Working and Learning in Collaborative Groups: What’s Key to Mathematics Teachers?

Abstract

This chapter presents and examines the accounts of four mathematics teachers focused on improving their professional practice through collaborative group experiences. The teachers work in very different contexts and were involved in very different types of collaborative groups, yet their accounts reveal that there are key elements of their collaborative group experiences, and outcomes of those experiences, that resonate across locations and activities. The chapter is presented in three sections. First, the aims of examining the collection of teacher accounts are identified and some observations related to common design features of the four collaborative groups are presented. Second, the four teachers’ stories are provided. And third, a synthesis of lessons learned from the experience of these teachers is presented, and questions are proposed to provoke consideration of how these might inform directions for mathematics teachers working and learning in collaborative groups in the future.

1 Introduction

Research suggests that collaborative professional development has the potential for impactful effects on teaching practices and student achievement (Borko, 2004; Jensen, 2014; Opfer, 2016). Yet, the OECD’s most recent Teaching and Learning International Survey (TALIS) data show a comparatively low percentage of teachers participating in collaborative professional learning activities (OECD, 2019).

This chapter examines the involvement of four mathematics teachers in collaborative groups, with the aim of: (i) understanding the kinds of opportunities for learning that these groups provided and the impact that they had; (ii) synthesising insights from the teachers’ participation in these groups to inform the ways that mathematics teachers might work and learn in collaborative groups in the future. The four teachers were invited to participate in a Plenary Panel at the ICMI Study 25 Conference, where they shared accounts of their collaborative group work and learning. Each teacher was asked to report on three areas, including: the context, purpose and design of the collaboration; outcomes of the collaboration; particular lessons learned (for example, in terms of: any factors that supported or limited the collaboration; any surprises or challenges encountered by participants; specific components of the collaboration that they believe provoked sustained changes in their own approach to teaching mathematics; ways they might reimagine the collaboration to make it more effective).

The four teachers work in different continents (Africa, Asia, Europe, North America), teach at different levels (primary, secondary, tertiary) and were involved in collaborative groups that have different forms (a pre-service, Lesson Study group, an international teacher exchange program, a teacher/researcher network, an online teacher network). Table 10.1 provides an overview of the different collaborative groups, including their location, form, participants and foci.

Table 10.1 Overview of the four teachers’ collaborative groups

Interestingly, despite the diverse collaborative group locations, forms, participants and foci, there were several elements in common to the design of the teachers’ collaborative group experiences, including: a consistent overarching purpose for the collaborative work; the involvement of group members in setting goals for the work; a focus on increasing teachers’ technical capability; the provision of access to specialist expertise. Some observations related to each of these design elements are presented next.

2 An Overarching Purpose

An overarching purpose for the work of the collaborative groups is consistently articulated across the teachers’ accounts. Each group has a strong focus on the improvement of teaching practice with the subsequent outcome of improved student learning outcomes. The teacher from Malawi, for example, reports:

The overall aim was to improve the quality of mathematics teaching in Malawi. As such, Lesson Study was introduced as a way to achieving the aim. It was thought that Lesson Study will be a scaffold for student teachers and teacher educators learning in Malawi.

In the Malawi context, the collaborative group Lesson Study focused on improving the quality of teaching of mathematics educators working in the important area of initial teacher preparation.

The teacher from China reports that a strong focus of their teacher exchange program is on improving UK teachers’ mathematics knowledge and pedagogy:

The goal of the programme is to raise curriculum standards in the UK in mathematics by improving teachers’ pedagogical and subject knowledge, and to refine the curriculum to ensure that all pupils achieve their full potential in mathematics without anyone ‘left behind’.

The teacher from France reports that their collaborative group is part of the network of Institutes for Research on Mathematics Teaching (IREM). These institutes provide collaboration between mathematics teachers from schools to universities, focusing their work on issues related to improving mathematics teaching, disseminating outcomes and conducting professional development for teachers.

The teacher from the USA reports that thousands of mathematics teachers from around the globe are engaging in an online Professional Learning Network (PLN), “actively working to promote quality mathematics instruction, mentorships for new teachers, and curriculum development”. She notes that, although activities associated with the network have diversified over time:

Throughout these efforts, the main goal has remained the same—a grassroots ‘for teachers, by teachers’ professional learning network to improve the quality of mathematics instruction for our students.

The different groups’ focus on developing teaching quality is not surprising, given that this is a priority consistently articulated in education policies, guidelines and initiatives globally and, as such, it frames the profession’s improvement agenda. As suggested by Emeritus Professor Dylan Wiliam during a keynote presented in 2012, “every teacher needs to improve, not because they are not good enough, but because they can be even better” (Wiliam, 2012). The teachers’ accounts clearly show their commitment to continuous professional learning and improvement of their teaching practice.

3 Participation in Goal Setting

A design feature common to the collaborative groups is the involvement of participants in determining the focus of, and explicit goals for, their work together. Each group used particular approaches to identify their foci and goals. In the context of the mathematics educators’ Lesson Study in Malawi, Lesson Study members worked with one another to set a long-term goal to increase the pass rate of mathematics student teachers by 2021. In the teacher exchange program between Shanghai and England, different core content topics were purposefully selected by group members for use in each round of Lesson Study, with guidance from an expert from Shanghai Normal University.

The teacher–researcher network in France first collectively defined their main objectives, which were to focus on “developing classroom settings and tools”, related to logic and logical reasoning, and “disseminating these tools to teachers”. Then, over a period of 6 months, “each participant worked freely choosing his line of work”, presenting their work, thoughts and questions at group meetings where they then identified agreed areas of work, “based on the points of convergence and divergence”. And, the online teacher network in the USA formed different goals according to the particular work and learning tasks they engaged in, for example: virtual mentoring; book studies; sharing research articles; sharing of mathematics problems on websites; inviting the public into classrooms virtually.

Potential benefits of the approaches used to set goals in these collaborative groups include authentic engagement of teachers and a sense of ownership associated with the process and outcomes of the work. In addition, as noted by the teachers from Malawi, France and the USA, participants were able to define and receive differentiated support specific to their individual needs.

4 A Focus on Increasing Technical Capability

There is a consistent focus across all four collaborative groups on improving mathematics teachers’ technical teaching knowledge and skills. As mentioned earlier, in the case of the French teacher–researcher network, the particular focus was on supporting increased capability related to the teaching of logic and logical reasoning, and this focus targeted not only secondary mathematics teachers, but also a teacher of French language and university teachers. In Malawi, teacher educators had the opportunity to plan in great detail a research lesson, and then engage in observing, analysing and reflecting on the teaching of the lesson, as well as refining the lesson design. The teacher from Malawi emphasises the importance of improving his teaching practice through a fine-grained examination of teaching:

Planning for a research lesson involves putting forward teachers’ content knowledge of a particular topic and the best teaching practices that could be used. There is inclusion of critical thinking approaches, probing questions and challenging tasks. This has become my common practice when planning my lessons. More time is spent figuring out how students will think and learn a particular concept than the teaching itself.

Shanghai teachers involved in the China–UK teacher exchange were similarly very focused on increasing their technical ability when they explored how to teach selected topics in reform-oriented ways, in order to share with UK teachers. And the teacher from the USA describes various ways that teachers collaborate in their online network to increase their technical capability, including engaging in discussions about content and pedagogy, supporting one another with learning activity design and sharing effective classroom management techniques.

This focus on increasing technical knowledge and skills suggests that the collaborative groups that the four teachers were involved in are, by design, examples of what Jensen (2014), drawing on the research of Clement and Vandenberghe (2000) and Rosenholtz (1989), refers to as ‘active collaboration’.

Active collaboration, in which teachers learn from each other through team teaching, joint research projects and classroom observation and feedback, has a positive impact on students. Collaboration that concentrates on administrative issues does not. (p. 7)

5 Access to Specialist Expertise

All four teachers express that they greatly value the way that their collaborative groups enabled them to access specialist expertise. This includes the expertise of other teachers, researchers, mathematics educators and other education professionals. The mathematics teacher from France reports the tremendous opportunities provided to their teacher–researcher network by working with a specialist teacher of French language, as well as university teachers. The French teacher, for example, “helped the mathematics teachers at the secondary school to define words that did not have the same meaning in mathematics and French” and supported the analysis of language used in teaching sessions. The teacher from Shanghai involved in the teacher exchange program reports that the importance of the collaborative partnership between Shanghai and UK teachers unfolded over time as the program developed, and teachers from both locations stated that they learned much from one another.

The teacher in Malawi notes, “I have learned a lot of teaching techniques through observing fellow educators teach”. And, the secondary mathematics teacher from the USA strongly emphasises the ways that the online teacher network she is a part of creates opportunities to share and learn with others. She reports:

Due to the rural nature of Oklahoma, there are minimal opportunities for professional development as it relates to mathematics education. […] when teaching a specialised curriculum such as AP [Advanced Placement] Statistics, the opportunities for traditional professional learning workshops are limited, often making it necessary to look for non-traditional methods of collaboration and networking […] One of the major benefits of collaboration via social media is the 24/7 access to teachers around the globe. With a single Facebook post or Twitter tweet, you can easily receive responses from teachers with a variety of teaching experience, backgrounds and geographic locations in a matter of minutes.

In addition to some shared design features, there were also some common elements related to the outcomes of the teachers’ collaborative group work and learning. A synthesis of lessons learned from the experience of the teachers and some reflections on future directions related to mathematics teachers working and learning in collaborative groups are provided after the presentation of the teachers’ stories that follow.

6 China–England Mathematics Teacher Exchange (Yiyi Chen)

6.1 Context, Purpose and Design of the Collaboration

The outstanding performance of Shanghai students in mathematics and science on the 2012 PISA assessment has attracted United Kingdom (UK) educators’ attention to the Shanghai mathematics curriculum (called Shanghai Maths), as well as to mathematics teaching and learning in Shanghai (called Teaching for Mastering). Shanghai students achieved a mean score of 613 (119 points above the OECD average) on the PISA in mathematics with an excellence rate of 55.4%. Since 2012, education authorities in China and the UK have had frequent contacts to explore mechanisms of collaboration. In February 2014, a delegation, including a representative from Ofsted (Office for Standards in Education) and other UK educational experts, visited Shanghai. During this visit, the Department of Education in the UK and the Education Commission in Shanghai agreed to launch a teacher exchange program, called the ‘China–England Mathematics Teacher Exchange Programme (MTE)’.

The goal of the programme is to raise curriculum standards in the UK in mathematics by improving teachers’ pedagogical and subject knowledge, and to refine the curriculum to ensure that all pupils achieve their full potential in mathematics without anyone ‘left behind’. The programme is funded by a UK federal grant and is jointly led by the United Kingdom’s Ministry of Education and the Shanghai Municipal Education Commission. Shanghai Normal University, UK National College for Teaching and Leadership (NCTL) and UK National Centre for Excellence in the Teaching of Mathematics (NCETM) are responsible for carrying out the programme.

By 2018, there have been three rounds of exchanges between Chinese and British mathematics teachers with more than 500 teachers from Shanghai and the UK directly participating. Additionally, approximately 12,000 British teachers have participated in Teaching for Mastering which disseminates what has been learned from the exchange programme. The programme will continue until 2023, with a goal of benefiting teachers from at least 9300 primary and 1700 secondary schools in the UK (Boylan et al., 2019). Major activities include reciprocal school visits and collaborative workshops. Throughout the programme, groups of UK teachers (Year 1–Year 9) led by experts from NCETM visit Shanghai schools. During these two-week visits, they observe classroom teaching, participate in school-based teaching research activities and attend workshops on developing Lesson Study.

Similarly, groups of Shanghai teachers, led by Dr. Xingfeng Huang from Shanghai Normal University, visit England. During the two-week stay in England, the Shanghai teachers teach lessons in local schools to explore how Shanghai teaching methods could be adapted and implemented in England. The details of the implementation have been evolving according to the goals in each round of exchange. My school (Cao Guangbiao Primary School) is one of the programme base schools that has served as the platform for the exchange programme in Shanghai over the past 4 years.

Before the exchange visit, the Shanghai participating teachers are divided into collaboration groups to prepare for the UK teachers’ visit, to learn about the UK education system and to update their knowledge about mathematics teaching in Shanghai. Some teachers who taught in the UK on their previous visit share lesson plans used, as well as their observations. When collaboratively planning lessons, teachers often simulate situations which may arise in UK classes. In addition, a Lesson Study group with ten to twelve teachers from programme-based schools explores how to teach a purposefully selected topic in a reform-oriented way, in order to share with UK teachers. With the leadership of Dr. Huang from Shanghai Normal University, each Lesson Study focuses on a different core content topic, such as equivalent fractions or fractions on a number line.

The collaboration not only takes place among teachers in Shanghai, but also between Chinese and British teachers. When the English teachers from the UK maths hub visit Shanghai, they are invited to teach a math lesson to the Shanghai pupils with the UK teacher and a partnering Shanghai teacher working together to develop the lesson plan. Similarly, when a Chinese teacher teaches in England, the partnering English teacher works with the Chinese teacher to develop the plan.

6.2 Overview of the Collaboration Outcomes

The programme has had a substantial impact on both British and Shanghai teachers and schools. In the UK, the Sheffield Institute of Education was commissioned by the Department of Education in December 2014 to undertake a longitudinal evaluation of the programme. Mixed methods have been used to analyse data collected over three academic years. Data from student testing has shown that in schools most directly involved in the exchange program, there has been an increase in pupils’ KS1 mathematics attainment. (The UK national curriculum is organised into blocks of years called ‘key stages’ (KS). KS1 refers to pupils at age 5–6, including Year 1 and Year 2.)

Moreover, survey and interview data has revealed that cohort I teachers (exchange participants in 2014–2015) improved in their beliefs about mathematics teaching and commitment to learning from Chinese mastery teaching methods. Observing the mastery teaching in Shanghai classrooms was perceived to have been particularly impactful. Cohort II teachers (2016–2017) particularly appreciated their visit to Shanghai, which deepened or challenged their previous understanding of Chinese mastery teaching methods. The visit by the Chinese teachers to England supported UK teachers’ implementation of teaching for mastering mathematics (Boylan et al., 2019).

To reflect on Shanghai teachers’ observation and understanding of mathematics teaching in Britain, Professor Huang edited a book called I Teach Mathematics in Britain, which includes chapters by programme participants from Shanghai. The book includes four sections and highlights the differences between Chinese and English mathematics teaching as observed by the teacher authors. One salient point noticed is that homogeneous grouping is prevalent in English primary school, which does not enable weaker students to progress as desired. Some teacher authors used vivid examples to describe differences in language and culture, and how best to address and learn from these instances. Additionally, some teacher authors highlighted their development as mathematics teachers, particularly in their use of hands-on activities.

6.3 Description of What Was Learned

Through participating in this project over the past 3 years, I have learned so much. I would like to highlight what I have learned: (1) learning to collaborate with UK teachers; (2) learning about cultural differences in defining mathematics concepts; (3) learning about differences in learning progressions; (4) learning to use research-based teaching practices.

6.3.1 Learning to Collaborate with UK Teachers

At the beginning of the collaboration, we thought it would be easy for us to know the learning situation in UK classrooms, simply by having the British teachers tell us what pupils had previously learned from the UK curriculum. But, we found it was often hard for us to design lessons based just on this information, and pupils having already been taught something does not mean that they have mastered it. Teachers from both countries realised that it is necessary to have deeper and more extensive discussions about students’ readiness to have effective collaboration. Now, teachers from both sides jointly select the teaching content and share more extensively about school culture and student learning. For example, my partner teacher, Mrs. Louis, gave me information about her school, Caroline Haslett Primary School, through videos and photos. These artefacts helped me to understand her classroom environment, as well as student homework and exercises, so that I could understand their learning situation in advance.

6.3.2 Learning About Cultural Differences in Defining Mathematics Concepts

Different cultural and educational backgrounds increase the difficulty of teaching in another country. Through collaboration, Chinese and British teachers are learning how to minimise the negative impact of these differences. For example, students recognise a rectangle and square at an early age, but when exploring the relationship between these shapes conceptually, UK pupils struggled to realise that a square is a specific rectangle. I was surprised because this had never happened in the Shanghai classrooms where I taught. My British partner teacher shared that, in British primary schools, attention is paid to the visual characteristics of the shapes of square and rectangle separately, without establishing conceptual connections between them. Furthermore, we checked the English National Curriculum which says that pupils should be taught to recognise rectangles (including squares) in Year 1, which is the age of five. This means that students in the class which I will teach were taught the concept 3 years ago. Moreover, the curriculum does not give a clear definition of rectangle, which surprised the Shanghai teachers.

After asking several teachers and pupils in the UK school, we found that the problem largely stemmed from ambiguous definitions. In the UK, most people think that the rectangle is a shape with four right-angles and two pairs of opposite sides equal but different in length. In Shanghai textbooks, a rectangle is defined as a quadrilateral with four right angles and two pairs of opposite sides equal. To understand further how a rectangle is defined in British textbooks, we consulted the textbook Maths—No Problem, which is recommended by the government. It says that the opposite sides of a rectangle are always parallel and equal. Based on this, a rectangle should include square. Thus, we believe that the student learning difficulty could be solved by appropriate practice.

After some deliberate practice, we were delighted to see that pupils could understand the concept very well. This makes me feel deeply that the difference between the pupils’ mathematical achievement in the two countries is not due to pupils’ different learning abilities, but to cultural differences. Reflecting on my own teaching, I now realise how important it is to focus on the knowledge that students have already learned and the context in which they learned it. This new idea leads me to think further about lesson planning.

6.3.3 Learning About Differences in Learning Progressions

Through the China–UK collaboration, we have discovered substantial differences in learning progressions in mathematics content between Shanghai and the UK. For example, when I was teaching addition and subtraction within 100, I found that UK pupils had a weak foundation of addition and subtraction within 20, an issue which will likely result in them having difficulty learning addition and subtraction with larger numbers. This weak foundation may be caused by the school teaching plan which showed they have spent almost half of the term learning multiplication involving 2 s, 5 s and 10 s, without any review of addition or subtraction. Nevertheless, British teachers insisted on continuing to teach this content.

To seek a solution to this dilemma, I posed the following in the WeChat group of the programme team: how to design the lessons about addition and subtraction within 100 when pupils are not fluent with operations within 20. (WeChat is the most popular social media platform in China, and is routinely used by Shanghai teachers to share teaching resources and ideas, and to discuss problems in teaching.) Teachers in the group provided various suggestions. For example, some teachers suggested using songs to help pupils remember the number bonds of 10. These ideas prompted me to study further the teaching content of addition and subtraction within 100 and clarify the relevant content in the Shanghai textbook. After comparing the presentations in Shanghai and British textbooks, it was found that the initial operation relies on the fluency of decomposition of number. As a result, I added a warm-up activity of reciting Make 10 songs in the teaching design. While students were excited and interested, they also naturally made sense of the decomposition. Based on this design, my teaching in the UK went well.

6.3.4 Learning to Use Research-Based Teaching Practices

Teachers in Shanghai conduct self-evaluations and reflect on the reform of mathematics teaching in China–UK workshops. I will illustrate with the example of an on-going Lesson Study of fractions on the number line which occurs in two stages. First, the learning trajectory of the content (fractions on number line) across grades in the Shanghai textbook was examined and the ways of presenting the content in different textbooks were compared. To understand student readiness, we gave a pre-test. To broaden our understanding of the presentation of the topic, we also consulted textbooks from other countries.

Finally, learning goals were set: (1) to find the position of the proper fraction; (2) to use the number line as a learning tool to compare the size of the fraction; (3) to have a preliminary experience in the integration of numerical and pictorial representations. The second stage includes the cycles of design, implementation, reflection and revision. After teaching the first class, teachers shared their thoughts about this lesson one by one, such as the large capacity of the whole lesson and insufficient teaching time. For language sentence patterns, such as two one-thirds is two-thirds, the meaning of fraction can be emphasised.

Drawing on the feedback from other teachers, and my self-reflection on the lesson, I changed my lesson plan mainly to focus on understanding how to locate fractions on a number line while using comparison of fractions to help students build links between fractions and integers on the number line. The lesson was taught a second time using the revised lesson plan. Pupils had more time for discussion and communication, and naturally established the relationship between the previous knowledge of fractions and number lines. For example, I asked pupils to find more fractions on the number line after they had found the fractions with denominators 2, 3 and 4. Students discussed in groups and then shared their group’s ideas in the whole class. They realised that all fractions can be found on the number line, because they could go on forever. Furthermore, with the help of language sentence patterns like four-thirds means four one-thirds, they could locate improper fractions on the number line.

7 Collaborative Work Within an IREM (Christelle Fitamant)

7.1 Context, Purpose and Design of the Collaboration

The IREM of Brest is part of the network of Institutes for Research on Mathematics Teaching. These institutes provide a collaborative organisation between mathematics teachers from schools to universities. They can work together on issues of mathematics teaching, disseminate their outcomes and conduct professional development for teachers. The meetings take place on the premises of the IREM at Brest University.

7.1.1 Birth of the Group

In 2009, new mathematics curricula were set up. During the annual conference planned by the Brest IREM, there were informal discussions about the teaching of logic in secondary school. These discussions led to an observation: on the one hand, the students receive little teaching of logic, while, on the other, the teachers feel a lack of resources to develop a practical teaching of logic as described in the curricula. Subsequently, a working group was formed to focus on this subject in September 2010.

The group consisted of six people: two mathematics teachers from Brest University, three mathematics teachers from secondary schools and a French teacher, working both at a secondary school and at a teacher-training university institute. In a rather traditional way, the role of university teachers is to bring theoretical content and analyse the activities proposed to the students. Teachers at the secondary school design class sessions, test them in their classrooms and analyse them. The French teacher designs class sessions for French courses and points out the different language elements used in mathematics and in French.

7.1.2 The Collaboration Purpose and Design

During the first meeting, we collectively defined our main objectives: at first, developing classroom settings and tools (like exercises, test, mind mapping, ...) wherein students can practice logic and logical reasoning without theoretical courses and, secondly, disseminating these tools to teachers.

One of the university teachers proposed to lead the group. We decided to meet once a month, Friday afternoon, at Brest University and we communicated regularly by e-mail. One of the participants (often the same) took care of the minutes of the meeting and reminded the work to be done by each participant with an e-mail just after the meeting and just before the next meeting. During the first 6 months, each participant worked freely, choosing his line of work. At each meeting, the participants presented their work, their thoughts and their questions. This allowed the group’s leader to identify areas of work based on the points of convergence and divergence upon which we agreed to work.

For the secondary teachers, the goal is not to teach theoretical logic, but the university teachers use theoretical logic with their students from the first year. So, we needed to design practice activities for secondary school that prepare students for the academic logic of the university and design additional practice activities for the university students that connect with the secondary school.

One of the objectives of our collaboration was to create class activities that can be used in mathematics courses, French courses in secondary school and university. Together, we created the same written tests for each of these levels. We asked, in our school, teachers who did not participate in the group to take the test to their students (fewer university teachers were involved in this phase).

With the data from the results of the test, we were able to assess the needs of our students. In order to prepare students for the teaching of logic at the university, the secondary teachers suggested that theoretical logic professional development be set up for all mathematics teachers by the university teachers.

During our meetings, we found that the new mathematics curricula and schoolbooks did not provide any progression in logic learning. Consequently, we had to plan a teaching of logic from the first year to the last year of secondary school. We have designed practice activities fitted to the level of our students, which are sequenced to progress from simple to more complex logic concepts over time. The French teacher helped the mathematics teachers at the secondary school to define words that did not have the same meaning in mathematics and French, and the secondary mathematics teachers recorded their course sessions so the whole group could analyse the sessions.

7.2 Overview of the Collaboration Outcomes

Written tests were conducted in all educational levels from first year of high school to first year of university. The activities for secondary school students were tested by secondary teachers and analysed by the whole group. The statistical results from the test and the analysis of the class sessions were published in the logique au fil de l’eau brochure.

The needs of secondary teachers regarding the notions of logic were identified and gave rise to a professional development. It was set up with the help of the Brest IREM Director during the annual conference. This professional development lasted one afternoon and could not address all of the secondary teachers’ questions. Some logic concepts (the most significant for secondary teachers) were reviewed and a list of resources was given to the participating secondary teachers, in order to supplement their learning from the day.

Our work dissemination was made thanks to the logique au fil de l’eau, a brochure published by the French APMEP (Association des professeurs de mathématiques de l’enseignement publique). In this brochure, the mathematics secondary school teachers described the sessions and their analysis. A university teacher wrote a logic course for the secondary school mathematics teachers. The French teacher and university teacher wrote a text on the links between logic and language. Moreover, secondary mathematics teachers conducted logic workshops at conferences organised either by several IREMs or the APMEP.

The Inter-IREM Committee of secondary schools is a commission which includes secondary teachers and university teachers from several IREMs. This commission has meetings in Paris five times per year. Several IREMs have also worked on logic teaching, so a group named ‘logic’ was set up within the commission. A broader collaborative work has been put in place with several IREMs. Our group chose a representative, a secondary teacher, who goes to Paris and has exchanges with the representatives of other IREMs about logic teaching during the Inter-IREM Commission meetings in Paris. Currently, members of the commission are writing a brochure summarising work on logic teaching from several IREMs.

7.3 Description of What Was Learned

7.3.1 Factors that Supported or Limited the Collaboration

The existence of an IREM was the key element in setting up this collaborative work. The IREM is a place of discussions about mathematics teaching, known to all the teachers from primary school to the university and easily accessible. The annual conference provides an opportunity to discuss without the institutional hierarchy. Additionally, an IREM has a library of teaching resources to which all teachers have access, so that when teachers want to improve their practice or have a question on mathematics teaching, it is natural to look to an IREM for collaborative work. Furthermore, the institution recognises IREMs, so the schedule of teachers who work at an IREM can be arranged so they can attend the meetings.

An issue was the organisation of remote work: often, we want to present a perfect document to the group, but it takes a lot of times to prepare and, even then, our document was not perfect for the group. It is difficult to present an unfinished work to the group, but the discussions are more open. Furthermore, collaborative tools have not been fully explored. Some of us did not use these tools and the others used different tools (Google Drive, Dropbox, email, ...). This situation has limited our ability to exchange our work between the meetings. One of the lessons learned was that the group members should take time to choose and learn how to use the same digital tools.

7.3.2 Challenges Encountered by the Participants

Another secondary teacher and I were to lead professional development for teachers for the first time. We were apprehensive about facilitating this experience for other teachers. The teachers who attended the professional development did not have a lot of knowledge about the teaching of logic, but they did teach the same level that we do in mathematics. Some participants did not want to teach logic without theoretical logic. We needed to convince them that our practical activities have a good impact on students’ progress and can prepare secondary school students for the academic logic at university.

All the participants of our group prepared this first professional development. We chose to prepare a debate and presentation of our work rather than a course of logic. During the session, one participant of our group was present with the teachers attending the professional development, in order to feed the debate between the attending teachers and the leaders of the professional development, if necessary. Finally, teachers who participated in the professional development worked well during the sessions with us, but it is difficult to determine how they use this training in their classroom. After this first experience, we continue to lead additional trainings with less apprehension.

During our work, we discussed with other IREM groups outside of ours at Brest. I was appointed to be the reporter of our work during the Inter-IREM Commission meetings in Paris. On this occasion, I met specialists in mathematical logic who work together in a committee with secondary teachers. They needed the point of view of several secondary teachers, so I accepted to join this committee. Since then, I continue to meet them regularly, and now I plan the committee meetings with another secondary teacher regarding mathematics learning in secondary school.

7.3.3 Changes in My Approach to Teaching Mathematics

This collaborative work provoked some changes in my mathematics teaching practice. It forced me to have more perspectives on my teaching. In a collaborative work, there are different points of view of the participants, sometimes contradictory. Each participant has to argue, defend his or her point of view and be able to make it evolve through the others. For example, during our work, I had to explain to the other participants why I chose some activities. I learned how to analyse the activities to be convincing, and now I continue to analyse the activities that I propose to my students. We also recorded course sessions and we listened to the recordings together to analyse the students’ reactions. Since the study of these recordings, I believe I am more attentive to the reactions of my students.

Teachers are alone in class in front of their students; collaborative work is an interesting way to improve teaching practice.

8 The Improving Quality and Capacity of Mathematics Teacher Education in Malawi Project: A Norwegian and Malawian Collaboration (Lameck Dition Sandram)

8.1 Context, Purpose and Design of the Collaboration

Collaboration in mathematics takes different forms and is practised in different contexts. This section, discusses experiences of collaboration in Lesson Study. This is a case of Malawi, a Southern African country of an area of 118,484 square kilometres and with a population of about 18.7 million people. The collaboration is actually taking place at Machinga Teacher Training College, one of the eight public teacher training colleges for primary school teachers in Malawi.

It all started at a national level as both a professional development program and a network of teacher educators from three Teacher Training Colleges. The program was under the Improving Quality and Capacity of Mathematics Teacher Education in Malawi Project, with funding from the Norwegian Program for Capacity Building in Higher Education and Research for Development (NORHED). The project was a collaboration between the University of Malawi and the University of Stavanger, and the overall aim was to improve the quality of mathematics teaching in Malawi. As such, Lesson Study was introduced as a way to achieving their aim. It was thought that Lesson Study will be a scaffold for student teachers and teacher educators learning in Malawi. Forty-six participants were involved in the program.

From the national level, Lesson Study activities trickled down to college level. At Machinga Teacher Training College, nine mathematics teacher educators were involved. A Lesson Study group was formulated and, in the group, there were five male educators and four female educators. I took the leading role of the group and, with the other educators as members, a goal was set. A long-term goal was to increase students’ achievement by improving the pass rate from the current 40 percent to 95 percent by 2021. Generally, students’ performance in mathematics was not impressive and, as a mathematics section of the department of mathematics and science at the college, we thought of putting in place strategies that could assist to rectify the problem, so we welcomed Lesson Study as one strategy.

Eight mathematics educators collaborated in the planning of the research lesson by developing a lesson plan, identifying teaching and learning resources, and observation tools to be used during the teaching of the research lesson. One of the eight mathematics educators taught the research lesson, while the other seven observed and collected data. Thereafter, a reflection process was initiated and the teacher educators discussed the research lesson and shared their experiences. The experiences were also shared at a national level during another professional development workshop, with teacher educators from two other colleges and experts from the project.

8.2 Overview of the Collaboration Outcomes

The first outcome of the collaboration is change of attitude. As indicated in one of the limitations to collaboration, the attitude of some educators who were not ready to have their lessons observed by fellow educators greatly changed. Educators are increasing their flexibility, accommodating presence of observers and taking part in the sharing of teaching experiences.

The second outcome is upon improvement in instruction. Lewis and Hurd (2011) argue that, “If you want to improve instruction, what could be more obvious than collaborating with fellow teachers to plan instruction and examine its impact on students?” (p. 3). Indeed, through collaboration, educators are able to develop lessons rich in critical and problem-solving strategies. This is helpful both to students and to teachers. Students are challenged with activities that keep them active throughout the lesson. For instance, in one of the research lessons, students were asked to model the addition of 45 and 16. The students came up with different ideas like using place-value boxes and tins or using an abacus and counters. Of interest was the use of stones in the place-value box instead of sticks. At the same time, student teachers are also developing teaching skills and chances for them to use critical thinking approaches. CORD (1999) argues that many teachers tend to interpret the learning environment according to their own experience as students − that is, they teach the way they have been taught. So, the likelihood that student teachers will use ideas of Lesson Study in their teaching, using critical thinking and problem-solving approaches to be specific, after having experienced it themselves, is very likely.

8.3 Description of What Was Learned

8.3.1 Factors That Supported Collaboration

There were some factors that supported collaboration in the teaching of mathematics. The first one was environmental in nature: that is, the context in which the collaboration was taking place. The college had everything the teacher educators needed to carry out Lesson Study. Rooms and curriculum materials were available, and students were also in college. The college administration gave the group a go-ahead and made teaching resources available.

Culturally, teachers are lifelong learners. It is in their tradition to seek knowledge. They would always want to learn to update their knowledge base and, when such a chance unveils itself, they go for it. Any initiative that proves to be productive in improving achievement of students is often taken seriously by teachers. Lesson Study came at a time when it was needed most. There was a need to understand a reviewed teacher curriculum. This called for a collective effort of teacher educators to understand its contents. Lesson Study was the timely solution and motivator to that cause.

8.3.2 Factors That Limited Collaboration

On limitations, the size of the class involved was big. There were about 40 student teachers involved during the research lesson. That affected mobility of the teacher educator, as well as the students during the lesson, because of limited space in the classroom. It also became difficult for the teacher educator to reach every student and give individual assistance. Resources were also inadequate for every student to be in contact with them. For an effective follow-up on each and every student during a lesson, it could be good to have not more than 20 students in one class. A Lesson Study lesson requires full understanding of how instructions are influencing learning in each and every learner, and this is only possible where the size of the class is small.

Practically, most schools in Malawi have large classes and that will take some years to be solved. I see this as the greatest challenge, and it cannot be overlooked when planning for lesson study. However, there are a number of aspects with research lessons that can be accomplished and improve learning other than focusing on the learning of individuals: for example, team planning, collective reflection and use of critical-thinking approaches can enhance learning. Hence, I feel modifying some areas of the Lesson Study process to suit the Malawian context can help teachers to carry out Lesson Study in highly populated classrooms.

Another limiting factor, which came as a surprise, was the unwillingness of some teacher educators to participate in some stages of the Lesson Study. They could neither make themselves available during planning, nor accept the role of teaching the research lesson. The thinking that they were the best teacher educators kept some participants away from the collaboration process. The solution to this has been geared towards attitude change. This is being coupled by allocating two educators to one class, so that they can plan and teach as a pair. By working in pairs, people will see the relevance of sharing ideas and working as a team.

It was also not all that easy for the leader of the collaboration group to lead through a model which was new to all of them. This demanded more time for the leader to study and search for more information, so that a right track could be followed. This helped the leader to become more knowledgeable about the Lesson Study.

8.3.3 Components of the Collaboration That Provoked Sustainable Change

The following paragraphs describe three components of Lesson Study that provoked sustainable change in my approach to the teaching of mathematics. The first component is planning. Lesson planning is a daily activity that teachers do as they prepare for their lessons. It is in its natural context to see a teacher planning for lessons. Success and failure of a lesson depends heavily on its planning. However, planning a research lesson collaboratively becomes more rewarding than planning it individually. Planning for a research lesson involves putting forward teachers’ content knowledge of a particular topic and the best teaching practices that could be used. There is an inclusion of critical thinking approaches, probing questions and challenging tasks. This has become my common practice when planning my lessons. More time is spent figuring out how students will think and learn a particular concept than the teaching itself.

Furthermore, I have learned a lot of teaching techniques by observing fellow educators teach. The way they approach their lessons and taking it through developmental steps is an important practice: for example, starting a lesson by asking students a challenging question, then building the lesson on students’ responses, until the objectives of the lesson are met. This was also the case with other educators who were involved in the collaboration.

The third component is about conducting a research lesson. In this stage, one member of the lesson study group teaches a research lesson, while the rest of the group members observe and collect data. A data-collection tool is used where experts observe the lesson and collect relevant data. This is the data that inform instruction and bring improvement. When one teaches a lesson individually, without colleagues monitoring the proceedings, very little data is obtained. However, the practice of collecting data when teaching is what is very important. I now treat my lessons as sources for data collection for my learning about my students’ learning.

I am able to identify gaps in my teaching and learning of students. For instance, I was teaching about subtraction of mixed numbers, e.g. 7$\frac{1}{3}$ − 3$\frac{2}{5}$. I asked students to explain how they could solve the problem. One student explained, “First subtract three from seven and get four, then subtract two-fifths from one-third”. The student proceeded up to this stage: 4 $\frac{5-6}{15}$. And then the student said, “We take one from four, the whole number, and add to five (minuend) to make fifteen and then subtract six from the fifteen, which means we have now 3 $\frac{15-6}{15}$”. This assisted me very much because I was able to understand the student’s thinking on the problem. The gap was identified and ways of handling the problem were shared with other educators.

Reflection forms an integral component of the Lesson Study circle. This is the stage where the teacher and members of the Lesson Study group share data from a research lesson. Members share what they feel are the successes and the challenges of the lesson they observed, as well as what they learn about the students’ learning. Collectively, they once again plan the lesson, fusing in new ideas and approaches and eradicating elements of the lesson that are not significant in realising the objectives of the lesson. There is power in reflection and every time a lesson is being reflected upon, new insights are realised. No wonder reflection has become part of the Teacher Education Philosophy of the reviewed Initial Primary Teacher Education, which states, “to produce a reflective, autonomous, lifelong learning teacher, able to display moral values and embrace learners’ diversity” (Malawi Institute of Education, 2017, p. ix), and is being implemented now.

It is my wish that, 1 day, in-service primary school teachers be introduced to Lesson Study as a form of collaboration. This will greatly assist to improve instruction and the performance of learners in mathematics. That might take a long time, but it will be a good undertaking. The challenge I anticipate is a lack of research skills in the primary school teachers. Lesson Study lessons are research lessons and research skills are very crucial to the Lesson Study process.

9 Professional Learning and Collaboration Via Social Networks (Shelli Temple)

9.1 Context, Purpose and Design of the Collaboration

Jenks Public Schools is a suburban school district serving approximately 12,000 students in grades Pre-K to 12 in Northeast Oklahoma. Oklahoma is a mostly rural state, located in the Central Plains of the United States, with two main metropolitan areas, Tulsa and Oklahoma City. Jenks is a southwestern suburb of Tulsa, and the school boundaries cover the city of Jenks, as well as a section of the southern city limits of Tulsa. Jenks High School, serving grades 10–12, has a graduating class of approximately 750 students, with a mathematics department of 15 regular education and five special education teachers, teaching classes ranging from Algebra 1 to Calculus 3. During my 20-year tenure at Jenks High School, I have taught a variety of classes, with a current teaching assignment of Advanced Placement (AP) Statistics, Geometry, and Forensic Science and Data Analysis. In addition, I serve as our site Professional Development Co-ordinator, as well as on our Leadership Team.

Due to the rural nature of Oklahoma, there are minimal opportunities for professional development, as it relates to mathematics education. In many districts, there may only be one or two mathematics teachers, so Jenks is fortunate to have a large department of educators. However, when teaching a specialised curriculum such as AP Statistics, the opportunities for traditional professional learning workshops are limited, often making it necessary to look for non-traditional methods of collaboration and networking.

In the late 1990s, teacher message boards and email listservs were vital elements to online teacher collaboration, but in the mid-2000s, online teacher journals, called blogs, started to become more popular, followed soon by the use of social media, such as Twitter, Facebook and Instagram, as a way to connect these teachers together and create real-time collaborative conversations revolving around lesson ideas and pedagogy. There are now thousands of mathematics teachers around the globe who are active participants in an online Professional Learning Network (PLN) called the #MTBoS, or the Math Twitter Blog-o-Sphere. Through these online connections and social networks, the members of the #MTBoS are actively working to promote quality mathematics instruction, mentorships for new teachers and curriculum development.

In general, the collaborative nature of the #MTBoS is fairly informal, using social media hashtags and Facebook groups to connect subject-area teachers. However, there have been organised efforts regarding book studies, outreach at national professional learning events and even a face-to-face, multi-day, math teacher conference, called Twitter Math Camp (TMC), during the summers of 2012 through 2018. Throughout these efforts, the main goal has remained the same—a grassroots ‘for teachers, by teachers’ professional learning network to improve the quality of mathematics instruction for our students.

9.2 Overview of the Collaboration Outcomes

The nature of social media as a medium for collaboration lends itself to opportunities for discussions with a wide reach, both geographically and longitudinally. A single tweet can create a multi-hour or even a multi-day discussion with contributors around the globe, all sharing their input and guidance on an activity, lesson plan or classroom management advice. Collaborations via the #MTBoS have resulted in pedagogical books being written, open-source software and curriculum, free sharing of lessons and Desmos activities, and even public outreach programs such as ‘Math on a Stick’ at the Minnesota State Fair.

The impact of the online teacher collaboration is, in many ways, difficult to measure, but the effects are far-reaching. One example of this can be seen with the success of Twitter Math Camp (TMC), drawing both presenters and participants from the greater #MTBoS community, as well as from the local hosting region. In 2012, the original TMC hosted approximately 40 teachers from a variety of teaching experiences and backgrounds for a three-and-a-half-day workshop. In 2014, Jenks High School hosted TMC and the workshop had grown close to 150 teachers. At its end, in 2018, TMC had impacted close to 600 teachers and classrooms through in-person attendance plus an additional unknown number through virtual interactions.

While the physical TMC conference lasted three-and-a-half days each year, the virtual portion of the conference lasted year-round. In the weeks and months preceding each TMC, the conference presenters were hard at work preparing for their sessions. Since many of these presenters were not in geographic proximity, they organised their presentations using online collaboration tools, such as Google Docs and Skype calls to hash out the details. During the actual conference, the whole-group sessions, such as the keynote speakers and the ‘My Favorites’ portions, were videoed and shared via the YouTube channel, plus participants were encouraged to ‘live-tweet’ from each session using a social media hashtag, so that people not in attendance could follow along. In the days and weeks following the conference, the conversations continued as teachers shared their learning experience through blog posts and Twitter discussions.

9.3 Description of What Was Learned

One of the major benefits of collaboration via social media is the 24/7 access to teachers around the globe. With a single Facebook post or Twitter tweet, you can easily receive responses from teachers with a variety of teaching experience, backgrounds and geographic locations in a matter of minutes. The exposure to teachers from different cultures and teaching environments enriches the personal professional learning experience, which can lead to richer experiences for students, from both a pedagogical and a social justice aspect. With traditional professional learning opportunities, teachers tend to be limited due to geographic proximity and, as a result, the participants generally come from very similar backgrounds and teaching experiences. In contrast, developing a PLN via social media allows for a diversity of perspective, which in turn creates a robust and responsive professional learning experience as classrooms and social environments evolve.

Within the AP Statistics community, one limiting factor of traditional collaboration is isolation, with most AP Statistics teachers being the only person in their district and surrounding area that teaches the course. Through the power of social media, these teachers, including myself, are no longer alone. By reaching out through a Facebook post or via Twitter, new AP Statistics teachers have ready access to experienced teachers to help guide and mentor them through the course and how best to teach challenging content. Around 15 years ago, a young teacher from Hattiesburg, Mississippi, reached out on a then-active teacher message board looking for another AP Statistics teacher to discuss course content and share teaching ideas.

My response to that post and the virtual mentorship that resulted is a key reason why I am so invested in the power of social media for teacher collaboration. This commitment to helping new AP Statistics teachers has continued throughout the years, including the development of a Facebook group in 2015 called the ‘AP Stat Teachers Support Group’ and through the online AP Statistics community on Twitter. While every teaching context is unique, these online partnerships are very empowering to teachers, as they seek to best prepare their students for the standardised AP exam given each year in May—a test that can earn students with college credit for specific scores.

Throughout the history of the #MTBoS, educational trends can be seen, often before they show up in traditional professional learning opportunities. One of the most powerful movements that I have been involved in was in the area of student assessment. Approximately 10 years ago, several prominent teacher bloggers started implementing Standards Based Grading (SBG) in the mathematics classroom, based on works by Robert Marzano, Dylan Wiliam, Ken O’Connor and others. During this same time frame, I had become disillusioned with traditional grading methods and the inability of the grading system to communicate clearly what my students knew. The desire to read the works of these authors and discuss thoughts with my virtual colleagues led to the creation of an online book club via Twitter, with weekly group chats to support the use of formative and summative assessment in the classroom using hashtags of #sbarbook and #eduread for easy curation. The change from traditional grading systems and appropriate use of formative assessment tools is one that has been slow to take off in mainstream educational circles, but is quite common within the online teacher community.

A more recent collaboration of the #MTBoS is the use of instructional strategies that truly inform and transform student learning. Through the use of rich mathematical tasks, teachers and students alike are growing as mathematical learners and thinkers. The online teacher community regularly shares these ‘low floor—high ceiling’ or ‘open middle’ problems with each other, presenting them freely for feedback and use by teachers around the globe. Several websites have been developed and crowd-sourced by the #MTBoS, including Visual Patterns (www.visualpatterns.org), Which One Doesn’t Belong (https://wodb.ca) and Open Middle (www.openmiddle.com).

Within my own classroom, these rich tasks have been vital in helping students see themselves as mathematical knowers and doers. In the past, there has been a disconnect between the mathematics classroom and what mathematicians actually do—look for patterns, explore curiosities and enjoy challenging problems. By utilising these tasks, students are able to showcase their thinking and reasoning skills and truly to see the joy and beauty of mathematics. While lengthy conversations with distant colleagues and websites full of tasks can definitely have a positive impact on the classroom and student learning, another powerful influence can be found through the collaborative efforts of the ‘180 blog’.

In the United States, an average school year consists of 180 days, so, several years ago, a few teachers decided to use social media platforms as a way to invite the public into their classrooms virtually to observe the day-to-day learning that takes place. Originally, the ‘180 blog’ utilised online blogging platforms, such as WordPress or Blogger, to journal these daily activities, but, over time, this idea has morphed to the micro-blogging platforms of Twitter and Instagram. By using the social media hashtag of #teach180, teachers are able easily to share photos each day of student work and learning activities with their parents and local stakeholders, as well as with the greater #MTBoS community. This initiative, whether through a traditional blog or through Twitter or Instagram, is an excellent way for teachers to receive a daily dose of inspiration and to spark new ideas for the classroom.

All of the above initiatives are important to the improvement of mathematical instruction, but, by far, the most powerful outcome of the #MTBoS is the relationships formed by teachers who would otherwise not know each other. The exposure to teachers from a variety of teaching environments, with diverse student populations, the ability to get teaching and learning advice from experienced educators and the real-time feedback for lesson development are the most valuable aspects of the #MTBoS community. By forming friendships across time zones and geographic boundaries, teachers are no longer limited by the size of their physical mathematics department within their district or surrounding area; they now have infinite opportunities for learning and collaboration within the virtual world.

10 A Synthesis of Lessons Learned

As noted in the introduction, and evidenced in the teachers’ accounts, the four collaborative groups were very different from one another, and there is much that can be learned about the nature, design and implementation of collaborative groups from the teachers’ experiences in these groups. Following is a synthesis of some observations related to lessons learned from the teachers’ experiences, together with questions to provoke consideration of how these might inform mathematics teachers’ participation in collaborative groups in the future.

10.1 Factors Supporting Collaboration

The four teachers identified and described a number of factors that supported the work and learning that took place in their collaborative groups. These included cultural, social, environmental and physical factors. Some of the key supporting factors were:

10.1.1 A Culture of Learning

The teacher from Malawi proposed that there is a strong tradition among teachers to see themselves as life-long learners and this positively influences their participation in professional learning:

Culturally, teachers are life-long learners. It is in their tradition to seek knowledge. They would always want to learn to update their knowledge base, and when such a chance unveils itself, they go for it. Any initiative that proves to be productive in improving achievement of students is often taken seriously by teachers.

In the French example, the connection to an existing organisation, the Institutes for Research on Mathematics Teaching (IREM), was considered to be a key element in setting up their collaborative group. The IREM is familiar and accessible to teachers, respected by them, well-resourced, and “when teachers want to improve their practice or have a question on mathematics teaching, it is natural to look to an IREM for collaborative work”. Participation of teachers in the collaborative group was encouraged and the scheduling of their attendance supported. Implicit support of this kind can be invaluable to the success of collaborative activities.

10.1.2 Motivation and Timing

For participants in the Malawi Lesson Study collaborative group, there was high motivation to be involved as the mathematics educators needed to understand a revised curriculum:

Lesson Study came at a time when it was needed most. There was a need to understand a reviewed teacher curriculum. This called for a collective effort of teacher educators to understand its contents. Lesson Study was the timely solution and motivator to that cause.

The teacher from the USA reported that her motivation for collaborating with others online was sparked by a young teacher reaching out for support:

Around fifteen years ago, a young teacher from Hattiesburg, Mississippi, reached out on a then-active teacher message board looking for another AP Statistics teacher to discuss course content and share teaching ideas. My response to that post and the virtual mentorship that resulted is a key reason why I am so invested in the power of social media for teacher collaboration. This commitment to helping new AP Statistics teachers has continued throughout the years, …

She also described how, at various points across her teaching career, she has had opportunities to source information and discuss ideas with virtual colleagues and these occasions have motivated her to reflect on and transform different aspects of her teaching.

10.1.3 Available Resources

The availability of needed resources, including physical space and materials, student-participants’ time and approval from administrators, was seen as key to implementing the collaborative group Lesson Study in Malawi:

The first one was environmental in nature. That is the context in which the collaboration was taking place. The college had everything the teacher educators needed to carry out a Lesson Study. Rooms and curriculum materials were available, and students were also in college. The college administration gave the group a go-ahead and made teaching resources available.

10.1.4 Partnership

The teacher involved in the China–UK exchange highlighted the importance of a genuine partnership in their collaborative group. She noted that, as their exchange program unfolded, the teachers from both locations realised they needed to work closely together to ensure deeper understanding about teaching and learning in the two countries, and she reported that now, “both sides jointly select the teaching content and share more extensively about school culture and student learning”.

10.1.5 Connection to Others

The teacher from the USA reported that a positive aspect of online teacher collaboration is that it connects teachers together to “create real-time collaborative conversations revolving around lesson ideas and pedagogy”. She noted:

The nature of social media as a medium for collaboration lends itself to opportunities for discussions with a wide reach, both geographically and longitudinally. A single tweet can create a multi-hour or even multi-day discussion with contributors around the globe, all sharing their input …

She described benefits of the exposure to teachers from different cultures and teaching environments through online platforms, contrasting these with traditional professional learning opportunities:

With traditional professional learning opportunities, teachers tend to be limited due to geographic proximity and, as a result, the participants generally come from very similar backgrounds and teaching experiences. In contrast, developing a PLN [Professional Learning Network] via social media allows for a diversity of perspective, which in turn creates a robust and responsive professional learning experience as classrooms and social environments evolve.

She also highlighted the important role that online collaboration can play for mathematics teachers who specialise in less common courses, or who may be working in less-populous areas:

one limiting factor of traditional collaboration is isolation, with most AP [Advanced Placement] Statistics teachers being the only person in their district and surrounding area that teaches the course. Through the power of social media, these teachers, including myself, are no longer alone.

10.2 Factors Limiting Collaboration

The teachers also identified and described factors that placed limitations on their collaborative group outcomes. These included:

• 1. Participation avoidance

  The teacher from Malawi reported that some mathematics educators were initially unwilling to participate in components of the Lesson Study process, requiring the group to implement strategies to provoke attitude change. A practical strategy that they applied involved pairing educators to work together, so that those who were reluctant would see the relevance of sharing ideas and working as a team.

• 2. Resource constraints

  In Malawi, when teaching the Lesson Study research lessons, student class sizes were not conducive to the particular instructional approaches they were trying to implement, and lesson materials for students were limited. The teacher from Malawi has signalled that there may need to be some modifications to Lesson Study approaches for the Malawi context, because most schools in Malawi have large classes.

• 3. Leading ‘new’ ideas and approaches

  Leading change in areas and approaches that are new is challenging. The teacher in Malawi noted that it was not easy for him “to lead through a model that was new to all of them”, and he needed additional time to study and research relevant information to support the group. Similarly, the French teacher reported that, when she was to lead professional development for teachers together with a colleague for the first time, they were apprehensive. She noted the pressure she felt when working with peers, and the need “to convince them that our practical activities have a good impact on students’ progress and prepare secondary school students for the academic logic of university”. She also reported, however, that she continues to provide training with less apprehension and, additionally, she contributes to a committee working with mathematics specialists.

• 4. Communication protocols and tools

  The French teacher reported that one issue arising in their context related to the ways that their group members organised and shared their work remotely. She noted that they were reluctant to share documents that were not ‘perfect’ with one another, and the preparation of such documents requires time. She also noted that there was a lack of consistency in members’ use of collaborative tools, limiting the ability of the group to exchange work between meetings. A lesson learned, she suggested, is that, “group members should take time to choose and learn how to use the same digital tools”.

10.3 Provoking and Sustaining Personal Professional Learning and Growth

In their accounts of their collaborative group work and learning, each of the four teachers mentioned aspects of personal professional learning and growth that were sustained over time. The teacher from Malawi reported three particular components of Lesson Study that “provoked sustainable change” in his approach to the teaching of mathematics. The first involves collaborative planning. He noted that, “planning a research lesson collaboratively becomes more rewarding than planning it individually”, because you need to contribute and scrutinise knowledge and ideas, and you learn much from what others contribute. The second component involves conducting the research lesson. He suggested that, because colleagues monitor the lesson proceedings, there is an opportunity for collecting data about one’s teaching:

When one teaches a lesson individually without colleagues monitoring the proceedings, very little data is obtained. However, the practice of collecting data when teaching is what is very important. I now treat my lessons as sources for data collection for my learning about my students’ learning.

The third component involves reflecting on the teaching of the research lesson and refining the lesson. The teacher noted, “There is power in reflection and every time a lesson is being reflected upon, new insights are realised”.

The teacher from Shanghai noted that the following key aspects influenced her professional learning over time: close collaboration with exchange partners; the detailed examination of mathematics curriculum and teaching approaches; thorough planning of Lesson Study research lessons; close examination of lessons taught (including receiving feedback about the lesson, reflecting on the lesson and refining the lesson). The exchange program that she is involved in has been in place for several years and, as she noted, “the details of the implementation have been evolving according to the goals in each round of exchange”. The program duration and its evolving nature (facilitating program relevance and currency) appear to have contributed to her sustained learning over time.

The teacher from France suggested that her involvement in the collaborative group provoked some lasting change in her mathematics teaching practice. She believes it forced her to consider more perspectives about teaching:

In a collaborative work, there are different points of view of the participants, sometimes contradictory. Each participant has to argue, defend his point of view and be able to make it evolve through the others. For example, during our work, I had to explain to the other participants why I chose some activities. I learned how to analyse the activities to be convincing, and now I continue to analyse the activities that I propose to my students.

She also noted the impact that analysing recorded teaching sessions had on her:

We also recorded course sessions; we listened to the recordings together to analyse the students’ reactions. Since the study of these recordings, I believe I am more attentive to the reactions of my students.

The teacher from the USA reported a variety of ways that different online opportunities have stimulated her collaborative activity and improved her mathematics instruction. However, she suggested that the most powerful outcome of her involvement in her collaborative online network is the relationships formed with other teachers. It is these, she suggests, that provoke and sustain her commitment to on-going learning and improved teaching practice:

The exposure to teachers from a variety of teaching environments, with diverse student populations, the ability to get teaching and learning advice from experienced educators, and the real-time feedback for lesson development are the most valuable aspects of the #MTBoS community. By forming friendships across time zones and geographic boundaries, teachers are no longer limited by the size of their physical mathematics department within their district or surrounding area, they now have infinite opportunities for learning and collaboration within the virtual world.

10.4 Reflection and Questions

The stories of the four teachers included in this chapter provide evidence of considerable work and learning in different collaborative contexts around the globe. Their experiences provide some insights into what is key to mathematics teachers working and learning in collaborative groups, and the kinds of professional learning and growth that can be provoked by, and sustained following, participation in collaborative groups.

A question of interest arising from the examination of the four teachers’ experiences is: how can the professional growth of teachers within and across collaborative groups be described, understood and compared? Teacher learning is very complex and any model selected to understand and describe the process must acknowledge this complexity. One model potentially useful for thinking about the sustained learning and growth of teachers working in collaborative groups is the Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002).

The Interconnected Model recognises both professional growth as an inevitable and continuing process of learning and the complexity of professional growth through the identification of multiple growth pathways between four domains in which ‘change’ might be located: the External Domain (change in external sources of information or stimuli); the Personal Domain (change in professional knowledge, beliefs and attitudes); the Domain of Practice (change in practice through professional experimentation); the Domain of Consequence (change in perceived salient outcomes related to classroom practice). Change in one domain is translated into another through the mediating processes of reflection and enactment as shown in Fig. 10.1.

Fig. 10.1

The Interconnected Model of Professional Growth. (Clarke & Hollingsworth, 2002, p. 951)

The Interconnected Model asserts that any processes of professional growth occur within the constraints and affordances of the enveloping change environment. The four teachers in this chapter had change environments particular to their locations and their collaborative groups. The teachers were provided with ‘external sources of information or stimulus’, as they experienced new things in their collaborative groups. Then, as they engaged in their collaborative group activities, they individually reflected upon and experimented with different aspects of their mathematics teaching practice, sometimes building new knowledge or adjusting beliefs and attitudes about their teaching.

The teacher from Malawi, for example, reported that one aspect of his professional growth that has been sustained following his participation in the collaborative Lesson Study at his teacher training college relates to his use of ‘new’ teaching techniques. Figure 10.2 displays the pathway—or growth network—that represents his learning using the Interconnected Model of Professional Growth. The teacher reported that he: observed colleagues using teaching techniques that were new to him (External Domain); applied some of these in his classroom (enactment—arrow 1); reflected on the implementation of these (reflection—arrow 2); continued experimenting with some techniques in his classroom (enactment—arrow 3); reflected on the outcomes associated with his use of the new techniques (reflection—arrow 4); reflected on the value of the techniques, establishing new beliefs about their efficacy (reflection—arrow 5).

Fig. 10.2

An example growth network

While the learning journey of each of the four teachers was undoubtedly unique, it is possible that some of the pathways that led to their professional growth—as can be represented by the Interconnected Model of Professional Growth—might be similar. Further exploration of these pathways (for these teachers and others) could provide opportunities to develop our understanding of teacher professional growth within and across collaborative group contexts, in particular the different growth pathways experienced by participating teachers and the factors that might promote or inhibit professional growth.

The experiences of the four teachers also provoke other questions about mathematics teachers’ work and learning in collaborative groups. Such questions include:

• Which support elements are absolutely critical to prove effective for collaborative groups?

• What kinds of processes facilitate authentic partnership roles in collaborative groups?

• How might flexibility and responsiveness be effectively incorporated in collaborative group work and learning?

• How might competing professional learning needs of collaborative group members be effectively managed?

• How might the processes and products of collaborative group work be effectively shared?

• How might effective collaborative group activities and outcomes ‘reach’ more mathematics teachers?

• How might cross-cultural insights related to mathematics teachers working and learning in collaborative groups be effectively shared?

Depending on how each of these questions is interpreted, they could relate to one or more of the four study themes examined in this volume: Theoretical Perspectives on Studying Mathematics Teacher Collaboration; Contexts, Forms and Outcomes of Mathematics Teacher Collaboration; Roles, Identities and Interactions of Various Participants in Mathematics Teacher Collaboration; Tools and Resources Used/Designed for Mathematics Teacher Collaboration.

For example, the question ‘What kinds of processes facilitate authentic partnership roles in collaborative groups?’ could be investigated in relation to the theme Contexts, Forms and Outcomes of Mathematics Teacher Collaboration, with a particular focus on the processes and form of the collaborative groups, or in relation to the theme Roles, Identities and Interactions of Various Participants in Mathematics Teacher Collaboration, with a focus on partnership roles. The association between the different study themes was observed during the ICMI Study 25 Conference, with several participants noting that, although their papers were located in one study theme group, they were also relevant to other themes.

It is anticipated that consideration of questions such as those listed above, as well as further exploration of teachers’ growth pathways using the Interconnected Model of Professional Growth, might usefully inform directions for mathematics teachers working and learning in collaborative groups in the future.