Process Designing Mathematical Tasks on Addition of Teachers Using Lesson Study ()

Katanyuta Bangtho^{1}, Narumol Inprasitha^{2}, Maitree Inprasitha^{2}

^{1}Centre of Excellence in Mathematics, CHE, Bangkok, Thailand.

^{2}Center for Research in Mathematics Education, Faculty of Education, Khon Kaen University, Khon Kaen,
Thailand.

**DOI: **10.4236/ce.2015.616170
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This research was aimed at investigating the process of designing mathematical tasks on addition of teachers using lesson study. It employed the qualitative research method. An analysis was performed on data obtained from discussion protocols of the Lesson Study team during the development of lesson plan of Unit 8: Addition (2) and questionnaire on the designing of mathematical tasks. Results of the research indicated that the lesson study team utilized the following processes for designing mathematical tasks: 1) determining the lesson goal; 2) interpreting the mathematical tasks; 3) identifying the use of mathematical tasks; and 4) anticipating students’ approaches.

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Bangtho, K. , Inprasitha, N. and Inprasitha, M. (2015) Process Designing Mathematical Tasks on Addition of Teachers Using Lesson Study. *Creative Education*, **6**, 1691-1696. doi: 10.4236/ce.2015.616170.

1. Introduction

Mathematical tasks are important to instructions and students’ learning nature. The nature of students’ learning is determined by type of task and the way it is used ( Clarke & Roche, 2010 ). A study reveals that there are differences in the structure of lessons and the types of tasks used in the mathematics classroom. American teachers focus on developing skills with most time spent on practicing routine procedures, whereas Japanese teachers focus on developing conceptual understanding with as much time spent on solving challenging problems as practicing skills. Japanese teachers used only a single task throughout the whole class period, valued a variety of solution methods, and also encouraged students to pose new tasks ( Stigler & Hiebert, 1999 ). With regard to instructional approach of Thailand, most classroom activities are executed with teachers describing relevant knowledge, rules, and formulas. Then, students are given a large number of exercises to practice as a means to test their knowledge obtained from the lesson earlier taught to them ( Phu-Udom, 2001 ). It can be seen that American and Thai students are different from Japanese ones in a way that the former have to practice mathematical exercises, but Japanese students have the opportunity to do more than mathematical questions. They have a creative way to solve mathematical tasks by inventing new methods and presenting such creative mathematical tasks ( Shimizu et al., 2010 ).

Choosing and determining appropriate tasks are considered the key to the success in mathematics class and essential for teachers when they are to design tasks (Doyle, 1988; Hiebert & Wearne, 1993; Stein & Lane, 1996; Takahashi, 2006; Martin 2007 cited in Shimizu et al., 2010 ). The tool that enables teachers to develop and study their own teaching practices is lesson study ( Baba, 2007 ). Designing mathematical tasks is a key component of lesson study ( Lieberman, 2009 ). For some teachers, it is very difficult to see that there may be more than one correct solution to a task or they may not be able to see other solution methods ( Roth & Ames, 2014 ). Designing mathematical tasks is not an easy job for teachers and to develop good tasks, especially open-ended ones, is the most difficult; therefore, teachers have to help each other design open-ended tasks (Nohda, 1982 cited in Maitree Inprasitha (2004) ).

2. Data Collection

The researcher collected data from the lesson study team of the first grade of KhookhamPittayasan School in the second semester of 2014 academic year for Learning Unit 8: Addition (2). The researcher conducted participant observation on three steps of lesson study. The first step collaboratively design research lesson. The second step collaboratively observing the research lesson and the third one collaboratively reflection on teaching practices. Materials used for data collection include project schoolbooks, field note, questionnaire, and video records.

3. Data Analysis

Data analysis was based on Lesson Planning Framework developed by Watanabe et al. (2008). Data used for the analysis were obtained from discussion protocols of the lesson study team during the designing process of mathematical tasks of Unit 8: Addition (2). The results proposed in this paper were obtained from data on Learning Unit 8: Addition (2). Examples of the mathematical tasks were presented in detail as follows: Data Analysis of Period 1: Playground (Figure 1).

3.1. Determining the Lesson Goal

The Lesson Study team began with mathematical tasks taught in the previous year, which were interpreted from Japan’s first-grade mathematics textbooks (Thai translation) by Gakkhotosho CO., LTD., Japan. The team jointly discussed the goal to be achieved by the students and scope of the lesson as shown in the protocol (Table 1).

From the protocol, the lesson study team jointly identified the lesson goal by interpreting mathematical tasks contained in the textbook and focused the discussion on what the students would learn from this period. It can be seen that with respect to the topic of addition, the first graders had to learn about two types of addition: combination and supplementation. For the first period of addition topic, the students would learn about addition in the form of combination. Upon reaching mutual understanding of the lesson goal, the lesson study team set the goal for the first period as “to be able to understand and solve the addition situation”.

3.2. Interpreting Mathematical Tasks

Due to the fact that the lesson study team had experiences in teaching and designing mathematical tasks, during this process there were old tasks as the basis of the team’s thinking process. The team members used the figure on Page 77 of the project mathematics textbook to jointly discuss the tasks and try to figure out what teachers should try to communicate with students based on the tasks and picture, provided that such communication had to be consistent with the establish goal of the class period as shown in the following protocol (Table 2).

From the protocol, it can be seen that the lesson study team jointly interpreted the mathematical tasks form the picture in the textbook to figure out how to enable students to understand from the playground picture. What

Figure 1. Mathematical tasks 9 + 4 ( Gakkohtosho, 2005 ).

Table 1. Protocol determining the lesson goal of lesson study team.

Table 2. Protocol interpreting mathematical tasks of lesson study team.

discussed by the team showed that they tried to understand the mathematical tasks and search for the issue hidden in the picture.

3.3. Identifying the Use of Mathematical Tasks

Upon reaching a mutual understanding from interpretation of the tasks, the lesson study team jointly created an instruction to communicate with the students about what they wanted the students to do. The instruction used was “there are nine children playing at the sand tray and four children at the slider, how many children are there in total?” What the lesson study team was concerned about was how to present the mathematical task. According to their discussions, it was about where the students would access the problem. The mathematical task in this period was divided into two phases: understanding the task from the picture and how to get the answer on how many children in total, as shown in the protocol (Table 3).

The protocol showed that the lesson study team discussed how to present a mathematical task in order to lead the students to the established goal of understanding and solving the mathematical task. The team decided that the presentation would be divided into two phases. The first one was to help students understand that the picture shown to them suggested the situation of addition, which meant combination of two things. The second phase involved with how they would find the answer after writing down a sentence using mathematical symbols. It can be seen that the lesson study team was aware of the use of a mathematical task in the classroom to enable the students to get to the task on their own.

3.4. Anticipating the Students’ Approaches

What the lesson study team always kept in mind in task design was anticipation of the students’ approaches in classroom based on the approaches of students from the past years and the students’ application of knowledge obtained in the previous class period to the picture shown to them. It can be seen that, with respect to the topic of addition (2), an important means was to base ten (make it a full amount of 10 first). According to the lesson study team’s observations, it was found that the students preferred using the number of 10 in the topic of addition. This information was important for the lesson study team to anticipate the student’s approaches. The illustrations (Table 4) were used to represent the students’ approaches.

This showed that the teachers understood that being able to understand the students’ approaches would make classroom management easier since the teachers knew how the students thought and what approaches they had. Important approaches could be used, discussed, and shared in the classroom. Anticipating different approaches used would enable the teachers to discuss similarities and differences of strategies used in classroom, which would be a means to promote the students’ comparative analysis.

4. Conclusion

The resulting data were analyzed with the design process. There are four steps that can be shown instead of the process designing mathematical of teachers using lesson study as shown in Figure 2.

5. Discussion

The outstanding attribute of the mathematical task design was that the teachers anticipated the students’ approaches in advance. This did not only indicate how much the teachers knew their students’ learning nature, but also helped the teachers manage the classroom. Fujii (2013) stated that mathematical task design was not only based on appropriateness of the task, but the lesson study team was also aware of inserting a mathematical les-

Table 3. Protocol identifying the use of mathematical tasks of Lesson Study team.

Table 4. Protocol anticipating the students’ approaches of lesson study team.

Figure 2. Process designing mathematical tasks of Lesson Study team.

son into the students’ fun activities. The ultimate goal of the lesson was not only to promote the students’ ma- thematical understanding, but also to give them profound understanding and freedom of thinking.

Acknowledgements

This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education Thailand, and Graduate School of KhonKaen University, and was supported by the Center for Research in Mathematics Education (CRME), Faculty of Education, KhonKaen University.

Conflicts of Interest

The authors declare no conflicts of interest.

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