Sociology Mind
2013. Vol.3, No.4, 284-289
Published Online October 2013 in SciRes (http://www.scirp.org/journal/sm) http://dx.doi.org/10.4236/sm.2013.34038
284
Contextual Factors in the Open Approach-Based Mathematics
Classroom Affecting Development of Students’ Metacognitive
Strategies
Ariya Suriyon1, Maitree Inprasitha2, Kiat Sangaroon3
1Department Doctoral Program in Mathematics Education, Khon Kaen Un iversity, Khon Kaen, Thailand
2Center for Research in Mathematics Education, Khon Kaen Univ ersity, Khon Kaen, Thailand
3Department of Mathematics, Faculty of Science, Khon Kaen Unive rsity, Khon Kaen, Thailand
Received July 8th, 2013; revised August 16 th, 2013; accepted August 29th, 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.
The objective of this research was to study the effect that contextual factors have on the development of
students’ metacognitive strategies in the open approach-based mathematics classroom: the framework for
learning and teaching activities in the class, the teacher’s role, and students’ role. The methodology was
based on ethnographic research and Begle’s conceptual framework (1969), which focused on observation
and study on the nature of occurrences. In the context, the researcher conducted participatory classroom
obse rvatio n. The ta rget gr oups wer e a mathe matics teacher , who is a s tudent a s a math teaching practi-
tioner, and four elementary school students at Grade 1 ranging from 6 to 7 years of age from Koo Kham
Pittayasan School. Data were collected from 3 learning units totaling 6 study periods. Qualitative data
analysis procedures were based on analyzing videos, protocols, students’ written work, and time units for
dealing with activities and narrative description. The concept of 4 open approach-based teaching steps
(Inprasitha, 2010) was considered for the analysis of the teacher’s teaching behavior and students’ prob-
lem solving behavior. The study findings suggest that contextual factors in the open approach-based
mathematics classroom affect the development of students’ metacognitive strategies in which the teacher
has planned learning management related to learning unit structures and focused on instructional activities
allowing students “to create knowledge from learning how to solve problems by themselves”. In addition,
the study demonstrates that the teacher and students have different roles in each teaching step.
Keywords: Contextual Factors; Metacognitive Strategies; Lesson Study; Open Approach
Introduction
A review of research papers on mathematical problem solv-
ing with regard to metacognition yields findings that fall under
the fundamental concept of Flavell (1976) regarding monitoring
and regulation. The importance of research into the teaching of
problem solving has been acknowledged since the 1980s (Les-
ter, 1994), the researchers attempted to find explanations for
various aspects due to the belief that metacognt ion is what makes a
problem solver successful in solving problems, corresponding
to Lesh (1982), Silver (1982) and Schoenfeld (1982) showing
that metacognitive actions as “a driving force” in problem
solving.
Use of metacogntive strategies is considered a strategy that a
problem solver applies to solving problems with various aims
besides that of finding answers only. In other words, it is a
strategy that a problem solver uses to monitor his or her goal in
problem solving, or it can be said that he or she is a problem
solver with characteristics of good thinking. Monitoring as men-
tioned above can be seen from monitoring behavior and reflec-
tion on a problem solver’s thinking process from work which
he or she has alr eady done. As for arran ging learning and teach-
ing activities in a class to stimulate or prompt students to apply
metacognitive strategies, it is considered difficult and compli-
cated. Allowing students to have a chance to participate in
mathematical problem solving is vital for encouraging students
to have a chance to create and develop metacognitive strategies;
therefore, it requires conditions and contextual factors related to
the classroom and learning and teaching activity design which
is based on thorough and careful planning including considera-
tion of the teacher’s and students’ roles with an emphasis on
practice guidelines leading to students’ participation by “creat-
ing knowledge from learning how to solve problems by them-
selves”. The teacher and students should consider these issues
and work together to find practice guidelines for creating good
classroom contexts, leading to the development of students’
metacognitive strategies as an outcome.
Another aspect of Silver’s research (1985) suggests that s tudy
concerning metacognition is an important issue and should be
considered for further research on mathematical problem solv-
ing, especially the study on development of a person or a group
of people in age ranges related to ones’ ability to solve prob-
lems, which is a necessity. According to Silver’s belief, study-
ing that aspect is fundamental for knowledge seeking, used by

A. SURIYON ET AL.
researchers for understanding mathematical learning and teach-
ing processes. In addition, the process of considering and de-
termining research issues on that aspect is important as a driv-
ing force in the future problem solving theory. Moreover, Les-
ter’s study (1994) stated that there are 3 study findings accept-
able concerning successful influences of metacognition in prob-
lem solving. The first finding, effective activities related to
metacognition during problem solving, was that students needed
not only to know something and when to monitor it but also to
know how to monitor it, meaning that teaching students how to
monitor their behavior was considered a difficult task. The sec-
ond finding was that teaching students to realize what happened
as they knew and monitored their performance in better prob-
lem solving should occur in the context of learning mathemati-
cal concepts and techniques, in particular for learning and teach-
ing in general which could take place but less efficiently. The
third finding was that complete metacognition development was
difficult and sometimes required stopping inappropriate behav-
ior development from previous experiences (Schoenfeld, 1992).
Those issues indicate that study on metacognition contains an
important aspect that should be examined and explained more
in research, especially finding ways of learning and teaching
management as w e ll as elements and conditions of devel o pment
of students’ metacognitive strategies leading to efficiency.
The study was conducted at a school which has participated
in the Teacher Professional Development Project with innova-
tions in lesson study and open approach since 2006. The fol-
lowing 3 steps instituted as a method of lesson study in the
process underlying collaboration among a teacher or a student
as a teaching practitioner, an observing teacher, a school coor-
dinator, and the researcher were illustrated as in Figure 1: 1)
participation in learning management planning; 2) collaborative
class observation; and 3) mutual result reflection on teaching
practice.
An issue of importance mentioned above has brought about a
research question concerning how practice guidelines on class-
room action affecting development of students’ metacognitive
strategies in the open approach-based mathematics classroom
were represented in three issues: the framework for learning
and teaching activities in the class, the teacher’s role, and the
students’ role.
Objective
The research aimed at studying practice guidelines in the
Figure 1.
Lesson study cycle (Inprasitha, 2004).
classroom as a contextual factor affecting the development of
students’ metacognitive strategies in the mathematics classroom
using the open approach on 3 issues: the framework for learn-
ing and teaching activities in the class, the teacher’s role, and
the students’ role.
Method
As for the research methodology, ethnographic research was
conducted, and Begle’s conceptual framework (1969), which
focused on observing the nature of occurrences, was employed.
The researcher had conducted participatory classroom observa-
tions from the academic years 2008 to 2010. Data were col-
lected in the academic year of 2010 in order to analyze findings.
The target groups consisted of one teacher who was a student as
a mathematics teaching practitioner at a school from Khon
Kaen University and four elementary school students in grade 1
aged 6 to 7 years (1 male and 3 females) from Koo Kham Pit-
tayasan School. Data were collected from the following 3
learning units totaling 6 study periods: addition (2), subtraction
(2), and addition or subtraction? Qualitative data analysis pro-
cedures were based on analyzing videos, protocols, students’
written work, and time units for dealing with activities and
narrative description. The analysis on the teacher’s teaching
behavior and students’ problem solving behavior was based on
the following 4 open approach-based teaching steps (Inprasitha,
2010).
1) Posing open-ended problems;
2) Students’ self learning;
3) Whole class discussion and comparison;
4) Summarization through connecting students’ mathemati-
cal ideas emerging in the classroom.
The research tools included a learning management plan de-
veloped from a lesson study process which comprised 6 study
periods one of which totaled 60 minutes, field notes, and vid-
eos.
Results
Practice guidelines as a contextual factor affecting the de-
velopment of students’ metacognitive strategies in the open ap-
proach-based mathematics classroom were addressed on 3 cri-
teria: the framework for learning and teaching activities in the
class, the teacher’s role, and the students’ role. Based on the use
of analytical description, the obtained results are demonstrated
hereinafter.
Framework for Learning and Teaching Activities in
the Class
1) Learning management planning in connection with
learning unit structures
The lesson study team began the learning management plan-
ning by working together to design learning unit structures by
determining purposes and a number of study periods for each
learning unit. What was taken into account was what students
could learn after they finished each learning unit. To study that
issue, the lesson study team used mathematics textbooks at-
tached to the Teacher Professional Development Project with
the innovation of lesson study and open approach as a main
document for reference and as a guideline for design. Subse-
quently, towards the planning of learning management for each
study period, the team determined purposes of learning con-

A. SURIYON ET AL.
286
nected with previously planned purposes of learning. Based on
field notes gathered from mutual learning management plan-
ning, the learning and teaching planning structure was designed
by the Center for Research in Mathematics Education, Khon
Kaen University. The structure of learning and teaching activi-
ties with an emphasis on 4 open approach-based teaching steps
and the concept of students’ approach to solving problems were
used for the planning of learning management for each study
period. The results of the procedure showed that the learning
unit structure had impacts on students’ thinking structures. For
example, in the learning unit on addition (2), most of the ideas
that students applied to problem solving were the product of
accumulative recording of previous learning experiences, in-
dicative that the learning unit structure designed for helping
students apply what they learned to further utilization was in-
deed implemented.
2) Structure of learning and teaching activities for each
period, as a part of time taken from arranging activities
With regard to learning and teaching activities emerging in
each study period, the teacher planned activities emphasizing 4
open approach-based steps. Table 1 shows the time used for
managing activities for each step.
From the data shown in Table 1, the researcher calculated
the average time used for each open approach-based teaching
step, time elapsed from actual classroom action is shown in
Table 2.
Table 1.
Time used for instructional activity management for each open approach-based teaching step.
Learning unit
(Gakkoh Tosho, 1999) Period Activity name Open approach-based teaching step Start time-end time (hour) Total time taken (hour)
Step 1 00:00:00 - 00:18:00 00:18:00
Step 2 00:18:01 - 00:42:47 00:24:47
Step 3 00:42:48 - 00:59:07 00:16:20
Step 4 00:59:08 - 01:18:30 00:19:23
1/12 Children pla ying in sa n dboxes and o n slides
Total time 00:00:00 - 01:18:30 01:18:30
Step 1 00:02:10 - 00:07:50 00:05:40
Step 2 00:07:50 - 00:33:58 00:26:08
Step 3 00:33:58 - 00:49:16 00:15:18
Step 4 00:49:16 - 00:52:58 00:03:42
3/12 Buying eggs to make omele t s
Total time 00:02:10 - 00:52:58 00:50:48
Step 1 00:02:15 - 00:06:15 00:04:00
Step 2 00:06:15 - 00:31:12 00:24:57
Step 3 00:31:12 - 00:56:23 00:25:11
Step 4 00:56:23 - 01:08:58 00:12:35
Learning unit 8:
addition (2)
6/12 Delighted Natalie
Total time 00:02:15 - 01:08:58 01:06:43
Step 1 00:00:00 - 00:01:50 00:01:50
Step 2 00:01:50 - 00:46:03 00:44:13
Step 3 00:46:03 - 00:58:35 00:12:32
Step 4 00:58:35 - 0:01:07:08 00:08:33
11/12 Review exercises
Total time 00:00:00 - 0:01:07:08 01:07:08
Step 1 00:14:45 - 00:15:52 00:01:07
Step 2 00:15:52 - 00:40:30 00:24:38
Step 3 00:40:30 - 01:04:25 00:23:55
Step 4 01:04:25 - 01:07:56 00:03:31
Learning unit 9:
subtraction (2)
11/13 A cockerel and his chicks
Total time 00:14:45 - 01:07:56 00:53:11
Step 1 00:00:00 - 00:05:15 00:05:15
Step 2 00:05:15 - 00:38:14 00:32:59
Step 3 00:38:14 - 00:57:50 00:19:36
Step 4 00:57:50 - 01:14:11 00:16:21
Learning unit 10:
addition or subt ra c tion ? 5/5 Coming train
Total time 00:00:00 - 01:14:11 01:14:11
Table 2.
Average time used in each open approach-based teaching st e p.
Step Sequence of teaching Average time (hour)
1 Posing open-ended problems 00:05:59
2 Students’ self learning 00:29:37
3 Whole class discussion and comparison 00:18:49
4 Summarization through co n nect ing students’ mathematical ideas eme rging in the classroom 00:10:41
Total time 01:05:05
Note: The total number of study periods was 6.

A. SURIYON ET AL.
The data analysis results from Table 2 show the average time
used in each teaching step from the total number of study peri-
ods, 6. The study results also show that the time used differs in
each step, and there is an accounting provided of the time used
from most to least. The first order was the second step taking
approximately 29 minutes, 37 seconds. The third step taking 18
minutes, 49 seconds was ranked second. The fourth step took
10 minutes, 41 seconds, and the first took the least time, 5 min-
utes, 59 seconds. The total time used towards activity manage-
ment was 1 hour, 5 minutes, 5 seconds per study period.
3) Structure of students’ performing activities
Students participated in performing activities in 4 open ap-
proach-based teaching steps with different aims depending on
the intended purposes of learning for each study period, mathe-
matical contents, as well as the aim of monitoring students’
ideas. The structure of students performing activities is charac-
terized by 3 kinds of activities.
Individual activity is defined as an activity in which a teacher
requires each student to demonstrate ideas and methods of p rob-
lem solving by writing ideas from documents or writing ideas
on a piece of paper and then presenting these ideas (one person
per one piece of work). This activity emphasizes completing
exercises at the end of a study period for each learning unit
including activities in the learning unit 8 on addition (2) in the
study period 11/12 and review exercises.
Sub-group activity is an activity in which the teacher requests
a student who is a member of his or her group to show ideas
and ways of problem solving by writing ideas from documents
or writing ideas on a piece of paper and then presenting those
ideas (one group per 1-2 pieces of work). The number of mem-
bers of each group was between 3 and 5 people. Students de-
termined tasks for each member, and members of each group
studied together and presented their work in front of the class.
This kind of activity mainly emphasizes solving problems to-
gether. The first step consisted of presenting open-ended situa-
tion problems which could be taken from the activities in the
learning unit 8 on addition (2), in period 1/12, Children Playing
in Sandboxes and on Slides, in period 3/12, Buying Eggs to
Make Omelets, in period 6/12, Delighted Natalie including
activities in the learning unit 9 on subtraction (2), in period
11/13, and the learning unit 9 on addition or subtraction, in
period 5/5, Coming Train.
Whole class activity is an activity in which the teacher re-
quests a student who is a member of the class to show ideas and
ways of problem solving by writing ideas from documents or
writing ideas on a piece of paper and then presenting ideas (one
group per one piece of work or one group per 1-2 pieces of
work). Individual activities or group activities may be used for
whole class activities. Whole class activities in the research
were characterized by competition games included in activities
in the learning unit 8 on addition (2), in period 9/12, Let’s Ar-
range Cards, in period 10/12, Let’s Play Cards on Addition,
and in period 12/12, Wheel Ring of Addition.
4) Structure of student work presentation
The structure of the student work presentation in the mathe-
matic classroom using the open approach is described as fol-
lows.
a) The teacher was tasked to assign a group to give a presen-
tation with instructions provided for putting the presentations in
correct order based on incorrect ideas, uncomplicated ones, or
the ones that most students could perform. First, the teacher
presented the aforementioned ideas in order to illustrate the
required tasks. Next, the teacher chose complicated ideas and
the ideas that a small number of students could perform, which
were the concepts that reflected advances in achievement ac-
cording to purposes of each study period before entering the
next step.
b) After a person on behalf of his group finished giving a
presentation in front of the class, the audience asked questions
by raising their hands to show their intention to set problems or
ask questions.
c) When the person who gave a presentation got a question,
he then answered the question, or the teacher prompted mem-
bers in each group to help each other determine answers or
participate in showing opinions.
d) When there was no question, the person who gave a pres-
entation went back to his group. For group tasks posted on the
black board, the person who gave a presentation could not take
his group task back to his group because the specific task would
then be used for comparing ideas from each group and for
drawing conclusions to connect with ideas emerging in the next
step.
The Roles of Teacher in the Classroom
The data analysis findings on the teacher’s teaching behavior
in the open approach-based mathematics classroom illustrated
that in each teaching step, the teacher played an important role
in the development of students’ metacognitive strategies. Spe-
cifics for each of the teaching steps are detailed below.
Step 1 Posing open-ended problems: In this step, the teacher
was tasked as “a motivator” in order to allow students the op-
portunity to participate in problem solving and better under-
stand problems with an emphasis on students’ interpretations of
pictures or media used for presenting problem situations and
the teacher’s use of motivating questions such as the following
conversation in the learning unit 8, in period 1/12, Buying Eggs
to Make Omelets.
Teacher: “Well, look at this (posting the picture on the
blackboard). What is it?”
Students: Saying “Wow!” (all together, the whole class)
Student A: “It is a picture of people playing on swings”
Student B: “and playing in the sand”
(The student describes the picture as he sees it on the
board).
Moreover, the teacher encouraged students to take on more
participation as a demonstrator or as a person who took the
initiative or used role playing by calling on students in the class
to act out the proposed situation.
Step 2 Students’ self learning: In this step, the teacher was
tasked as “a supporter and a facilitator” with the intent to help
students more effectively and to realize her role of getting in-
volved in students’ problem solving. The aim of the second step
was that students learned to solve problems by themselves; that
is to say, the teacher could help students when they needed help
or asked clarification questions which could arise after they
encountered difficulties in problem solving. The teacher could
give advice to students so that they could solve problems and
overcome difficulties in problem solving by themselves. How-
ever, a teacher’s role did not include providing ways of solving
problems or giving answers to students. As for the teacher’s

A. SURIYON ET AL.
288
role of motivating students to continually apply themselves and
progress in problem solving, the teacher encouraged students to
this end by prompting them during problem solving, which was
evidenced by the teacher’s elicitations as provided below from
unit 8, in period 1/12, Buying Eggs to Make Omelets, “Try to
think in different ways”.
Step 3 Whole class discussion and comparison: In this step,
students monitored one another and reflected on problem solv-
ing, and the teacher had an important role as an initiator of
classroom discussion by proposing issues for whole class con-
sideration. In other words, the teacher played the role of “a
coordinator of understanding” by creating an atmosphere of
discussion in order that students could consider the opinions
and suggestions of their classmates.
Step 4 Summarization through connecting students’ mathe-
matical ideas emerging in the classroom: The teacher was t ask e d
in this step as “a connector” to summarize students’ ideas by
connecting students’ ideas in 2 ways.
1) Drawing conclusions through synthesizing student ideas:
The teacher was tasked to propose a problem to students to in
order to ascertain the spectrum of ideas and approaches to mee t-
ing task objectives during the whole class discussion stage so
that students could evaluate ideas and ways which helped them
to solve problems effectively; that is, solving problems easily,
quickly, and correctly. Based on the analysis results, students
came to the conclusion that producing 10 was a factor that mo-
tivated them solve problems effectively, and other concepts
such as counting, adding, and counting one for each item also
helped them solve problems but were quite slow ways some-
times resulting in miscounting.
2) Drawing conclusions through synthesizing ideas that stu-
dents applied to problem solving and initial situations or prob-
lems: In this step, the teacher was tasked with preparing media
and organizing media systems. Tools used in each study period
from beginning of activities included pictures and instructions
used for initial situations, work showing students’ ideas, and
media. These tools were then used to check students’ under-
standing of whether or not the ideas used in problem solving
were consistent and rational with initial problems.
The Roles of Student in the Classroom
The analysis results on students’ problem solving behavior in
the open approach-based mathematics classroom illustrated that
in each teaching step, a student was tasked to know how to
solve problems himself, which could lead to development of
metacognitive strategies. Student tasks for each of the teaching
steps are detailed below.
Step 1 Posing open-ended problems: In this step, students
were tasked as “participants trying to understand situation prob-
lems” by making observations of what they saw from pictures
or media used for presenting problem situations including the
teacher’s answering of questions. Examples of students’ answe rs
from observations and the teacher’s answering of questions are
in the following conversation in the learning unit 8, in period
1/12, Buying Eggs to Make Omelets.
Teacher: “Well, look at this (posting the picture on the
blackboard). What is it?”
Students: Saying “Wow!” (all together, the whole class)
Student A: “It is a picture of people playing on swings”
Student B: “and playing in the sand”
(The student describes the picture as he sees it on the
board).
Moreover, students were tasked as “demonstrators” or “ex-
perimenters” relevant to media that the teacher presented or that
of a role of “an actor in role playing” in the situation problem
presented by the te acher.
Step 2 Students’ self learning: In this step, students were
tasked as “problem solvers” with regard to learning how to
solve problem themselves; that is, they had to encounter diffi-
culties in self-problem solving, be cognizant of ideas or ways
that they previously learned and used as problem solving tools.
Students’ roles while problem solving in sub-groups were that
of “idea recorders”, “observers of situation problems”, and
“examiners”. For these roles, any student who was influential in
his group often had the privilege to choose roles before other
members in the group. Mostly, he was a student who demon-
strated greater abilities than others. Furthermore, when the
teacher prompted students during problem solving by saying,
for example, “Try to think in different ways”, thereafter stu-
dents usually tried to find various other ways to solve problems.
Step 3 Whole class discussion and comparison: In this step,
students were tasked as “persons who give presentations” and
“an audience of a presentation of work concerning the collabo-
ration of students in problem solving activities. Students in the
whole class monitored one another and reflected on the prob-
lem solving process including discussion with members in the
class.
Step 4 Summarization through connecting students’ mathe-
matical ideas emerging in the classroom: In this step, students
were tasked as “evaluators” as they were required to answer the
teacher’s questions in order to compare the effectiveness of
ideas and approaches to presentation creation during the whole
class discussion stage, including examining whether or not and
how ideas used for problem solving were consistent and ra-
tional with initial expectations.
Discussion and Conclusion
Contextual factors related to classroom action affecting the
development of students’ metacognitive strategies in the open
approach-based mathematics classroom are detailed in the fol-
lowing three issues.
1) Structure of learning and teaching activities in the class
As for the structure of learning and teaching activities in the
class, in the study, the research considered the following 4 is-
sues: learning management planning related to the following
structures: learning units, periodic instructional activities con-
sidered from time used for arranging learning and teaching
activities, students’ performing activities, and students’ work
presentation. The study findings indicated the importance of
each issue concerning emerging structures of instructional ac-
tivities, especially activities underlining problem solving proc-
esses which could prompt students to develop metacognitive
strategies as well as results obtained from a time study used in
arranging activities, showing that students spent the most time
engaged in the second step of self-learning. These results con-
firmed students’ ability to perform more tasks than simply find-
ing answers only, which was considered evidence proving that
students had indeed furthered the development of their meta-
cognitive strategies.

A. SURIYON ET AL.
2) The roles of teacher in the classroom
As for teacher’s roles with regard to helping students to de-
velop metacognitive strategies, what teachers should be most
cognizant of was their role in getting involved in problem solv-
ing to help students at the right time so that students could then
best help themselves. The teacher was tasked as the person
responsible for determining the directions of activities emerg-
ing in the class. In other words, students could be empowered
toward self-learning in the future. The fundamental practice
guideline was that the teacher had to understand and know
when to get involved in students’ problem solving at the appro-
priate time, corresponding to Polya (1957) in which the teacher
had to rely on experiences in classroom observations until she
could interpret students’ thinking processes in any activity as
well as surrounding factors emerging in the classroom, for ex-
ample, situation problems that the teacher presented to students,
instructions, and instructional media used in activities. When
students could solve problems by themselves, the outcome was
that a variety of ideas in problem solving emerged.
3) The roles of students in the classroom
For student’s roles in the mathematics classroom using the
open approach, students were responsible for carrying out
various important tasks in the class. Receiving emerging dif-
ferent roles while performing instructional activities helped stu-
dents to evolve their roles differently. The outcome was that
students had a chance to develop learning skills and process
extensively: problem solving, mathematical communication,
expressions showing thinking, linking, and reasoning. In par-
ticular, in the aspect of problem solving, students could learn
from their actions, leading to accumulative recording of “re-
sources” gained from experiences according to roles that stu-
dents received as in Schoenfeld (1985) suggesting that these
existent resources are fundamental elements related to success
and failure in problem solving.
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 the Center for
Research in Mathematics Education, Thailand.
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