the type of open-ended problem, the unfamiliar problems were

used. In addition, the problem should provide open process,

open product, and guidelines for developing the open-ended

problem which was called open-ended problem; 3) the evalua-

tion in students’ guidelines of answers should be various ways

(Nohda, 2000). The students were courage to think, do, express

themselves, and concluded knowledge by themselves (Inpra-

sitha & Loipha, 2007). In the Open Approach, communication

among students in classroom was developed with worth (Isoda,

Shimizu, & Ohtani, 2007). Many researchers stated that the

mathematical communication was necessary for mathematics

learning (Sierpinska, 1998) gave an importance to characteris-

tics of mathematical communication and interaction partici-

pated by students in learning (Emori, 2005; Sierpinska, 1998)

Furthermore, it was depended on basis of communication

(Emori, 2005). Mathematics communication was mathematical

learning process as an important part of techniques for sharing

one’s ideas which could help students to learn meaningfully

(National Council of Teacher of Mathematics, 2000). Conse-

quently, the teachers should encourage their students to discuss

as well as shared their ideas with each other (Cooke & B uchholz,

2005).

The researchers and teacher didn’t give an importance to

Mathematical Communication much since they paid their atten-

tion to the students’ number of speaking rather in classroom

without considering the quality of thinking, and expression

technique (Emori, 2005). The students communicated their

comprehension through speaking and gestures in expressing for

sharing the meaning from the work task (Pire, 1998). The Ges-

ture was a language leading to thinking (McNeill, 1992 cited in

Arzarello & Edwards, 2005). It was a part of communication

which the sender using for learning very well, and decreasing

the mistaken (Lozano & Tversky, 2006) and extending the

communication to be successful (Thurston, 1994). It was one’

body movement considered as the extension part of human’s

attention (Kendon, 2000; Rasmussen, Stephan & Allen, 2004).

It might include the writing of symbols, graph, formula, table,

chart, picture drawing, calculating, etc. (Radford, 2005; T h ur s to n ,

1994). If there was a systematic observation, the students’ ges-

tures were not only to fill the gap of speaking, but also to pre-

sent worthy information of thinking (Kendon, 1997; Scherr,

2008). It was like a bridge connecting the speaking, and associ-

ating the action, viewing, memory, language, and written de-

scription (Bjuland, Cestare, & Bergensen, 2007; Edwards,

2005). Most of people expressed while they were speaking. So,

the gestures included one’s thought and language (Nunez, 2004)

as well as the stimulator for speaking expression (Wu & Coul-

son, 2007). When the students had obstacle in speaking for

communicating their ideas with the others, the gesture was a

part in expressing that approach of student (So, Kita, & Goldin

Meadow, 2009). It could be seen that the gesture had strong

point in development of human beings, perception, learning,

and communication. But, it was surprising that there was very

little number of research studies regarding to gestures in learn-

ing and teaching area (Roth, 2001). Since the past to present,

the gestures were overlooked in communication (Bjuland et al.,

2007; Edwards, 2005).

It would be viewed that the knowledge couldn’t be directly

transferred to the others. So, the lecture wasn’t successful in

learning and teaching. Since both of communication and ges-

ture were important in learning meaningfully. Furthermore, the

communication could complete the gap of speaking. But, it was

surprising that there was very little number of research studies

regarding to gestures in learning and teaching area. But, in

classroom using process of Lesson Study and Open Approach

at the Center for Research in Mathematics Education, Faculty

of Education, Khon Kaen University, were used in classroom,

providing opportunity for students to learn based on their own

potentiality. Therefore, the researcher was interested in survey-

ing the mathematics communication by students’ gestures un-

der context of Lesson Study and Open Approach context.

Research Question

How many kinds were there in Mathematical Communica-

tion by the 5th grade students’ gestures in Lesson Study and

Open Approach Context?

Research Objective

The objective of this research was to explore the Mathemati-

cal Communication by 5th grade students’ gestures in Lesson

Study and Open Approach context.

Methodology

The target group of this study included 27 fifth grade stu-

dents, Nong-tum-nong-ngu-lerm School and 33 fifth grade

students, Beung-neum-beung-krai-noon School which were the

schools participating in Project for Professional development of

Mathematics teachers through Lesson Study and Open Ap-

proach, under supervision of the Center for Research in

Mathematics Education, Faculty of Education, Khon Kaen

University, and Mathematics Communication by students’ ges-

tures as follows:

Phase 1: Collaboratively design research lessons (plan), term

consisted of teachers, researchers, school coordinator, and un-

der supervision by coach, starting from determination of activi-

ties in mathematics problems by using Open-ended Problem

from Mathematics Textbooks using in the project in aligned

with designing and establishing the teaching media and material ,

and discussing the teaching sequence through Open Approach

by considering students’ gestures as well as mathematics

Communication which would occur in teaching sequence by

Copyright © 2012 SciRes. 633

Y. KONGTHIP ET AL.

Open Approach which the lesson plans were written together

every week.

Phase 2: Collaboratively observing the research lessons (do),

during this session, the details of phase 1 in writing the lesson

management plans, were used in classroom. A teacher in team

was a representative of teaching. The rest of members were

classroom observers or witnesses in teaching sequence using

the 4 phases of Open Approach including: 1) Posing open-ende d

problem; 2) Students’ self learning through problem solving

while the teacher take notes students’ idea for later discussion;

3) Whole class discussion and comparison; and 4) Summariza-

tion through connecting students’ mathematical ideas emerged

in the classroom aiming to observe mathematical approach of

students communicating by gestures expressing during the

phase of self learning, and group discussion. The teacher’s

teaching ability wasn’t considered. The audio tape and video

tape were recorded during sequence of teaching.

Phase 3: Collaboratively Reflection or Post-discussion (see),

the collaboration in discussion was performed after teaching

practice in order to consider the findings after observing the

learning management for improving the lesson planning, and

teaching in next year. This session, was performed every week.

The reflections were ranked in order by allowing the teacher

reflect one’s own teaching for the first person. Then, the class-

room observation team discussed the existing approach in

teaching sequence through the Open Approach, gestures, and

Mathematics Communication occurring in classroo m.

Since 2007, Center for Research in Mathematics Education

had started implementing the lesson study and open approach in

Nongtoom-nong-ngu-lerm School, and Ban-Beung-neum-be-

ung-krai-noon School were funded by the Office of The Basic

Education Commission in Research Program on “Model for

Fostering Students’ Mathematical Thinking by Implementing

Lesson Study and Open Approach”. I and team introduced our

information to teachers and teacher introduced team to students

in the classroom. The team could implement following proc-

esses of lesson study and open approach, observe behavior of

students both inside and outside classroom as participation

observation, record video and audio tap in the classroom and

reflect students’ thinking in school meeting every week. In

2009, these schools were funded by the Office of The Basic

Education Commission in Project for Professional development

of Mathematics teachers through Lesson Study and Open Ap-

proach. I used participation observation and informal interview

to find target group in this research. I record my data following

my framework at Nongtoom-nong-ngu-lerm School in 2009

and at Ban-Beung-neum-beung-krai-noon School in 2010 by

recording video and audio taps. Video provided a more com-

prehensive understanding of the students’ learning. I reviewed

the video and audio taps to select data and posing problem

which brought opening them and asked students in interviewing

students. Data analysis used video analysis supported by pro-

tocol analysis, and used analytical description for students’

behavior in mathematical communication by gesture to analysis

data.

Instruments Using in the Study

The implementation of this study, the instruments for data

collection as well as data analysis was used as follows:

Instruments using for data collection included the lesson

plans developed by Open Approach, Filed Note form, Video

Tape Recorder, Audio Tape Recorder, Notebook Computer,

and students’ work pieces.

Instruments using for data analysis included the analysis of

video tape supported protocol, findings of filed note, in-depth

interview, target group’s demographic data, students’ perform-

ances.

Data Analysis

For Qualitative Data Analysis, Video Analysis supported by

Protocol was an opening of Video Tape based on teaching steps

of Open Approach in order to see movement as well as speak-

ing by teacher and students while the students were solving the

open-ended problem. Then, they were deciphered into Protocol.

The word “Protocol” referred to deciphering the behavior ob-

taining from audio tape and video tape into narration including

pictures and word describing the occurred gestures in classroom

according to the teaching incidence by Open Approach by us-

ing the word “Item” which referred to one’s behaviors includ-

ing each one’s spoken language and gestures, writing, and body

movement. The word “Episode” referred to the behavior groups

expressed by the students during their Mathematical communi-

cation, and used in analytical describing the students and

teacher’s behaviors expressing in classroom by analyzing the

students’ gestures under context of Lesson Study and Open

Approach respectively through the steps of Open Approach and

being based on data analysis unit as basic cycle of dyad feed-

back, by adapting from Emori (2005) in order to consider the

students’ gesture that it was a message sending or receiving by

students while they were solving mathematical problem. So, the

sent or received message might be either spoken words or ges-

tures using in investigating the students’ common understand-

ing whether they had.

The characteristics of mathematical communication: Rigor-

ousness, Economy, or Freedom which they participated in

communication by deictic, iconic, beat, or metaphoric gestures.

The characteristics of mathematical communication includ-

ing Rigorousness which referred to one’s opinion expression,

speaking and talking, and discussing for sharing one’s mathe-

matical ideas expressing step by step in mathematical problem

solving, and being able to send and receive message congru-

ently with one’s idea. Economy referred to one’s opinion ex-

pression for sharing the mathematical ideas concisely to the

others in mathematical problem solving, and being able to send

and receive the concise message as well as make the communi-

cation participants have common understanding. Freedom re-

ferred to one’s opinion expression, speaking and talking, and

discussing for sharing various or new mathematical ideas in

mathematical problem solving.

Students’ Gestures referred to the students’ observable body

movement including: Deictic gesture as their body movement

showing the mathematical approach in determining the existing

or visible objects. Iconic gesture referred to one’s body move-

ment expressing the mathematical ideas in drawing picture

referring to the lesson content. Beat gesture referred to one’s

repeated or rhythm body movement to emphasize that idea.

Metaphoric gesture referred to one’s body movement based on

mathematical ideas regarding to the abstract content.

We used basic cycle of dyad feedback which was unit of

analysis messages as gesture and verbal language of sender and

receiver communicated mathematical ideas following mathe-

matical communication framewo rk.

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634

Y. KONGTHIP ET AL.

Example Findings

Mathematical communication by students’ gestures found all

steps of Open approach as teaching approach. Example of re-

search findings found following: Freedom Mathematical com-

munication by students’ deictic and iconic gestures in Class-

room Discussion and Comparison step and Economically Ma-

thematical communication by students’ iconic gesture in Sum-

marization through connecting students’ mathematical ideas

emerged in the classroom step of Open approach.

Classroom Discussion and Comparison Step.

In this stage, the students expressed their opinion by various

communication techniques in Mathematics Communication

through gestures as follows:

Item 459-469, is video analysis supported by Protocol in

which the teachers asking the students’ performances. The stu-

dents in group responded by many techniques.

Item Name Messages

459 Teacher: Then, what is going on? It is like this. What’s

going on?

460 M: (Point at the picture.)

461 Teacher: I t comes close. It comes close, quick!

462 M: (Express gesture of parallel in vertical line and hori-

zontal line.

463 Students: (Laugh)

464 J: (Point at her performance.) These two lines could be

vertical pattern. This line could be horizontal one. This line

would be a diagonal pattern. It the cross with each other, there

would be a rectangular.

465 M: These two lines could be a vertical pattern. These

two lines could be a horizontal pattern. These two lines could

be a diagonal pattern. These two lines could be a horizontal

pattern.

466 J: (Pick her performance)

467 M: These two lines make a diagonal pattern.

468 J: (Bring microphone to her friend’s mouth, her friend

escaped it.) Well, this line is an inclined pattern. It’s the longest.

Well, this one is a short diagonal pattern. These two lines make

a short diagonal pattern. They cross each other as a rectangular.

469 Teacher: How many patterns do we have?

According to the Episode in Item 459-468 as the above, for

Item 459, the teacher asks what characteristic this group made.

The teacher sends a me ssa ge to students to answer the questions

that: 1) Item 460 M, points at the picture. But, in Item 461, the

teacher stimulates “It comes close. It comes close, quick.” The

students respond; 2) Item 462 M, use her thumb and index

make the claw of a crab. Then, she draws it in vertical line, and

horizontal line; 3) Item 464 J, points to the group’s perform-

ance and said that: “These two lines would make vertical pat-

tern. This line would make a horizontal line. This line would

make a diagonal pattern. When, they are crossed, it would be a

rectangular pattern”; 4) Item 465 M, point at the line as group’s

work; and 5) item 468 J points at the line which is group’s per-

formance, in Item 460 Item 462 Item 465 and Item 468 by spe-

cific gesture, for Item 464 includes a physical gesture in re-

sponding the teacher’s question that the made rectangular as 2

parallel lines in diagonal pattern, horizontal pattern, or vertical

line including various patterns. Then, the teacher passes this

issue into other ones. According to the above Episode, it could

be viewed that the student uses gesture as pointing as well as

picture in Mathematical Communication in various approaches.

It could be concluded that it was Freedom Mathematical Com-

munication by students’ deictic and iconic gestures.

Summarization through connecting students’ mathematical

ideas emerged in the classroom step.

In this stage, the student expresses her various opinions as

well as techniques in Mathematical Communication by gestures

while they were concluding the lessons as follows:

Item 393-398, is video analysis supported by Protocol in

which the teacher asking student to explain why six channels

were obtained the same. The student explains picture (d).

Item Name Messages

393 Teacher: Well, are there six channels? One, two, three,

four, five, and six. There are six channels. How about this one?

Picture b. For this one, some students count them as six chan-

nels. Some students cut this part, and put it this side. We still

count the same. For this one, c, bring it here (the left hand).

This one is informed by a friend to cut at this point, and put it

here. (The right hand) or we can cut it here (the right hand). Put

it here. How many channels we would get? Six channels again.

For this one. For (the left hand), this one, we can cut it here,

can’t we? Isn’t it? How can we cut it? Let this group show how

to cut it, it will be better. Chicken Group, please cut it. Where

would you draw? Where do you draw it? Write it down.

394 Student:

395 Teacher: The channels are full the same.

396 Student: Count the channels drawn by her friend.

397 Student: Walk to see her Group’s work.

398Teacher: This picture, point to the student’s work.

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Y. KONGTHIP ET AL.

We get this picture, don’t we? Then, we can put it here. Is it

a rectangular? Well, let’s see our friend’s. At first, our friend

cut it like this. How many channels are there? There are six

channels.

According to the Episode in Item 393-398 as the above, in

Item 393, the teacher asks the students to explain why do they

get six channels the same. The student explain picture (d) as

Item 394, (cross the picture. Then paint it). The gesture is

iconic. As the Item 398, the teacher shows the Group’s per-

formance for their friends to consider again. According to the

above Episode, it could be seen that the student uses gesture in

concise messages in Mathematical Communication. So, the

participants have common understanding. It could be concluded

that it is Economically Mathematical Communication by stu-

dent’s iconic gestures.

Conclusion and Discussions

The research findings found that there were 7 kinds of stu-

dents’ Mathematical Communication by students’ Gestures

including 1) rigorousness by students’ beat gesture-Mathematical

Communication in which the students wanted to emphasize the

statements or pictures as their own ideas, or communicate with

their friends or teacher; 2) rigorousness by students’ metaphoric

gestures—gesture referring to content or concepts of the lesson;

3) economy by students’ deictic gesture—this kind of gesture

could be easily performed economically with common under-

standing; 4) economy by students’ iconic gesture—this kind of

gesture was to draw a picture to communicate economically

with each other for common understanding; 5) freedom by

stu dents’ deictic gesture—the students used various kinds of point-

ing, and had freedom to express their mathematical ideas; 6)

freedom by students’ iconic gesture—the students could use

many kinds in drawing picture as well as have freedom to ex-

press their mathematical ideas; and 7) freedom by students’

deictic and iconic gestures—the students used both of pointing,

and drawing picture to communicate their ideas as well as had

freedom in express their mathematical ideas, and the most

commonly used the characteristic of economically mathematical

communication by deictic gestures, and Students’ self learning

through problem solving while the teacher take notes students’

idea for later discussion in Open Approach. The teachers used

gestures for observing students’ learning in the classroom and

students had happiness in the classroom. Furthermore, the

schools in Lesson Study and Open Approach context, the stu-

dents had opportunity in learning based on their potentiality,

being able to think, perform, and express. They preferred to

express divergent think.

Recommendations

The teachers and Educational Staffs could use the behavioral

or gestures observation technique in learning of students during

they expressed Mathematical Communication with each other

meaningfully which was an evaluation from the students’

working process. In addition, their gestures could be able to be

used in reasoning, explaining, and analyzing their performance,

concluding their approaches, and evaluating their emotion or

feeling during learning. Specifically, under Lesson Study, and

Open Classroom situations, the students had freedom in ex-

pressing their ideas in various ways including both of verbal,

and gestures meaningfully with themselves. We implement this

result through pre-service teachers to bring this knowledge

working group with teachers in schools and teacher learned

these knowledge and I will write article to publish this result.

The studies of students’ Mathematical Communication by ges-

tures in other class levels should be studied further. And each

academic year had trained pre-service teachers about new

knowledge to bring this knowledge working group with teach-

ers in schools.

Acknowledgements

This study was fund by the Commission on Higher Educa-

tion, Thailand, and Graduate school, Khon Kaen University,

and was supported by Center for Research in Mathematics

Education (CRME) and Srinakharinwirot University.

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