Students’ Abstraction Process Based on Compression into Thinkable Concept from Blending Embodiment and Symbolism under Context Lesson Study and Open Approach

doi:10.4236/psych.2012.39110

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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

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: [email protected]

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

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

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

732

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-

N. SUTHISUNG, M. INPRASIT HA



Threewaysofthinkingwere

dividedto3stepsasfollo w:

Step1Decomposethefirstand

secondnumbersandco mpo s e

thenumberstobuild10and

compos e10wit hsumofthe

others.

Step2Decompose10intotwo

numbersanddecompos ethe

otheradditi on of10,sumtwo

numbers.

Step3Decomposenumbers

fromstep2.

Thestudents’thinkin g useddiagram

fordecomposing,composingand

recomposingbasedonbaseten.

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

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)

734

N. SUTHISUNG, M. INPRASIT HA

embodiment

Realworld

Mathematicalworld

Figu r e5

.

Theproc essof abstraction:focusingoncompressiontothinkableconcept inbl ending embodimentandsymbol ism



III

symbolism

Embodimentand sy m b oli s m

Eff ectisused Revisethink able conce

Per ception



Action

Reflection



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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