Students’ Metacognitive Strategies in the Mathematics Classroom Using Open Approach

doi:10.4236/psych.2013.47084

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Psychology

2013. Vol.4, No.7, 585-591

Published Online July 2013 in SciRes (http://www.scirp.org/journal/psych) http://dx.doi.org/10.4236/psych.2013.47084

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

Email: [email protected]

Received March 24th, 2013; revised April 25th, 2013; accepted May 22nd, 2013

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

Introduction

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

A. SURIYON ET AL.

management.

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.

Method

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

586

A. SURIYON ET AL.

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.

Results

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

works.

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.

A. SURIYON ET AL.

lowing are the details of the four steps through the open ap-

proach:

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.

588

A. SURIYON ET AL.

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

process.)

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

problems.

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.

A. SURIYON ET AL.

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.

Acknowledgements

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.

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