Enhancing the pedagogy of mathematics teachers to emphasize understanding, reasoning and communication in their classrooms (EPMT)

ENHANCING THE PEDAGOGY OF MATHEMATICS TEACHERS TO EMPHASIZE UNDERSTANDING, REASONING AND COMMUNICATION IN THEIR CLASSROOMS (EPMT)

Berinderjeet Kaur, Yeap Ban Har and Low Hooi Kiam

Abstract This intervention project was a special focus project of CRPP. This project focused on professional development of primary and secondary schools to enhance the teachers‟ pedagogy of mathematics in the areas of designing of mathematical learning tasks and of lessons that emphasize understanding, reasoning and communication in teachers‟ classrooms. The professional development (PD) involved experts working with teachers for a period of two years, during which the experts involved teachers in crafting mathematical learning tasks and designing mathematics lessons that aim to teach for understanding. Throughout the PD teachers worked with fellow teachers, as part of a community of learners, to advance their knowledge and skills in their classroom practice. The project showcased a model of PD, the blended approach, which teachers certainly attested was a preferred model of teacher learning. The deliverables of the project, in particular the three professional books, have been whole heartedly welcomed by mathematics teachers in Singapore schools.

Introduction

This project was a special focus project (CRP 06/06 BK) of CRPP‟s research program. It was an intervention and the goal of the project was to enhance the pedagogy of mathematics teachers‟ to emphasize understanding, reasoning and communication in their classrooms so as to improve student learning.

Professional Development of Mathematics Teachers

In 1997, Mr Goh Chok Tong the Prime Minister of Singapore in his speech (Goh, 1997) at the opening of the Seventh International Conference on Thinking held in Singapore noted that Singapore has a strong education system, one that is widely recognized for having produced high levels of achievements among pupils of all abilities. However, he also cautioned that what may have worked well in the past will not work well for the future as the old formulae for success are unlikely to prepare the young Singaporeans for the new circumstances and new problems they will face in the new millennium. He emphasized that we must ensure our young can think for themselves, so that the next and future generations can find their own solutions to whatever new problems they may encounter. He announced at the opening of the conference that Singapore‟s vision for meeting this challenge is encapsulated in four words: THINKING SCHOOLS, LEARNING NATION (TSLN).

FINAL RESEARCH REPORT Project No. CRP 6/06 BK

March 2010

E

FINAL RESEARCH REPORT

Page 2

With the unveiling of the TSLN vision, it was realized that teachers are the key to the success of the mission and hence their on-going professional development (PD) is critical. Since 1998 all teachers in Singapore are entitled to 100 hours of training and core-upgrading courses each year to keep abreast with the current knowledge and skills. The PD is funded by the Ministry of Education. Yet another subsequent development that has accorded teachers the responsibility of their own professional development is the Enhanced Performance Management System (EPMS) (Ministry of Education, undated) put in place by the Ministry of Education (MOE) in 2005. The EPMS is an appraisal system that contains rubrics pertaining to fields of excellence in the education system be it teaching, leadership or senior specialist. Over the past couple of years, mathematics teachers have been focused on excellence in their mathematics classrooms.

What Counts as Professional Development of Teachers?

Upon the completion of pre-service education, teachers continue their learning journey through participation in many types of PD activities. For a long while the most common traditional type of PD in Singapore has been in-service courses. These courses are conducted for about 3 hours each day either for about 10 days consecutive days or spread over a number of weeks. They are conducted by experts in the field and are “off-line” forms of knowledge production. After the completion of the course there is no follow up with the teachers about the use of the knowledge acquired and any impact that knowledge may have had on student achievement. Over time the nature and scope of PD has expanded and at present it includes any activity and interaction that may increase the knowledge and skills and improve teaching practice. These experiences can range from formal, structured topic-specific seminars, workshops to school-based activities involving curriculum design, discussions on instruction techniques, day to day collaborative activities that enhance teachers‟ knowledge and skills, co-teaching, peer observation, mentoring, etc.

Characteristics of High-Quality Professional Development

High-quality PD programs are activities that increase teacher knowledge and skills, impacting their instructional practice and ultimately resulting in improved student achievement. Stiff (2002) stated that according to a large scale study conducted by the American Institutes for Research (AIR) in Washington, D.C. it was found that professional development programmes were effective and improved instruction if they had the following six characteristics:

Form – traditional classes or workshops were less effective than reform types of activities, such as teacher networks or study groups

Duration – longer professional development programs are more likely to make an impact. Sustained and intensive programs are better than shorter ones.

Collective participation – activities designed for teachers in the same school, grade, or subject are better than professional development programs that do not target groups of teachers who work together.

Content – PD course that focus on how to teach and what to teach – the substance and the subject matter – are key.

Active learning – teachers learn best when observing, being observed, planning for classroom implementation, reviewing student work, and presenting, leading and writing.

E


Page 3

Coherence – teachers need to perceive PD as part of coherent programs of teacher learning and development that support other activities at their school, such as adoption of new standards and textbooks.

Desimone (2009) stated that recent research does reflect a consensus about at least some of the characteristics of PD that are critical to increasing teacher knowledge and skills and improving their practice, and which hold promise for increasing student achievement. Drawing on pertinent case studies, co-relational analyses conducted with nationally representative teacher data, longitudinal studies of teachers, meta-analysis and experimental designs, Desimone (2009) affirmed that characteristics of high-quality PD programs have the following characteristics: content focus, active learning, coherence, duration and collective participation. Desimone (2009) noted that:

The content focus of teacher learning may be the most influential feature as evidence in the past decade points to the link between activities that focus on subject matter content and how students learn that content with increases in teacher knowledge and skills and improvements in practice, and, to a more limited extent increases in student achievement.

Opportunities for teachers to engage in active learning, i.e. engaging in activities such as observing expert teachers or being observed followed by interactive feedback and discussion, leading discussions, etc., are also related to the effectiveness of professional development.

There must be coherence in PD programmes, i.e teacher learning is consistent with teachers‟ knowledge and beliefs. Importantly, there must be consistency of school, district, and state reforms and policies with what is taught in PD.

As intellectual and pedagogical change takes time to set in, PD programmes must be of sufficient duration. However, research has not indicated an exact “tipping point” for duration but shows support for activities that are spread over a semester and include 20 hours or more of contact time.

Collective participation is critical and participation by teachers from the same school, grade or department allow for powerful form of teacher learning through prolonged interaction and discourse.

Wilson and Berne (1999) suggested three features of effective PD: “communities of learning”, teachers playing an active role, and “critical colleagueship” where trust and critique were present. Although these features do not expand the above list characteristics and belong to active learning and collective participation as delineated by Desimone (2009), of interest is the nature of interactions that contribute towards effective PD. In the field of mathematics education, one of the most influential PD efforts of the last two decades in the US has been the work of the Cognitively Guided Instruction (CGI) Project (Carpenter, Fennema, Franke, Levi & Empson, 1999). In particular two features of the CGI PD effort and associated research initiative are particularly noteworthy. First, encouraging teachers to flexibly apply general principles in designing instruction for their students made it possible for teachers to adapt and use materials readily available in their local settings. Second, the CGI researchers situated PD in the everyday work of the teaching (i.e., adapting curricular materials, analyzing student thinking, conducting interviews). Ball and Cohen (1999), have argued that “teachers‟ everyday work could become a source for constructive PD” (p.6) through the development of a curriculum for professional learning that is grounded in the tasks, questions, and problems of practice.

E


Page 4

How Should We Measure Teachers’ Experiences in PD Activities and Subsequent Changes in Practice?

Often, evaluating PD is a matter of administering a satisfaction survey at the end of the course or workshop and for PD this is a common mode of assessing teachers learning in Singapore. According to the research literature, the three most commonly used and debated methods of data collection for empirical descriptive, correlational, and causal studies of teachers are observation, interviews and surveys / questionnaires (Desimone, 2009). However, like all methods, these three also have their supposed strengths and weaknesses. According to Wragg (1999), observation is often heralded as the most unbiased form of data collection as it allows a clear look into what is actually occurring during a PD activity and, subsequently, in the classroom as the teacher implements new content and strategies. However, observation is time-consuming and an expensive method of collecting data for the evaluation of PD and teaching. Using video observation to assess both classroom instruction and teacher learning experiences has the potential to offer rich data that capture the complexity of interactions (Stigler, Gallimore, & Hiebert, 2000). However, the research literature tells us that there is still much work to be done in exploring the logistics and practicality of using video observation to assist in the understanding of PD and its effects (Desimone, 2009). Wengraf (2004) states that interviews allow for the development of a trusting relationship between the researcher and the subject thereby making it possible to elicit comprehensive and truthful information. In this case the only shortcoming may be interviewer bias. Surveys are lauded for being the only feasible mechanism for collecting data on large samples, but survey research have also received the most criticism in published research (Desimone, 2009). Desimone (2009) has examined in depth the reliability and validity of observation, interviews, and surveys used in relation to the study of PD and teacher learning and claimed that for behaviour-based constructs, when data collection is confidential and not linked to the teacher‟s own evaluation (Mayer, 1999), such as PD activities and behavioural aspects of classroom instruction, well-constructed and administered observation, interviews, and surveys can elicit much the same information. Most importantly, the research questions of a study need to drive the method of data collection. Having collected data through appropriate means teacher change may be classified using the 4I model (Yeap & Ho, 2009). The 4 I model delineates types of teacher change. Table 1 provides a summary of the four types of teacher change under the 4-I model.

E


Page 5

Table 1: Teacher change in an Education reform

Type of Teacher

Change Characteristics

Ignore The teachers ignore the given examples to implement the reform. They may at best, use the given examples, if required to do so.

Imitate The teachers use the given examples to implement the reform. They may, at best, change superficial features of the examples to generate their own.

Integrate The teachers use and adapt the given examples to implement the reform. They are able to change more than the superficial features of the examples to generate their own. In traditional situations, they may revert to actions that are not consistent with reform principles. At best, they recognise this inconsistency when it is made explicit to them.

Internalise The teachers use and adapt the given examples to implement the reform. They are able to apply reform principles even in traditional situations. They do not revert to actions that are not consistent with the reform principles.

The EPMT Project

Rationale

This intervention project, in the area of mathematics classroom pedagogy, addressed two main issues: the nature of mathematical learning tasks that enhance students‟ reasoning and communication in mathematics classrooms, so as to help them develop habits of mind necessary for higher order thinking, and teaching for understanding rather than assessment. The impetus for this project arose from three main sources. The sources were: i) The findings of a project “Student perspective on effective mathematics pedagogy:

stimulated recall approach” (Kaur, 2010) that showed that Singapore teachers were generally bound in their choice of “learning tasks” (tasks used by the teacher during instruction to develop a concept or demonstrate a skill or process) available in the textbook used by the school and that these tasks are not suitable to engage students in reasoning (logical, deductive or inductive) and communication (explaining the process / thinking either during oral presentations or in writing). In addition, the lessons observed as part of the project did not make explicit the need to understand but rather placed emphasis on procedural knowledge, i.e. to remember algorithms and use them correctly to pass tests and examinations (Kaur, Seah & Low, 2005).

ii) The revised framework for mathematics implemented in 2007 by the Ministry of Education (Ministry of Education, 2006a, 2006b) expanded the scope of Processes to include

Mathematical reasoning, communication and connections

Thinking Skills

Heuristics Teachers were familiar with thinking skills and heuristics as both have been apart of the framework for the last decade. As mathematical reasoning, communication and connections were new attributes in the framework implemented in 2007, there was a need to work with teachers in this area.

iii) The criticism raised in the American Institutes for Research (AIR) Study comparing the quality of US elementary school mathematics instruction with that of Singapore‟s, a recognized world leader, about primary school mathematics instruction in Singapore schools lacking emphasis on 21

st century thinking skills, such as reasoning and

communication (Ginsburg, Leinwand, Anstrom & Pollock, 2005). The AIR study was one that compared textbooks used in US elementary schools and Singapore primary schools. It is evident from this study that the nature of mathematical tasks present in Singapore

E


Page 6

school textbooks lack emphasis on reasoning and communication which facilitate higher order thinking skills but rather highlight practice exercises that emphasize procedural knowledge and readiness for examinations.

Theoretical Framework and Design of Project

The design of this intervention professional development (PD) project was guided by research findings of effective PD programmes (Ball & Cohen, 1999; Wilson & Berne, 1999; Carpenter, et al., 1999; Stiff, 2002; Desimone, 2009). The five significant feature of EPMT were: Content focus - the project was specific and in the content area of mathematics Coherence - The project was coherent with the needs of the teachers: i) Revised math curriculum of 2007 placed emphasis on reasoning and communication in

math lessons. Textbook questions were inadequate for the purpose, therefore need to learn how to craft mathematical tasks that facilitate reasoning and communication during math lessons.

ii) teachers rely very heavily on textbooks for their daily work, therefore the need to learn how to use a textbook question as a starting point and craft a task that would engage students in reasoning and communication.

iii) with TLLM in place, more emphasis on teaching for understanding therefore the need to learn about lessons that are engaging and how to plan for such lessons.

Active learning - teachers were engaged in hands on work, they crafted mathematical tasks and planned lessons, worked in pairs to video tape their lessons, critique their lessons, revise their plans, thereby engaging in iterative cycles of planning and implementing. Duration - the duration of the project was 2 years [teachers attended 60 hours of instruction spread over 6 months, these sessions were conducted by “experts” in the field one of which is the author of this paper followed by 6 months of school based work guided and monitored by the researchers of the project, followed by another year of self-directed school based work by teachers in the project]. Collective participation - at least 4 teachers, with 2 teachers teaching the same grade year and math programme, participated from each school, worked together in implementing their learning in classrooms and formed a “learning community” within the school. These teachers worked collectively, building their knowledge, putting it into practice, critiquing their peer‟s work, participating in sessions organized by the “experts” (for the entire duration of the project) during which teachers shared their experiences and difficulties encountered during the implementation of their newly gained knowledge, showed to others video‟s of their students interactions in class and collectively planned for conference presentations and on resources they would like to put together for fellow teachers not in the project in the hope of lighting many more fires across the educational system.

Research Questions

The main research question that guided the EPMT project was “How effective was a blended approach to Professional Development for mathematics teachers in Singapore schools?” In the context of the question “blended” means an integration of expert knowledge into the practice of teachers.

E


Page 7

The sub-questions were: i) Did the teacher participants of the project find the strategies taught to engage students in

reasoning and communication useful? ii) Did the teacher participants of the project find the theoretical framework for lessons that

teach for understanding useful? iii) Were teacher participants of the project able to infuse their learning into their classroom

work with students during and after participation in the project? iv) What were the features of the EPMT project that facilitated teachers to integrate their

learning into their classroom practice? v) Did the teacher participants of the project find the model of professional development as

modelled by the EPMT project an innovation? How was it different from traditional in-service courses they normally attended?

vi) How did the strategies explored during the PD for “reasoning and communication” impact student learning?

vii) What pedagogy were the teachers in the EPMT Project capable of before and after the intervention which was aimed at enhancing teachers‟ pedagogy?

Subjects

Table 2 shows the numbers of schools and teachers who participated in the project from January 2007 till December 2008. A requirement for participation in the project was that a group of at least 4 teachers per school had to participate.

Table 2: Number of schools and teachers in the project

Primary Secondary

Number of schools in the project 5 5 Number of teachers in the project for the 1

st year

[Jan – Dec 2007] 20 28

Number of teachers in the project for the entire duration [Jan 2007 – Dec 2008]

18 22

During the second year of the project 2 teachers from the primary schools and 2 teachers from the secondary schools were on maternity and child care leave. Four teachers from the secondary schools moved schools at the beginning of the second year and hence were unable to continue with the project.

Development of the project - the learning journey of the teacher Jan 2007

Teachers were administered a pre-intervention teacher questionnaire (see Appendix A). Teachers were given resources to video tape a good lesson of theirs and do a reflection of it using a guide provided by the researchers (See Appendix B).

Jan 2007– Mar 2007 Module I (30 hr) – Design of Tasks [Pathways to Reasoning & Communication] See Appendix C and Appendix D for schedule and content of the module respectively. i) Weekly 3 hr workshops conducted by experts ii) Teachers worked in groups and created tasks iii) Teachers shared their tasks with fellow participants in the project via an e-portal iv) Teachers tried some of their tasks and shared their experiences with the rest of the

project participants

E


Page 8

v) Researchers collated teachers work for subsequent use in resource books to be produced by participants for fellow teachers.

Apr 2007 – May 2007 Module II (30 hr) – Teaching for Understanding See Appendix E for the schedule and outline of the module. i) Weekly 3 hr workshops conducted by experts ii) Teachers discussed in groups, ideas presented by experts iii) Teachers worked in groups to make sense of “frameworks” such as Understanding

by Design (Wiggins & McTighe, 2005), Teaching for Understanding (Blythe & Associates, 1998) and examine the feasibility of such frameworks in their classrooms

iv) Teachers tried some of their ideas and shared their experiences with the rest of the project participants

July 2007 – Nov 2007 Teachers were assigned the following tasks to complete in their own time collaboratively with their fellow project participants in the school. Plan, design and teach, i) [A] at least 1 lesson that infuse reasoning tasks and facilitate communication in your

mathematics classrooms over a period of the next 10 - 15 weeks. ii) [B] at least 1 lesson that teach for understanding in your mathematics classrooms

over a period of the next 10 - 15 weeks. iii) Video-tape anyone of the above lessons in A or B. For all lessons, you must submit the lesson plan, samples of student work and your

reflections about the lesson. See Appendix F for guide used to do reflection of the lesson. While teachers were working on their assignments, the researchers facilitated monthly

sharing sessions during which teachers shared their work with the others and invited critique.

Jan 2008 – Nov 2008 Researchers kept in touch with participants through e-mail contact. Teachers were

encouraged to continue working in their schools to put into practice their learning and share their knowledge with others who may be interested in it. Researchers supported teachers who did school-wide sharing, cluster sharing, about the project and their learning. Two whole group meetings were organized for project participants to catch up with each other and also work on project matters.

July 2008 – participants met to review the drafts of the resource books in which their work was showcased. See Appendix G and Appendix H for briefs on resource books.

Sept 2008 – participants met for the last time formally. During this meeting the researchers took them down memory lane and administered their final survey to collect data. See Appendix I and Appendix J for post-intervention teacher questionnaires.

The following data were collected from the participants of the project.

E


Page 9

Type

Instrument / Description of data

Survey data Pre-Intervention Teacher Questionnaire

Post-Intervention Teacher Questionnaire

Journals

Reflective journal of one mathematics lesson prior to participation in the project.

Reflective journal of Pre & Post-Intervention video (compare and contrast analysis) of mathematics lesson

Journals of three lessons after participating in the project

Video data Video record of a lesson prior to participation in the project.

Video record of lesson after participating in the project.

Lesson plans Lesson plans of three lessons after participating in the project.

Data analysis and findings

A total of 33 participants of the project, 16 from primary schools and 17 from secondary schools completed the final survey questionnaire of the project in September 2008. The overall response rate was 82.5%. In this section, the analysis of the data and findings are presented in order of the research questions. Research Question (i)

Did the teacher participants find the strategies taught to engage students in reasoning and communication useful? Table 3 shows feedback about the PD module: Pathways to Reasoning. It is apparent that at least 75% of both the primary and secondary teachers felt that the PD module was useful, it‟s content comprehensive, the knowledge gained from the module helpful in planning their lessons, improving student learning and they would continue to infuse the knowledge and skills that have acquired during the PD in their lessons.

E


Page 10

Table 3: Feedback about the PD module: Pathways to Reasoning

Level of Agreement - % of teachers

P-Primary and S-Secondary N(P) = 16; n(S) = 17

PD Module – Pathways to Reasoning

Strongly Agree

Agree Neutral Disagree Strongly Disagree

I found the PD module: Pathways to reasoning comprising of strategies such as: what number makes sense, what‟s wrong, etc., useful.

P[44%] S[59%]

P[56%] S[35%]

S[6%]

I found the PD module content comprehensive.

P[31%] S[24%]

P[69%] S[76%]

I found the PD module helpful for planning my lessons / activities.

P[31%] S[24%]

P[44%] S[71%]

P[25%] S[5%]

I found the PD module useful for improving student learning.

P[38%] S[35%]

P[38%] S[41%]

P[24%] S[24%]

I would continue to infuse the knowledge and skills that were acquired during the PD

P[31%] S[41%]

P[63%] S[53%]

P[6%] S[6%]

Table 4 shows the perceived usefulness of each of the strategies explored in the PD module: Pathways to Reasoning. From the table it is apparent that teachers found some strategies more useful than others. They did not find any strategy that was not useful at all.

Table 4: Feedback about the usefulness of strategies explored in the PD module: Pathways to Reasoning

Usefulness of each strategy / approach

Level of Agreement - % of teachers P-Primary and S-Secondary

N(P) = 16; n(S) = 17

How useful do you find the following strategies for planning

your lessons / activities?

Very Useful

Useful Moderately Useful

Of Little Use

Not useful At All

Strategy 1: Reasoning by Analogy P[6%] S[18%]

P[63%] S[41%]

P[31%] S[35%]

S[6%]

Strategy 2: Inductive and Deductive Reasoning

P[12%] S[35%]

P[38%] S[35%]

P[50%] S[24%]

S[6%]

Strategy 3: What number makes sense?

P[50%] S[29%]

P[25%] S[65%]

P[25%] S[6%]

Strategy 4: What‟s wrong? P[44%] S[82%]

P[38%] S[18%]

P[18%]

Strategy 5: What would you do? S[24%] P[56%] S[24%]

P[44%] S[52%]

Strategy 6: What questions can you answer?

P[2] S[29%]

P[5] S[24%]

P[9] S[41%]

S[6%]

Strategy 7: What‟s missing? P[19%] S[24%]

P[50%] S[65%]

P[31%] S[11%]

E


Page 11

Strategy 8: What if? P[31%] S[53%]

P[31%] S[47%]

P[38%]

Strategy 9: What‟s redundant? [Applicable to secondary level only]

S[18%] S[35%] S[41%] S[6%]

Strategy 10a: What‟s the question if you know the answer? [context is given]

P[13%] S[3]

P[31%] S[8]

P[50%] S[5]

P[6%] S[6%]

Strategy 10b: What‟s the question? [context is absent]

P[6%] S[6%]

P[19%] S[47%]

P[63%] S[41%]

P[12%] S[6%]

Research Question (ii)

Did the teacher participants of the EPMT project find the theoretical framework for lessons that teach for understanding useful? Table 5 shows feedback about the PD module: Teaching for Understanding. It is apparent that at least 50% of both the primary and secondary teachers felt that the PD module was useful, it‟s content comprehensive, the knowledge gained from the module helpful in planning their lessons, improving student learning and they would continue to infuse the knowledge and skills that have acquired during the PD in their lessons.

Table 5: Feedback about the PD module: Teaching for Understanding

Level of Agreement - % of teachers

P-Primary and S-Secondary N(P) = 16; n(S) = 17

PD Module – Teaching for Understanding Strongl

y Agree

Agree Neutral Disagree Strongly Disagree

I found the PD module: Teaching for Understanding comprising of approaches such as: infusing reasoning tasks and facilitating communication in the mathematics classroom, what is understanding, etc., useful.

P[19%] S[12%]

P[69%] S[65%]

P[12%] S[23%]

I found the PD module content comprehensive.

P[13%] P[50%] S[65%]

P[31%] S[29%]

P[6%] S[6%]

I found the PD module helpful for planning my lessons / activities.

P[18%] P[38%] S[53%]

P[44%] S[47%]

I found the PD module useful for improving student learning.

P[12%] S[12%]

P[44%] S[41%]

P[44%] S[47%]

I would continue infuse the knowledge and skills that are acquired during the PD module course.

S[12%] P[56%] S[65%]

P[44%] S[23%]

Table 6 shows the perceived usefulness of each of the strategies / approaches explored in the PD module: Teaching for Understanding. From the table it is apparent that all the teachers found the strategies / approaches useful.

E


Page 12

Table 6: Feedback about the usefulness of strategies explored in the PD module: Teaching for Understanding

Usefulness of each strategy / approach


N(P) = 16; n(S) = 17

How useful do you find the following strategies / approaches for planning

your lessons / activities?

Very Useful

Useful Moderately Useful

Of Little Use

Not Useful At All

Having clear objectives about teaching for understanding.

P[38%] S[35%]

P[56%] S[59%]

P[6%] S[6%]

Having clear performance indicates as outcomes to demonstrate understanding.

P[25%] S[29%]

P[63%] S[47%]

P[12%] S[24%]

The three step design of lessons that teach for understanding: Step 1: Identify desired results / outcomes Step 2: Determine acceptable evidence Step 3: Plan Learning Experiences & Instruction.

P[25%] S[24%]

P[50%] S[41%]

P[25%] S[35%]

Research Question (iii)

Were teacher participants of the project able to infuse their learning into their classroom work with students during and after participation in the project? Table 7 shows the frequency of teachers using the strategies in their lessons. From the table it is apparent that some strategies were more frequently used than others. Also, some strategies were found more useful in primary school than secondary school and vice-versa.

Table 7: Frequency of teacher using each strategy in their classrooms

Frequency of each strategy used


N(P) = 16; n(S) = 17

How often have you used the following strategies in your

classrooms? Always

Very Often

Sometimes

Rarely Never

Strategy 1: Reasoning by Analogy

P[6%] P[12%] S[47%]

P[46%] S[41%]

P[18%] S[6%]

P[18%] S[6%]


S[6%] P[12%] S[18%]

P[64%] S[64%]

P[18%] S[6%]

P[6%] S[6%]


S[6%] P[38%] S[18%]

P[50%] S[58%]

P[6%] S[12%]

P[6%] S[6%]

Strategy 4: What‟s wrong? P[6%] S[35%]

P[50%] S[53%]

P[38%] S[12%]

P[6%]

Strategy 5: What would you do? P[18%] S[12%]

P[58%] S[47%]

P[12%] S[29%]

P[12%] S[12%]

E


Page 13


P[12%] P[32%] S[64%]

P[50%] S[24%]

P[6%] S[12%]

Strategy 7: What‟s missing? P[18%] S[18%]

P[44%] S[52%]

P[38%] S[24%]

S[6%]

Strategy 8: What if? P[12%] S[18%]

P[18%] S[36%]

P[58%] S[40%]

P[12%] S[6%]

Strategy 9: What‟s redundant? [Applicable to secondary level only]

S[6%] S[82%] S[12%]


P[6%] S[6%]

P[31%] S[41%]

P[57%] S[53%]

P[6%]


S[12%]

P[32%] S[29%]

P[50%] S[35%]

P[18%] S[24%]

Table 8 shows the frequency of teachers using the approaches that helped to teach for understanding. From the table it is apparent that teachers, both in the primary and secondary schools, were grappling clear performance indicators as outcomes to demonstrate understanding.

Table 8: Frequency of teacher using the approaches that helped to teach for understanding

Frequency of each approach used


N(P) = 16; n(S) = 17

How often have you used the following approaches when

planning your lessons / activities?

Always Very Often

Sometimes

Rarely Never


P[18%] S[24%]

P[44%] S[35%]

P[38%] S[41%]

Having clear performance indicators as outcomes to demonstrate understanding.

P[18%] S[6%]

P[18%] S[29%]

P[46%] S[41%]

P[18%] S[18%]

S[6%]

c.The three step design of lessons that teach for understanding: Step 1: Identify desired results / outcomes Step 2: Determine acceptable evidence Step 3: Plan Learning Experiences & Instruction.

P[12%] P[31%] S[24%]

P[18%] S[64%]

P[39%] S[6%]

S[6%]

Research Question (iv)

What were the features of the EPMT project that facilitated teachers to integrate their learning into their classroom practice?

E


Page 14

Specific to this question, the data of the item, What helped you to infuse the knowledge and skills that you acquired during the PD modules conducted by the experts in your lessons? is analysed. This item was part of the post-intervention survey conducted towards the end of the second year of the project. The qualitative responses of the items were analysed using content analysis. The responses to each question were first scanned through for common themes, following which codes were generated and the data coded. Inevitably “a progressive process of sorting and defining and defining and sorting” (Glesne, 1999, p. 135) led to the establishment of the final list of codes for the themes. Table 10, shows examples of the responses to the above question and inferences made.

Table 10: Content analysis of data

Teacher

code Response Inferences

P- 12 The crafting of questions for use in class during the PD sessions

Work during PD directly related to class work Hands on experience

P-15 Team teaching with fellow colleague who was also in the project

School-based peer support

S-2 The handholding sessions and sharing with colleagues and teachers from the other schools have been great.

Support from expert Participation as community of learners

S-12 My understanding of the knowledge and skills explored during the PD modules. A better understanding means more willingness to try these strategies out. Secondly, the positive reinforcement of good lessons conducted using these strategies.

Good grasp of knowledge and skills Positive reinforcement of good lessons

Table 11, shows all the conditions and factors that aided the infusion of knowledge and skills acquired during the PD modules, inferred from the qualitative data of the post-intervention survey. Table 11: Conditions / Factors that aided the infusion of knowledge and skills acquired during the

PD modules

Aspect Conditions / factors

PD Course Great ideas, notes, resources shared by experts Hands on work – crafting of tasks, planning of lessons for immediate use in class

Support On-going support from experts School-based peer support Support from the community of learners

Motivation Encouragement from peers in the project Positive reinforcement of lessons that infused learning from PD modules

Learning of Maths To make the learning of math more meaningful and fun for my students To enhance the reasoning and communication of my students during math lessons.

E


Page 15

Research Question (v)

Did the teacher participants of the project find the model of professional development as modelled by the EPMT project an innovation? How was it different from traditional in-service courses they normally attended? Specific to this question, the data of the item, Tell us how different or similar has it been participating in the project compared to attending a traditional in-service course? Which would you prefer to participate or attend in the future? is analysed. This item was part of the post-intervention survey conducted towards the end of the second year of the project. Tell us how different or similar has it been participating in the project compared to attending a traditional in-service course? Which would you prefer to participate or attend in the future? Table 12, shows examples of the responses to the above question and inferences made.


Teacher


P-6 Participating in the project as a participant has its merits. Being with colleagues, we were able to work and learn together. It was also good to attend the 2 PD courses with colleagues as we got to encourage on one another and share our experiences. We were also able to see how the new skills could be applied to our pupils at large. The two year-long project was a journey of reflection and improving on my teaching methodologies. Attending the PD courses during term time was apt as I was able to apply the strategies at school and to see how the pupils respond to the strategies. I was glad to be able to ‘stretch’ them more and probe further into their understanding. The process on reflection was beneficial for my personal growth as I reflected upon the good and „bad‟ instances of the lessons. Attending an in-service course is another viable way of improving content/pedagogical knowledge. However, they tend to be during the holidays or a one-off incident (as in 24 hours). Usually, they will be a mini assignment to „show‟ our understanding. Hence, the impact may not be that significant. I would prefer to be part of a project where there is a system of reflection, PD courses and the opportunity to work with colleagues (which is like a small community). I believe the impact will be greater as through the project, the community of practice (formed by the teachers in the project) can go on to influence and excite the others.

More useful Merits Collaborative work – learning community Put into practice almost immediately the learning and evaluate outcomes – impact on student learning Engaged in reflection Able to contribute towards the development of other teachers

S-12 Participating in the project has been more fruitful as the outcomes are more specific. We are able to work on assignments together as a group and then try out the tasks in our classes. Sometimes we are excited by

More useful Merits Outcomes are specific Collaborative work

E


Page 16

ideas when attending a course but are unable to bring these ideas to xxx due to time or manpower constraints in school. Another difference is that Prof. Kaur is able to guide us throughout the project whereas a trainer in a course may not be able to provide any other support after the workshop.

Implement learning Sustained support from expert

Table 13: Usefulness of the project

Response No of Responses (%)

Primary n=16

Secondary n=17

Total n=33

More useful 12 (75%) 16 (94%) 28 (85%) No preference 1(6%) 1 (6%) 2 (6%) Not useful - - - No response 3 (19%) - 3 (9%)

Table 13 shows that 85% of the teachers who participated in the project found it more useful, compared to traditional in-service courses. Table 14 shows the differences between the project and traditional in-service courses, as perceived by the teachers in the project.

Table 14: Differences between the project and traditional in-service courses

Dimension Project Traditional In-service Courses

Content Knowledge

Co- construction of knowledge by participants and expert.

Mainly dissemination of knowledge by an expert.

Duration Substantive Very short

Participation Collaborative – community of learners A few teachers from a school amongst participants – intra school peer support Like minded teachers from several schools

Mostly individual participation and may not know fellow participants

Learning Active More transfer of learning Reflection – a must Learning from each other during scheduled sharing sessions Critiquing work of peers

Mostly passive Less transfer of learning

Implementation Participants are required to implement their learning almost immediately

Not much scope for implementation

Evaluation Participants able to evaluate their learning; check for affirmation from students and fellow colleagues

Seldom there is an opportunity for participants to evaluate their learning.

End product Resource package – contributions from all teachers in the project Continue work beyond project and contribute towards the development of other teachers

Assignment, usually done individually – seldom have access to the other participants assignments.

Coherence Specific, relevant and support the needs of teachers

May not directly support the needs of teachers

Support from Expert

On-going One time

E


Page 17

Research Question (vi)

How did the strategies explored during the PD for “reasoning and communication” impact student learning? Specific to this question, the data and analysis of the item: In what ways did the strategies explored during the PD for “reasoning and communication” help to improve student learning in your classrooms? is analysed. This item was part of the final survey conducted for the participants towards the second year of the project. In what ways did the strategies explored during the PD for “reasoning and communication” help to improve student learning in your classrooms? Table 15, shows examples of the responses to the above question and inferences made.


Teacher


P-2 Pupils were more engaged and they found the Math lesson interesting.

Lesson – interesting Pupils – engaged

P-14 Students could verbalise using the correct mathematical language (at most times). Students became more critical of their answers.

Students – verbalise thoughts; critical of their answers

S-8 It took their focus away from memorizing formula, and to how formula are derived. The process makes them verbalized their thinking and increase retention of knowledge.

Lesson – shift of focus from rote learning to conceptual understanding Students – verbalise their thinking

S-11 The students were more engaged. Students – engaged

Table 16, shows the changes in the behaviours of students and likely causes

Table 16: Change in the behaviours and likely cause

Primary pupils Secondary students

Change Due to Change Due to

More alert, Motivated, Engaged

Lessons were fun and interesting because of activities designed by teachers

Motivated Engaged

Lessons had activities that were varied, fun, and thought provoking

Explored their understanding of mathematical concepts Explored alternative approaches to solve a task Verbalised their thoughts, often using mathematical language, and

The nature of tasks they were given to do

Questioned their understanding of concepts Shift of focus from rote learning to how the formula/generalisation came about Verbalised their thoughts , clarified their understanding and increased retention of

The nature of tasks they were given to do

E


Page 18

clarified their understanding

knowledge Increased use of logical thinking, analytical thinking Confidence of weak students improved

Critical of their answers More aware of likely mistakes Were more reflective

Emphasis on nurturing good habits of mind

Self assessed their learning Were more reflective

Emphasis on nurturing good habits of mind

Research Question (vii)

What pedagogy were the teachers in the EPMT Project capable of before and after the intervention which was aimed at enhancing teachers‟ pedagogy? Table 17 shows the number of teachers who submitted their lesson videos.

Table 17: Number of Teachers who Submitted Lesson Videos

Primary School

Number of Teachers in

Pre-Intervention

Lesson Video


Post- Intervention

Lesson Video

Secondary School


Pre- Intervention

Lesson Video


Post- Intervention

Lesson Video

A 1 2 P 3 6 B 1 4 Q 2 3 C 3 4 R 1 1 D 3 4 S 4 0 E 3 3 T 10 10

A total of 11 primary teachers and 16 secondary teachers submitted complete sets of pre-intervention and post-intervention lesson videos. There were a total of 17 primary-level teachers and 20 secondary teachers who submitted post-intervention lesson videos. The general research question explored using the video data was:

What pedagogy were the teachers in the EPMT Project capable of before and after the intervention which was aimed at enhancing teachers‟ pedagogy?

The specific research questions were:

Were there evidence that teachers participating in the EPMT project were using pedagogy that encourages reasoning and communication?

Were there evidence that teachers participating in the EPMT project were using pedagogy that encourages understanding beyond procedural understanding?

What were the effects of introducing strategies and frameworks to the teachers on their pedagogical practices?

How the data was analysed Each lesson video was transcribed. Verbatim transcription was not done for two reasons. Firstly, the sound quality of the video was not always good and verbatim transcription became impossible although it was still possible to describe the event that took place and paraphrased the dialogue. Secondly, often the dialogue was difficult to comprehend without understanding the context of the conversation. For each video lesson, a transcription which

E


Page 19

consists of a description of classroom activities supported by essential dialogue was produced as primary data. See the next section for an example of the primary data. The data analysis consisted of two main stages. In the first stage, a lesson video was cut up into segments. In the second stage, the segments were viewed if there was (a) teaching for reasoning and communication, (b) teaching for understanding and (c) engaging pedagogy. Defining a Segment A segment is defined as the duration of a lesson when there is a consistent classroom activity. The following lesson video transcript illustrates the meaning of a lesson segment. It is a transcript for an entire lesson which was 60 minutes in duration.

Table 18: Primary Video Data for Teacher E2

Segment Number & Duration

Segment Classroom Activity

Segment 11 2.5 min

In this segment, students were asked to individually show their understanding of area and perimeter on the diagrams of a rectangle.

The teacher asked the students who were seated in groups of four to show on diagrams of a rectangle the meaning of area and perimeter. She asked, “What is meant by area and what is meant by perimeter? What‟s the definition? What do you understand by area and what do you understand by perimeter?”

Segment 12 1.5 min

A student, Jamie, was asked to share her response on the board.

Jamie‟s response was to show that area is the product of the length of two adjacent sides (“B × C”) and perimeter is the sum of the lengths of all the sides (“A + B + C + D”).

Segment 13 1 min

The teacher reviewed the meaning of area and perimeter of a figure. Students‟ responses were chorus responses.

Referring to the student‟s response, the teacher asked if area can be found by the product of the other pair of adjacent sides, “Can I have A and D? (Yes) Can I do so with A and C? (No). She then went on to say that some students shaded the diagram to show area. She emphasized this by shading the area within the sides of the rectangle. The teacher then defined area, “Area means the number of square units it takes to cover a flat surface.” Similarly, she defined perimeter, “Perimeter is actually distance around any shape or figure given to you…the total distance…”

Segment 21 5 min

The teacher presented a task and conducted a whole-class discussion on the possibility of completing the task. Eight students gave or were asked to give their responses. Students were able to detect the fact

The teacher presented a task that has missing information. The task included this diagram. The width of the rectangle was given as 6 cm. The task was to find the shaded area.

E


Page 20

that there was missing information. One student argued that more than one piece of information was missing. Some students suggested that the area of the circle had to be calculated but the teacher was quick to exclude that because this is a skill to be taught two years later.

The teacher asked, “What is the question asking you to do?” Rena said that it was to find the area of the shaded part. The teacher asked further, “What information do you know about this question…?” Two students responded. The first mentioned that the breadth of the rectangle was given as 6 cm. The second said that a circle is cut from the rectangle. The teacher asked if it was possible to complete the task, “Can you solve this problem?” (No) and the reason why the students thought it was not possible. “Why can‟t you solve the problem?”, she asked. Ariel explained that the length of the rectangle was not given and “without the length you can‟t find the area of the figure”. Anisha said that “They must give you the area of the circle.” And the teacher wanted to know why. Kimberly was asked to explain this and she said that it was because “the whole area minus the figure in the centre” gives the shaded area. Another student argued that even if the length of the rectangle was given “you still can‟t find the (shaded) area”. She wanted to find the area of the circle. The teacher said that, “You come across it (finding area of circle) in, if I am not wrong, P6.” Rachel said that it was necessary to know the “length” of the circle referring to the diameter of the circle. Again the teacher excluded the need to calculate the area of the circle by saying that “For today‟s purpose … we will not do that. We will do what you have been taught.” She said that the class may “explore it again when we have time (in) term four”.

Segment 22 8 min

Teacher gave instruction for students to complete the task in pairs. The task was essentially identifying missing information and solving the given problem based on values students selected for the missing information. Students completed the task in pairs while the teacher reminded them to

Students were given a worksheet with the task and a list of questions. What is the question? What information do you know from the problem? What else do you need to know to solve the problem? Pick a number that represents the length of the rectangular photograph. Pick a number that represents the area of the circle. What is the area of the shaded part? There was an expectation that students would be asked to present their responses later. They teacher asked them to work with their “twelve-o‟clock partner”. The teacher said, “There should be discussion. This is not individual work.” It was evident that students were busy working on the tasks. In obtaining the answers to the given questions, students had to make logical

E


Page 21

discuss. arguments such as the length of the rectangle could be 14 cm because of the proportion of its length and width as depicted by the diagram.

Segment 23 8 min

The teacher randomly selected a student to present the pair‟s work, questioned her on the choice of numbers and asked the other students to give their views on the choice.

Using a classroom device, a student, Meena, was randomly selected. The teacher asked, “What numbers did you use? Why?” Meena said that “for the length I used 12 cm because it is a small number.” When asked for the reason for choosing a small number, she said that “It is easier to multiply and minus.” Meena went on to describe the steps used. The pair chose 9 square cm as the area of the circle. The teacher asked the other students, “Do you agree with the numbers that she used?” One student said that she “could have used a larger number for the area which is not shaded, the circle.” She disagreed with the choice of 9 square units as the area of the circle because “the circle is so big”. A second girl gave a similar view that 9 was “not realistic”.

The teacher randomly selected another student to present.

Rajini was selected. This pair also chose 12 cm as the length of the rectangle. When asked why 12 cm was a common choice, she said that the length “seems double the breadth”. For the area of the circle, they chose 10 square cm which the class felt was not realistic. The teacher then requested a different response for the area of the circle.

The teacher randomly selected a third student to present.

Linyue was selected and she came to the front with her “12-o‟clock partner”. They had selected 14 cm because Linyue “measured it”. The ratio of the lengths of the sides is actually 3 : 7 They had selected 24 cm

2 as the area of the circle

because the rectangle‟s area is 84 cm2 and 24 “is smaller

than” 84. Three students responded to this pair. One student commented on the way the unit is read (cm squared rather than square cm). Meena thought the value for the circle was “more realistic” without elaborating. Another followed up on Liyue‟s method of using proportion to determine the length of the rectangle.

Segment 24 4 min

The teacher reviewed the activity.

Did anyone put the length as 6? Did anyone here put the length as less than 6? The teacher asked for the reasons why not. A girl said that the length should be longer than the breadth and another mentioned properties of rectangle.

E


Page 22

The teacher reviewed that some of the values proposed were reasonable but others were not. She also said that the area of the shaded area obtained by different groups were different because the values chosen were different. A girl raised a point that two of the groups presented earlier had the same value for the shaded area even though they had chosen different values for the missing information. The teacher extended the discussion to get students to think of the reason for that observation.

Segment 31 4 min

The teacher gave instructions for students to pose their own What‟s Missing? Task and allowed students to clarify the task requirements.

Segment 32 19 min

Students worked collaboratively to complete the required task.

Students were actively involved in the tasks. There was evidence of experimentation (e.g. a girls drew on the mini whiteboard a rectangular within a circle). They argued and attempted to defend their views. They invented a name for their figure (“cirtangular – how do you spell that?” for a figure that includes both a circle and a rectangle). Discussion was often animated.

Segment 33 4 min

Teacher concluded the lesson.

She briefly shared two interesting examples and used the rest of the time to collect students‟ work. She invited them to try to solve their friends‟ tasks.

Within this 60-minute lesson, there were ten lesson segments. As previously defined, a segment is defined as the duration of a lesson when there is a consistent classroom activity. For example, this part of the lesson is considered as a segment:

Teacher E8 Segment 11: In this segment, students were asked to individually show their understanding of area and perimeter on the diagrams of a rectangle.

There was one consistent classroom activity – showing one‟s understanding of area and perimeter of a rectangle. The activities in this segment were different from those in the next segment where one of the students was presenting her solution:

Teacher E8 Segment 12: A student, Jamie, was asked to share her response on the board.

Both segments were based on the same tasks, hence, the first digit in the code are the same. To identify segments that are based on the same task, a coding nomenclature is used. The first digit identifies the task which the segments were based on. Thus, segments 11, 12 and 13 are all based on the first task while segments 21 to 24 were all based on the second task in this lesson.

E


Page 23

Coding the Segments The segmented primary data were then analysed to identify segments where there was (a) teaching for reasoning and communication, (b) teaching for understanding and (c) engaging pedagogy. The following sections explain the coding criteria. In the Singapore mathematics curriculum documents (Ministry of Education, 2006a, 2006b) mathematical reasoning and communication are defined as such: Reasoning refers to “the ability to analyse mathematical situations and construct logical arguments” (p. 5, Ministry of Education, 2006a, 2006b). Communication refers to “the ability to use mathematical language to express mathematical ideas and arguments precisely, concisely and logically” p. 5, Ministry of Education, 2006a, 2006b). When a teacher attempts to get students to construct logical arguments, we say that the teacher teaching for reasoning. When he attempts to get students to express mathematical ideas, we say that the teacher is teaching for communication. This is the criteria used to determine if teaching for reasoning and communication is present in a lesson segment. The following section shows the application of these analysis frameworks on the ten lesson segments of one complete lesson previously given in the Table 18. To determine if teaching for reasoning characterizes a lesson, each segment was coded as having reasoning or otherwise where reasoning is defined as “constructing logical arguments”. The segment that contains this

Teacher E2 Segment 21: Ariel explained that the length of the rectangle was not given and “without the length you can‟t find the area of the figure”. Anisha said that “They must give you the area of the circle.” And the teacher wanted to know why. Kimberly was asked to explain this and she said that it was because “the whole area minus the figure in the centre” gives the shaded area. Another student argued that even if the length of the rectangle was given “you still can‟t find the (shaded) area”. She wanted to find the area of the circle.

involved students constructing logical arguments and was coded as having reasoning. Table 20 shows the lesson segments that were coded as having reasoning. To determine if teaching for reasoning characterizes a lesson, each segment was coded as having communication or otherwise where communication is defined as “expressing mathematical ideas orally, in written form or other forms”. Table 21 shows the lesson segments that were coded as having communication. A summary table such as Table 19 shows the segments in each lesson video which were coded as having reasoning and communication.

E


Page 24

Table 19: Lesson segments where there was evidence of students engaging in reasoning and communication

Lesson Segment Teaching for Reasoning Teaching for Communication

E2-11

E2-12

E2-13 E2-21

E2-22

E2-23

E2-24

E2-31 E2-32

E2-33

Table 20: Segments where students were reasoning with the evidence for students constructing

logical arguments are in bold



Segment 21 5 min

The teacher presented a task and conducted a whole-class discussion on the possibility of completing the task. Eight students gave or were asked to give their responses. Students were able to detect the fact that there was missing information. One student argued that more than one piece of information was missing. Some students suggested that the area of the circle had to be calculated but the teacher was quick to exclude that because this is a skill to be taught two years later.

The teacher presented a task that has missing information. The task included this diagram. The width of the rectangle was given as 6 cm. The task was to find the shaded area. The teacher asked, “What is the question asking you to do?” Rena said that it was to find the area of the shaded part. The teacher asked further, “What information do you know about this question…?” Two students responded. The first mentioned that the breadth of the rectangle was given as 6 cm. The second said that a circle is cut from the rectangle. The teacher asked if it was possible to complete the task, “Can you solve this problem?” (No) and the reason why the students thought it was not possible. “Why can‟t you solve the problem?”, she asked. Ariel explained that the length of the rectangle was not given and “without the length you can’t find the area of the figure”. Anisha said that “They must give you the area of the circle.” And the teacher wanted to know why. Kimberly was asked to explain this and she said that it was because “the whole area minus the figure in the centre” gives the shaded area. Another student argued that even if the length of the rectangle was given “you still can’t find the (shaded)

E


Page 25

area”. She wanted to find the area of the circle. The teacher said that, “You come across it (finding area of circle) in, if I am not wrong, P6.” Rachel said that it was necessary to know the “length” of the circle referring to the diameter of the circle. Again the teacher excluded the need to calculate the area of the circle by saying that “For today‟s purpose … we will not do that. We will do what you have been taught.” She said that the class may “explore it again when we have time (in) term four”.

Segment 22 8 min

Teacher gave instruction for students to complete the task in pairs. The task was essentially identifying missing information and solving the given problem based on values students selected for the missing information. Students completed the task in pairs while the teacher reminded them to discuss.

Students were given a worksheet with the task and a list of questions. What is the question? What information do you know from the problem? What else do you need to know to solve the problem? Pick a number that represents the length of the rectangular photograph. Pick a number that represents the area of the circle. What is the area of the shaded part? There was an expectation that students would be asked to present their responses later. They teacher asked them to work with their “twelve-o‟clock partner”. The teacher said, “There should be discussion. This is not individual work.” It was evident that students were busy working on the tasks. In obtaining the answers to the given questions, students had to make logical arguments such as the length of the rectangle could be 14 cm because of the proportion of its length and width as depicted by the diagram.

Segment 23 8 min

The teacher randomly selected a student to present the pair‟s work, questioned her on the choice of numbers and asked the other students to give their views on the choice.

Using a classroom device, a student, Meena, was randomly selected. The teacher asked, “What numbers did you use? Why?” Meena said that “for the length I used 12 cm because it is a small number.” When asked for the reason for choosing a small number, she said that “It is easier to multiply and minus.” Meena went on to describe the steps used. The pair chose 9 square cm as the area of the circle. The teacher asked the other students, “Do you agree with the numbers that she used?” One student said that she “could have used a larger number for the area which is not shaded, the circle.” She disagreed with the choice of 9 square units as the area of the circle

E


Page 26

because “the circle is so big”. A second girl gave a similar view that 9 was “not realistic”.






because the rectangle’s area is 84 cm2 and 24 “is

smaller than” 84. Three students responded to this pair. One student commented on the way the unit is read (cm squared rather than square cm). Meena thought the value for the circle was “more realistic” without elaborating. Another followed up on Liyue‟s method of using proportion to determine the length of the rectangle.

Segment 24 4 min


Did anyone put the length as 6? Did anyone here put the length as less than 6? The teacher asked for the reasons why not. A girl said that the length should be longer than the breadth and another mentioned properties of rectangle. The teacher reviewed that some of the values proposed were reasonable but others were not. She also said that the area of the shaded area obtained by different groups were different because the values chosen were different. A girl raised a point that two of the groups presented earlier had the same value for the shaded area even though they had chosen different values for the missing information. The teacher extended the discussion to get students to think of the reason for that observation.

Segment 32

Students worked collaboratively to

Students were actively involved in the tasks. There was evidence of experimentation (e.g. a girls drew on the mini

E


Page 27

19 min complete the required task.

whiteboard a rectangular within a circle). They argued and attempted to defend their views. They invented a name for their figure (“cirtangular – how do you spell that?” for a figure that includes both a circle and a rectangle). Discussion was often animated.

Table 21: Segments where students were communicating with the evidence for students

expressing ideas in some forms in bold



Segment 11 2.5 min

In this segment, students were asked to individually show their understanding of area and perimeter on the diagrams of a rectangle.

The teacher asked the students who were seated in groups of four to show on diagrams of a rectangle the meaning of area and perimeter. She asked, “What is meant by area and what is meant by perimeter? What‟s the definition? What do you understand by area and what do you understand by perimeter?”

Segment 12 1.5 min

A student, Jamie, was asked to share her response on the board.

Jamie‟s response was to show that area is the product of the length of two adjacent sides (“B × C”) and perimeter is the sum of the lengths of all the sides (“A + B + C + D”).

Segment 21 5 min

The teacher presented a task and conducted a whole-class discussion on the possibility of completing the task. Eight students gave or were asked to give their responses. Students were able to detect the fact that there was missing information. One student argued that more than one piece of information was missing. Some students suggested that the area of the circle had to be calculated but the teacher was quick to exclude that because this is a skill to be taught two years later.

The teacher presented a task that has missing information. The task included this diagram. The width of the rectangle was given as 6 cm. The task was to find the shaded area. The teacher asked, “What is the question asking you to do?” Rena said that it was to find the area of the shaded part. The teacher asked further, “What information do you know about this question…?” Two students responded. The first mentioned that the breadth of the rectangle was given as 6 cm. The second said that a circle is cut from the rectangle. The teacher asked if it was possible to complete the task, “Can you solve this problem?” (No) and the reason why the students thought it was not possible. “Why can‟t you solve the problem?”, she asked. Ariel explained that the length of the rectangle was not given and “without the length you can‟t find the area of the figure”. Anisha said that “They must give you the area of the circle.” And the teacher wanted to know why. Kimberly was asked to explain this and she said that it

E


Page 28

was because “the whole area minus the figure in the centre” gives the shaded area. Another student argued that even if the length of the rectangle was given “you still can‟t find the (shaded) area”. She wanted to find the area of the circle. The teacher said that, “You come across it (finding area of circle) in, if I am not wrong, P6.” Rachel said that it was necessary to know the “length” of the circle referring to the diameter of the circle. Again the teacher excluded the need to calculate the area of the circle by saying that “For today‟s purpose … we will not do that. We will do what you have been taught.” She said that the class may “explore it again when we have time (in) term four”.

Segment 22 8 min

Teacher gave instruction for students to complete the task in pairs. The task was essentially identifying missing information and solving the given problem based on values students selected for the missing information. Students completed the task in pairs while the teacher reminded them to discuss.

Students were given a worksheet with the task and a list of questions. What is the question? What information do you know from the problem? What else do you need to know to solve the problem? Pick a number that represents the length of the rectangular photograph. Pick a number that represents the area of the circle. What is the area of the shaded part? There was an expectation that students would be asked to present their responses later. They teacher asked them to work with their “twelve-o‟clock partner”. The teacher said, “There should be discussion. This is not individual work.” It was evident that students were busy working on the tasks. In obtaining the answers to the given questions, students had to make logical arguments such as the length of the rectangle could be 14 cm because of the proportion of its length and width as depicted by the diagram.

Segment 23 8 min

The teacher randomly selected a student to present the pair’s work, questioned her on the choice of numbers and asked the other students to give their views on the choice.

Using a classroom device, a student, Meena, was randomly selected. The teacher asked, “What numbers did you use? Why?” Meena said that “for the length I used 12 cm because it is a small number.” When asked for the reason for choosing a small number, she said that “It is easier to multiply and minus.” Meena went on to describe the steps used. The pair chose 9 square cm as the area of the circle. The teacher asked the other students, “Do you agree with the numbers that she used?” One student said that she “could have used a larger number for the area which

E


Page 29

is not shaded, the circle.” She disagreed with the choice of 9 square units as the area of the circle because “the circle is so big”. A second girl gave a similar view that 9 was “not realistic”.






because the rectangle‟s area is 84 cm2 and 24 “is

smaller than” 84. Three students responded to this pair. One student commented on the way the unit is read (cm squared rather than square cm). Meena thought the value for the circle was “more realistic” without elaborating. Another followed up on Liyue‟s method of using proportion to determine the length of the rectangle.

Segment 24 4 min


Did anyone put the length as 6? Did anyone here put the length as less than 6? The teacher asked for the reasons why not. A girl said that the length should be longer than the breadth and another mentioned properties of rectangle. The teacher reviewed that some of the values proposed were reasonable but others were not. She also said that the area of the shaded area obtained by different groups were different because the values chosen were different. A girl raised a point that two of the groups presented earlier had the same value for the shaded area even though they had chosen different values for the missing information. The teacher extended the discussion to get students to think of the reason for that observation.

Segment 32 Students worked Students were actively involved in the tasks. There was

E


Page 30

19 min

collaboratively to complete the required task.

evidence of experimentation (e.g. a girls drew on the mini whiteboard a rectangular within a circle). They argued and attempted to defend their views. They invented a name for their figure (“cirtangular – how do you spell that?” for a figure that includes both a circle and a rectangle). Discussion was often animated.

Rating a Lesson

Based on these analysis frameworks, each lesson was rated as having little, some or significant teaching for reasoning and communication, teaching for understanding and engaging pedagogy. In the case of Teacher E2, it was evident that students were communicating in all the three lesson segments and were reasoning in two of the three segments. A lesson is rated as having little teaching for reasoning or communicating if less than 10% of the lesson time was devoted to students reasoning or communicating. It is rated as having some teaching for reasoning or communicating if more than 10% or less than half the time was devoted for reasoning or communication. It is rated as having significant teaching for reasoning or communication if more than half the lesson time was devoted to these processes. In the case of this lesson by Teacher E2, the lesson had significant opportunities for students to engage in both reasoning and communication.

Table 22: Lesson segments where students were reasoning and communicating

Lesson Segment Time in minutes Teaching for Reasoning Teaching for

Communication

E2-11 2.5

E2-12 1.5

E2-13 1 E2-21 5

E2-22 8

E2-23 8

E2-24 4

E2-31 4 E2-32 19

E2-33 4

Findings and Discussion In this section, we discuss the findings of the seven sub research questions that guided the project. From the data presented in the last section we make the following inferences. Firstly the teacher participants found the strategies taught to engage students in reasoning and communication helpful. They also found, generally, ideas and the framework about teaching for understanding helpful in planning their lessons. They were able to infuse their knowledge and skills acquired during the two PD modules in their classroom practice. The features of the EPMT project that facilitated teachers to integrate their learning into their classrooms were the content and nature of the PD modules, support for teachers, motivation resulting from positive experiences and a desire to make the learning of mathematics both meaningful and fun. 85% of the teacher participants found the EPMT project more useful than traditional in-service courses that normally attended. They also found it distinctively different from the traditional in-service courses they usually attended. The differences highlighted by the teachers addressed

E


Page 31

i) The coherence and duration of the project – address their needs and provide for adequate time to work through their newly acquired knowledge

ii) the manner in which the content knowledge was dealt – teachers preferred to be co-constructors of knowledge versus passive receivers;

iii) the nature of participation – preference was for collaborative versus individual; iv) the mode of learning – they wanted to be actively involved, i.e. discussing, doing,

sharing, critiquing, reflecting; v) the scope of implementation of their learning – they wanted to experiment with their

learning in the classrooms almost immediately; vi) the impact of their learning on student outcomes – they wanted affirmation from their

students, fellow teachers; vii) support from experts in the field – preferred sustained support versus one-off kind of

encounter viii) contribution towards the learning of fellow teachers at large – teachers wanted to be

empowered to contribute towards the development of teachers after their participation in the project.

In view of the differences that the project participants highlighted, it may be claimed that the teacher participants found the EPMT project an innovation. It was certainly different from the traditional in-service course they normally attended and one that addressed their needs much better. The EPMT adopted a blended approach to PD is Singapore schools thereby integrating expert knowledge into the practice of teachers. It is also apparent that students found lessons engaging when their teachers infused their learning about how to enhance students reasoning and communication in mathematics classrooms. They found the lessons interesting, as the activities were more engaging, required them to “think about what they were doing”, talk about it with fellow classmates and present their work to the class. In so doing, they verbalised their thoughts, clarified their thinking and used mathematical language. Pupils in primary schools had the opportunity to explore various approaches to solve a task, in particular word problems. Students in secondary schools shifted their focus from “memorising the formulae” to how the formulae came about, were engaging in logical thinking and analytical thinking more frequently. Teachers also noted that the confidence of weak students improved when they worked in groups on tasks that demanded more “reasoning” than procedural work. Most importantly, there were signs that good habits of mind specific to the learning of mathematics were being nurtured, such as being reflective, aware of possible mistakes, being critical and regulation of learning through self assessment. From the video data it was found that nearly all the post-intervention lesson videos had some emphasis on reasoning and communication. More than half had significant emphasis on these processes. It was more likely to find a post-intervention lesson that had significant emphasis on reasoning and communication in a primary-level lesson video than in a secondary-level lesson video

Conclusions Based on the findings from the seven sub questions of the project it may be concluded that the EPMT project was an innovative professional development project for engaged learning. It was significantly different from traditional in-service courses that teachers normally attended. The project was a novel attempt to integrate expert knowledge into the classroom practice of teachers. Hence it is befitting to say that it blended two aspects of professional development of mathematics teachers in Singapore, i.e. expert knowledge and classroom practice. In addition the project successfully fulfilled the three aims it set out to, viz-a-viz,

E


Page 32

engage mathematics teachers in professional development to improve their classroom pedagogy, create teacher practitioner learning communities at the school level and enthuse and support teachers to put together their work in print form and showcase it to other fellow teachers. A significant milestone of the project has been the production of three publications, Kaur and Yeap (2009b, 2009c) and Yeap and Kaur (2010), by which teachers have successfully contributed towards the development of fellow teachers in Singapore. Some indicators that the project achieved it’s goals Deliverables of the Project

3 workshops by PI and teachers of the project for the last two CRPP conferences in 2007 and 2009

Production of two resource books for teachers by teachers (See Appendix G and Appendix H)

Production of the book: Pedagogy for engaged mathematics learning by the PI and Co-PI of the project (See Appendix K). This book is based on the data collected from the video records submitted by the teachers.

Sharing of the project and distribution of the books to all schools in Singapore through the zonal math HOD meetings this year (2009).

Teachers from one project school (Marymount Convent School) have been successful in obtaining the TLLM IGNITE 2010 Award to implement their work on a larger scale in their school

Teachers from one project school (Orchid Park Secondary School), carrying out an action research project based on EPMT in their school.

References

Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practioners: Towards a practice-based theory of professional education. In L. Darling-Hammond & G. Sykes (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 3–32). San Francisco: Jossey-Bass.

Blythe, T., & Asociates. (1998). The teaching for understanding guide. USA: Jossey-Bass. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S.B. (1999). Children’s

mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann. Desimone, L. M. (2009). Improving impact studies on teachers‟ professional development:

Toward better conceptualisations and measures. Educational Researcher, 38(3), 181–199.

Ginsburg, A., Leinwand, S., Anstrom, T., & Pollock, E. (2005). What the United States can learn from Singapore’s world-class mathematics system and what Singapore can learn from the United States: An exploratory study. Washington, DC: American Institutes for Research.

Goh,C. T. (1997). Shaping our future: “Thinking Schools” and a “Learning Nation”. Speeches, 21(3), 12–20. Singapore: Ministry of Information and the Arts.

Kaur, B., Seah, L. H., & Low, H. K. (2005, June). A window to a mathematics classroom in Singapore–Some preliminary findings. In Proceedings of the Redesigning pedagogy: Research, policy International Conference. Singapore: Centre for Research in Pedagogy and Practice, National Institute of Education. http: //conference.nie.edu.sg/crppp

Kaur, B. (2010). A study of mathematical tasks from three classrooms in Singapore schools. In Y. Shimizu, B. Kaur, & D. Clarke (Eds.), Mathematical Tasks in Classrooms around the World, 15–33. Sense Publishers.

Mayer, D. P. (1999). Measuring instructional practice: Can policymakers trust survey data? Educational Evaluation and Policy Analysis, 21(1), 29–45.

E


Page 33

Ministry of Education. (undated). Enhanced Performance Management System. Singapore: Ministry of Education.

Ministry of Education (2006a). Mathematics syllabus–Secondary. Singapore: Author Ministry of Education (2006b). Mathematics syllabus–Primary. Singapore: Author Stiff, L. V. (2002, March). Study shows high-quality professional development helps teachers

most. NCTM News Bulletin, 38(7), 7. Stigler. J. W., Gallimore, R., & Hiebert, J. (2000). Using video surveys to compare

classrooms and teaching across cultures: Examples and lessons from the TIMSS video studies. Educational Psychologist, 35(2), 87–100.

Wengraf, T. (2004). Qualitative research interviewing. Thousand Oaks, CA: Sage. Wilson, S. M., & Berne, J. (1999). Teacher learning and the acquisition of professional

knowledge: An examination of research on contemporary professional development. Review of Research in Education, 24, 173–209.

Wragg, E. C. (1999). An introduction to classroom observations (2nd ed.). New York: RoutledgeFalmer.

Wiggins, G., & McTighe, J. (2005). Understanding by design (Expanded 2nd ed.). Alexandria, Va: Association for Supervision and Curriculum Development.

Yeap, B. H., & Ho, S.Y. (2009). Teacher change in an informal professional development programme: The 4-I model. In K. Y. Wong, P. Y. Lee, B. Kaur, P. Y. Foong, & S. F. Ng (Eds.), Mathematics education–The Singapore journey (pp. 130–149). Singapore: World Scientific.

Yeap, B. H., & Kaur, B. (2010). Pedagogy for engaged mathematics learning. Singapore: National Institute of Education.

Publications arising from the project

(A) Professional Books for Teachers Kaur, B. & Yeap, B. H. (2009). Pathways to reasoning and communication in the primary

school mathematics classroom. Singapore: National Institute of Education. Kaur, B., & Yeap, B. H. (2009). Pathways to reasoning and communication in the secondary

school mathematics classroom. Singapore: National Institute of Education. [2000 copies of each book were printed and distributed to all primary and secondary schools

during the zonal Mathematics HOD meetings in April/May 2009] Yeap, B. H., & Kaur, B. (2010). Engaged pedagogy for mathematics learning. Singapore:

National Institute of Education. [5000 copies of each book were printed and distributed to all primary and secondary schools

during the Mathematics HOD meetings in Feb / April 2010] (B) Book Chapter Kaur, B. (2009). Reasoning and communication in the mathematics classroom–Some „What‟

strategies. In D. Martin, T. Fitzpatrick, R. Hunting, D. Itter, C. Lenard, T. Mills, & L. Milne (Eds.), Mathematics–of Prime Importance (pp. 102–110). Australia, Victoria: Mathematical Association of Victoria.

(C) Journal Paper Kaur, B. (2009). Enhancing the pedagogy of mathematics teachers (EPMT): An innovative

professional development project for engaged learning. The Mathematics Educator, 12(1), 33–48.

E


Page 34

(D) Conference Papers Kaur, B. (2009, November). Enhancing the pedagogy of mathematics Teachers (EPMT): An

innovative professional development project for engaged learning. Paper presented at the Educational Research Association of Singapore Conference 2009: Unpacking Teaching and Learning through Educational Research, National Institute of Education, Singapore.

Low, H. K. (2009, November). Traditional versus community-based professional

development–what teachers have to say? Paper presented at the Educational Research Association of Singapore Conference 2009: Unpacking Teaching and Learning through Educational Research, National Institute of Education, Singapore.

Yeap, B. H., & Kaur, B. (2009, November). From competency to mastery: A model of

professional development for mathematics teachers. Paper presented at International Conference on Teacher professional Development, Organised by Kasetsart University, Bangkok, Thailand.

(E) Conference Presentations (Workshops conducted) Kaur, B., Ng, Y.M., Tan, K.I., Fernando, C. & Lim, S.C. (2007, May). Mathematical tasks for

engaging primary pupils in reasoning and communication. Workshop presented at the Conference Redesigning Pedagogy : Culture, Knowledge and Understanding, National Institute of Education, Singapore.

Kaur, B., Leow, H. F., Ang, C. T., Ng, Y. L., & Kumari, R. (2007, May). Mathematical tasks

for engaging primary pupils in reasoning and communication. Workshop presented at the Conference Redesigning Pedagogy: Culture, Knowledge and Understanding, National Institute of Education, Singapore.

Kaur, B., & Low, H.K. (2009, June). Mathematical tasks for engaging primary pupils in

reasoning and communication. Workshop presented at the Conference for Redesigning Pedagogy: Designing New Learning Contexts for a Globalizing World, National Institute of Education, Singapore.

(F) Presentations for Ministry of Education, Singapore Kaur, B. (2009, April 28). Strategies for engaging secondary school students in reasoning

and communication in the math classroom. Presentation to North Zone mathematics HODs at Anderson Secondary School, Singapore.

Kaur, B. (2009, April 29). Strategies for engaging secondary school students in reasoning

and communication in the math classroom. Presentation to South Zone mathematics HODs at CHIJ (Toa Payoh) Secondary School, Singapore.

Kaur, B. (2009, May 6). Strategies for engaging secondary school students in reasoning and

communication in the math classroom. Presentation to West Zone mathematics HODs at Clementi Town Secondary School, Singapore.

Kaur, B. (2009, May 7). Strategies for engaging secondary school students in reasoning and

communication in the math classroom. Presentation to East Zone mathematics HODs at Geylang Methodist Secondary School, Singapore.

E


Page 35

Kaur, B. (2009, April 16). Strategies for engaging primary school students in reasoning and communication in the math classroom. Presentation to West Zone mathematics HODs at New Towsn Primary School, Singapore.

Kaur, B. (2009, April 17). Strategies for engaging primary school students in reasoning and

communication in the math classroom. Presentation to East Zone mathematics HODs at Bedok Green Primary School, Singapore.


communication in the math classroom. Presentation to North Zone mathematics HODs at Anderson Primary School, Singapore.


communication in the math classroom. Presentation to South Zone mathematics HODs at Keming Primary School, Singapore.

Kaur, B. (2009, Sept 16). An intervention project - Enhancing the pedagogy of math teachers

to emphasize reasoning and communication in their classrooms. Presentation to senior management of MOE, Singapore.

Kaur, B. (2010, Mar 8). Pedagogy for engaged mathematics learning. Presentation to

mathematics HODs (primary) at MOE HQ, Singapore. Kaur, B. (2010, Apr 21 & 22). Pedagogy for engaged mathematics learning. Presentation to

mathematics HODs (secondary) at MOE HQ, Singapore.

E


Page 36

About the authors

Berinderjeet Kaur is an Associate Professor with the Mathematics and Mathematics Education AG and Head of the Centre for International Comparative Studies (CICS) at the National Institute of Education (NIE) in Singapore. Yeap Ban Har is an Assistant Professor with the Mathematics and Mathematics Education AG and Centre for Research in Pedagogy and Practice (CRPP) at the National Institute of Education (NIE) in Singapore. Low Hooi Kiam is a research assistant with the Centre for Research in Pedagogy and Practice (CRPP) at the National Institute of Education (NIE) in Singapore. For further information

Contact us

For further information, please email: [email protected] You could also contact: A/P (Dr) Berinderjeet Kaur Office of Education Research National Institute of Education 1 Nanyang Walk Singapore 637616 Tel: +65 6790 3895

Centre for Research in Pedagogy and Practice

National Institute of Education 1 Nanyang Walk

Singapore 637616 http://www.crpp.nie.edu.sg

mailto:[email protected]

E


Page 37

Appendix A

Pre Intervention Teacher Questionnaire designed by research team (For Primary and Secondary Teacher Participants)

________________________________________________________________________

Enhancing The Pedagogy of Mathematics Teachers

PRE-INTERVENTION TEACHER QUESTIONNAIRE

Name: _________________________________ Date: __________________ School‟s Name: ______________________________________________ Name of Class being videotaped: ________________________________ PART ONE: PLEASE PROVIDE SOME INFORMATION ABOUT YOURSELF 1. Gender: _________________ 2. Grade Level(s) being taught in 2007: ____________________________________ 3. Number of years teaching primary school mathematics : _____________________ 4. Number of years teaching secondary school mathematics: ___________________ 5. Professional experiences other than school teaching (please describe):

____________________________________________________________________ ____________________________________________________________________

6. Other professional development in mathematics attended other than this current one.

(a) ________________________________________________________________ (b) ________________________________________________________________ (c) ________________________________________________________________ (d) _________________________________________________________________

E


Page 38

PART TWO: IN THIS SECTION, YOU ARE ASKED TO EXPLAIN HOW YOUR LESSON WAS USUALLY DESIGNED.

7. What guides you in your mathematics lesson planning? (Please describe)

____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

8. What is the source of activities or/and materials you use in your mathematics lessons? (Please describe) ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

9. Please indicate the title, author, publisher and year of publication one textbook/resource used

most often by you in planning your mathematics lessons.

Title: ________________________________________________________________ Author(s): ____________________________________________________________

Publisher: ____________________________________________________________ Publication year: ___________________ Edition: __________

10. Did you occasionally modify the questions / activities that you chose from available sources?

Yes. What was the reason(s) for doing so? ____________________________________________________________________ ____________________________________________________________________

No. What was the reason(s) for not doing so? ____________________________________________________________________ ____________________________________________________________________

E


Page 39

PART THREE: IN THIS SECTION, WE ARE INTERESTED TO FIND OUT MORE ABOUT YOUR MATHEMATICS INSTRUCTION.

11. Please describe the typical form of your mathematics lesson. Check the relevant boxes and state

the percentage of time you usually spend on it during a 60 minute lesson. The total should add to 100%).

Follow-up on Homework - correct common mistakes/Go through problems they could not do

__________% Review past content knowledge through teacher talk __________% Introduce new content knowledge through teacher talk __________% Engage the whole class in discussion __________% Demonstrate to students how to solve problems __________% Seat work - students complete work in class __________% Use student work done in class [on board / individually/ in groups]

to highlight common misconceptions __________%

Give students tests and quizzes __________% Other (please describe) ______________________________ __________% Other (please describe) ______________________________ __________%

12. (a) Do you assign mathematical questions / problems for your students to work on in class?

No (skip to 13) Yes (go to 12(b))

(b) Please describe the typical form of the mathematical questions / problems. Check the relevant boxes and state what percentage of time they may occur in a semester (20 weeks) of instruction time. The total should add to 100%).

Applying new procedures taught __________ % Practicing new procedures taught __________ % Consolidating content from previous lessons __________ %

Open-ended investigation __________ % Completing work started in the lesson __________ % Other (please describe) ________________________ __________ %

E


Page 40

13. Considering your training and experience in both mathematics content and instruction, how comfortable do you currently feel you are to teach your students to

(a) explain their reasoning when solving a question or a problem

Not comfortable Somewhat comfortable Very comfortable

(b) work on question or problem for which there is no immediately obvious method of solution

Not comfortable Somewhat comfortable Very comfortable

(c) write explanations about what was taught or observed (e.g., in a journal) Not comfortable Somewhat comfortable Very comfortable

PART FOUR: TEACHER OPINIONS 14. How aware do you feel you are of the ideas about teaching reasoning and communicate

mathematically? Not aware at all Not very aware Somewhat aware Very aware

15. How familiar are you with the current ideas about teaching for understanding?

Not familiar at all Somewhat familiar Fairly familiar Very familiar

16. How often do you have the following types of interactions with other mathematics teachers?

(Check one box only in each row) Once or Once or A few twice twice twice a week a month a year Never

(a) Discussion about how to teach a particular concept ……………………………

(b) Working on preparing

instructional materials ………………………

E


Page 41

(c) Visit to another teacher‟s classroom to observe his/her teaching …………………

(d) Observation of my classroom by another teacher ……………………………..

E


Page 42

Appendix B Pre-Intervention guide for reflection of vido-taped lesson

Reflection on Videotaped Lesson (This is to be completed before participant involves in professional development courses)

As you write each reflection, consider the following: This reflective writing is a documentation of the lesson you taught to help you lay

the foundation for the need to undertake the professional development courses. You are honoured to express your frank and honest personal views and opinions. Please take opportunity to view your own teaching objectively and not find faults

with yourself or anyone else.

Teacher Participant: _________________________Date of lesson: ___________________ School: __________________________________________________________________ Grade level being videotaped: _________________________________________________ Topic/content of the lesson: ___________________________________________________

A. In this section, we would like you to give your view on the introduction of lesson.

Give a brief description on what was the main concept/skill you wanted students to learn from this lesson and why you think it is important for students to learn this?

How did you begin the lesson and why did you do so?

E


Page 43

B. In this section, we would like you to look at the “mathematical tasks” you used in the lesson.

Give one or two examples of the tasks that you used in the lesson and comment on them. For instance, How did you select the task(s)? What were the mathematical concepts/skills these assigned task(s) were developing? How did you use the task(s) to engage students in reasoning and communication? Were there any part(s) of the lesson where you felt that students were engaged in

thinking/reasoning and mathematical communication? What have you learnt from students‟ responses? Do you think you have selected the appropriate task(s)?

(i) If so, why? (ii) If not, please explain why?

An example of a task: Example 1:

Find the value of 12 + 4 × 7 2.

Example 2: Let‟s find the area of the shaded figure.

E


Page 44

C. In this section, you may want to give an overview of the lesson in terms of your own teaching and learning objectives.

Did you achieve your teaching and learning objective(s) for this lesson? (i) If so, what have you achieved? (ii) If no, why was it so and what would you do to resolve it?

D. In this section, we would like you to share with us your understanding of “teaching for understanding”.

E


Page 45

Appendix C

Enhancing the Pedagogy of Mathematics Teachers Overview of Professional Development Module 1 – Pathways to Reasoning

Primary Secondary

Week Date Content Week Date Content

1

29 Jan

Introduction – what is reasoning & Outline of module 1. Reasoning by Analogy? Method of models – overview. 2. Inductive and Deductive

Reasoning

1

30 Jan

Introduction – what is reasoning & Outline of module 1. Reasoning by Analogy? Questions with similar structures. 2. Inductive and Deductive

Reasoning

2

5 Feb

3. What number makes

sense? 4. What‟s wrong?

2

6 Feb

3. What number makes

sense? 4. What‟s wrong?

3

12 Feb

5. What would you do? 6. What questions can you

answer?

3

13 Feb

5. What would you do? 6. What questions can you

answer?

4

26 Feb

7. What‟s missing? 8. What if?

4

27 Feb

7. What‟s missing? 8. What if?

5

5 Mar

9. What‟s the question if you

know the answer? [context is given]

10. What‟s the question? [context is absent]

5

6 Mar

9. What‟s redundant? 10a. What‟s the question if

you know the answer? [context is given]

10b. What‟s the question? [context is absent]

6

19 Mar

Completion of work by participants and collection of work by instructor. Schools to check that their portfolios are complete

6

20 Mar

Completion of work by participants and collection of work by instructor. Schools to check that their portfolios are complete

© Kaur, B. 2009

E


Page 46

Appendix D – Part I

Overview of Eight “What…” Strategies of Professional Development Module 1 – Pathways to Reasoning: Primary Mathematics

Strategy 1: What number makes sense? In “What number makes sense?” pupils are presented with a mathematics version of a cloze passage,

many pupils would be familiar with in their Language lessons. Pupils are presented with problem

situations from which numerical data is missing. A set of numbers is provided and pupils determine

where to place each number so the situation makes sense. The steps given as part of the problem

sheet help to focus the pupils on the steps they need to take and also explain their thinking. The

teacher must ensure that group interaction followed by class discussion occurs so that pupils have the

opportunity to explain their thinking and also learn of ways of solving problems that differ from their

own. As pupils work through tasks of this nature, they practice computation and increase their

repertoire of problem-solving skills. Reasoning skills are improved by being exposed to a variety of

ways to solve a problem (Krulik and Rudnick, 2001). Such a task can be very easily crafted from a

typical textbook question.

EXAMPLE - My Study Table Read the problem. Look at the numbers in the box. Put the numbers in the blanks where you think they fit best. Read the problem again, do the numbers make sense? My study table has a rectangular table-top.

It is _______ cm long and _________ cm wide.

The area of the table-top is ____________square cm.

My exercise book is rectangular in shape too. It is _______ cm

long and _____ cm wide. To completely cover the top of my

table with exercise books, I need __________ exercise books.

15 16 20 60 80 4800

© Kaur, B. (2009)

E


Page 47

Strategy 2: What’s wrong?

In, “What‟s wrong?” the pupils are provided with an opportunity to use their critical thinking skills. They

are presented with a problem and its solution. However the solution contains an error, either

conceptual or computational. The pupil‟s task is to discover the error, correct it and then explain what

was wrong, why it was wrong and what was done to correct the error (Krulik and Rudnick, 1999). The

teacher must ensure that pupils are engaged in class discussion after completing the task either in

small groups or individually so that they hear ways of solving problems that differ from their own

Furthermore the group interaction that occurs during these discussions often leads to deeper

mathematical understanding (Krulik and Rudnick, 2001). Such tasks are not difficult for teachers to

craft as they are constantly exposed to such errors pupils make in class and in their written

assignments.

EXAMPLE - Prize Money

John and Henry won a prize of $500 at a Charity Fair. With the money, John bought a bicycle for

$140. On their way home they decided to

share the prize money equally.

John‟s thinking:- $500 - $140 = $360 $360 ÷ 2 = $180 Each person gets $180 There is something wrong with John‟s thinking. 1. Show how you would find the answer to the problem.

2. Explain the mistake in John‟s thinking. © Kaur, B. (2009)

E


Page 48

Strategy 3: What would you do? In “What would you do” kind of tasks pupils are provided with problem situations that stimulate creative

thinking skills and also engage them in decision making. Their decisions can be based on personal

ideas, personal experiences, or whatever the pupil wishes to call into play. However, the pupil must

explain the mathematics that influenced his or her decision. The teacher must ensure that pupils are

engaged in class discussion after completing the task so that pupils get an opportunity to learn how

their friends solved the task and also appreciate the multitude of creative solutions justified by

reasoning based on differing assumptions.

EXAMPLE - Birthday Candles

Raju‟s has 8 large and 5 small candles.

He has to put candles on a birthday cake to celebrate his grandfather‟s 64th birthday.

1. How many candles of each type could Raju put on the birthday cake? 2. Explain your reasoning.

© Kaur, B. (2009)

E


Page 49


In “What questions can you answer?” kind of tasks pupils are provided with situations that include

numerical data and / or geometrical figures and are asked to generate questions that can be answered

using the given information. This activity is both creative (as pupils have to pose more than one

question and hence stretch beyond the obvious) and critical (as pupils have to make sure that the

questions they pose are solvable). The teacher must ensure that after completing the task pupils show

case their questions together with solutions and engage in class discussion so that they realize the

breadth and depth of questions that can be constructed with the information. The sophistication of the

questions posed by individual pupils show their developmental level and this is excellent feedback for

the teacher.

EXAMPLE - The Exhibition

An average of 215 people visited a 4-day exhibition on the first three days. Another 310 people visited

the exhibition on the fourth day.

Write two questions you can answer about the visitors to the exhibition. 1. Question 1 ____________________________________________________________ ____________________________________________________________ 2. Question 2 ____________________________________________________________ ____________________________________________________________ 3. Find the answer to your first question.

Show your work.

© Kaur, B. (2009)

E


Page 50

Strategy 5: What’s Missing?

In “What‟s missing?” kind of tasks pupils are presented with tasks that cannot be solved

because an important piece of information has been omitted. Pupils must identify what is

missing, supply appropriate data, and then solve the problem. Such tasks provide an

opportunity for pupils to engage in both critical thinking and creative thinking skills. Whole

class discussion must precede individuals working on such tasks because there is a wide

range of data that pupils can supply to solve each problem. As each different piece of

missing information supplied by a pupil produces a different problem, interesting discussions

based on the specific data chosen are possible.

EXAMPLE – Donuts

Mary bought 7 boxes of donuts for her class party.

She paid $35 for the 7 boxes.

How much did each donut cost?

1. What is the question?

2. What information do you know from the problem?

3. What else do you need to know to solve the problem?

4. Pick a number that shows how many donuts might have been in a box.

5. How much would each donut cost?

© Kaur, B. (2009)

E


Page 51

Strategy 6: What if?

In “What if?” kind of tasks two kinds of demand are made on the pupils‟ cognition. The first is

when the given information is changed. This modification permits pupils to reexamine the

task and see what effect these changes have on the solution process as well as the answer.

In this way pupils are reinforcing their critical thinking as they analyze what is taking place

(Krulik and Rudnick, 1999). The second is the generation of “what if” questions after they

have solved a given task. This draws on the creative thinking skills of the pupil and engages

him or her in problem posing (Brown and Walter, 1985). Problem posing is the generation of

new problems and the reformulation of given ones (Silver, 1994). Whole class discussion

must precede individuals working on such tasks because pupils need to share the “what if?”

tasks they created with the others as well as make their thinking visible.

EXAMPLE – Cookies and Boxes

Mrs Tan baked 24 cookies. Each box holds 4 cookies. At least how many boxes are needed to hold all the cookies? What if Mrs Tan baked 30 cookies? What if each box can hold 5 cookies? What if each box can hold up to 4 cookies?

Generate another 3 “What if” tasks and answer them.

Look out for any interesting observation/pattern.

© Kaur, B. (2009)

E


Page 52

Strategy 7: What’s the Question if you Know the Answer?

In “What‟s the question if you know the answer?” kind of tasks pupils are presented with the

context and data of a problem but with the question/s missing. They are given a solution and

asked to write a question that matches it. Such tasks provide an opportunity for pupils to

engage in critical thinking skills. Whole class discussion must precede individuals working on

the task as it is important for pupils to recognize that there may be several questions that

have the same answer.

EXAMPLE – Eggs

Eggs sold by Mr Ali

Day Monday Tuesday Wednesday Thursday

Number of eggs sold 135 150 110 185

1. What‟s the question if the answer is Wednesday? ___________________________________________________________________ 2. What‟s the question if the answer is 580? ____________________________________________________________________ 3. What‟s the question if the answer is 75? ____________________________________________________________________ 4. What‟s the question if the answer is 145? ____________________________________________________________________

© Kaur, B. (2009)

E


Page 53

Strategy 8: What’s the Question?

In “What‟s the question” kind of tasks the question is missing. These tasks require pupils to

write possible questions that correspond to a given solution. Such tasks encourage

reasoning and the ability to work backward from a specific answer. Pupils will draw on their

creative thinking skills to craft the question while using their critical thinking skills to ensure

that the question matches the given solution. When pupils do such tasks they get an

opportunity to take stock of and use their existing knowledge. For example in the task, below

taken from Yeap and Kaur (1997), the answer [One-Five-Four] provides a stimulus for pupils

to summarise all they know about the area of figures. Whole class discussion must precede

individuals working on such tasks because pupils need to share with one another the

questions they have crafted. During the presentation of the questions by the pupils the

teacher could also engage the pupils in some sort of classification of the questions which

may be used latter for revision work.

EXAMPLE – One-Five-Four

Topic: Area of plane figures

1. What could the question be?

___________________________________________________________________

2. My solution is:

___________________________________________________________________

© Kaur, B. (2009)

The area is 154 cm2

E


Page 54

Appendix D – Part II

Overview of Nine “What…” Strategies of Professional Module 1 – Pathways to Reasoning: Secondary Mathematics Strategy 1: What number makes sense? In “What number makes sense?” students are presented with a mathematics version of a

cloze passage, many students would be familiar with in their Language lessons. Students

are presented with problem situations from which numerical data is missing. A set of

numbers is provided and students determine where to place each number so the situation

makes sense. The steps given as part of the problem sheet help to focus the students on the

steps they need to take and also explain their thinking. The teacher must ensure that group

interaction followed by class discussion occurs so that students have the opportunity to

explain their thinking and also learn of ways of solving problems that differ from their own. As

students work through tasks of this nature, they practice computation and increase their

repertoire of problem-solving skills. Reasoning skills are improved by being exposed to a

variety of ways to solve a problem (Krulik and Rudnick, 2001). Such a task can be very

easily crafted from a typical textbook question.

EXAMPLE - A Toy Locomotive A typical textbook question Source: New Syllabus Mathematics Book 3 (Shinglee publishers pte Ltd) Page: 119 Q 14 (a) A locomotive is 10 m long and weighs 72 tonnes.

A similar model, made of the same material is 40 cm long, find the mass of the model.

(b) Suppose the tank of the model locomotive contains 0.85 litres of water when full, find the capacity of the tank of the locomotive.

A toy locomotive, made of the same material and density, is an exact model of a real one. The locomotive is ____________ long and weighs _______________. The toy locomotive is ___________ long and weighs _____________. The capacity of the locomotive‟s oil tank is __________________ and this is ___________ times the capacity of the toy locomotive‟s oil tank.

4.8 kg 10 m 40 cm 75 tonnes 3125 litres 253

© Kaur, B. (2009)

E


Page 55

Strategy 2: What’s wrong?

In, “What‟s wrong?” the students are provided with an opportunity to use their critical thinking

skills. They are presented with a problem and its solution. However the solution contains an

error, either conceptual or computational. The student‟s task is to discover the error, correct

it and then explain what was wrong, why it was wrong and what was done to correct the error

(Krulik and Rudnick, 1999). The teacher must ensure that students are engaged in class

discussion after completing the task either in small groups or individually so that they hear

ways of solving problems that differ from their own Furthermore the group interaction that

occurs during these discussions often leads to deeper mathematical understanding (Krulik

and Rudnick, 2001). Such tasks are not difficult for teachers to craft as they are constantly

exposed to such errors students make in class and in their written assignments.

EXAMPLE - Triangular Poster Level: Secondary Two/ Three (Special / Express) Topic: Similar figures Rani made a triangular poster for the coming math competition. It was in the shape of an equilateral triangle with side 30 cm. Her teacher liked the poster very much. She asked Rani to make another similar poster but with an area twice that of the first poster.

To make the new poster Rani decides to double the sides of the equilateral triangle.

Rani‟s thinking:- Original side of poster = 30 cm Multiply each side by 2 30 cm x 2 = 60 cm therefore,

the new poster is an equilateral triangle of side 60 cm

There is something wrong with Rani‟s thinking.

1. Show how you would solve the problem.

2. Explain the error in Rani‟s thinking. © Kaur, B. (2009)

E


Page 56

Strategy 3: What would you do? In “What would you do” kind of tasks students are provided with problem situations that

stimulate creative thinking skills and also engage them in decision making. Their decisions

can be based on personal ideas, personal experiences, or whatever the student wishes to

call into play. However, the student must explain the mathematics that influenced his or her

decision. The teacher must ensure that students are engaged in class discussion after

completing the task so that students get an opportunity to learn how their friends solved the

task and also appreciate the multitude of creative solutions justified by reasoning based on

differing assumptions.

EXAMPLE - Visit to Jurong BirdPark Level: Upper Secondary (Special / Express / Normal) Skill: Logical Reasoning

Your class is planning a day at the Jurong BirdPark. The students will be grouped in fours or fives. You‟ll arrive at the Park by 9.00 am and leave promptly at 4.00 pm. The members of each group will stay together all day and will follow the schedule of activities that they set up in advance.

Information available from the BirdPark: Opening Hours and Show Times – see attached Map of the Park – see attached

Your task is: 1) to plan the schedule of activities for your group, and 2) explain your choices.

© Kaur, B. (2009)

E


Page 57


In “What questions can you answer?” kind of tasks students are provided with situations

that include numerical data and / or geometrical figures and are asked to generate

questions that can be answered using the given information. This activity is both creative

(as students have to pose more than one question and hence stretch beyond the obvious)

and critical (as students have to make sure that the questions they pose are solvable). The

teacher must ensure that after completing the task students show case their questions

together with solutions and engage in class discussion so that they realize the breadth and

depth of questions that can be constructed with the information. The sophistication of the

questions posed by individual students show their developmental level and this is excellent

feedback for the teacher.

EXAMPLE - Figure It Out Level: Lower Secondary (Special / Express / Normal) Skill: Congruent Shapes

The figure below is made up of seven congruent rectangles. The figure is rectangular in shape. Its length is 50 cm.

Write two questions that you can answer about parts of the figure?

1. Question 1 _______________________________________________________________________

2. Question 2 _______________________________________________________________________ 3. Find the answer to your first question. Show your work.

© Kaur, B. (2009)

E


Page 58

Strategy 5: What’s Missing ?

In “What‟s missing?” kind of tasks students are presented with tasks that cannot be solved

because an important piece of information has been omitted. Students must identify what

is missing, supply appropriate data, and then solve the problem. Such tasks provide an

opportunity for students to engage in both critical thinking and creative thinking skills.

Whole class discussion must precede individuals working on such tasks because there is a

wide range of data that students can supply to solve each problem. As each different piece

of missing information supplied by a student produces a different problem, interesting

discussions based on the specific data chosen are possible.

EXAMPLE - Who Earns More Level: Secondary One (Special / Express / Normal) Topic: Arithmetic Problems Mr Tan and Mr Samy are salesmen in a Computer store. Mr Tan‟s monthly salary is $1500. Mr Samy‟s monthly salary is made up of a basic amount of $500 and a commission of 16 % on the sales he make in a month. Which salesman earned more last month? How much more? 1. What is the question? 2. What information do you know from the problem? 3. What else do you need to know to solve the problem? 4. Pick a reasonable number for the information you need. 5. Who earned more? How much more?

© Kaur, B. (2009)

E


Page 59

Strategy 6: What if ?

In “What if?” kind of tasks two kinds of demand are made on the students‟ cognition. The

first is when the given information is changed. This modification permits students to

reexamine the task and see what effect these changes have on the solution process as

well as the answer. In this way students are reinforcing their critical thinking as they

analyze what is taking place (Krulik and Rudnick, 1999). The second is the generation of

“what if” questions after they have solved a given task. This draws on the creative thinking

skills of the student and engages him or her in problem posing (Brown and Walter, 1985).

Problem posing is the generation of new problems and the reformulation of given ones

(Silver, 1994). Whole class discussion must precede individuals working on such tasks

because students need to share the “what if?” tasks they created with the others as well as

make their thinking visible.

EXAMPLE - Let‟s dee y dee x! Level: Secondary Four (Additional Mathematics) Topic: Differentiation Find dy if y = sin 3x. dx What if

o y = 3 sin 3x?

o y = sin3 3x?

o y = sin 3x3?

o y = sin (3

x)?

o y = sin (x

3)?

o x = sin 3y?

Generate another 10 “what if” questions and answer them. Look out for any interesting observation / patterns. The “what if” questions you generate must allow you to show your ability to use the chain rule, product rule and implicit differentiation.

© Kaur, B. (2009)

E


Page 60

Strategy 7: What’s Redundant? In “What‟s redundant?” kind of tasks students are presented with tasks that contain

extraneous information that is not needed for the solution of the task. Students must

identify what is extraneous and exclude it when solving the problem. Such tasks provide

an opportunity for students to engage in critical thinking skills. Whole class discussion

must precede individuals working on the task as it is important for students to recognize

the uniqueness or non-uniqueness of the redundant data depending on the nature of the

task.

EXAMPLE – Points and Lines Level: Secondary Three (Special / Express) Topic: Co-ordinate Geometry Find the equation of a line passing through the points P (0, -4), Q (3, -2) and parallel to the line 2x – 3y = 14. 1. What are you asked to find? ___________________________________________________________________ 2. What do you need to know to find an equation of a line? ____________________________________________________________________ 3. What‟s redundant (not needed) in the question? ____________________________________________________________________ 4. Show how you would find what you are asked to? ____________________________________________________________________

© Kaur, B. (2009)

E


Page 61

Strategy 8: What’s the Question if you Know the Answer? In “What‟s the question if you know the answer?” kind of tasks students are presented with

the context and data of a problem but with the question/s missing. They are given a solution

and asked to write a question that matches it. Such tasks provide an opportunity for students

to engage in critical thinking skills. Whole class discussion must precede individuals working

on the task as it is important for students to recognize that there may be several questions

that have the same answer.

EXAMPLE – Just one card Level: Secondary Four (Special / Express) Topic: Probability Eleven cards numbered 11, 12, 13, 14, ……, 21 are placed in a box. A card is removed at random from the box.

1. What‟s the question if the answer is 11

5?

___________________________________________________________________


4?

____________________________________________________________________


9?

____________________________________________________________________


6?

____________________________________________________________________


3?

____________________________________________________________________

© Kaur, B. (2009)

E


Page 62

Strategy 9: What’s the Question?

In “What‟s the question” kind of tasks the question is missing. These tasks require students

to write possible questions that correspond to a given solution. Such tasks encourage

reasoning and the ability to work backward from a specific answer. Students will draw on

their creative thinking skills to craft the question while using their critical thinking skills to

ensure that the question matches the given solution. When students do such tasks they get

an opportunity to take stock of and use their existing knowledge. For example in the task,

below taken from Yeap and Kaur (1997), the answer [One-Five-Four] provides a stimulus for

students to summarise all they know about the area of figures. Whole class discussion must

precede individuals working on such tasks because students need to share with one another

the questions they have crafted and the solutions they have completed. During the

presentation of the questions by the students the teacher could also engage the students in

some sort of classification of the questions which may be used later for revision work.

EXAMPLE – One-Five-Four Level: Secondary One (Special / Express) Topic: Mensuration

1. What could the question be?

___________________________________________________________________

2. My solution is:

___________________________________________________________________

© Kaur, B. (2009)

The area is 154 cm2

E


Page 63

Appendix E

Enhancing the Pedagogy of Mathematics Teachers Overview of Professional Development Module 2 – Teaching for Understanding

Primary Secondary

Week Date Content Week Date Content

1 2 April Infusing Reasoning Tasks and Facilitating Communication in the Mathematics Classroom Examples - 2 lesson plans of such lessons

1 3 April

Infusing Reasoning Tasks and Facilitating Communication in the Mathematics Classroom Examples 1 & 2 – Lower secondary topic [lesson] Examples 3 & 4 – Upper Secondary topic [lesson]

2 9 April What is understanding? [Chapter 2 textbook] Characteristics of lessons that teach for understanding? [Paper: Teaching & Learning Math with Understanding]

2 10 April

What is understanding? [Chapter 2 textbook] Characteristics of lessons that teach for understanding? [Paper: Teaching & Learning Math with Understanding]

3 16 April

The evidence of understanding? The six facets of understanding. [Chapter 4 textbook] Crafting of tasks that collect evidence of understanding. [Paper: Assessment in Classrooms that promote understanding] Examples of some primary math items that test understanding.

3 17 April

The evidence of understanding? The six facets of understanding. [Chapter 4 textbook] Crafting of tasks that collect evidence of understanding. [Paper: Assessment in Classrooms that promote understanding] Examples of some secondary math items that test understanding.

4 23 April

The three step design of lessons that teach for understanding: Step 1: Identify desired results / outcomes Step 2: Determine acceptable evidence Step 3: Plan Learning Experiences & Instruction. Hands on work – design a lesson that teach for understanding

4 24 April

The three step design of lessons that teach for understanding: Step 1: Identify desired results / outcomes Step 2: Determine acceptable evidence Step 3: Plan Learning Experiences & Instruction. Hands on work – design a lesson that teach for understanding

© Kaur, B. 2009

E


Page 64

Appendix F

Enhancing the Pedagogy of Mathematics Teachers Pre & Post Intervention Reflections designed by the research team

(For Primary and Secondary Participants) First we suggest you do this exercise with a colleague, who is also participating in the project. You need to view the taped lessons with the following questions in mind.

Planning of your lesson [what changes if any were there between the pre and post intervention lessons]

The source of questions that you used in your lessons The sequence of activities during the lessons [e.g. teacher talk (demonstration), seat

work, discussion? / teacher talk (instructions), group-work, student presentations, whole class discussion, etc….]

Student engagement [passive, active, problem solving, explaining, problem posing, etc….]

Evaluation of lesson [lesson objectives – were they met, difficulties encountered, etc….]

You may like to use the template on the next page to record your observations and comments.

E


Page 65

School: ______________________________________________ Name of Teacher: _______________________________________

Focus Pre – intervention lesson Post – intervention lesson

Lesson Preparation [what guided you in your objectives, topic, etc.., what materials or resources did you rely on?]

Source of materials used during lesson, e.g. mathematical tasks [directly used from sources or crafted by yourself, strategies, e.g. what if?, what‟s wrong?

Example/s of task/s and how it was used during the lesson

Lesson Delivery [sequence of activities, teacher talk, student work, etc…]

Student Participation [describe it,

E


Page 66

e.g. students solved problems in groups, presented their solutions, students peer evaluated their friend‟s work, etc..]

Your evaluation of the lesson; goals – achieved or not, student engagement, assessment for learning – what evidence could you use to make a judgment about the success or failure of your lesson.

Your reflections about [feel free to share with us anything not asked for above]

Pre-intervention lesson…..

Post-intervention lesson…

E


Page 67

Appendix G

The Cover Page for Primary Resource Book

E


Page 68

The Content Page for Primary Resource Book

E


Page 69

Appendix H

The Cover Page for Secondary Resource Book

E


Page 70

The Content Page for Secondary Resource Book

E


Page 71

Appendix I

Enhancing the Pedagogy of Mathematics Teachers

Teacher Participation Feedback Questionnaire (For Primary Teacher Participants)

Enhancing the Pedagogy of Mathematics Teachers (Primary)

Teacher Participation Feedback Questionnaire Name of Participant (Optional): ______________________________________________ School: _________________________________________________________________ Class videotaped: Primary __________________________________________________ Overview

The EPMT (primary) project was carried out during the period January 2007 – December 2008. As part of the project you attended two professional development modules during the first half of 2007 and also put into practice your learning during the second half of the year. During several follow-up meetings during the second half of the year, lessons showcasing the infusion of the strategies developed during the first module as well as the approach to teach for understanding were demonstrated by participants of the project. In December 2007, you submitted a video recording of one of your lessons demonstrating the infusion of your learning into your classroom pedagogy. A selection of the tasks, you have all crafted during the first module are now ready for publishing in the form of a resource book titled: Pathways to reasoning and communication in mathematics (primary) and will be available next year. We hope to launch the book early next year (2009) so that your work can light many more small fires in primary schools across Singapore. The whole of this year, you have been left to work on your own and we hope you have made several attempts to infuse your learning into your classroom pedagogy so as to engage your pupils in fruitful mathematics lessons. Also, we hope as participants of the project from a school you have worked as a small community and supported each other in your efforts. Please help us to evaluate the success of the project by answering the following questionnaire truthfully. Instruction

This questionnaire consists of three evaluation sections and two sections for comments. Please give us your opinion of participating in this project.

E


Page 72

Part I: Circle the number that corresponds with your level of agreement with each statement.

1. First PD Module: Pathways to Reasoning Strongly

Agree Agree Neutral Disagree Strongly Disagree

a. I found the first PD module: Pathways to reasoning comprising of strategies such as: what number makes sense, what‟s wrong, etc., useful.

b. I found the first PD module content comprehensive.

c. I found the first PD module presentations clear and understandable.

d. I found each session (2.5 hours) was just the right length.

e. I felt engaged during the first PD module course.

f. I found the first PD module helpful for planning my lessons / activities.

g. I found the first PD module useful for improving student learning.

h. I would continue infuse the knowledge and skills that are acquired during the first PD module course.

2. Second PD Module: Teaching for Understanding

a. I found the second PD module: Teaching for Understanding comprising of approaches such as: infusing reasoning tasks and facilitating communication in the mathematics classroom, what is understanding, etc., useful.

b. I found the second PD module content comprehensive.

c. I found the second PD module presentations clear and understandable.


e. I felt engaged during the second PD module course.

E


Page 73

f. I found the second PD module helpful for planning my lessons / activities.

g. I found the second module useful for improving student learning.

h. I would continue infuse the knowledge and skills that are acquired during the second PD module course.

Part II: Tick the box that corresponds with the respective usefulness.

1. How useful do you find the following strategies for planning your lessons / activities?

Very Useful Useful

Moderat-ely

Useful Of Little Useful

Not Useful At All




Strategy 4: What‟s wrong?

Strategy 5: What would you do?


Strategy 7: What‟s missing?


Strategy 9: What‟s the question if you know the answer? [context is given]

Strategy 10: What‟s the question? [context is absent]

E


Page 74

2. How useful do you find the following approaches for planning your lessons / activities?




Part III: Tick the box that corresponds with the respective frequency.

1. How often have you used the following strategies in your classrooms? Always

Very Often

Some-times Rarely Never









Strategy 9: What‟s the question if you know the answer? [context is given]

Strategy 10: What‟s the question? [context is absent]

E


Page 75

2. How often have you used the following approaches when planning your lessons / activities?




Part IV: Please give your comment. 1. What have you done so far this year about the strategies you learnt in the first PD

module?

2. What helped you to infuse the knowledge and skills that acquired during both modules in

your classrooms?

3. What hindered you from infusing the knowledge and skills that acquired during both

modules in your classrooms?

4. In what ways did the strategies help to improve the student learning in your classrooms?

5. In what ways did the approaches of teaching for understanding help you to improve in


6. Tell us how different or similar has it been participating in the project compared to

attending an in-service course where there is no working together with participants. Which

would you prefer to participate or attend in future?

7. Are there any other comments that you would like us to take note of?

E


Page 76

Part V: Please provide us some insights of your learning journey while participating in this project. My learning journey:

E


Page 77

Appendix J

Enhancing the Pedagogy of Mathematics Teachers Teacher Participation Feedback Questionnaire

(For Secondary Teacher Participants)

Enhancing the Pedagogy of Mathematics Teachers (Secondary)

Teacher Participation Feedback Questionnaire

Name of Participant (Optional): ____________________________________ School: _______________________________________________________ Class videotaped: Secondary ______________________________________ Overview

The EPMT (secondary) project was carried out during the period January 2007 – December 2008. As part of the project you attended two professional development modules during the first half of 2007 and also put into practice your learning during the second half of the year. During several follow-up meetings during the second half of the year, lessons showcasing the infusion of the strategies developed during the first module as well as the approach to teach for understanding were demonstrated by participants of the project. In December 2007, you submitted a video recording of one of your lessons demonstrating the infusion of your learning into your classroom pedagogy. A selection of the tasks, you have all crafted during the first module are now ready for publishing in the form of a resource book titled: Pathways to reasoning and communication in mathematics (secondary) and will be available next year. We hope to launch the book early next year (2009) so that your work can light many more small fires in secondary schools across Singapore. The whole of this year, you have been left to work on your own and we hope you have made several attempts to infuse your learning into your classroom pedagogy so as to engage your pupils in fruitful mathematics lessons. Also, we hope as participants of the project from a school you have worked as a small community and supported each other in your efforts. Please help us to evaluate the success of the project by answering the following questionnaire truthfully. Instruction

This questionnaire consists of three evaluation sections and two sections for comments. Please give us your opinion of participating in this project.

E


Page 78

Part I: Circle the number that corresponds with your level of agreement with each statement.

1. Fisrt PD Module: Pathways to Reasoning

Strongly Agree Agree Neutral Disagree

Strongly Disagree

a. I found the first PD module: Pathways to reasoning comprising of strategies such as: what number makes sense, what‟s wrong, etc., useful.

1 2 3 4 5

b. I found the first PD module content comprehensive.

1 2 3 4 5

c. I found the first PD module presentations clear and understandable.

1 2 3 4 5


1 2 3 4 5

e. I felt engaged during the first PD module course.

1 2 3 4 5

f. I found the first PD module helpful for planning my lessons / activities.

1 2 3 4 5

g. I found the first PD module useful for improving student learning.

1 2 3 4 5

h. I would continue infuse the knowledge and skills that are acquired during the first PD module course.

1 2 3 4 5

2. Second PD Module: Teaching for Understanding

a. I found the second PD module: Teaching for Understanding comprising of approaches such as: infusing reasoning tasks and facilitating communication in the mathematics classroom, what is understanding, etc., useful.

1 2 3 4 5

b. I found the second PD module content comprehensive.

1 2 3 4 5

c. I found the second PD module presentations clear and understandable.

1 2 3 4 5

E


Page 79


1 2 3 4 5

e. I felt engaged during the second PD module course.

1 2 3 4 5

f. I found the second PD module helpful for planning my lessons / activities.

1 2 3 4 5

g. I found the second module useful for improving student learning.

1 2 3 4 5

h. I would continue infuse the knowledge and skills that are acquired during the second PD module course.

1 2 3 4 5

Part II: Tick the box that corresponds with the respective usefulness.

3. How useful do you find the following strategies for planning your lessons / activities?

Very Useful Useful

Moderately Useful

Of Little Useful

Not Useful At All









Strategy 9: What‟s redundant?



E


Page 80

4. How useful do you find the following approaches for planning your lessons / activities?




Part III: Tick the box that corresponds with the respective frequency.

3. How often have you used the following strategies in your classrooms? Always

Very Often Sometimes Rarely Never









Strategy 9: What‟s redundant?

Strategy 10a: What‟s the question if you know the answer? [context is absent]

E


Page 81


4. How often have you used the following approaches when planning your lessons / activities?




Part IV: Please give your comment. 1. What have you done so far this year about the strategies you learnt in the PD

module: Pathways to Reasoning?

2. What have you done so far this year about the approaches you learnt in the

PD module: Teaching for Understanding?

3. What helped you to infuse the knowledge and skills that you acquired during the PD modules conducted by the experts in your lessons?

4. What hindered you from infusing the knowledge and skills that you acquired during the

PD modules conducted by the experts in your lessons?

5. In what ways did the strategies help to improve the student learning in your classrooms?

6. In what ways did the approaches of teaching for understanding help you to improve in


E


Page 82

7. Tell us how different or similar has it been participating in the project compared to

attending an in-service course where there is no working together with participants.

Which would you prefer to participate or attend in future?

8. Are there any other comments that you would like us to take note of?

Part V: Please provide us some insights of your learning journey while participating in this project. My learning journey:

E


Page 83

Appendix K

The Cover Page for Pedagogy Engaged Mathematics Learning

E


Page 84

The Content Page for Pedagogy Engaged Mathematics Learning

Documents

Enhancing the pedagogy of mathematics teachers to emphasize understanding, reasoning and communication in their classrooms (EPMT)