Psychology
2012. Vol.3, No.9, 729-736
Published Online September 2012 in SciRes (http://www.SciRP.org/journal/psych) http://dx.doi.org/10.4236/psych.2012.39110
Copyright © 2012 SciRes. 729
Students’ Abstraction Process Based on Compression into
Thinkable Concept from Blending Embodiment and Symbolism
under Context Lesson Study and Open Approach
Nisara Suthisung1, Maitree Inprasitha2
1Doctoral Program in Math ematics Education, Faculty of Ed u c at i o n, Khon Kean University,
Khon Kean, Thailand
2Center for Research in Mathematics Education, Facult y of Education, Khon Kean University,
Khon Kean, Thailand
Email: muinisara@hotmail.com
Received June 26th, 2012; re v ised July 29th, 2012; accepted August 26th, 2012
The objective of mathematics learning and teaching was to develop students’ concept in content so that
the persons as teachers, educators, and researchers would try to search for instruments, and know for
comprehending the students’ existing concepts. The study abstraction process based on compression into
thinkable concept was to be a major guideline in considering and findings answer in order to understand
the students’ concept (Gray & Tall, 2007). In Thai classroom using Lesson Study and Open Approach
produced the students’ mathematical thinking by using open-ended problems with designed material for
supporting interaction of students and their problem solving arithmetic (Inprasitha, 1997). According to
view of Tall (2007a) suggested a parallel construction of compression in the symbolism and embodiment
to thinkable concept. This paper aim to analyze the students’ abstraction process based on compression
into thinkable concept from blending embodiment and symbolism. The four first grader as targeted group
at Kookham Pittayasan School, a project school with supported by CRME, Faculty of Education, Khon
Kaen University, Thailand. For research design used ethnographic study and teaching experiment. The
collected data were used video analysis, interviewing, and students’ task analysis. The research revealed
that, the students’ thinking shift steadily from performing sequence of parallel compression from actions
being linked together in increasingly sophisticated ways to thinkable concept in embodiment and symbol-
ism. This research revealed that Lesson Study incorporated Open Approach as teaching approach pro-
vided to students’ abstraction process from considering that they manipulated with designed materials for
supporting and checking their various symbolic thinking before into same effect on arithmetic operation.
Keywords: Lesson Study; Open Approach; Abstraction Process; Embodiment and Symbolism;
Compression to Thinkable Concept
Introduction
The objective of Mathematics Learning and Teaching was to
develop students’ concept in content so that the person as
teachers, educators and researchers would try to search for in-
struments and know for comprehending the students’ existing
concepts (Gray & Tall, 2007). Teachers need to know, to sup-
port and develop students’ thinkable concept from learning
activities, to bridge real world and mathematical world using
problem solving as well as designing materials and tools. Stu-
dents manipulated with designed materials for supporting, cal-
culating, checking their r es ul ts a nd t he y built think ab le concept.
To consider thinkable concept, Skemp (1987) suggested
process of abstraction as a tool to build thinkable concept.
Skemp (1971) considered the fundamental human thinking
activities to be perception, action and reflection. Tall (2004)
considered that the progress of students’ mathematical thinking
based on two fundamental human activities: perception and
action which is develop through compression to thinkable con-
cept in accordance with Gray and Tall (2007).
Gray and Tall (2007) suggested that the abstraction process
through compression to thinkable concept is the key to develop
increasingly powerful thinking. Tall (2007a) noted that think-
able concept must be integrated in the curriculum. However,
there was no empirical evidence. So the researchers and educa-
tors should study and make it clear for teachers, students and
parents (Tall, 2007a).
The abstraction process on an operation problem solving
arithmetic using procedures in thinking are compressed to
thinkable concept, according to Gray and Tall (2007) suggested
that for long term learning concept formation, thinkable con-
cept must be integrated in the curriculum. Gray and Tall (1994)
suggested the abstraction process of compression operation
arithmetic using procedures in problem solving to same effect.
Tall (2004) explained that the changing process from proce-
dures to thinkable concept cannot be seen easily. So, this is a
point of view important of Tall and Isoda (2007) said that for
classroom developed through Lesson Study provided to com-
pression through four steps based on Tall (2006b) to be five
steps of compression to thinkable concept.
In Thailand, under context of Lesson Study and Open Ap-
proach, Inprasitha et al. (2007) explained that Lesson Study
focused on changing to develop the learners’ progress in real
class with 3 steps: team collaboration, observation and reflec-
N. SUTHISUNG, M. INPRASIT HA
tion. And Open Approach as a teaching approach is an innova-
tion for learning and developing students’ abilities. According
to Inprasitha (2010), Open Approach is a teaching approach to
support and help the students to learn to the best of their abili-
ties including 4 steps as following: posing open-ended problem,
students self learning, whole class discussion and comparison,
and summary through connection. The students’ perceived real
world embedded in the open-ended problems through atten-
tively interacting with designed materials for supporting vari-
ous students’ thinking way.
To survey teachers’ opinions 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 Open approach had found that the teachers con-
cerned and required to provide the students’ concept formation.
Inprasitha (1997) and Shimada and Becker (1997) suggested
problem situation in mathematical activities that bridge real
world and mathematical world. Nohda (1998) viewed that Open
Approach is a teaching activity used for integrating mathemati-
cal activity and students’ thinking, and explained that the
mathematical activity is considered to be an activity in the ab-
straction process from real world to mathematical world. This
study used Lesson Study and Open Approach to produce the
students’ mathematical thinking by focusing on open-ended
problems with designed material for supporting interaction of
students and their problem solving arithmetic. According to
Poynter (2004), and Tall (2007a) noted that to a parallel con-
struction of compression in the symbolism and embodiment to
thinkable concept. In addition to this, Tall (2007b) suggested
that Lesson Study provided area of the students’ compression to
thinkable concept.
From this point of view, the researcher interested to study the
students’ abstraction process through compression to thinkable
concept from blending embodiment and symbolism in context
using Lesson Study and Open Approach, which is a part in my
research.
Hypothesis
Students’ abstraction process is to be step up sequence of
thinking to develop into concept.
Research Question
How is the students’ abstraction process based on compres-
sion into thinkable concept from blending embodiment and
symbolism?
Research Objective
To analyze the students’ abstraction process based on com-
pression into thinkable concept from blending embodiment and
symbolism
Methodology
This research was conducted by Ethnographic Study and
Teaching Experiment. Researcher treated Lesson Study as in-
novation instructional in classroom, Open Approach as se-
quence of teaching. Researcher as observation participates to
describe school instructional culture which researcher embed-
ded to study for 3 years.
And the teaching experiment as adopted in the study which
incorporated Open Approach as teaching approach is targeted at
and is hoped to throwing light on the understanding of students’
abstraction process from connecting the action of a hand ma-
nipulative using designed materials to the effect of that action
on the learning of arithmetic operation.
Used Teaching Experiment to see the students’ process of
abstraction based on action in compression to thinkable concept,
observed several ways they use to solve problem and chose
important concept to build thinkable concept. Researcher
treated Open Approach as a sequence of teaching to provide
various the natural thinking way in class to study target group
using video, photographs, tape recording, field notes, inter-
viewing teachers, teacher trainees and collaboratively observed
in class to analyze the data as in framework. (It will be mention
later.) Researcher 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 school with only one class in each grade. The first grade
students were used Lesson Study and Open Approach in 3 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, research team wrote
teaching plan in units and periods, learning activities, objectives
and open-ended problems using Japanese textbook. It was team
collaboration of 4 schools. During semester, there were tea-
ching plans on Tuesdays for this school, using students’ con-
cept in class students’ background knowledge, experiences as
well as expecting students’ ideas in doing mathematical activi-
ties, 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 teach-
ing focused on 4 steps of Open Approach: posing open-ended
problem situations, student’s self learning, whole class discus-
sion and comparison, and summary through connection. The
data was collected by tape recording and analyzed with the
other steps.
At the teaching step, teachers taught in class after team plan-
ning, focused on the important of thinking time, and the stu-
dents presented their work in front of class. Teachers walked
around to see the students’ concept, to arouse them showing
their way of thinking, and help them in class presentation by
using authentic teaching materials. Observer team (teachers,
internship mathematics student teachers, school coordinators,
and researcher) participated at this step in class by observing
students’ ideas and oral presentation in the classroom. Observer
teachers, teachers, internship mathematics student teachers,
research team, school administrators participated at this step.
They observed students’ tasks: oral and action to build think-
able concept according to four steps of Open Approach. Ob-
server teachers, teachers, internship mathematics student teach-
ers, research team, school administrators and experts partici-
pated at the reflecting step in each classroom from grade 1 to
grade 9. They observed students’ concept and their tasks. The
data was collected by tape recording, video and analyzed.
At the collaboration observation step, collected data from
teaching experiment in class to see the procedures of 4 targeted
students’ abstraction process through compression to thinkable
concept with conceptual analysis, using video recordings, field
notes, pictures, interviewing witnesses in instruction back-
ground assembles (teachers, observer teachers and internship
Copyright © 2012 SciRes.
730
N. SUTHISUNG, M. INPRASIT HA
mathematics student teachers) and analyzing students’ tasks
with triangulation. The data was from class observing, inter-
viewing and students’ task. Exemplar analyzing students’ con-
cept using problem situation “get on the train” (9 + 5 – 7 = 7)
from team collaboration to build and analyze classroom teach-
ing from planning lesson focused an open-ended problem situa-
tion as mention above. Observed students’ oral and action pre-
sentation and analyzed their tasks. Analyzed empirical evidence
in teaching scenes to understand how the students formulated
the concept of “addition and subtraction”. The purpose of ana-
lyzing teaching scenes was to study the students’ abstraction
process through compression to thinkable concept under class-
room using Lesson Study and Open Approach. The data was
analyzed under the framework that proposed by Tall and Isoda
(2007). The analysis were divided into three parts: 1) analyzing
students’ way of thinking in solving problem; 2) Analyzing of
compression into thinkable concept in students’ abstraction pro-
cess from calculation symbolic and interaction designed mate-
rial; 3) Analyzing the students’ abstraction process based on
compression into thinkable concept from blending embodiment
and symbolism under context using Lesson Study and Open
Approach.
Conceptual Framework
Thinkable concept is the teaching and learning goals. In
achieving that, teachers should provide appropriate learning
experiences for learners. Open Approach as a teaching ap-
proach which cooperates with Lesson Study is the effective
way to develop mathematical activity using open-ended prob-
lems for promoting and developing students’ concept. To con-
sider process of abstraction focused on compression to thinkable
concept of interacting with learning materials and symbolic cal-
culation in students’ real world under the views as Figure 1.
Context of Study: Lesson Study and Open Approach
Lesson Study
At present Lesson Study has been used widely. Many educa-
tors state compatible the important of Lesson Study. Lewis
(2002) viewed that Lesson Study is teachers’ collaboration for
learning aim and developing students’ progress, designing les-
son plans to support teaching and changing classroom. Tall
(2007b) suggested that Lesson Study is pattern for construction,
analyzing classroom and developing students’ abilities thinking
which teachers and researchers working together. And mathe-
matics Lesson Study is individual lesson design to long-term
teaching approach. Tall (2008) explained Lesson Study is a
complete preparation lesson which learners construct mathe-
matics concept and develop their own thinking concept.
For Thailand, Lesson Study is implied from Japan to im-
prove and develop classroom teaching in real class, focusing on
students’ progress in class and exchanging knowledge and
problems with other teachers. And teachers in group realize the
teaching aim and adapt lesson to context in 3 steps: 1) Using
Open Approach collaboratively for designing research lesson;
2) Observing; 3) reflection (Inprasitha et al., 2007).
This study used and treated Lesson study as pattern for con-
structing and analyzing classroom with based on learning and
teaching with team (teachers, observers, internship mathematics
student teachers, research team) collaboration to develop learn-
ing progress and the process of thinking as following.
Collaboratively
Plan
Collaboratively
Observing
Collaboratively
Reflection
Figure 1.
Cycle of lesson study including 3 phase s.
Collaboratively for designing lesson plan, using Open Ap-
proach form problem situation in students’ real life, create de-
signed materials to support students concept. Focused on les-
son’s goal, learn how to learn, timing for each period, designing
4 steps of teaching.
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.
Collaboratively reflect, discussing problems and obstacles 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.
Open Approach
Nohda (1998, 2000) viewed that Open Approach is used for
supporting various kinds of students activities and students
mathematics thinking to solve problem. Open Approach is
teaching approach which we can adapt to many students ways
of thinking or we can say that students mathematical thinking
and developing teaching approach should be integrated. Open
Approach is expected to be a tool for changing classroom,
helping students to learn from their abilities. In Thailand, Les-
son Study has been used with Open Approach as a teaching
approach in four steps according to Inprasitha (2010). It is
started from posing open-ended problem situation, student’s
self learning, whole class discussion and comparison, and
summary through connection. Students learn and understand
the contents by solving problems.
This research focused on Open Approach at the step of stu-
dents’ self-learning, to analyze the students’ abstraction pro-
cess. Students manipulated with designed materials, calculat-
ing, and using Lesson Study to build thinkable concept.
1) Posing open-ended problem: A teacher posed problem to
encourage 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)).
4) Summary through connection: At this step, the teacher
summarized the students’ concept to help them understand the
content (Figure 2(d)).
Inprasitha (2010) suggested in classroom using Lesson Study
Copyright © 2012 SciRes. 731
N. SUTHISUNG, M. INPRASIT HA
is integrated Open Approach provided and supported various
students’ way mathematical thinking, that is in first phase (col-
laboratively plan)—team lesson planned lesson and teaching
sequence based on 4 steps Open Approach: posing open-ended
problem, students’ self learning, whole class discussion and
comparison, and summary through connection. In second phase
(collaboratively observed)—team lesson observed students’ ma-
thematical thinking following 4 steps of Open Approach. In
finally phase (collaboratively reflect)-team lesson reflect stu-
dents’ mathematical thinking happened following team lesson’s
expecting and coming out to develop lesson later (Figure 3).
Conceptual Framework for Analysis the Students’
Abstraction Process Based on Compression into
Thinkable Concept from Blending Embodiment and
Symbolism
Abstraction Process
Skemp (1971, 1987) considered the fundamental human
thinking activities to be perception, action and reflection. Tall
(2004) described the process of abstraction through perception
and action and Gray and Tall (2007) viewed that the abstraction
process to develop through compression to thinkable concepts
too. They described that the abstraction process through com-
pression to thinkable concept is the key to develop increasingly
powerful thinking. Lesson Study and Open Appr oach is an im-
portant teaching approach that arouses the students to think and
act, to solve problem from blending embodiment with symbol-
ism, so it is used in this research.
(a)
(b)
(c)
(d)
Figure 2.
4 steps of open approach.
Lesson St udy cycle
Collaboratively
Plan
Collaboratively
Observing
Collaboratively
Refl ec ti on
Open Ap pr o ach
I. posing open- e nded problem
II. Student’s self le arn ing
III. Whole class discussion and
Comparison
IV. Summary through connection
Figure 3.
Lesson study cycle and implementing open approach by focus on open-
ended problem situation.
Blending Embodiment and Symbolism
Tall (2008) said that Lesson Study is lesson preparation for
developing students’ mathematical thinking from embodiment
to symbolism or vice versa. Tall (2006a) found the parallel con-
struction of embodiment and symbolism, the first could be seen
but the second based on human thinking from symbolic calcu-
lation through thinkable concept, and Tall (2007b) developed
research framework through conceptual embodiment and pro-
ceptual symbolism. In addition to this, Tall (2007a) noted that
to a parallel construction of compression in the symbolism and
embodiment to thinkable concept (Figure 4).
This study presented the process of abstraction focused on
action; it can be seen through compression to thinkable concept
in blending embodiment and symbolism from empirical evi-
dence. Using Lesson Study and Open Approach produced the
students’ mathematical thinking by focusing on open-ended
problems with designed material for supporting interaction of
students and their problem solving arithmetic.
Compression to Thinkable Concept
Tall and Isoda (2007) said that for classroom developed
through 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 following:
1) a procedure; 2) multi-procedure (and used the most effi-
cient); 3) an overall process: to recognized the different ways
that related in each steps to same effect; 4) a thinkable concep t
or procept according to Gray and Tall (1994): it is dually char-
acteristics of process in calculation to same effect through com-
pression to thinkable (Figure 5).
The above concept based on Tall (2006b), developed the con-
tinuously 5 steps through compression: 1) pre procedure; 2) a
procedure; 3) multi-procedure; 4) a process; and 5) thinkable
concept.
In addition Poynter (2004) explained the abstraction process
combined manipulation on physical objects and symbols to
support students’ mathematical thinking: 1) action; 2) multi-
acti on; 3) same effect; and 4) effect a s protot ype. Tall (2007a)
noted that to a parallel construction of compression in the sym-
bolism and embodiment to thinkable concept. This study pre-
sented the students’ abstraction process in specific problem
situations to thinkable concept in blending embodiment (learn-
ing materials) with (symbolism) written symbol (Figure 6).
Analysis and Results
Exemplar in analyzing grade 1 activity “get on the train”. A
Copyright © 2012 SciRes.
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N. SUTHISUNG, M. INPRASIT HA
Figure 4.
Blending embodiment and symbolism in mathematical development
(Tall, 2007b).
Figure 5.
Compression t o t hi nk a bl e c on ce pt in s ymbolism (Tall, 2006b).
Figure 6.
A pararell compression to thinkable concept in embodiment and
symbolism (Tall, 2006b).
teacher presented problem situation and stuck the designed
material instruction on the blackboard for students. They read
There are 9 students at Khon Kaen station, 5 students get on
the train at Ban Pai and 7 students get off at Muang Phol sta-
tion, so how many students are there on the train?” Learning
materials were some paper, a picture of running train and a
picture of each student on the train. Students background
knowledge was construct 10 from decomposing and compos-
ing, using diagram as thinking tool. This situation focused on
writing symbols addition and subtraction using diagram and
base 10 based on analyzing data under the theory of Tall and
Isoda (2007).
Problem situation “get on the train” was closed to students’
daily lives and used a picture as teaching tool to arouse students
to solve problem on open-ended problem situation. To find
answer and use Open Approach as teaching tool for supporting
and promoting students’ abstraction process to thinkable con-
cept.
The teacher read problem
situation
The students read
proble m situ ation
The teacher told story
Students used base 10 and diagram as thinking tool to same
result. Students decompose the first and second number and
compose numbers to build 10 and decompose 10 with the third
number. Students understood + for addition, – for subtraction (9
+ 5 – 7 = 7). They checked the result by picking the learning
materials (As in Figure 7).
Analyzing Students’ Way of Thinking
The focus switches to the number of children on the train,
which can be found by the action of counting. Looking at dif-
ferent ways of performing the operation, as 9 + 5 then take
away 7 or 9 + 1 making ten, plus 4 and taking away 7, and so
on. This is the operational world of mathematics in which dif-
ferent operations can have the same effect. It is the effect, the
total number, which 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-
Copyright © 2012 SciRes. 733
N. SUTHISUNG, M. INPRASIT HA
Threewaysofthinkingwere
dividedto3stepsasfollo w:
Step1Decomposethefirstand
secondnumbersandco mpo s e
thenumberstobuild10and
compos e10wit hsumofthe
others.
Step2Decompose10intotwo
numbersanddecompos ethe
otheradditi on of10,sumtwo
numbers.
Step3Decomposenumbers
fromstep2.
I
I
I
Thestudents’thinkin g useddiagram
fordecomposing,composingand
recomposingbasedonbaseten.
Take 1 from 5 to give 9, 5
is le ft 4
9 is 10, bring 10 plus 4 is 14
take 7 from 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
Figure 7.
The students’ thinking using diagram for decomposing, composing and
recomposing ba sed on base ten .
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 decompose the
first and second number 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 7 as from sym-
bolic sentence, and the students’ way of thinking was divided
into three steps to the same effect: building 10 with other num-
bers decompose 10 to subtract from the other number and
compose number from step two.
Analyzing of compression into thinkable concept in students
abstraction process from calculation symbolic and interaction
designed material:
Considering from procedure in problem to thinkable concept,
students had several thinking methods in solving problem based
on Tall and Isoda (2007), especially in final step the students
revised and checked, they recognized concept formation and
this concept was built to utilize later for extending mathemati-
cal structure (Suthisung & Sangaroon, 2011a, 2011b, 2011c),
which will be analyzed in area the students’ abstraction process.
Gray and Tall (2007) suggested that for classroom developed
through Lesson Study in four steps of compression to thinkable
concept: I. A procedure, II. Multi-procedure, III. An overall
process, IV. A thinkable concept and Suthisung and Sangaroon
(2011a, 2011b, 2011c) studied according to the concept men-
tioned earlier, had found the fifth step revise thinkable concept.
From the above concept, students used 3 methods from com-
pression to thinkable concept to same effect and they knew that
9 + 5 – 7 = 7 were several ways of thinking. According to
Howat (2005) suggested students used base 10 as thinkable
concept. They used 10 and wrote symbolic sentences to same
effect: 9 + 5 – 7 = 7, 9 + 1 + 4 – 7, 7 + 3 + 4 – 7, 4 + 10 – 7, 4 +
5 + 5 – 7, 2 + 2 + 5 + 5 – 7, 4 + 7 + 3 – 7. The students’ way of
thinking was divided into three steps to the same effect: build-
ing 10 with other numbers decompose 10 to subtract from the
other number and compose number from step two.
And learning designed materials were related to symbolic
calculation, used to check the result from problem situation:
there were 9 students on the train and then 5 students got on,
there were 14 students on the train and after that 7 students got
off, so there were 7 students on the train. In addition, the stu-
dents’ abstraction process in “action”, the students used learn-
ing tools to support their thinking. According to Poynter (2004)
explained the abstraction process combined manipulation on
physical objects and compression to thinkable concept in 4
steps: I. Action; II. Multi-action; III. Same effect; IV. Effect is
prototype, and Suthisung and Sangaroon (2011a, 2011b, 2011c)
studied and found V. Effect is used (Summarized in Table 1).
The study revealed that the students thought and checked
their answers by using designed materials step by step in solv-
ing problem as in Table 1 and in accordance with Tall (2007a)
Table 1.
Action of abstraction process focusing on compression to thinkable
concept from b lend ing embodiment and symbolism.
Embodiment Symbolism
effect is used
(the effect is utilized later ) revise thinkable concepts
Using base ten to bring construc
t
new concept
(The effect is extended, the precise
effect)
effec t as pro totype
(think effect as prototype)
a thinkable co ncepts
9 + 5 – 7 = 7
10 as thinkable concepts, using
decomposing, composing and
recomposing
(The effect is considered as a
concept in itself)
same effect
(The same effect from different
action)
process of calculation
from procedures to same effect
9 + 1 + 4 – 7, 3 + 7 + 0 + 4 – 7,
5 + 5 + 4 – 7, 5 + 5 + 2 + 2 – 7
(The realization that the different
procedures may involve different
sequence of steps, but they all
achieve ‘the same effect’)
multi-action
(Several different action on
physical object to carry out
the same e ffect)
multi- proc edu re
(Several different procedures, to
choose the most efficient)
action
(A movement step-by-step on
physical objects to carry out
the effect)
procedure
(A single step-by-step procedure
to carry out the operation)
Copyright © 2012 SciRes.
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N. SUTHISUNG, M. INPRASIT HA
Copyright © 2012 SciRes. 735
embodiment
Realworld
Mathematicalworld
Figu r e5
.
Theproc essof abstraction:focusingoncompressiontothinkableconcept inbl ending embodimentandsymbol ism
I
II
III
symbolism
Embodimentand sy m b oli s m
Eff ectisused Revisethink able conce
p
t
Per ception
Action
Figure 8.
The process of abstraction: focusing o n compression to t hinkable concept in blending embodime nt a nd s ymbolism.
noted that to a parallel construction of compression in the sym-
bolism and embodiment to thinkable concept.
Students used learning designed materials to support and
promote their action in problem solving. They used multi-pro-
cedures to solve problems to same effect. The integration be-
tween learning designed materials and symbolic thinking sup-
ports the students to think quickly and efficiently.
Besides, the students used base 10 in addition and subtract-
tion to find the answer. They used 10 for decomposing, com-
posing and recomposing according to Gray and Tall (1994) to
build thinkable concept. Tall and Isoda (2007) said that the
different procedures to the key concept and the students recog-
nized and used various methods to build thinkable concept.
The students’ abstraction process in action through compres-
sion to thinkable concept happened step by step in interacting
with designed materials and symbolism.
Poynter (2004) and Tall (2007a) said that the compression to
thinkable concept combined manipulation on physical objects
and symbols to support students’ mathematical thinking. The
symbolic thinking and using learning designed materials to the
same effect based on Tall (2007b) suggested a parallel con-
struction of compression to thinkable concept from blending
embodiment and symbolism.
Analyzing the students abstraction process based on com-
pression into thinkable concept from blending embodiment and
symbolism under context using Lesson Study and Open Ap-
proach:
The students manipulated with designed materials for sup-
porting and checking their various symbolic thinking before
into same effect on arithmetic operation. In the students’ ab-
straction process based on action, the students’ thinking shift
steadily from performing sequence of compression from actions
being linked together in increasingly sophisticated ways to
thinkable concept in embodiment and symbolism. According to
Poynter (2004) purposed that compression in embodiment in-
cluding action, multi-action, same effect, effect as prototype
and expanded to effect is used. According to Tall and Isoda
(2007) purposed that compression in symbolism including pro-
cedure, multi-procedure, process, a thinkable concept (procept)
and expanded to revise thinkable concept as in solving mathe-
matics problem.
Moreover, the classroom using teaching with Open Approach
which is driven by Lesson Study, the implementation of ab-
straction process started in step of Posing open ended prob-
lem—the students had perception through seeing and interact-
ing with designed materials as well as problem situations. Con-
sequently, the students were interested in and persistent in
meaningful problem solving leading to duration in step of Stu-
dents’ self-learning—the students tried to use various proce-
dures in problem solving from the designed material, mathe-
matical symbolic thinking, language usage in explaining as well
as writing mathematical symbols systematically. Consequently,
the students used considered the way of thinking and refined
the importance ideas by critical thinking and careful reflective
thinking in order to develop major approaches from different
problem solving procedures through 5 steps of compression to
thinkable concept. In step of Whole class discussion and com-
parison—the students investigated their own thinking way and
share ideas from their friend and teacher. In step of Summary
through connection—the students had opportunity in reviewing
as well as viewing the value of the occurred tools of thinking
from their problem solving procedures (Figure 8).
Conclusion and Discussion
The research revealed that, Lesson Study incorporated Open
Approach as teaching approach provided to students’ abstract-
tion process in step students’ self learning from considering that
they manipulated with designed materials for supporting and
checking their various symbolic thinking before into same ef-
fect on arithmetic operation. The students’ thinking shift stea-
dily from performing sequence of compression from actions
being linked together in increasingly sophisticated ways to
thinkable concept in embodiment from action to effect onto
effect is used and symbolism from process to concept onto
revise thinkable concept respectively.
N. SUTHISUNG, M. INPRASIT HA
Especially, the parallel compression to thinkable concepts in
embodiment and symbolism that is compression in embodiment
including action, multi-action, same effect, effect as prototype
and expanded to effect is used. Compression in symbolism in-
cluding procedure, multi-procedure, process, a thinkable con-
cept (procept) and expanded to revise thinkable concepts as in
solving mathematics problem. It is shifting steadily from per-
forming sequence of compression in students’ thinking from
actions being linked together increasingly sophisticated ways:
accumulation students’ way thinking in 1 - 3 steps to refine im-
portant 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 can be carried at in va-
rious ways, to compressing it as a thinkable concept.
The further study, my research will analyze the function of
students’ abstraction process as operation of compression to
thinkable concept from blending embodiment and symbolism,
which is micro scale in analyzing mechanism students’ mathe-
matical thinking.
Acknowledgements
This work was supported by the Higher Education Research
Promotion and National Research University Project of Thai-
land, Office of the Higher Education Commission, through the
Cluster of Research to Enhance the Quality of Basic Education
and Center for Research in Mathematics Education, Faculty of
Education, Khon Kaen University, Thailand.
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