Creative Education 2012. Vol.3, No.7, 11881196 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 Email: muinisara@hotmail.com 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 openended 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 openended problems, students’ selflearning, 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 openended 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 openended 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 problemsolving. 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 openended problem situations, student’s selflearning, 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. Copyright © 2012 SciRe s . 1189
N. SUTHISUNG ET AL. 1) Posing Openended problem: A teacher posed to encour age students to solve problem (Figure 2 (a)). 2) Students’ selflearning: They made goaldirected 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 selflearning. Student: The students solve the problem by them selves and used symbolic calculations. They cre ate d v ari ous way and goaldirected 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’ selflearning, 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 subunit 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 Copyright © 2012 SciRe s . 1190
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 KookKham 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 subsequence 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… Selfcontained Bigg and ollis (1980s Gray & Tall single procedureRelational Extended abstract (1990s) Procedure… ecific algorithm Conceive a whole, irrespective o 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 Copyright © 2012 SciRe s . 1191
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 problemsolving 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) Multiprocedure; 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) preprocedure; 2) a procedure; 3) procedures; 4) multiprocedure and 5) thinkable concept. This study considered increasingly sophisticated ways of mathematical problemsolving 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 openended 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, openended 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 openended problem situations, student’s selflearning, 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 openended 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 Copyright © 2012 SciRe s . 1192
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 openended 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 Threewaysofthinkingwere dividedto3stepsasfollow: Step1Decomposethefirstand secondnumbersandcompose thenumberstobuild10and compo se10withsumofthe others. Step2Decompose10intotwo numbersanddecomposethe otheradditionof10,sumtwo numbers. Step3Decomposenumbers fromstep2. Thestudents’thinkinguseddiagram fordecomposing,composingand recom posing basedonbaseten. 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 multipro 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 multiprocedures. 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 thinkProtocol 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 stepbystep 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 multiprocedures 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 bystep 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 proceduremulti procedure processprocept 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 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+47, 3+7+0+47, 5+5+47, 5+5+2+27 10 as how to, using decomposing, composing and recomposing Brin 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 partpart or partwhole. 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 (metbefore) 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. land , Office of thession, through the Cluster of SOLO taxonomy. New York: Academic Press. Davis, R. B. (1984). Le cognitive science ap proach to mathematicJ: Ablex. atical thinking (pp. H: An analy Gudy with your friends MATHEMATICS Ghmetic. Journal for Research in Higher Education Commi Research to Enhance the Quality of Basic Education and Center for Research in Mathematics Education, Faculty of Education, Khon Kaen University, Thailand. REFERENCES Biggs, J., & Collis, K. (1982). Evaluating the quality of learning: The arning mathematics: The s education. Norwo od, N Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.), Advanced mathem 95123). Dorfre c h t: Kluwer. owat, H. (2005). Participation in elementary mathematics sis of engagement, attainment and intervention. Ph.D. Thesis, War wick: University of Warwick. akkotosho Co., Ltd. (2005). St for elementary school 1st grade. Toky o: G ak k o to s ho Co., LTD. ray, E., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arit Mathematics Education, 25, 115141. doi:10.2307/749505 Gray, E., & Tall, D. (2007). Abstraction as a natural process of mental compression. Mathematics Education Research Journal, 19, 2340. doi:10.1007/BF03217454 prasitha, M. (1997). P Inroblem solving: A basis to reform mathematics In., & Thasarin, P. (2007). To prepare con 11 September 2011 . Nngs of ICMIEARCOME 1, N the 24thConference of the Interna P athe instruction. Reprinted from the Journal of the National Research Council of Thailand, 2 9 , 221259. prasitha, M., Pattanajak, A text for leading the teacher professional development of Japan to be called “Lesson Study” implemented in Thailand. Bangkok: Tham masat University, 152163. Inprasitha, M. (2010). One feature of adaptive lesson study in Thailand: Designing a learning unit. Proceedings of the 45th Korean National Meeting of Mathematics Education. Gyeongju: Dongkook University, 89 October 2010, 193206. Isoda, M. (2010). The principles for problem solving approach and open approach: As a product of lesson study. International Confe rence on Educational Research (ICER 2010), Learning Communities for Sustainable Development, 10 Leslie, P. S., & Patrick, W. T. (2000). Teaching experiment methodol ogy: Underlying principles and essential elements. Handbook of re search design in mathematics and science education (pp. 267306). Mahwah: Lawrence Erlbaum Associate. ohda, N. (1998). Mathematics teaching by “openapproach method” in Japanese classroom activities. Proceedi 1721 August 1998, 185192. ohda, N. (2000). Teaching by approach method in Japanese mathe mati cs cla ssroo m. Proceeding of tional Group for the Psychology of Mathematics Ed ucation, 1139. oynter, A. (2004). Effect as a pivot between actions and symbols: The case of vector. Ph.D. Thesis, Warwick: University of Warwick. Pegg, J., & Tall, D. O. (2005). The fundamental cycle of concept co struction underlying various theoretical framework. ZDM (M matics Education), 37, 468475. doi:10.1007/BF02655855 fard, A. (1991). On the dual nature of mathematical conceptions: Re flections on processes and objects as different sides of the s Same coin. Educational Studies in Mathematics, 22, 1 36. doi:10.1007/BF00302715 kemp, R. R. (1971). The psychology of learning Pengin. Smathematics. London: rlbaum Associated, Inc., 921. ent’s abstraction process. S process through compression to thinkable concept. Procee T T of mathematical growth through embodi de Sciences Ts1185800600103 Skemp, R. R. (1987). The psychology of learning mathematics. London: Lawrence E Suthisung, N. and Sangaroon, K. (2011a). The steps up of compression to thinkable concept inaction of the stud The 16th Annual Meeting in Mathematics (AMM 2011), 1011 March 2011. uthisung, N., & Sangaroon, K. (2011b). “How to” in the students’ ab straction dings of the 35th Conference of the International Group for the Psy chology of Mathematics Education (Developing Mathematical Thin king). Ankara, 1015 July 2011, 1 398. all, D. O. (2004). The nature of mathematical growth. URL (last checked 23 March 2010). http://www.tallfamily.co.uk/david/mathematicalgrowth all. D. O. (2006). A theory ment, symbolism and proof. Annales de Didactique et Cognitives, 1, 195 215. all, D. O. (2007a). Developing a theory of mathematical growth. ZDM, 39, 145154. doi:10.1007/ in Thailand, 117. ng Book on Tsulation of a process? Journal of Mathematical Be Tall, D. O. (2007b). Setting lesson study within a longterm framework of learning. APEC Conference on Lesson Study Tall, D. O. (2008). Using Japanese lesson study in teaching mathemat ics. Scottish Mathematical Council Journal, 38, 4550. Tall, D. O., & Isoda, M. (2007). Longterm development of mathemat ical thinking and lesson study. Chapter for a Forthcomi Lesson Study. all, D. O, Thomas, M., Davis, G., & Gray, E. (2000). What is the ob ject of the encap havior, 18, 2232 41. doi:10.1016/S07323123(99)000292 Copyright © 2012 SciRe s . 1196
