Contextual Factors in the Open Approach-Based Mathematics Classroom Affecting Development of Students’ Metacognitive Strategies

doi:10.4236/sm.2013.34038

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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

Email: [email protected]

Received July 8th, 2013; revised August 16 th, 2013; accepted August 29th, 2013

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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