cycle of lesson study. A student as a teaching
practitioner, the observing teacher, the researcher as a school
coordinator, and external experts participated in design and
teaching planning, collaborative class observation and discus-
sion and mutual result reflection (Inprasitha & Loipha, 2007)
every week. The principal aim was to create and develop open-
ended situations based on students’ ideas.
Figure 1 shows the process of lesson study every week, at
Koo Kham Pittayasan School, in which the student as a teach-
ing practitioner at school, the observing teacher, and the re-
searcher as a school coordinator collaborated in the design and
teaching planning on Tuesday and participated in observation
based on the teaching schedule assigned from the school direc-
tor. They discussed results on Thursday. This process has con-
tinued since 2006 at the school. In the mathematical class, there
was a plan in arranging learning and teaching activities high-
lighting the open approach-based problem solving with the aim
that students could participate in activities and show potential
in mathematical thinking with all of their ability. From the re-
searchers’ participation in each step of lesson study and the
continuous observation of elementary student at grade 1 since
the academic year of 2007, it found that while solving problems,
for most of their behavior, the students focused on writing
along with thinking: writing diagram, writing expressions sen-
tences, and writing description on a self-thinking process with
letters. Students tried to write description of self-thinking by
writing and spelling to communicate with other people (Suri-
yon et al., 2011). Rose (1989 cited in Pugalee, 2004) described
the process as “Thinking aloud on paper.” Writing is not only
describing what they think but also providing evidence of
thinking that helps students to be aware of their thoughts and
shows how they solve problems. Writing is thinking evidence
that problem solvers can use to investigate their self-thinking
process. Also, NCTM (2000: p. 61) suggested, “in writing,
mathematics can help students gather their thoughts since writ-
ing requires students to reflect on their work results and clari-
Figure 1.
Lesson study cycle (Inprasitha, 2004).
fies their thoughts about concepts.” Pugalee (2001) suggested
that students should develop an association between writing,
metacognition, and mathematical problem solving. Therefore,
the research considered students’ written works as important
evidence in this st udy.
The methodology was based on qualitative research in which
the research aimed at investigating and finding emerging facts
related to students’ use of metacognitive strategies while solv-
ing problems in the mathematics class using the open approach.
Considering students’ behavior and abilities to solve problems
was based on information on classroom observation and par-
ticipation in the academic years of 2008 and 2009 as criteria for
inclusion of target groups. The target group was a group of 1st
grade students (1 male and 3 females) in the academic year of
2010 from Koo Kham Pittayasan School with the age range
from 6 to 7 years. The students have been studying at grade 1
for 7 months. The researcher, the teacher, and the observing
teacher worked together in the consideration by determining
attributes of the target group concerning the following behavior
and abilities.
Abilities to speak, read, and write explanations of their own
thinking ways and their groups’ ideas.
Behavior of monitoring their own thinking process and their
groups’ ideas.
Abilities to think differently.
Helping and working together with other people.
Talking and explaining reasons to illustrate their own
thinking ways and ideas from their groups.
Participating in the classroom continuously using the open
approach from the beginning of the first semester in the
academic year of 2010.
The teacher was a mathematics intern student from Faculty
of Education, Khon Kaen University. She practiced teaching in
the academic year of 2010, focusing on the open approach and
use of the open-ended problem situation developed from stu-
dents’ ideas. In addition, she took part in lesson planning with
teaching staffs, the school coordinator, the researcher, and other
intern students (as observers). According to the lesson study
cycle, the open approach was used during instructional practice
with emphasis on the problem solving process. The research
team has participated progressively in all steps of the lesson
study process from 2008 to the present.
For this research, qualitative methodology was applied with
Copyright © 2013 SciRes.
an emphasis on the three-year observational study based on
Begle’s conceptual framework (1969) with ways of observation
and consideration on studying the nature of occurrences which
starts with extensive, careful, and empirical observations of
mathematics teaching and learning. In 2008 and 2009, any
trends noted in these observations would lead to the formula-
tion of hypotheses. In 2010, these hypotheses could then be
checked agains t further observ ations and refined and sharpened.
There was data collection in the learning unit on “addition (2)”,
the first activity “Children playing in sandboxes and on slides”
and the second activity “Buying eggs to make omelets”, which
were developed from the lesson study process. All important
qualitative data came from class observations, video recording
while students were solving problems, field notes, and students’
written works which were analyzed. This process was based on
triangulation from three data sources: video recording while
students were solving problems, field notes, and analyzing stu-
dents’ written works.
From the data analysis, it showed students’ metacognitive
strategies by analyzing students’ problem solving behavior in
the class, corresponding to the teaching steps in the open ap-
proach. The data from the first activity “Children playing in
sandboxes and on slides” was used to reflect on images of the
previous class in order to analyze the data in the next activity
“Buying eggs to make omelets.” From the second activity, data
were interpreted and determined as an explanation related to the
students’ problem solving behavior in the classroom in order to
show existing consistency in each teaching step through the
open approach. In this research, students’ behavior and abilities
showing use of metacognitive strategies in each open ap-
proach-based teaching step were considered under the follow-
ing definition of metacognitive strategies. Metacognitive strate-
gies could be defined as thinking ability causing behavior that a
problem solver can control, monitor, and reflect his own think-
ing process, based on an idea or a way which he values from
existent resources—accumulative recording of previous learn-
ing experiences and which he then uses as a problem solving
tool which will function as a determinant of thinking ways and
keep continuous problem solving for advance in problem solv-
ing. Furthermore, metacognitive strategies are used for exam-
ining his own thinking and ensuring that he has already
achieved his goal. According to the study, the researcher has
obtained the following results.
Activity 1: Children playing in sandboxes and on slides
Problem situation: There are 9 children playing in the
sandbox and 4 children playing on the slide. How many chil-
dren are there (see Figure 2)?
Instructions: 1) Students find out how many children there
Figure 2.
Picture for the activity 1 (Gakkoh Tosho, 1999).
are and explain their ways of thinking, 2) Students present their
This activity occurred during the first period in the unit on
“addition (2)”. At the beginning of the activity, the teacher
reviewed students’ previously learned ways of thinking in
preparation for students’ readiness in problem solving for the
next activity “Children playing sandboxes and on slides”. After
that, when students learned problem solving on their own
(learning how to learn), the target student group used ways of
counting at the beginning of problem solving and then ways of
problem solving: Student remembered previously learned ideas
and strategies: how to make ten, decomposing, writing block
diagram and arrow diagrams to describe thinking process in-
cluding explanation of their thinking processes in their own
words. The previous ideas and ways were used as tools to solve
problems they were encountering. Students were able to moni-
tor and to reflect on the thinking process with their own words
as shown in Figure 3. Students tried to create problem solving
strategies that showed different ways of thinking besides only
finding an answer: students used how to make ten as a way to
solve the problem. Then students decided to apply that idea as a
problem-solving tool, making the problem solving process car-
ries on progressively and students succeed in problem solving
in the following situation.
Activity 2: Buying eggs to make omelets
Problem situation: Ms. Pha had three eggs in the egg tray.
She wanted to make some omelets, but the number of eggs was
not enough. Then she went out to buy an egg tray containing 9
eggs from Pop, egg seller. From this activity, students need to
find out the total number of eggs Ms. Pha has (see Figure 4).
Instructions: 1) Students show the thinking way “3 + 9” and
other ways of thinking, 2) Then students show their thinking.
To look for students’ metacognitive strategies which ap-
peared in the four teaching steps through the open approach in
the activity 2 “Buying eggs to make omelets”, the research
team examined activity management by video recording and
considered the interpretation by checking data from field notes
and students’ written works. The research team examined
con-sistency with the definition of metacognitive strategies. Fol
Figure 3.
Evidence from students’ written works in the activity 1.
Figure Egg trays
Figure 4.
Materials for the activity 2.
Copyright © 2013 SciRes. 587
lowing are the details of the four steps through the open ap-
1) Posing open-ended problems
This step was the beginning of problem solving: posing
open-ended problem related to a problem situation presented by
the teacher as shown in Figure 5. Students’ behavior and use of
metacognition, which connected with the first step related to
attempting to understand the problem situation; students
thought those problems were their problems (students’ prob-
lematic), showing their desire in proving or finding solutions by
themselves. This conclusion was drawn from the observed en-
thusiasm to solve problems by themselves or saying something
to show acceptance , for example, “I can do”, or “I want to do”,
and expressing their happiness when the teacher asked students
to participate in problem solving as a group. In this step, the
teacher posed the problem situation to students by using pic-
tures and telling stories in order to lead to the mentioned in-
structions. The teacher began her class by greeting students,
and after that, she put the pictures on the magnet board and
chose two volunteer students to act as supporting characters in
the stories by using the pictures. The teacher allowed students
to observe and consider the pictures.
In this step, it demonstrated that students tried to understand
the problem by showing enthusiasm, concentrated on the pic-
tures, and described what they had observed. Students’ attempts
to find answer and make predictions were the beginning, which
led to proof of finding facts of conclusions by students for the
next steps.
2) Students’ self learning
The students’ self learning began after the teacher presented
the open-ended problem. Behavior and students’ metacognitive
strategies were regarding students’ learning how to learn by
participating in problem solving in subgroups. One student was
the recorder of ideas on papers to present, and three students
participated in showing ideas by expressing their ideas. While
the student recorded ideas, the members examined ways of
thinking by taking an egg from the tray of 3 eggs to put on the
tray of 9 eggs so that 9 could became 10. In the first tray, 2
eggs were left, as shown in Figure 6.
Figure 5.
The teacher posed the problem situation to students by using
pictures and telling stori es .
Figure 6.
Examining ideas by exper iment from egg trays.
After finishing writing their first ideas, members in each
group helped each other check by reviewing the ideas, showed
their opinions towards and improved what they had done. Dur-
ing the problem solving step, the students changed roles within
their groups. The strategies and ideas of problem solving which
students used as a thinking tool were considered as previously
learned: how to make ten, decomposing by writing blocks,
arrows to show thinking process, and descriptions of thinking
process using their own words. Students tried to show different
and various ways of thinking by writing to show the thinking
process as the Table 1. In addition, they studied the problem
together and asked questions in their groups while solving
problems. For students in each group who did not work on re-
cording, they worked on checking works instead from written
works of the groups, questioning and reasoning to make a mu-
tual conclusion.
3) Whole class discussion and comparison
Whole class discussion was the relevant step to behavior and
abilities of students showing metacognitive strategies. In this
step, it included examining problem solving strategies together
from the teacher and classmates, accepting suggestions from
the teacher and friends, including correcting mistakes immedi-
ately. To begin with, students who were volunteers of each
group presented works in front of the class, introduced them-
selves, and presented the ideas from their groups. For presenta-
tion from each group, students participated in asking a question,
explaining, and comparing and contrasting their friends’ strate-
gies, with their own. There was consideration of works from
many student groups in terms of formats of writing messages
and expressions. Presenting students listened to suggestions
from their friends and corrected mistakes for some ideas. The
teacher and students in the class worked together to solve prob-
lem for mutual understanding as illustrated in the Figure 7.
4) Summarization through connecting students’ mathe-
matical ideas em er gi ng in the classroo m
The final step was conclusion of connecting students’ ideas.
Behavior and students’ matacognitive strategies were regarding
evaluating validity and correctness of ideas and ways that stu-
dents performed corresponding to the initial problem situation.
For assessment of ideas and strategies, the students valued ef-
fective idea and ways, for example, applying the idea of how to
make ten, which was considered as a simple way to solve prob-
lems and take less time. For considering choice and making a
decision, students used the data from making choices in prob-
lem solving through various ways and ideas and from experi-
ences in solving problems by themselves in the step of self-
learning. Moreover, it included
Comparison of efficiency of ideas and ways in the step of
Figure 7.
Whole classroom discussion and co mparison.
Copyright © 2013 SciRes.
Copyright © 2013 SciRes. 589
Table 1.
Students’ strategies in problem solving and evidence from students’ written works.
Students’ thinking ways of problem solv ing Students’ written works
1) Additio n by writing blo ck diagram with n umbers in each bl ock and
writing an arrow to show the thinking process based on how to make
ten with written descriptions of own thinking process, which show
self-correc tion of mista kes
Strategy 1
3 gives 1 to 9, then 2 was left from 3. Then combine 2 with 10 beco mes 12.
2) Addition by writing block diagram and drawing an arrow to show
their own thinking process based on how to make ten with written
descriptions of own th inking process, which show correcting mistakes
Strategy 2
3 gives 1 to 9. Take 9 combine with 1 become 10. 2 was left from 3. Then
take 2 combine with 10 be comes 12.
3) Addition by writing symbolic sentences and showing answers
Strategy 3
4) Addition by decomposing of addend (Students found that writing
one number nearly close to another one led to confusion. To avoid this
problem, students should use strategies 5 and 6.)
Strategy 4
5) Addition by decomposing of top line (This way is based on how to
make ten, decomposing, and linking an arrow to describe the thinking
process. This way is based on the same ideas as in the first way, but it
is different in showing thinking process.)
Strategy 5
6) Addition by decomposing of top line (This way is based on how to
make ten, decomposing, and linking an arrow to explain the thinking
Strategy 6
whole class discussion and valuing ideas as shown in the Fig-
ure 8. Selected ways were recorded as meaningful resources for
students which were then used as a thinking tool to solve other
The aforementioned data caused compiling and organizing
data to show association between teacher teaching behavior and
students’ problem solving behavior in the open approach-based
mathematic class. This method encouraged students to show
behavior and abilities, reflecting metacognitive strategies
through the open approach.
Discussion and Conclusion
These results illustrate the importance of metacognitive
strategies, which could bring about successful student mathe-
matical problem solving. It could be seen that students could
solve problems successfully; they tried to find various problem
solving strategies and could continue solving problems without
Figure 8.
Summarization through connecting students’ matheatical ideas
emerging in the classroom.
giving up their efforts to create new problem solving ap-
proaches and to express various ways of thinking by using
problem solving tools of previously learned ideas and strategies.
These findings are in line with Schoenfeld’s conclusion (1985)
that a good problem-solver constantly questions his or her
achievement. S/he generates a number of possible candidates to
the method of solution, but is not seduced by them. By making
careful moves such as pursuing productive leads and abandon-
ing, fruitless path, s/he solves the problem successfully.
Secondly, the study showed association between the open
approach-based teaching and students’ problem solving process.
The open approach-based teaching underlining problem solving
in the mathematic class consisted of the four teaching steps: 1)
posing open-ended problem, 2) students’ self learning, 3) whole
class discussion and comparison, and 4) summarization through
connecting students’ mathematical ideas emerging in the class-
room. The aforementioned relation could be seen from recipro-
cal assimilation between the teacher’ s teaching behavior and
students’ problem solving behavior, leading to planned objec-
tives. Each teaching step promoted students’ learning in many
skills and processes, for example, ability of connecting their
previously learned ideas with new situations, ability to commu-
nicate with other people, open-mindedness, ability to work with
other people, and especially the emphasis that student could
learn and solve problems by themselves. The study results are
consistent with the study of Kongthip et al. (2012) which
showed that the open approach-based mathematics class in the
lesson study context a l l owed the student s to have opportunity in
learning base d on t h eir potentiality , being able to think, perform,
and express. They preferred to e xpress divergent think.
In addition, the findings indicated the importance of open-
ended problem solving situations, planning teacher orders for
learning units and planning order of activities in each study
period according to objectives in each unit and in each study
period. Those plans were developed from the process of lesson
study with an emphasis on preparation for important learning
experience depending on recording and combining what stu-
dents learned and especially tools for students’ thinking as a
way or an idea of thinking for problem solving which the stu-
dents could apply in the future and could do by themselves. The
teacher’ s teaching and learning activity management corre-
sponded to the open approach based teaching steps to create a
class highlighting the problem solving process. This classroom
environment could help motivate students to participate in
problem solving and to express various thinking ways. Also,
the students could apply their previously learned knowledge
and experiences to solving new problems. Students’ problem
solving behavior with monitoring and reflecting on their own
problem solving process showed students’ efficient metacogni-
tive strategies as a good trait of a good problem solver which
should be cultivated in students beginning at the earliest school
grade as recommended by NCTM (2000).
According to the study results, what the research team is in-
terested in further research is developing the aforementioned
findings into creating tools for exploring students’ metacogni-
tive strategies in order to survey and study how students devel-
oped metacognitive strategies in open-ended problem situations.
In addition, it includes contextual factors affecting development
of students’ metacognitive strategies in the mathematic class-
room, using the innovation of lesson study and open approach
in three areas: the structure of teaching and learning activities in
the class, the teacher’ s intervention and interaction with stu-
dents, and interaction between students. The research team
plans to explore these areas for further study.
This research was supported by the Higher Education Re-
search Promotion and National Research University Project of
Thailand, Office of the Higher Education Commission, through
the Cluster of Research to Enhance the Quality of Basic Educa-
tion. This research was partially supported by Center for Re-
search in Mathematics Education, Thailand.
Begle, E. G. (1969). The role of research in the improvement of
mathematics education. Educational Studies in Mathematics, 2, 232-
244. doi:10.1007/BF00303460
Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. B.
Resnick (Ed.), The nature of intelligence (pp. 231-236). Hillsdale, NJ:
Gakkoh Tosho Co., LTD. (1999). Study with your friends MATHE-
MATICS for elementary school 1st grade Gakkoh Tosho. Tokyo:
Gakkotosho Co., LTD.
Goos, M., & Galbraith, P. (1996). Do it this way! Metacognitive strate-
gies in collaborative mathematical problem soving. Educational
Studies in Mathematics, 3 0 , 229-260. doi:10.1007/BF00304567
Inprasitha, M. (1997). Problem solving: A basis to reform mathematics
instruction. The Journal of the National Research Council of Thai-
land, 29, 221-259.
Inprasitha, M. (2003). Reform process of learning mathematics in
schools by focusing on mathematical proces se s. Khon Kaen.
Inprasitha, M. (2004). Teaching by open-approach method in japanese
mathematics classroom. KKU Journal of Mathematics Education, 1.
Inprasitha, M., & Loipha, S. (2007). Developing student’ s mathemati-
cal thinking through lesson study in Thailand. Progress Report of the
the APEC Project: “Collaborative Studies on Innovations for
Teaching and Learning Mathematics in Different Cultures
(II)-Lesson Study focusing on Mathematical Thinking” Center for
Research on International Cooperation in Educaitonal Development
(CRI CE D) (pp. 255-264). Tsukuba.
Inprasitha, M. (2010). One feature of adaptive lesson study in Thailand:
Designing learning unit. Proceeding of the 45th Korean National
Meeting of Mathematics Education (pp. 193-206). Gyeongju: Dong-
kook University.
Kongthip, Y., Inprasitha, M., Pattanajak, A., & Inprasitha, N. (2012).
Mathematical communication by 5th grade students’ gestures in les-
son study and open approach context . Psychology, 3, 632-637.
Kroll, D. L., & Miller, T. (1993). Insights from research on mathemati-
cal problem solving in the middle grades. In D. T. Owens (Ed.), Re-
search ideas for the classroom: Middle grades mathematics. Reston:
National Council of Teachers of Mathematics (NCTM) (1980). An
agenda for action. Reston, VA: National Council of Teachers of
National Council of Teachers of Mathematics (NCTM) (1989). Cur-
riculum and evaluation standards for school mathematics. Reston,
VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (NCTM) (2000). Princi-
ples and Standards. Reston, VA: NCTM.
Pugalee, D. K. (2001). Writing, mathematics, and metacognition:
Looking for connections through students’ work in mathematical
problem solving. School Science and Mathematics , 101, 236-245.
Pugalee, D. K. (2004). A Comparison of verbal and written descriptions
of students’ problem solving processes. Educational Studies in
Mathematics, 55, 27-47.
Schoenfeld, A. H. (1985). Mathematical problem soving. New York:
Copyright © 2013 SciRes.
Copyright © 2013 SciRes. 591
Academic Press.
Suriyon, A., Sangaroon, K., & Inprasitha, M. (2011). Exploring stu-
dents’ metacognitive strategies during problem solving in a mathe-
matics classroom using the open approach. Proceeding of the 35th
PME Conference, 1, 397.
Veenman, M. V. J., Wilhelm, P., & Beishuizen, J. J. (2004). The rela-
tion between intellectual and metacognitive skills from a develop-
mental perspective. Learning and Instruction, 14, 89-109.