The Open Approach in Lesson Study- Enhancing Teachers Knowledge of Teaching Division of Fractions.

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1 The Open Approach in Lesson Study- Enhancing Teachers Knowledge of Teaching Division of Fractions. Abstract Lloyd Munroe Hiroshima University Fraction is a difficult topic for teachers to teach and for students to learn. This is partly because we do not allow students to explore the concept in their context. The open approach help students to explore mathematics through their own experience, to communicate those ideas to others which allows them to reflect and critically assess their thoughts as well as others creating a deeper understanding of concept. This paper reports on a lesson study conducted with six Jamaican teachers. The teachers developed a lesson to teach division of fractions through the use of open approach method to a grade six class. Their main aim was to develop a better way to help students understand division of fractions. The study highlights the pedagogical knowledge, content knowledge, and the interactions between teacher and students during discussion of the lesson. Most students displayed a deeper understanding of diving fractions and teachers expressed their increase in confidence to use the open approach method to teach mathematics. Introduction and Background Many reasons have been given as to why fractions are difficult for teachers to teach and for students to learn. Some reasons include; fractions represent relations between two quantities, young children have difficulty grasping the fact that two different whole numbers are used to form one, and because there are infinitely many fractions between any two numbers (Cheng-Yao Lin, 2013). There are different perspective of looking at fractions and one fraction can be used to represent different things. We can say we cut an orange in half, or that six is half of twelve or that nine square meters are half of an area of 18 square meters. Despite the many challenges, fractions would be easier to teach students if we allow them to explore fractions in their own context. Each student has a characteristic way of perceiving the world around him (Davis et. al. 1990). Constructivists argue that a person s knowledge will be formed and change through his own actions. This implies that students should be given the opportunity to investigate problems based on their own experiences. The National Council of Teachers of Mathematics (NCTM) in their book Principles and Standards for School mathematics have stated that effective

2 mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it. There are a range of professional development for mathematics teachers in Jamaica. These include; workshops which can last from one to three days. These professional development are not mandatory, therefore, few teachers participate in these sessions. Teachers at a Cornwall Primary School in Jamaica have weekly common planning sessions on Wednesdays. Students are sent home and all teachers meet to discuss their lessons. Teachers share information about best practices and methodology for teaching upcoming topics. The researcher introduced Lesson study to the teachers at one of these common planning sessions. All teachers agreed to participate in the study. The teachers decided to embark on a lesson study research employing the open approach to teach fractions. Theoretical Framework The theoretical framework is based on Vygotsky s preposition of social interaction. Vygotsky s zone of proximal development (ZPD) is the difference between what a learner can do without help and what he or she can do with help (Fani & Ghaemi, 2011). The leaners in this case are the teachers engaged in the lesson study activity. Vygotsky theory suggests that knowledge is constructed through social interaction, negotiation, reflection and discussion. Lin (2002) states that teachers learn through reflection, cognitive conflict and social interaction and Catherine Lewis listed individual professional knowledge (Lewis, 2000, p.13) where teachers learn from watching others teach as fundamental for teachers professional growth. The teachers in this study participated in an in-service research for professional development. The teachers were seeking new ways of improving their common planning session. Unlike the common planning session of simply discussing what can be done, lesson study provided the opportunity for teachers to develop deeper mathematical understanding and widen their pedagogical content knowledge of fractions through discussion, observation and reflection. Review of Literature. The open approach is a teaching method that is used to promote higher-order thinking skills in students (Becker & shimada,1997; Nohda,1986) The philosophy of the open approach in teaching mathematics is to encourage students to use their own way of thinking and to apply their previous knowledge to solve a given problem. Other countries have adopted the open approach to teach mathematics in school. These include England, Germany, America, Australia, and Thailand (Inprasitha, 2006; Yee, 2002; Stacey, 1995). The open approach engages students in exploring, zig-zagging, discussing, explaining, questioning, investigating, and using critical, creative and mathematical thinking skills. It allows students to develop their own way of solving the problem which

3 increases the chance of them remembering it in the future. Students are also encourage to communicate their ideas to others and to the teacher, this helps to improve their mathematical thinking and understanding (Nohda, 1997, 2000; Yee,2002, Sanchez,2013). The open approach emphases a five-component lesson: (1) introducing the problem; (2) understanding the problem; (3) Problem solving by students; (4) What is lesson study? Comparison and Discussion of students solutions; (5) teaching summarizing the lesson and students reflection (Becker & Shimada 1997). Lesson Study (LS) is a professional development activity started by teachers in Japan. Lesson Study is a cyclic activity which include a group of teachers working together to design, implement and improve a lesson. A member of the team teaches the lesson while the others observe and record the teachingleaning process. The team then meets again to evaluate and improve the lesson. A different member of the team reteaches the lesson while the others observe and take notes (Perry & Lewis, 2009; Robinson and Leikin, 2012; Cheng & Yee, 2012). Enacting and watching the lesson will produce ideas that may surprise and challenge the teacher during instruction (Clea, 2005). Research Design and Lesson Study Procedure Cornwall Primary School is a public school with one class per grade. Grade six was chosen as the research group. The six teachers (Ray, Joan, Gill, Wendy, Carla and Sue) and the researcher carried out a lesson study cycle within two weeks. Joan, the grade six teacher, was chosen as the team leader. The team decided on teaching division of fractions to the grade six class. The two main reasons for this topic were; (1) most of the teachers had difficulty teaching fractions and were in search of new information to deepen their mathematical and pedagogical content knowledge about it. (2) To provide enough time for discussing and planning the lesson- the team wanted the lesson study topic to be students first exposure to the concept. The teachers felt that most students struggle to understand division of fractions. The team formulated three questions to guide the study: 1. Can Lesson Study assist with teacher development and understanding of teaching division of fractions? 2. How will students respond to the use of the open approach method? 3. How different is Lesson Study to the Common Planning Session? Lesson study Format a. Planning 1&2: The first two meetings were used for discussion of the topic. Joan, Ray and the researcher guided the discussions. The researcher acted as the recorder of the meetings. b. Planning 3: The team developed the lesson plan in the third meeting. The lesson required three different manipulatives. Gill prepared strips of

4 paper of equal measurements. Wendy prepared ribbons; one 18 inches and the other 4 inches in length. And Sue prepared oranges as the manipulatives for the lesson. c. Planning 4. The team meet to have a short mock lesson focusing on different member s role and critical aspect of the lesson such as the use of the manipulatives. This meeting was the shortest and lasted for 82 minutes. d. Teaching 1: Ray taught the first lesson, the researcher and Carla videoed the lesson and the other teachers observed and took notes. e. Evaluation 1: The lesson was evaluated on the following day. Carla was absent for this meeting. Modifications of the lesson were made and Joan executed the revised lesson. Teaching 2: Joan taught the second lesson. Ray and the researcher videoed the lesson while the others observed and took notes. f. Evaluation 2: The researcher took the opportunity to do a group interview of the teachers. Individual interview was conducted in week three. The team reflected, critique and the lesson again. Ms. Joan was satisfied with the lesson and did not wish to reteach it. However, the suggestions given were adopted into the revised lesson plan for future use. This session was also use to interview teachers about their experience. The researcher returned the following week to interview teachers individually. Examples of students solutions. Richard s solution Blake s solution Data Collection and Analyses. A qualitative analysis would be more appropriate for this study. The focus questions of the study were: 1. Will the teachers learn from participating in Lesson Study (LS)? 2. How do Jamaican teachers view lesson study when compared with common planning?

5 3. How do teachers view the use of the open approach to teach mathematics? The sources of data collection were; observation, interview and audio-tape recordings of the meetings (not all meetings were audio taped). Observations were recorded in the form of fieldnotes. This was done throughout the LS process, i.e. before, during and after conducting the lessons. The purpose of the fieldnotes was to identify teachers perception of the planning and execution process. Individual interviews were used to capture teachers personal opinion of LS and the open approach method. Examples of questions asked were, how was the lesson study process different from your common planning time? How often do think you will use the open approach method in your lessons? Results and Discussion Teachers first discussion surrounded the difference between the common planning sessions they were familiar with and the lesson study format. They quickly realized that while they normally discuss the activity at each grade in common planning, only one grade and one lesson will be discussed in LS. The teachers also realized that the common planning was similar to the first stage of lesson study, however, lesson study takes it further to actually planning and executing the lesson and reflecting on it. This report will summarize the discussion surrounding teachers pedagogical and content knowledge as well as what they have learnt during each of the following: (a) Pedagogy: The discussion on the open approach first brought out the issues of questioning techniques which further evolved into use of words. Joan mentioned that the way a question is asked or worded is important as students may interpret words differently. Teachers were surprised at the different terminologies and expressions they each have about fractions. They realized that students may interpret statements such as 6 into 12 and divide 12 by 6 Differently. This came about during reflection when Sue pointed out that Ray had said 12 oranges divide by 6 but then later in his explanation he said 3 into 12. Ray was using the oranges to help students to understand the statement the more we divide a number the smaller it gets but the more parts we have this was said by Blake, one of the students in the class). Ray had written on the board 12 6 = 2, 12 3 = 4, 12 1 = 12, and 12 ½ = 24. In teaching the second lesson, Joan allowed students to state common phrases use in division.

6 Other aspects of pedagogy discussed was the use of the open approach and questioning techniques. Sue stated that she had used open questions and asked student to discuss different methodologies to solve a problem but she had never asked them to reflect on what they did. Questioning technique became a main point of discussion as teachers discuss how best to guide students during the main activity. Ray, Joan and Sue admitted their lack of patience when questioning students this led to a long discussion on questioning techniques and scaffolding strategies. (b) Content Knowledge Two planning sessions were allocated to discussing fractions and how to solve them. The teachers admitted that they had not used other methods of dividing fractions beside changing the sign and inverting the second fraction. Teachers researched on the topic and return to the second planning session with other methods of dividing fractions. The teachers discovered that there were different ways of viewing and representing fractions besides dividing one item. Example, fractions of a group and fraction of an area. Sue said that even though I had shown these representation to students I have never really thought about it. It was like I was doing it because it was in the textbook. (c) Learning from students: The teachers realized that being an observer in the lesson gave them a golden opportunity to listen to and observe students better and that focusing on students thinking process can help them to carefully listen to and analyze students reasoning. Joan speculated that when we are teaching we already have preconceived ideas in our minds which hinders us for looking deeply into what students are thinking. Gill added that this is one of the skills that the open approach teaches, to look at the concept from different approaches and to expect multiple solutions from students. Carla said, I was surprised by Blake s solution. I had no idea we could solve it (dividing fractions) in that way (see Blake s solution above). Sue added that Tyrone s method made the question look simpler than she had taught it to be. Carla said that even though Horace was saying he was actually, adding ½ + ½ + ½. She added that teachers should not only listen to what students are saying but look carefully at what they are doing. Conclusion Teachers concluded that while their common planning time offers insights into better practice, lesson study is a more meaningful and practical activity that gives them a deeper understanding of not only in pedagogy but also in content. They became aware of their instructional vocabulary including being consistent with their expressions. The teachers were also satisfied with students reaction to the open question that was used. While some students were hesitant and

7 gave up easily, others rose to the challenge and solve the questions in unexpected ways. Teachers expressed their increase in confidence to use the open approach method in teaching. References Becker,J.P., & Shimada,S. (1997). The open-ended approach: A new proposal for teaching mathematics. Reston, Va: National Council of teachers of Mathematics. Bogdan, R. C.,& Biklen, S. K. (1992). Qualitative research in education: An introduction to theory and methods. Boston: Allyn and Bacon. Cheng, L. P., & Yee, L. P. (2012). A Singapore Case of Lesson Study. Mathematics Educator, 21(2), Clea Fernanadez (2005) Lesson Study: A Means for Elementary Teachers to Develop the Knowledge of Mathematics needed for Reform-Minded Teaching?., Mathematical Thinking and Learning, 7:4, ,DOI: /s mtl Davis,R.B., Maher,C.A.& Noddings,N. (eds) Constructivist Views on the teaching and Learning of Mathematics. JRME Monograph Number 4. Reston (VA): NCTM Fani, T., & Ghaemi, F. (2011). Implications of Vygotsky's Zone of Proximal Development (ZPD) in Teacher Education: ZPTD and Self-scaffolding. Procedia-Social and Behavioral Sciences, 29, Nohda,N.( 1986) A study of open-approach method in school mathematics. Tsukuba Journal of Educational Study in Mathematics. 5, Lewis, C. (2000). Lesson Study: The Core of Japanese Professional Development. Robinson, N., & Leikin, R. (2012). One teacher, two lessons: The lesson study process. International Journal of Science and Mathematics Education, 10(1),