1

Improving Lessons on Mathematical Thinking through Lesson Study ~ 3 Case Studies

Peggy Foo

Marshall Cavendish Institute

Abstract Often, teachers have found it challenging to facilitate for enhanced mathematical thinking. The main purpose of this study is to investigate the different types of professional knowledge teachers acquired through Lesson Study with respect to promoting mathematical thinking in their own classrooms. Three case study lessons representative of preschool, primary and secondary levels will be discussed in this paper. Findings revealed that choice of tasks, use of variations, use of manipulatives and materials, as well as the use of effective questioning can help to enhance students’ mathematical thinking. Implications from the findings on lesson planning and refinement of textbook materials will be drawn. Introduction In 2004, Prime Minister Lee Hsien Loong in his inaugural National Day Rally Speech urged Singapore schools to “Teach Less” so that our students can “Learn More” (MOE Press Release, 2005). In 2005, the Ministry of Education (MOE) of Singapore adopted the Teach Less, Learn More (TLLM) movement which is about teaching better, to engage our learners and prepare them for life, rather than teaching for tests and examinations. There is more quality in terms of classroom interaction, room for expression, and the learning of life-long skills such as thinking through innovative and effective approaches. At the same time, there is less focus on rote-learning, repetitive tests, and following prescribed answers and set formulae. The emphasis on thinking permeates the teaching and learning of every subject including Mathematics. In 2006, the mathematic syllabus is revised to reflect the developments and trends in Mathematics education. The revised syllabus continues to emphasize the conceptual understanding, proficiencies of skills and thinking skills in the teaching and learning of Mathematics. These components are integral to the development of mathematical problem-solving ability, which is at the heart of the Mathematics framework. Teachers were asked to provide more opportunities for students to discover, reason and communicate Mathematics. Students were encouraged to engage in discussions and activities where they can explore possibilities and make connections. Such changes in the classrooms require a change in instructional approaches. In order to bring about a change in instructional approaches such as activity-based and learner-centred methodologies, MOE recognizes that teacher is the key to the change and is committed to enhance the professional development of teachers. In 2008, MOE announced a nation-wide effort to enhance the professional expertise of their teachers by developing each school into a Professional Learning Community (PLC).

2

The key objective of establishing schools as PLCs is to improve instructional practice that leads to improved student learning outcomes. Each PLC will embrace 3 Big Ideas, namely (DuFour & Eaker, 1998), 1) Focus on Student Learning 2) Focus on Collaborative Culture 3) Focus on Data-Driven Outcomes Within a PLC, teachers are grouped into learning teams. Each team is guided by 4 critical questions on improving instructional approaches in the classroom. They are 1. What is it we expect students to learn? 2. How will we know when they have learned it? 3. How will we respond when they don't learn? 4. How will we respond when they already know it? - www.allthingsplc.org In this PLC context, schools are encouraged to choose any professional development tools or platforms to embrace the 3 Big Ideas and 4 critical questions. One of the tools which is gaining popularity among Singapore schools is Lesson Study. In particular, Japan’s method of teaching Lesson Study is focused on because of Japanese students’ high levels of achievement and it being centred on the idea that teaching is a complex, cultural activity (Stifler & Hiebert, 1999).

Lesson Study, also termed as research study, is known as kenkyuu jugyou in Japanese (Lewis, 2002), where kenkyuu means research or study, and jugyou means lesson(s) or instruction. It is seen as a shift from teaching as telling to teaching for understanding in Japanese Mathematics and Science education (Lewis, 2002), valued by educators in Japan. The idea is simple: teachers coming together working in a group throughout the lesson study process, collaborating to plan, observe and reflect on lessons. However, developing and implementing effective lesson study can be complex, as there are factors at work (Stifler & Hiebert, 1999).

The lesson study process consists of different parts – goal setting, research lesson planning, lesson teaching and evaluation, and consolidation of learning.

For goal setting, a group of teachers identifies the research theme. It could be students’ weakness in an area of learning, or a topic which educators find challenging to teach. Based on the identified goal, the team develops a lesson, called a research lesson. The lesson goals are defined, with the teachers spending time investigating possible resources, considering available lesson plans to start with while tapping on their own experiences (Lewis, 2002). In the planning of lessons, it is also considered if any manipulative is needed to help students to understand the lesson. Liptak (2005) stated that manipulatives ‘can support students’ thinking about a concept, or, if used inappropriately, can merely distract them’. Suitable manipulatives help learners to relate processes to their conceptual underpinnings and give them concrete objects to explain and justify their thinking. The

3

thoughtful design of the lesson plan is the crux of an effective lesson – a commonality in the professional practice of effective teachers (Cowan, 2006).

The lesson planned is then taught by one teacher while the other team members are present to observe the lesson and make notes as well as collect evidence of student learning and thinking. There is then a discussion involving everyone in the lesson study group, evaluating and reflecting on the lesson. The original lesson plan is then revised.

With the revised plan, it is taught to another group of students. Meeting up after the lesson to evaluate, reflect and work on the lesson plan will take place and the cycle can be repeated with each new class the improved plan is delivered to (Appel, Leong, Mangan, Mitchell & Stepnaek, 2007).

Finally, there is a consolidation of learning for sharing purposes. This is seen as an important part of the lesson study process as the consolidating and sharing consist of the findings about teaching and learning, together with the teachers’ reflections. With the knowledge gained, teachers can use them in planning and conducting future lessons (Lewis, 2002). Research Question In this study, the research question being investigated is “What are the different types of professional knowledge teachers acquire through lesson study with respect to promoting mathematical thinking in their own classrooms?” Method Sample The participants of this study are Mathematics teachers in three case studies, namely from kindergarten, primary school and secondary school. The kindergarten case study comprises pre-school principals and teachers from various kindergartens; the primary school case study consists of teachers from the same school while the secondary school case study is made up of teachers from the same school. A trainer from Marshall Cavendish Institute served as an external facilitator and led 5-8 sessions (of 1.5 hour each) to complete the lesson study cycle in each school. Each cycle begins with a session to draft the research theme. It is followed by one or two lesson planning session(s). Thereafter there is another session to conduct the research lesson in the classroom accompanied by a post-lesson discussion (usually on the same day). The cycle usually ends with one or two sessions to refine the lesson and prepare for documentation.

4

The profiles of Mathematics teachers are listed in Table 1. Table 1: Case Number of teachers

involved in the lesson study team

Profile of team members

Kindergarten 8 Pre-school principals and teachers from different kindergartens

Primary 12 Head of departments (non-academic), subject heads, level managers and senior teachers from the same school

Secondary 15 Head of Department (Mathematics) and Mathematics Teachers from the same school

Procedure Stage 1: Craft Research Theme At the start of each lesson study cycle, each learning team came together to identify the research theme. They formulated their respective research themes based on their school’s mission, vision and curriculum goals. They also compared the desired outcomes of their students and the current reality to identify gaps. From the gaps, they would identify a broad research theme such as ‘Thinking Individuals’ and ‘Engaged Learning’. The research theme gave each learning team focus and direction as they commenced their lesson study cycle to enhance their collective professional knowledge.

Research theme Kindergarten How to help students explore different options, articulate thinking

and apply what they have learnt in new situations? Primary Thinking individuals and Self-directed learners Secondary Engaged learning All 3 learning teams want to build their professional knowledge on promoting mathematical thinking in their respective classrooms. Stage 2: Plan a Lesson Teachers collaboratively designed lesson plans which sought to promote mathematical thinking. Lesson Kindergarten (for K1, 5 years old)

How many different models can you make using 5 unifix cubes? (activity-based with unifix cubes, investigative in nature, group work)

Primary (for Primary 2, 8 years old)

How to compare and order unit fractions? (activity-based with fraction discs, group work)

5

Secondary (for Secondary 2, 14 years old)

How to solve a real-life problem using students’ own methods? The lesson eventually leads to the role of simultaneous equations, how to form and solve them? (activity-based, with the use of calculators, group work)

In the lesson planning sessions, the facilitator led the teams to consider the flow of the lesson, the type of instruction approaches and anticipate the types of students’ responses that might arise from the main task(s). They also discussed how the corresponding responses of the teacher should be in different scenarios. In particular, “How would the teacher respond if the students have learnt?” and “What if the students have not learnt?” These are two of the four critical questions provided by MOE to guide all learning teams. The team then incorporated inputs from different team members and finalised the lesson plan. Stage 3: Conduct the research lesson For the kindergarten research lesson, the lesson objectives are:

By the end of the lesson, pupils will be able to: (a) Arrange the cubes in group of 5 to make 3 models (b) Use simple language structures to express their opinions (c) Negotiate and work cooperatively with a partner to make models

It was conducted in a Kindergarten 1 class by their own class teacher, Teacher K. Teacher K started the lesson by making a model with three cubes, she later asked the children to make a model with four cubes. The last round of activity was for the children to make as many different models as they could with five cubes (conservation of numbers). For the primary research lesson, the lesson objectives are:

By the end of the lesson, pupils will be able to: (a) Compare and order unit fractions (b) Articulate their thinking

It was conducted in a Primary 2 class by their own Mathematics teacher, Teacher P. Teacher P started by using a real cake to revise the basic concepts of fraction (such as ½ is 1 out of 2 equal parts). Teacher P then put the children into groups of 3 to do the task of comparing and ordering unit fractions (given in a worksheet) using fraction discs. Each group also had to construct one problem at the end of the worksheet. Thereafter, four groups of children were selected to present and explain their answers before the class (with the help of the fraction discs). Teacher P then posed a challenging question to the whole class to stretch their mathematical thinking, i.e. Is ½ always greater than 1/4?

6

For the secondary research lesson, the lesson objectives are:

By the end of the lesson, pupils will be able to: (a) Explain the meaning and representation of simultaneous equations (b) Form simultaneous equations from word problems (c) Solve simple simultaneous equations

It was conducted in a Secondary 2 class (Normal Academic- a stream for the weaker learners) by their own Mathematics teacher, Teacher S. Teacher S started by using a real-life word problem and asked the students to find the answers to the question (in groups) using their own methods. Teacher S selected different groups to present their methods and answers. He then varied the numbers in the earlier question to increase the difficulty level of the same problem. The students were then led to see the role of a simultaneous equation which is one of the more efficient ways to arrive at the answer. Teacher S then explained how to form and solve simultaneous equations. Stage 4: Conduct a Post-Lesson Discussion After each research lesson, the learning teams engaged in a post-lesson discussion on the same day as the research lesson. The facilitator followed the protocols as follow:

• Recap the research theme • Recap the main chunks of the research lesson • Invite the research teacher to comment on the changes made to the lesson plan (if

any) • Invite the observers to share instances of thinking and identify factors which

promote or hinder mathematical thinking • Invite the knowledgeable other to comment and synthesize the observations and

attributing factors shared by the observers • Facilitator summarizes the tentative professional knowledge with respect to

mathematical thinking Knowledgeable other Kindergarten (K2, 6 years old)

The facilitator also acts as the knowledgeable other

Primary (Pri 2, 8 years old)

Head of Departments (Mathematics)

Secondary (Sec 2, 14 years old)

Former lecturer from National Institute of Education, Singapore

Stage 5: Revise the Lesson After the post-lesson discussion, one of the learning teams went on to revise the lesson plans, incorporating the inputs and suggestions discussed at the post-lesson discussions. The Kindergarten team proceeded to conduct the revised lesson in another class for the second cycle.

7

Data Collection Methods The study uses qualitative data collected during the lesson study stages. The data is collected from field notes such as observers’ reflections and minutes recorded at post-lesson discussions. The reflections and observations are sorted into categories of factors which the teachers deemed to have supported or hindered mathematical thinking. An analysis of the categories of factors from all case studies is made to sieve out recurring factors. Findings Based on the teachers’ reflections and minutes of post-lesson discussions, there are four recurring factors which surfaced from all the case studies. These four factors have been identified to play a significant role in promoting mathematical thinking in the respective research lessons. The first factor identified is the choice of tasks. Teachers recognized that the tasks used are investigative in nature. They are usually open-ended with either more than one answer or allow more than one method of solving it. In the Kindergarten case study, the task of asking pupils to construct different models using five cubes provided opportunities for divergent and creative thinking. In the Secondary case study, teacher S encouraged his pupils to use different methods to solve a real-life problem. Such a task allows pupils to reveal their preferred thinking processes and justify their solutions. Teachers identified the choice of such tasks as excellent platforms to enhance and promote different types of mathematical thinking. The second factor is the use of variations ~ use of one anchor task and vary its difficulty levels in subsequent parts. One common observation is that the first task posed has a relatively easy entry point. Starting with an easy task (consisting smaller numbers or less complex nature) seems to allow most pupils to experience success in solving and hence build confidence for solving subsequent parts of higher difficulty levels. This is particularly important for weaker learners who might need to experience successful moments in order to increase their motivation levels. Making variations to the anchor task also helps weaker pupils to make connections from the earlier part to the more challenging subsequent ones. Furthermore, the latter parts are usually used to ‘stretch’ the high ability learners. Hence, the use of variations in an anchor task serves as a form of differentiation for learners of different ability groups.

The use of manipulatives is the third factor. The teachers learnt that the use of concrete materials is crucial for developing mathematical thinking as the pupils were able to visualize the mathematical concepts embodied in the act of handling those manipulatives. For instance, the use of fractions discs used in the Primary school case study helped pupils to visualize the size of each fraction in relation to the whole. Hence, pupils were able to compare the size of each fraction physically (by putting a piece of the fraction disc on top of another to compare their sizes). Observations were also made that

8

manipulatives seem to help pupils articulate their mathematical thinking together with the use of language and/ or when the latter is limited. The fourth is scaffolding through the use of effective questions. In all case studies, teachers recognized that effective questioning is beneficial in supporting, leading, deepening and making connections in thinking, particularly for weaker learners. Examples of such questions used in the research lessons are “How are they the same?”, “How are they different?”, “What is the size of each fraction?” and “Can you think of a different way?” Such questions posed (by the teacher) during the course of problem-solving process seem to lead pupils to think about a critical element or step of a problem which may eventually lead them to solve the whole problem successfully. Limitations There are limitations to this study in the limited amount of field notes (teachers’ reflection and minutes). There is also an issue of irregular attendance of participants during the lesson study cycle in each case study (sample size). Discussion of Findings In line with the theme of this conference “Innovation on Problem-Based Textbook and e-Textbooks”, it is worthwhile to consider the extent to which the professional knowledge of these four factors has been incorporated in the design of Mathematics textbooks and e-textbooks. In this section, the paper seeks to highlight evidences of these factors in Singapore’s Primary Mathematic textbooks. Choice of Tasks Based on a comparison of some old and new Mathematics textbooks in Singapore, there are changes to the types of tasks from exercises, requiring the use of certain procedures to arrive at the answer to problems which are open-ended (usually with more than one answer or more than one method of solving it). In the current series of various textbooks in Singapore, a few of such open-ended problems are found at the end of each topic to challenge pupils of higher ability groups.

9

Examples of tasks taken from old Mathematics textbooks, Primary Mathematics 2A, Curriculum Development Institute of Singapore (Pg. 48)

10

Examples of tasks taken from newer textbooks, Mathematics in Action 2A, Pg. 58

11

Use of Manipulatives In most Singapore’s Primary Mathematics textbooks, the use of different types of manipulative is a distinctive feature. There is often a deliberate use of photographs of concrete materials in the beginning of a topic, followed by drawings of materials to eventually the use of other forms of pictorial representations such as model drawings. These varied forms of pictorial representations are then accompanied by abstract symbols and notations (introduced at a later stage). Examples on the use of manipulatives taken from My Pals are Here! 1B, Pg. 80

12

Use of Effective Questions Another feature which can be found in Singapore’s Primary Mathematics textbooks is the use of speech bubbles which consist of scaffolding questions. These questions posed at the interim stage of a problem-solving process lead pupils to think about the key element or step of a problem. This is particularly useful for scaffolding weaker pupils to help them arrive at the final answer successfully. Examples of scaffolding questions taken from My Pals are Here! 4B, Pg.60

13

Conclusion & Implications Lesson Study has become increasingly popular in Singapore schools as it provides opportunities for teachers to enhance their professional knowledge through collaborative efforts in designing a lesson plan, observing a real lesson and discussing observations of student learning. Such a collaborative learning platform allows teachers to identify critical factors or stimulus such as the choice of tasks, use of variations, use of manipulatives and use of effective questions to promote mathematical thinking for different ability learners. Such findings can be made more explicit in teacher training to help teachers plan and design their lessons (with a focus on mathematical thinking) more effectively. They are also useful in refining Mathematics textbooks since there is evidence that the four factors have already appeared in the Singapore’s Primary Mathematics textbooks but perhaps more can be done to improve the quantity and quality of open-ended tasks and good questions in textbooks for different levels (including preschool and secondary levels). This professional knowledge gained by teachers through lesson lessons is arguably more real, explicit and ‘accessible’ as it is backed up by actual observations of student learning in the real classroom settings. This calls for further research on the effectiveness of Lesson Study as an alternative form of professional development for teachers versus traditional workshops in training settings.

14

References Appel, G., Leong, M., Mangan, M.T., Mitchell, M. & Stepanek, J. (2007). Leading lesson

study: A practical guide for teachers and facilitators. Thousand Oaks, California:

Corwin Press.

Cowan, P. (2006). Teaching Mathematics: A handbook for primary and secondary school

teachers. New York: Routledge Taylor & Francis Group.

DuFour, R., & Eaker, R. (1998). Professional learning communities at work: Best

practices for enhancing student achievement. Bloomington, IN: Solution Tree

(formerly National Educational Service).

Lewis, C. C. (2002). Lesson study: a handbook of teacher-led instructional change.

Philadelphia, PA: Research for Better Schools, Inc.

Liptak, L. (2005). For principals: critical elements. In Wang_Iverson, P. & Yoshida, M.

(Eds.), Building our understanding of lesson study, 39 – 44. Philadelphia, PA:

Research for Better Schools, Inc.

MOE Press Release, 2005, December 29. Flexible School Design Concepts to Support

Teaching and Learning. Retrieved June 30, 2008, from

http://www.moe.gov.sg/press/2005/pr20051229.htm

Stifler, J.W. & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers

for improving education in the classroom. New York: The Free Press.