1">well as environment should be provided for students to learn; 2)
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.
Copyright © 2012 SciRes.
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.
Copyright © 2012 SciRes. 635
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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