© 2010 Khon Kaen University, Thailand

International Journal of EducationVol.33, No.3, July.-September., 2010

pp. 78-91

The Students’ Abstraction Process in Classroom by Using Lesson Study and Open Approach

Nisara Suthisung1 and Kiat Sangaroon2

1Doctoral Program in Mathematics Education, Faculty of Education, Khon Kaen University2 Department of Mathematics Khon Kaen University

Email: muinisara@hotmail.com

Abstract This study aims to present the students’ abstraction process through foundational human thinking activities in classroom using lesson study and open approach. For research design, it was a qualitative research using ethnographic study and teaching experiment method, with the first graders. The research findings revealed that, under context lesson study including the team collaboration to build and analyze classroom teaching for planning lesson focused making visible student’s mathematical thinking and using open approach as teaching approach including 4 steps for supporting the abstraction process as following: 1. Posing open-ended problem-students perceived real world embedded in the open-ended problems situation through attentively interacting with designed materials 2.Student’s self learning-they have goal-directed thinking, attempting to create multi representations into mathematical world before coming to mathematical concept 3. Whole class discussion-they realized and checked way of thinking in order to systematically explain their ideas 4. Summary through connection-they have opportunity to revise concept. Keywords: Lesson study, Open Approach, Mathematical activity, Process of abstraction

Introduction At present, lesson study has been used widely in classroom research. Many educators stated the important of lesson study in changing classroom and developing students’ thinking. Lewis (2002) viewed that lesson study is teachers’ collaboration for learning and developing students’ thinking, designed lessons for supporting teaching and changing classroom. Tall (2007) viewed that lesson study is one of constructing and analyzing classroom which teachers and researchers work together. Tall (2008) viewed that lesson study is a preparation of completed lessons for the students to build and develop concept formation. Tall&Isoda (2007) viewed that lesson study in mathematics is the design

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of lessons to long term teaching method to understand the nature of thinking in mathematics and it is useful for all students. Verhoef&Tall (2010) viewed that lesson study would be a starting point to make visible student’s mathematical thinking. In Thailand, under context of lesson study, Maitree, et.,al (2007) got the concept of lesson study from Japan. It is focused on changing to develop the learners’ progress in real class with team collaboration plan, observers and reflection. Open Approach as a teaching method is an innovation for learning and developing students’ abilities. According to Nohda (1998), the learning objectives of using Open Approach are to support various activities and mathematical thinking of students. Students are free to solve problem to their abilities, relations and interests. Nohda (2000) viewed that Open Approach is expected to be a tool for changing classroom, helping students to learn from their abilities. In Thailand, according to Inprasitha (2010), Open Approach is a teaching method to support and help the students to learn to the best of their abilities including 4 steps as following: posing open-ended problem, students’ self learning, whole class discussion and summary through connection. Students perceived real world embedded in the open-ended problems through attentively interacting with designed materials for supporting various students’ thinking way. Tall&Isoda (2007) viewed that using Open Approach through Lesson Study is not limited to one method of thinking, but supported the efficiency, accuracy and meaning of students’ thinking. Moreover, Nohda (1998) viewed that Open Approach is a teaching activity used for integrating mathematical activity and students’ thinking, and explained that the mathematical activity is considered to be an activity in the abstraction process from real world to mathematical world. In the other hand, Shimada&Becker (1997) viewed that focusing on developing the efficient mathematical activity as a problem leading to the space of thinking connected between real world and mathematical world by the abstraction process. Also, Shimizu (2009) viewed that mathematical activities were activities for using mathematics in daily life too. To survey teachers’ opinions in four schools, participating in the project under Center for Research in Mathematics Education, Faculty of Education, Khon Kaen University for four years using Open approach have found that the teachers concerned and needed to provide the students’ concept formation. However, the teachers didn’t have any tool to understand the students’ abstraction process. The abstraction process is new in Thailand and mostly focused on students’ concept formation in content. Teachers focused on using symbolic for calculation and memorizing to build concept formation in mathematical world, but disconnected with real world in solving mathematical problem. Based on Skemp (1971) viewed that the concept formation through human thinking foundational activities: perception, action and reflection and the researcher previous studies presented the abstraction process for connecting between real world and mathematical world, but those studies did not have the correspondent explanation of abstraction process which happen to under context of Lesson Study and Open Approach. There is a question from the above context: How in classroom using Lesson Study and Open Approach made the students’ abstraction process for concept formation? This study presents the abstraction process under classroom using Lesson Study and Open Approach.

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Objective To study that in classroom using lesson study and open approach can be made the students’ abstraction process for concept formation

Conceptual framwork Instruction in classroom using Open Approach as an innovative teaching approach which cooperates with Lesson Study is the effective way to develop mathematical activity using open-ended problems for promoting and developing students’ concept. To consider process of abstraction in classroom using lesson study and open approach focused concept formation of interacting with learning materials and symbolic calculation through human thinking foundational activities: perception, action and reflection under the views as following:

1. Lesson study: At present Lesson Study has been used widely. Many educators state the important of Lesson Study which is teachers’ collaboration for learning aim and developing students’ progress, designing lesson plans to support teaching and developing classroom. Thailand adopted Lesson Study from Japan. Inprasitha et al., 2007 viewed that Lesson Study in Thailand started from 2002. It is used for developing classroom teaching, improving lessons 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) reflecting. Use Open Approach in Lesson Study in order to present students different ways of thinking in mathematical activity. This study used and treated lesson study as a pattern for constructing and analyzing classroom with based on learning and teaching with team collaboration to develop learning progress and the process of thinking as following. Collaboratively for designing lesson plans, using Open Approach from problem situation in students’ real life, create learning materials to support students’ concept, using teachers’ experiences in designing lesson plans and predicting students’ concept. Focused on lesson’s goal, learn how to learn, timing for each period, designing 4 steps of teaching.Collaboratively observe in class, students’ reaction to learning 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 mathematical materials, students’ way of solving problem, students’ new ways of thinking, and the successful of using lesson plans.

2. Open approach: Nohda (1998) viewed that Open Approach is used for supporting various kinds of students’ activities and students’ mathematics thinking to solve problem. Open Approach is teaching activity used for solving different problem in 3 aspects: 1.Students Open activity: students create their own problem. 2. Mathematics activity is different way of thinking: using problem to learn in mathematical world. 3. Mathematics activity and students’ activity are

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integrated and they are essential for all students. Students are interested in solving problem. Nohda (2000) viewed that Open Approach is teaching method which we can adapt to many students ways of thinking or we can say that students’ mathematical thinking and developing teaching method should be integrated. Open Approach is expected to be a tool for changing classroom, helping students to learn from their abilities. Inprasitha (2010) viewed that Open Approach is a teaching method which is a tool to supporting students’ solve problem and understand the contents from solving problem. It is started from posing open-ended problem situation, student’s self learning, whole class discussion and summary through connection. Use Open Approach in collaborative planning lesson to support students ways of thinking in real life situation, students attempt to create multi representations from learning tool and write the symbol for calculation in real class and reflect the result. Use Open Approach in four steps as following: Step 1: posing open-ended problem: A teacher posed problem to encourage students to solve problem. Step 2: student’s self learning: They made goal-directed thinking, attempted to solve problem with different methods. Step 3: whole class discussion: The students presented their ideas in front of the class. They realized and checked way of thinking in order to systematically explain their ideas. Step 4: summary through connection: At this step, the teacher summarized the students’ concept to help them understand the content.

3. Mathematical activity: For mathematics educators, mathematical activity should be expanded in learning and teaching according to Shimizu (2009) viewed that now school curriculum is focus on mathematical activity, using mathematics in daily life and mathematical world, and it should be promoted to students by the teachers presented mathematical activity to students. Shimada & Becker (1997) described that mathematical activity involve and arouse students activity as grouped thinking area various which it developed students mathematical learning between world of reality and world of mathematics. Inprasitha (1997) viewed that the open-ended problems causing the problem forced to mathematical activity from real world to mathematical world, and mathematical activity based on process of abstraction from students’ idea for symbolizing. Nohda (1998) pointed that mathematical activity was an activity in the abstraction process from the concrete experience of real life to mathematical world.

4. Human thinking activities: Skemp (1971) considered the fundamental human activities to be perception, action and reflection. Perception involves input from sense, action involves output through interaction with perceived phenomena, and reflection is the process whereby we think about relationships between perception and action. Skemp (1989) talked about two distinct systems, ‘delta-one’ which involves perception of and action on the actual world we live in, and a second internal system, ‘delta-two’, whereby our brains imagine internal perceptions and actions and reflect on them. Here we have perceptions of and actions on

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objects with reflection producing thinkable concept. According to Tall (1995) viewed that children learning elementary mathematics this begins with ability to perceive things, to act on them and to reflect upon. He found it useful to separate out three components of human activity as input (perception), internal activity (thought) and output (action). My presentation at EARCOME5 found that the students’ abstraction process accorded with Nohda’s concept. Nohda(1998) viewed that the abstraction process caused to connect between real world and mathematical world, and I viewed that it depended on ability to perceive things on real world, to act on them leading to symbolic thinking and to reflect upon these to leading to concept formation (Suthisung, 2010). These presentations did not have the explanation clearly of correspondent the students’ abstraction process in classroom using Lesson Study and Open Approach. Then, this paper presents under classroom using Lesson Study and Open Approach is able to make student’s abstraction process for concept formation.

Methodology This is a qualitative research conducted by ethnographic study. Researcher treated Lesson Study as learning and describing school instructional culture which researcher embed to study for 3 years, 2 days per week (days of writing plan and reflecting). Target group was one of four schools in the project under Center fsor Research in Mathematics Education, Faculty of Education, Khon Kaen University. 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. Teaching plans were divided into two periods: before semester 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 teaching plans on Tuesdays for this school, using students’ concept in class students’ background knowledge, experiences as well as expecting students’ ideas for doing mathematical activity, open-ended problems. There was instruction for students to reveal thinking concept during doing mathematical activity and to create and do teaching material together. At the teaching step, teachers taught in class after team planning, focused on the important of thinking time, and the students 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 material. 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 front of the classroom. Observer teachers, teachers, internship mathematics student teachers, research team, school administrators participated at the reflecting step from grade 1 to secondary 3. They observed students’ concept and their tasks. Collected data from teaching experiment in class with conceptual analysis, video recordings, field notes, pictures, interviewing witnesses in instruction background assembles and analyzing students’ tasks. Analyzed students’ concept using problem situation “get on the train” (9+5-7 = 7) from team collaboration to build and analyze classroom teaching for

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planning lesson focused an open-ended problem situation as mention above. Observed stu-dents’ oral and action presentation and analyzed with their tasks. Analyzed teaching scenes to know how the students formulated the concept formation. What was the students’ abstrac-tion process like? Analyzed empirical evidence used Lesson Study and Open Approach in classroom.

Analysis and results Analyzed grade 1 activity “get on the train” A teacher presented informed problem situation and stick the material designs 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 station, so how many students are there on the train?” Learning materials were paper, a picture of running train and a picture of each student on the train.

Students background knowledge was construct 10 from decomposing and composing, using diagram as thinking tool. This situation focused on writing symbols addition and subtraction using diagram and base 10. Analyzed students’ concept using these framework: 1) to analyze the abstraction process based on views of Skemp (1971) 2) to analyze the open approach as teaching approach provide to student’s abstraction process in classroom context using lesson study. Problem situation “get on the train” was closed to students’ daily lives and used 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 concept.

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Figure 1: The abstraction process in Open Approach

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Table 1: The students’ abstraction process in Open Approach under context classroom using Lesson Study

The results of analysis the students’ abstraction process in Open Approach under context classroom using Lesson Study including 4 steps as following

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Figure 2: The students’ abstraction in Open Approach in context using Lesson Study

The Students’ abstraction process in Open Approach

The research results revealed that, the correspondence between the steps of teahcing and human activity thinking as following: Posing open-ended problem-perception, Student’s self learning-perception, action and relfection into concept, Whole class discussion-action and reflection, and Summary through connection-reflection into revise concept. Otherwise, the teacher posed an open-ended problem situation. The students perceived real world embedded in the open-ended problem through attentively interacting with designed materials. Then they made goal-directed thinking, using decomposing and composing for addition and subtraction, using diagram as a tool for calculation, seeing picture as listening to teacher’s story. And students used learning materials to check the result from problem situation: there are 9 students on the train, and then 5 students get on, there are 14 students on the train and after that 7 students get off, so there are 7 students on the train. Students used formal mathematics symbols and formal written language and then they presented their ideas in front of the classroom. At this step, students revised and checked their ways of thinking by peering. The last step, teacher summarized the students’ concept to help them understand the content. The abstraction process combined manipulation on physical objects and symbols to support students’ mathematical thinking into concept of the addition (increase) and subtraction (remainder) in meaning of increasing and take away by using diagram for decomposing composing, and using base ten as a tool of students’ thinking.

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Figure 3: The students’ thinking using diagram for decomposing, composing and recom-posing based on base ten

The students’ idea in action: evidence from written work

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 understood the symbol + for addition, - for subtraction from symbolic sentence (9+5-7=7). At last they checked the answer by designed materials. They acted differently but have the same result. It showed that the students have concept formation of sum and different. They knew that 9+5-7, 9+1+4-7, 3+7+0+4-7, 5+5+4-7, 5+5+2+2-7 had the same result=7 and 7 from various procedures.

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Table 2: The protocol show the student way of thinking for calculation Evidence from protocal

Status Items Speech Teacher 46 If you finished, you can write sentence symbol now. Let’s

think together. Ah write down sentence symbol, please. Is it better if you sit with friend here? Common, boy. You can know if he writes correctly or not. Let’s help.

Student 67 Five gives nine one. It is equal four. Therefore, nine is equal ten. Take ten and four together is fourteen. Ten is taken seven. It is equal three. Four is taken zero. It is equal four. Put seven and zero together is? Put three and four together is?

Teacher 68 We have calculated this. I want a new style. Try to calculate a new way again. Give me am paper. It will be like Pom. Calculate the new way again. Let’s see. Which way that you have you taken, don’t use it. Try to calculate the new way. Try to read and try to read friend’s work. Write it. Write it, please

Teacher 69 Do it again, please. Do it which is not like the old one again.

Teacher 70 What are you doing, Pom. What is the answer?

Student 71 Seven. Continue it.Teacher 72 Ten minutes left. We still have enough. Write it. If you can’t

explain now, explain it when presenting.Student 73 Nine is taken out five. Nine remain four. Five is equal ten.

Ten plus four is fourteen. Ten is taken out five. Four is taken out two. Four remain two. Ten remain five. Five plus two is seven.

The focus switches to the number of children on the train, which can be found by the action of counting. Looking at different 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 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 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.

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Conclusion and discussion Under classroom using Lesson Study and Open Approach forced student’s abstraction process into concept formation and they would be a starting point to make visible student’s mathematical thinking and are able to make sense for teachers as the teachers had a tool to understand the students’ abstraction process. Moreover, using Open Approach as teaching approach lead to correspondence between the steps of teaching and process of abstraction as following: 1) The teacher posed open-ended problem situation-the students perceived real world embedded in the open-ended problem 2) Student’s self learning-they developed goal-directed thinking and constructed many ways of thinking, used formal mathematics symbols and formal language to describe in mathematical world before coming to mathemati-cal concept 3) whole class discussion-they realized and checked way of thinking in order to systematically explain their ideas in front of the class 4) Summary through connection- they have opportunity to revise concept. Especially, in action step the students could develop their concept by recognized using procedures in problem solving into mathematical concept. When students used one successful way, they found others to solve problem, students used 10 as tools of thinking and as thinkable concept, according to Howat (2005) viewed that the concept formation of ‘ten’ as a thinkable concept students will cope with place value. So the further study, the research will verify action of abstraction process in context of Lesson Study and Open Approach focusing the student’s thinking procedures as well as it was an area of the study so on.

Figure 4: Lesson Study cycle and implementing Open Approach by focus on open-ended problem situation

Acknowledgement This research is granted by the Graduate School, Center for Research in Mathematics Education, Khon Kaen University Thailand .

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