Creative Education
2012. Vol.3, No.7, 1188-1196
Published Online November 2012 in SciRes (http://www.SciRP.org/journal/ce) http://dx.doi.org/10.4236/ce.2012.37177
Copyright © 2012 SciRe s .
1188
Students’ Abstraction Process through Compression to
Thinkable Concepts: Focusing on Using “How to” in Learning
Units of Lesson Sequences to Provide a Tool in
Conducting Students’ Concepts
Nisara Suthisung1, Maitree Inprasitha2, Kiat Sangaroon3
1Doctoral Program in Mathematics Education, Faculty of Education, Khon Kaen University,
Khon Kaen, Thailand
2Center for Research in Mathematics Education, Faculty of Education, Khon Kaen Univers ity, Khon Kaen, Thailand
3Centre of Excellence in Mathematics, CHE, Bangkok, Thailand
Received September 14th, 2012; revised October 17th, 2012; accepted Oct ober 28th, 2012
The purpose of this study is to analyze “how to” in the students’ abstraction process through compression
to thinkable concept under classroom using Lesson Study and Open Approach. Data for this study were
collected by using a teaching experiment, with the four of first graders as targeted. The research results
revealed that in the students’ abstraction process, they compressed computable symbols and conducted 10
as “how to” in their thinking and thinkable concept at the same time. It is shift steadily from performing
sequence of compression in students’ thinking from actions being linked together in increasingly sophis-
ticated ways.
Keywords: Lesson Study; Open Approach; Precept; Compression to Thinkable Concept; How to;
Learning Unit
Introduction
The objective of Learning and Teaching Mathematics is to
develop students’ concept in content. Teachers and researchers
try to search for instruments to comprehend student’s existing
concepts (Gray & Tall, 2007). In Thailand, instruction in the
classroom using an Open Approach as an innovative teaching
approach that cooperates with a Lesson Study is an effective
way to develop mathematical activity using open-ended prob-
lems for promoting the use of tools in students’ problem solv-
ing and in developing their concepts (Inprasitha, Pattanajak, &
Thasain, 2007). Therefore, to depend on the area of teaching
implementation in a classroom for studying the abstraction
process with natural occurrences is a major guideline in con-
sidering and finding answers to understand the concept forma-
tion of students.
Skemp (1987) explained the abstraction process as an im-
portant instrument in developing concepts and considered the
fundamental human activities to be perception, action and re-
flection. Tall (2004) considered students’ mathematical think-
ing growth based on perception and action through compression
to thinkable concept to develop their concept. Gray and Tall
(2007) viewed that the abstraction process through compression
to thinkable concepts is the key to developing increasingly
powerful thinking. This point of view focused that instructional
must be framed with an awareness of students’ abstraction
process to produce thinkable concept. Tall (2007a) noted that
thinkable concepts must be integrated in the curriculum. How-
ever, there was no empirical evidence. Therefore, the research-
ers and educators should study and make it clear for teachers,
students and parents.
Gray and Tall (1994) explained the abstraction process of
compression operation arithmetic using procedures in problem
solving to the same effect. Tall (2004) suggested that the chan-
ging process from procedures to thinkable concept cannot be
seen easily. Tall and Isoda (2007) described in classroom using
Lesson Study caused to the student’s abstraction process for
concept formation from considering compression to thinkable
concept through 4 steps of procedures in problem solving to
effect based on Tall (2006) five steps of thinkable concept.
Lesson Study and the Open Approach have been integrated
into Thai classrooms. It was a unique teaching for developing
students thinking process, continuing, analyzing teaching and
controlling classrooms. Inprasitha et al., (2007) adopt the con-
cept of Lesson Study from Japan. It is focused on changing to
develop the learners’ progress in real class with team collabora-
tion, observers and reflection, creating problem situation, de-
signing learning materials and steps of teaching. According to
Inprasitha (2010), the Open Approach is a teaching approach to
solve problems and understand the learning content of solving
problems, including four steps as follows: posing open-ended
problems, students’ self-learning, whole class discussions and
summary through connection.
Survey the opinions of teachers in four schools, participating
in the project under Center for Research in Mathematics Educa-
tion, Faculty of Education, Khon Kaen University for four
years using the Open Approach and a Lesson Study has found
that teachers are concerned and eager to help their students to
build thinkable concepts. The teachers used daily life problems
that the students had already known as well as designed touch-

N. SUTHISUNG ET AL.
able learning tools and designed problem situation focused on
using tools in students’ problem solving, and the teachers pro-
duced “how to” in learning unit of lessons sequence. So the
students could solve mathematical problems, wrote symbolic
sentences easily .
Tall (2007b) argued that Lesson Study provides an area for
the students’ compression to thinkable concepts. Moreover,
Tall (2008) suggested that Lesson Study is to be the major idea
to support students have “how to” in solving problem for com-
pression to thinkable concept. The purpose using Lesson Study
in Thai classroom is producing “how to” as a tool in thinking to
build students’ concept, which is designed in learning unit of
lessons sequence to support using as a tool in students’ solving
problem in step students’ self learning of Open Approach (In-
prasitha, 2010).
From the above rationale, the researchers was interested in
studying the students’ abstraction processes through compres-
sion to thinkable concepts focusing on empirical evidence in
context using Lesson Study and Open Approach, and using
their “how to” in problem solving and how can it be conducted
to thinkable concepts.
Objective
To analyze students’ abstraction process through compres-
sion to thinkable concepts focused on using “how to” in units of
lessons to provide a tool in conducting students’ concepts.
Context of Study
Thinkable concepts are the teaching and learning goals. In
achieving that, teachers should provide appropriate learning
experiences for students. Using Lesson Study with Open Ap-
proach from open-ended problem and interacting with learning
materials can support and develop students’ thinkable concept.
Students are able to think from their daily lives problems, in-
teract with learning materials, use symbolic for calculation.
Especially, considering “how to” is a tool in the students’ prob-
lem solving and is playing a key role to product thinkable con-
cept in their abstraction process through compression under the
views as following:
Lesson Study
Lesson Study is an innovative tool for building, analyzing
classrooms and developing students’ mathematical thinking.
Inprasitha et al., (2007) adapted the concept of Lesson Study
from Japan to be used in Thai classes. It consists of three steps
in planning, observing and reflectin g as follows:
Teachers, observers, internship mathematics student teachers,
research team wrote teaching plans in units and periods, learn-
ing activities, objectives and open-ended problems using a Ja-
panese textbook (Gakkoh Tosho, Study with Your Friends
mathematics for Elementary School 1st grade). It was team
collaboration consisting of designing learning materials, steps
of teaching, predicting students’ ways of thinking. Designed
learning materials for helping students to think, act and proc-
essed from well being plans.
The next step was to bring the team teaching plan to use with
the Open Approach (it will be mentioned later.) The team ob-
served a teacher, the students’ way of thinking, how they solved
problems, their interactions with learning materials, and their
expected and unexpected concepts.
At the reflection step, the team reflected on many aspects, the
students’ ways of thinking that happened in class.
By studying the Lesson Study as it is taught by the team, we
can observe the students’ ways of thinking through compres-
sion to thinkable concepts by using a Lesson Study and the
Open Approach from the above theories.
1) Collaboratively for designing lesson plan, using Open
Approach from problem situation in students’ real life, create
designed materials to support students’ concepts. Focused on
lesson’s goal, learn how to learn, timing for each period, de-
signing 4 steps of teaching (Figure 1).
2) Collaboratively observe in class, bring the team teaching
plan to use with Open Approach (It will be mentioned later).
The team observed a teacher, the students’ way of thinking,
how they solved problems, their reaction to designed materials
for using symbolic calculation to solve problem situation (Fig-
ure 1).
3) Collaboratively reflect, discussing problems and obsta-
cles in using lesson plans as well as considering the position of
using designed materials, students’ way of solving problem,
students’ new ways of thinking, and the successful of using
lesson plans (Figure 1).
In addition, using the Open Approach is an important teach-
ing approach that motivates the students to think, so it was used
in this research.
Open Approach
Nohda (1998) believed that the Open Approach could be
used for supporting various kinds of student activities and
mathematical problem-solving. The Open Approach is a teach-
ing approach that helps students to reflect on their own thinking,
to solve various kinds of problems, and it is essential for all
students to do their mathematical tasks to the best of their abili-
ties. Nohda (2000) mentioned that Open Approach can adjust
several ways of students thinking or students’ mathematics
thinking and the progress of teaching approach should be inte-
grated. Open Approach is expected to be a tool for changing
classroom, helping students to learn from their abilities. Open
Approach is aimed at the students can think on their own. In
Thailand, Lesson Study has been used with the Open Approach
as a teaching approach in four steps according to Inprasitha
(2010). It is started from posing open-ended problem situations,
student’s self-learning, whole class discussion and comparison,
and summary through connection. Students learn and under-
stand the contents by solving problems.
1. Collaboratively
Plan
2. Collaboratively 3. Collaboratively
Figure 1.
Cycle of Lesson Study i nc lud in g 3 phases.
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N. SUTHISUNG ET AL.
1) Posing Open-ended problem: A teacher posed to encour-
age students to solve problem (Figure 2 (a)).
2) Students’ self-learning: They made goal-directed thinking,
attempted to solve problem with different methods (Figure
2(b)).
3) Whole classroom discussion and comparison: The stu-
dents presented their ideas in front of the class. They realized
and checked way of thinking in order to systematically explain
their ideas (Figure 2(c)).
This research focused on the teaching steps: students’ self-
learning. The students used learning tools and different ways to
solve problems that led them to build thinkable concepts.
From the above framework, the related theories, the proce-
dures of “how to” in students’ abstraction process through com-
pression to thinkable concept are as follows.
Teacher: A teacher posed a problem situation that was close to stu-
dents real world problem. Student: Students perceived problem
situation through seeing and hearing. They paid attention and were
eager to solve that problem. The problem situation seemed to be
their problem.
(a)
Teacher: After posing the problem, the students thought and did
self-learning. Student: The students solve the problem by them-
selves and used symbolic calculations. They cre ate d v ari ous way and
goal-directed thinking, and tried to write formal mathematical sym-
bols and formal language into mathematical world before coming to
mathematical concepts.
(b)
(c)
Teacher: The teacher connected the students’ idea by presenting
main ideas to summarize the main points for understanding. Student:
The students realized the different ways of calculation. The teacher
summarized through connection to the main concept for giving stu-
dents. They have oppo rtun ity to revise concept.
(d)
Figure 2.
Four steps of Open App roach.
“How to”
Inprasitha (2010) explained that the Lesson Study teams
planned the study lesson with an emphasis on “how to” which
was a key influence for engaging students in the self learning
phase (i.e. students’ problem solving). Isoda (2010) viewed that
teachers plan the lesson and teach that children enable to learn
the value of mathematics and “how to” develop mathematics as
well as mathematical idea and skills.
Thus, designing learning unit in such a way the lesson study
team has to be concerned with what are important “how to”
within a unit and between units.
The purpose of using a Lesson Study In Thai classroom of
producing “how to” as a tool in thinking to build students’ con-
cept, which is designed in learning unit of lessons sequence to
be used as a tool in students’ problem solving. Moreover, Tall
(2008) suggested that the Lesson Study be the major idea to
support children having “how to” in solving problems for com-
pression to thinkable concept. Therefore, it is interesting for
studying how it can be conducted to thinkable concept.
The Designing Learning Unit
Inprasitha (2010) suggested that in the Japanese textbook of
the 1st grade mathematics textbook, the sequence of learning
units be as follows: number up to 10, decomposing, numerical
order, addition (1), subtraction (1), number larger than 10, addi-
tion (2), subtraction (2) then add or subtract (Gakkotosho Co.,
Ltd., 2005). The reasons why the Japanese textbook designs the
sequence of learning units as such as:
 Most of the first grade students have experience in “order
number” outside of the school. They can count by one be-
fore entering the school. However, it is difficult for them to
conceptualize the number 5 as the combination of each
number.
 Before making addition, they must see the number 9 as
(1,8), (2,7), (3,6), (4,5).
 They must see the value of the “base ten”, that is, they see
the number 8, they should combine with 2 to make it be-
comes 10, and
 They must use it as a tool in their problem solving and con-
structing concept.
The above mentioned “how to” appeared in the decomposing
unit of the Japanese textbook and prepared the tool that the
students were to use when they learn the addition, subtraction
and add or subtract unit. From this point of view, just designing
the learning units in the Lesson Study process are not a guaran-
tee for students’ self-learning, and this design should be con-
cerned with the teaching approach. The following example
illustrates this idea:
1) In the decomposing unit, the students learn the structure of
numbers 5, 6, 7, 8, 9 and 10, “number patterns among the com-
bination of those numbers”, and the value of base 10. Then,
they learn how to add numbers where the result is not more
than 10 (Figure 3).
2) In the addition (1), subtraction (2) and add or subtract
units, they use decomposing numbers and “base ten” as tools in
problem solving. Then, they learn how to add or subtract where
the result is more than 10 (Figures 4 and 5).
The following sub-unit extended the idea of addition and
subtraction in order that students uses those “how to” tools to
make addition and subtraction with a result of not more than 20.
The empirical data below were collected in the 2010 academic
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N. SUTHISUNG ET AL.
5 มาจาก 0 กับ 1
1 กับ 4
2 กับ 3
3 กับ 2
4 กับ 1
0 กับ 5
The number 5 as the combination of 0 and 5,1 and 4, 2 and 3, 3
and 4, 4 and 5, 5 and 0.
Figu re 3.
Decomposing unit.
Example in addition unit (2)
Problem situation 1: There are 9 children on the sand box
and 4 children on the seesaw. “How many children are there in
all?”
The students used diagram as thinking tools. They decomposed 4 to 1, 3
and composed 9 with 1 to 1 0 a n d adde d th em to 13.
Problem situation 2: There were 9 eggs yesterday and there
are 7 eggs today. From a question: “How many eggs are
there?”
Figu re 4. it.
Addition un
Example in subtraction unit (2) re, chicks or roosters?
Problem situation 3: Which is mo
The students’ thinking used a diagram for decomposing,
composing and recomposing bas ed on bas e ten.
Figu re 5.
ear from first grade students at Kook-Kham Pittayasan School
study
te
n Study In Thai classroom of
pr
Conceptual Framework for Analyzing
Precep
d Tall coined term the “precept” in 1994. It has dual
ch
se concepts are in harmony with the SOLO Model (Bigg
&
able 1. ent of precept.
Process … Concept
Subtraction unit.
y
in the Northeastern part of Thailand. This school has been im-
plemented Lesson Study and Open Approach since 2006.
Thus, designing learning unit in such a way the lesson
am has to be concerned with what are important “how to”
within a unit and between units.
The purpose of using a Lesso
oducing “how to” as a tool in thinking to build students’ con-
cept, which is designed in learning unit of lessons sequence to
be used as a tool in students’ problem solving. Moreover, Tall
(2008) suggested that the Lesson Study be the major idea to
support children having “how to” in solving problems for com-
pression to thinkable concept. Therefore, it is interesting for
studying how it can be conducted to thinkable concept.
Compression to Thinkable Concept
t
Gray an
aracteristics of process and object from the same symbol to
same effect through compression to thinkable concept. Using
process to precept is natural process compression sequencing
from process to concept formation. Precept is the changing pro-
cess from procedures to thinkable concept in accordance with
evolutionary development, according to Tall et al., 2000 (Table
1).
Tho
Collis, 1982), which mentions Unistructural, Multistructural,
Relational and Extended Abstract. Davis (1984) divided to pro-
cedure and integrated to process and entity. Sfard (1991) com-
prised of interiorization, condensation and reification. APOS of
Dubinsky (1991) comprised of action, process, and object and
expanded to schema according to Pegg and Tall (2005: p. 472)
as shown in the Table 2.
T
Developm
Piaget A
Op … T (1950s) ction(s), eration(s )…
hematized
object of
thought
Dienes
(1960s) Predicate… … Subject
Davis (198 0 s )Visually m o d erated Integrateduence… a thing, an
e
Greeno Procedure… Inp
Ac ep IEnc
Sfar) pReifie ct
C )
Sp
Process… d as
Sequence…
seq
Seen as a whol e, and
can be broke n into
sub-sequence
ut to another pro-
ntity, a noun
Conceptual
(1980s) cedure…
nteriorized process… entity apsulated Dubinsky
(1980s) tion… Each st triggers the next
Interiorized
ro ss
With consc i o us control object
d objed (1980scess… Proce
performed
Unistructural a
Condensed process…
Self-contained
Bigg and
ollis (1980s
Gray & Tall
single procedureRelational Extended abstract
(1990s) Procedure… ecific algorithm
Conceive
a whole, irrespective o
f
algorithm
Procept, symbol
evoking
process or
concept
able 2. ecept.
SOLO ModelDavis Sfard APOS of Gray & Tall
T
Step of pr
Dubinsky
[B]
Unist tural Procedure Interiorization Action
U
( )Refication Object Procept
ase Objects
ruc
Multistructural (VMS)
I
Procedure
Relational ntegrated
Process Condensation Process Process
nistructural
in a new cycleEntity
Schema
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N. SUTHISUNG ET AL.
Moreover, Tall 004) belied thating process
fr
lassroom developed
th
y
This research study Teaching Experiment
(L
used on the importance of thinking time, and the
st
team and school administrators partici-
pa
on process
th
yzed using the
pr
ncept in symbolic calculation and
assroom
us
lts
Example analyse train” from add
or subtract unit, atuation and stuck
th
(2ev the chang
om procedures to thinkable concept cannot be seen easily.
Therefore, we described the concept based on empirical evi-
dence according to the above theories in the students’ abstrac-
tion process through compression to thinkable concept. These
various underlying frameworks have a general development of
increasing flexibility and compression, which is introduced in
an overall problem-solving way in Lesson Study.
Compression to Thinkable Concept
Tall and Isoda (2007) suggested that a c
rough Lesson Study does not limit students to think, it helps
the students to think and act differently in solving problem to
same effect through four steps of compression to thinkable
concept as follows:
1) Aprocedure; 2) Multi-procedure; 3) An overall process: to
recognize the different ways that related in each steps to same
effect; 4) A thinkable concept or procept according to Gray and
Tall (1994): it has dual characteristics of a process in calcula-
tion to the same effect through compression to thinkable con-
cept.
The above concept based on Tall (2006), developed the five
continuous steps through compression: 1) pre-procedure; 2) a
procedure; 3) procedures; 4) multi-procedure and 5) thinkable
concept.
This study considered increasingly sophisticated ways of
mathematical problem-solving to the same effect, the students’
procedures using “how to” in the abstraction process through
compression to thinkable concept. This study presented the
students’ abstraction process in specific problem situations to
thinkable concept in blending the embodiment (learning mate-
rials) with the written symbol.
Methodolog
was conducted by
eslie & Patrick, 2000), to analyze students’ abstraction proc-
ess focused on several ways and “how to” they use to solve
problem and chose important concept to build thinkable con-
cept. The researchers treated Open Approach as a sequence of
teaching in class to study students’ mathematical thinking with
target group using video, photographs, protocol, tape recording,
field notes, interviewing teachers, teacher trainees and collabo-
ratively observed in class to analyze the data in framework (it
will be mentioned later.) The researchers embedded to study
learning and teaching culture for 3 years, target group was one
of four schools in the project under Center for Research in
Mathematics Education, Faculty of Education, Khon Kaen
University for 5 years. It was a small and typical school with
only one class in each grade. The first grade students were used
Lesson Study and Open Approach in three steps collecting data
as following:
Teaching plans were divided into two periods: before semes-
ter and after semester. Before semester, teachers, observers,
internship mathematics student teachers, and the research team
wrote the teaching plan in units and periods, learning activities,
objectives and open-ended problems using Japanese textbook.
It was a team collaboration of four schools. During the semester,
there were teaching plans on Tuesdays for this school, using
students’ concept in class students’ prior knowledge, experi-
ences as well as expecting students’ ideas in doing mathemati-
cal activities, open-ended problems. There was instruction for
students to reveal thinking concept during doing mathematical
activity and to create teaching plans and materials together. In
class teaching focused on four steps of teaching procedures:
posing open-ended problem situations, student’s self-learning,
whole class discussion and summary through connection. The
data was collected by tape recording and analyzed with the
other steps.
At the teaching step, the teachers taught in class after team
planning, foc
udents presented their work in front of the class. Teachers
walked around to see the students’ concept, to arouse them
showing their way of thinking, and help them in class presenta-
tion by using authentic teaching materials. Observer team
(teachers, internship mathematics student teachers, school co-
ordinators, and researchers) participated at this step in class by
observing students’ ideas and oral presentation in front of the
classroom. Observer teachers, teachers, internship mathematics
student teachers, research team, school administrators partici-
pated at this step. They observed students’ tasks: oral and ac-
tion to build thinkable concept. Used Open Approach to collect
and analyze the data.
Observer teachers, teachers, internship mathematics student
teachers, the research
ted at the reflecting step in each classroom. They observed
students’ concept and their tasks. The data was collected by
tape and video recording, and these were analyzed.
Collected data from the teaching experiment in class to see
the procedures of 4 targeted students’ abstracti
rough compression to thinkable concept with conceptual ana-
lysis, using video recordings, field notes, pictures, interviewing
witnesses in instruction background assembles (teachers, ob-
server teachers and internship mathematics student teachers)
and analyzing students’ tasks with triangulation.
The data was from class observing, protocol, interviewing
and students’ tasks. Students’ concepts were anal
oblem situation “get on the train” (9 + 5 – 7 = 7) from team
collaboration to build and analyze classroom teaching from
planning lessons focusing on an open-ended problem situation
as mention above. The students’ oral and active presentations
were observed and analyzed. Empirical evidence in teaching
scenes was analyzed to understand how the students formulated
the concept of “addition and subtraction”. The purpose of ana-
lyzing teaching scenes was to study “how to” as a tool in the
students’ abstraction process through compression to think-
able concept under classroom using Lesson Study and Open
Approach. The data was analyzed based on the framework that
proposed by Tall and Isoda (2007). The analyzing was divided
into three parts: 1) Analyzing students’ way of thinking in
solving problem
2) Analyzing students’ abstraction process through compres-
sion to thinkable co
3) Analyzing “how to” in the students’ abstraction process
through compression to thinkable concept under cl
ing Lesson Study and Open Approach.
Analysis and Resu
is grade 1 activity “get on th
teacher presented problem si
e material designed instruction on the blackboard for the stu-
dents. They read, “There are 9 students at Khon Kaen station, 5
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N. SUTHISUNG ET AL.
students get on the train at Ban Pai and 7 students get off at
Muang Phol station, so how many students are there on the
train?” Learning materials were some paper, a picture of run-
ning train and a picture of each student on the train. Students
prior knowledge was construct 10 from decomposing and com-
posing, using diagram as thinking tool. This situation focused
on writing symbols addition and subtraction using diagram and
base 10 under the theory of Tall and Isoda (2007).
The problem situation “get on the train” was closed to stu-
dents’ daily lives and used a picture as a teaching tool to moti-
vate students to solve problem on open-ended problem situation.
To find the answer and use the Open Approach as teaching tool
for supporting and promoting students’ abstraction process to
thinkable concept.
A teacher tells the story A teacher re a ds the
problem Students read the
problem situa tion
Students used base 10 and a diagram as a thinking tool to the
ame result. Students decomposed the first and second number
nd composed numbers to build 10 and decomposed 10 with
th
as found by counting. Looking at different ways of
pey 7 or 9 + 1
s
ae third number. Students understood the meaning of symbol
“+” for addition and “–” for subtraction (9 + 5 – 7 = 7). They
checked the result by picking the learning materials (as in Fig-
ure 6).
1) To analyze the students’ thinking process
The focus switches to the number of children on the train,
which w
rforming the operation, as 9 + 5 then take awa


Threewaysofthinkingwere
dividedto3stepsasfollow:
Step1Decomposethefirstand
secondnumbersandcompose
thenumberstobuild10and
compo se10withsumofthe
others.
Step2Decompose10intotwo
numbersanddecomposethe
otheradditionof10,sumtwo
numbers.
Step3Decomposenumbers
fromstep2.
I
I
I
Thestudents’thinkinguseddiagram
fordecomposing,composingand
recom posing basedonbaseten.
take 1 from 5 to give 9,
5 is left 4
9 is 10,
bring 10 plus 4 is 14
take 7 ftom 10
10 is left 3
Take 0 from 4
4 is left 4,
bring 7 plus 0 is 7
bring 4 plus 3 is 7
Figu re 6.
The students’ thinking using diagram for decomposing, composing and
recom p osi n g bas ed o n ba se ten.
, plus 4 and taking away 7, and so on. This is the
0
making ten
operational world of mathematics in which different operations
can have the same effect. It is the effect, the total number that
matters.
This is performed even more efficiently by simply focusing
on numbers and their operations and, in particular, the flexibil-
ity of those operations. It means not just knowing lots of dif-
ferent ways of doing something, it means simplifying the prob-
lem by choosing an efficient and meaningful way of getting the
answer, to make the arithmetic simpler.
Students used base 10 and diagram as thinking tools for the
same result. Students’ ways of thinking were to decompose the
first and second numbers and compose numbers to build 10 and
subtract from the third number to find answer. Students under-
stood the symbol + for addition, – for subtraction from sym-
bolic sentence (9 + 5 – 7 = 7). At last, they checked the answer
by picking designed materials. The answer was seven as from
the symbolic sentence, and the students’ way of thinking was
divided into three steps to the same effect: building 10 with
other numbers decompose 10 to subtract from the other number
and compose number from step two.
2) To analyze compression to thinkable concept in the
students’ abstraction proce ss fr om symbolic se ntence
Considering the procedure to thinkable concept of the stu-
dents three methods in solving problem based on Tall and Isoda
(2007), especially in final step the students revised and checked
way of thinking, they recognized concept formation and this
concept was built to utilize later for extending mathematical
structure (Suthisung, 2011a, 2011b). These can analyze in area
the students’ abstraction process. Considering students’ tools in
steps 3, 4 and 5 from procedure to thinkable concept, the stu-
dents recognized concept formation and this concept was built
to be utilized la ter.
Moreover, students used learning designed materials to check
the result from the problem situation: there were nine students
on the train and then five students got on, there were 14 stu-
dents on the train and after that seven students got off, so there
were seven students on the train. Students used formal mathe-
matics symbols and formal written language.
Action of abstraction process focusing on compression to
thinkable concept: in what level and how it happens (as in Ta-
ble 3).
Students used learning designed materials to support and
promote their action in problem solving. They used multi-pro-
cedures to solve problems to the same effect. They used base 10
and a diagram as learning tools for calculation in addition and
subtraction to thinkable concept as follows:
 Students used base 10 from diagram to decomposing, com-
posing and recomposing in accordance with Gray and Tall
(1994) the different symbol and process but same effect.
 Students used the form as in No. 1 to get the result. They
decomposed and recomposed to get 10 and subtracted from
10.
 The students used different ways to get the same effect.
They checked the result and chose the most efficient way to
solve the problem.
 Students got the result from multi-procedures. They used 10
by decomposing, composing and recomposing as flexible
concepts. Howat (205) described 10 as a thinkable concept
for providing place value.
 Students could create or construct new knowledge from
solving the mathematics problems. They used previous
Copyright © 2012 SciRe s . 1193

N. SUTHISUNG ET AL.
Table 3.
Using “how to” in abstraction process.
e step of compression to think-Protocol Th
able concept
 revise thinkable concept: Using
base ten to bring construct new
concept (The effect is extended, the
precise effect)
The students used base 10 and
decompose, from their
background kdge. They
compose
nowle
used 10 as concept in solving
problem and then checked several
methods in solving problem.
 a thinkable concept: 9 + 5 – 7 =
7, 10 as thinkable concept, using
decomposing, composi ng and
recomposing (The effect is
considered as a concept in itself)
Interviewing students: At first I
make 10, it is easy. The students
got concepts in solving problem.
process of calculation from pro-
cedures to same effect: 9 + 1 + 4 –
7, 3 + 7 + 0 + 4 – 7, 5 + 5 + 4 – 7, 5
They used 10 in addition and
subtraction to find answers. They
decomposed , composed and an
efficient and meaningful way of
getting the answer.
+ 5 + 2 + 2 – 7 (The realization that
the different procedures may involve
different sequence of steps, but they
all achieve ‘the same effect)
Student (I.90):9 + 5 – 7 = 7( nine
plus five minus 7 is equal to
seven)
Teacher (I.91): Anything else?
Student (I.92): There were 9
people on the train, then 5
students got on the train and
people got off to buy somethin
7
minus?
 multi- procedure: (Several dif-
ferent procedures, to choose the
most efficient)
g,
so how many people were there
on the tr ain? Symbol ic sent ence is
9 + 5 – 7 = 7 is it correct?
Student (I.118): Yes.
Teacher (I.172): Look at number
7. Think carefully. Do you know
which words mean plus or
Student (I.173): Got on the train.
Student (I.177): Got off the train.
Student (I.73): Take 5 from 9 to
give 5 is 10, 9 is left 4, bring 10
plus 4 is 14. Take 5 from 10. Take
2 from 4, 4 is left 2, 10 is left 5.
Bring 5 plus 2 is 7.
Student (I.121): 9 plus 5 equals to
7. Take 5 from 9 to give 5 is 10, 9
is left 4, Bring4 plus 10 equals
to14. Then I take 7 from 14…
Student (I.123) Take 7 from 10,
take 0 from 4, 10 is left 3 and 4 is
left 4. Bring 7 plus 0 equals to 7.
Bring 3 plus 4 equals to 7.
 procedure: (A single step-by-step
procedure to carry out the operation)
Student (I.67): Take 1 from 5 to
give 9, 5 is left 4,
9 is 10.Bring 10 plus4 equals to
14. Take 7 from 10, 10 is left 3.
Take 0 from 4, 4 is left 4, then
bring 7 plus 0 equals to 7, 3 plus 4
equals to 7.
knowledge to think and find answsituations.
Students used 10 to add and subtract. They used decompose,
ompose and recompose. Gray and Tall (1994) described action
Tall &
Is
how to” in the students’ abstraction pro-
ce
d even more
ef
ording to Gray and Tall (1994) in “action”, the stu-
de
’ way
th
ca
t is interesting that this lesson is about develop-
in
blem in the step
of
oblem based on Tall and Isoda
(2007) in the n process of
abstraction toept interacts
w
ents
ers in new
c
compression procedures of idea onto thinkable concept.
oda (2007) said that multi-procedures to solve problems and
thinkable concept.
Analysis of “action” in the students’ abstraction process
through compression to thinkable concept as in Figure 7.
3) To analyze “
ss through compression to thinkable concept under class-
room using Lesson Study and Open Approach
Focusing on the number of children on the train at any point
and calculating the changing number by adding and subtracting
the numbers getting on and off. This is performe
ficiently by simply focusing on numbers and their operations
and, in particular, the flexibility of those operations. It means
not just knowing lots of different ways of doing something, it
means simplifying the problem by choosing an efficient and
meaningful way of getting the answer, to make the arithmetic
simpler.
In the study, students used base 10 in addition and subtract-
tion to same effect. They decompose, compose and recompose
again acc
nts’ way of thinking through compression to thinkable con-
cept using learning tools in 5 steps as mention before. In the
fifth step, the students recognized the concept from solving
problem to construct new knowledge in new situations.
It is shifting steadily from performing sequence of compres-
sion in students’ thinking from actions being linked together
increasingly sophisticated ways: accumulation students
inking in 1 - 3 step to refine important ideas in step 4 and it is
realized to extend useable mathematical structure in step 5 also.
It happened clearly by compression of knowledge from step-
by-step procedure, to the possible choice of several different
procedures, to seen the overall effect as a general process that
n be carried at in various ways, to compressing it as a think-
able concept.
In terms of this Figure 8, for “process”, it can be said that
procedures such as 9 + 5 – 7, 10 + 4 – 7, 14 – 7 all have the
same effect’. I
g the way that the children are encouraged to think flexibly
from the start. Thus the sequence procedure-multi procedure-
process-procept occur in continuous steps, indeed, the lesson
focuses early on flexibility of arithmetic, so the idea of “proc-
ess” builds at the same time as the children play with multi-
procedures, while implicitly focusing on the flexibility required
for precept. This encouragement to think more flexibly leads
more naturally to more sophisticated thinking.
In addition, for the students’ abstraction process in “action”,
the students used learning tools to support their thinking. They
bridged real world problem to mathematics pro
whole class discussion. To check their symbolic thinking at
each step, they used learning tools in addition and subtraction
efficiently. According to Poynter (2004) and Tall (2007a), the
abstraction process combined manipulation on physical objects
and symbols to support students’ mathematical thinking based
on Poynter (2004) and Tall (2007a). For further study, the re-
search will present the integration of embodiment and sym-
bolic.
Conclusion and Discussion
Students’ concept to solve pr
fourth step of compression in actio
thinkable concept. Thinkable conc
ith thinking tools in action process of abstraction through
compression important ideas into thinkable concept.
Considering students’ thinking tools in steps 3, 4 and 5 from
procedure to thinkable concept, the students recognized concept
formation and this concept was built to utilize later.
Students used 10 as “how to” to build thinkable concept. They
understood the value of “how to” which help them to extend the
mathematics structure. Howat (2005) viewed that the stud
Copyright © 2012 SciRe s .
1194

N. SUTHISUNG ET AL.
Copyright © 2012 SciRe s . 1195
Real world Mathematical world
IIIIII
perception action reflection
10 as revise thinkable conce
p
t
effect is used
Figure7.
“how to” in the students’ abstraction process through compression to th inkable concept under classroom using Lesson Study and Open Approach.
Compression to thinkable concept
Procedure
Multi- procedure Process
Thinkable
concept
Revi se thi n kable
concept
How to
I II
III IV V
9+1+4-7,
3+7+0+4-7,
5+5+4-7,
5+5+2+2-7
10 as how to,
using
decomposing,
composing
and
recomposing
Brin
g
ho w to for
construction new
knowledge and
extension
mathematical
structure
Figure 8.
“how to” in the students’ a bstraction process through compres sion to thinkable concept.
w
f “ten” as a thinkable concept. This study found that “how to”
recompose for providing the part-part or part-whole.
ou
ffi-
ci
t to achieve flexibility and effectiveness of problem
solving effectively and quickly, whenever. To prepare using
“h
lation that is integrated
be
Promotion and National Research University Project of Thai-
ill not cope with place value if they cannot form the concept They use i
oow to” in learning units of lesson sequences is to provide a
tool in conducting students’ concepts.
The further study, the research will present the student’s ab-
straction process through compression to thinkable concept fo-
cused on the student’s thinking procedures in interacting with
learning materials and symbolic calcu
is important and it is a tool to build thinkable concept as fol-
lowing:
1) Students used “how to” and base 10 to decompose, com-
pose and
2) “how to” makes extension mathematical structure on base
10 through addition and subtraction. Students used their previ-
s (met-before) knowledge to construct new knowledge.
3) “how to” makes students to realize the different procedures
to solve math’s problem. Students used meaningful and e
tween embodiment and symbolism according to Tall (2007a).
Acknowledgements
This work was supported by the Higher Education Research
ent way to solve problem and they saw mathematical values.
In particular, “how to” is to be compared as a measure of
success in students’ solving problem and concepts formation.

N. SUTHISUNG ET AL.
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