Teachers Learning Mathematics: Professional development research

Introduction

The issue of mathematical knowledge of teachers has been documented in New Zealand for 80 years, but no effective long-term solution has been found; indeed, the situation has worsened. For example, the 2004 New Zealand Ministry of Education Teacher Census (Ministry of Education, 2005) showed that 25 percent of secondary school mathematics teachers had no university mathematics qualification—a rise from 21 percent in 1977.

This one-year research study, undertaken in 2007, aimed to investigate the development of teachers’ own mathematical knowledge for teaching. Seven secondary teachers used action research methodology to investigate, develop and evaluate some aspect of their mathematical knowledge, completing two cycles over the year. In addition, the seven teachers came together to reflect upon the mathematical development of practising secondary teachers from their collective experiences. Two researchers from the Department of Mathematics at The University of Auckland co-ordinated the project, supporting each teacher individually in their own study, and facilitating the collective reporting of experiences. Resources from the Department of Mathematics were available to the project.

The seven teachers were from secondary schools in the Auckland region. All were currently involved in teaching senior secondary mathematics classes, and all had had previous connections with The University of Auckland. A one-year project is minimal in terms of teacher change, so having a group of teachers in which mutual trust was already present was an advantage. The restriction to Auckland was a practical one.

The original description of the project was for two researchers to work with eight senior mathematics teachers on eight research studies associated with their classrooms. One teacher was promoted to head of department and withdrew from the project because of the pressures of work.

The project attracted attention from researchers at Oxford University (Dr Anne Watson and associates) and The University of Michigan (Dr Deborah Ball and associates). Using the support of the BeSTGRID network we held discussions with them on issues relating to the project. Their insights into our research contributed to both the conduct and analysis of the project.

The research project is reported on three levels. First the individual teachers’ cycles of experiences are described. Then we consider the role of the group as a community that supports professional learning. Finally we discuss the role that increasing the depth and breadth of understanding of mathematical knowledge may play in promoting effective teaching of secondary mathematics.

Aims and objectives

The study aimed to investigate the development of mathematical knowledge in teachers.

Research questions

1. What developments will a practising mathematics teacher be able to make in their own mathematical practice using an action research methodology supported by an external researcher?
2. What do teachers who have engaged in such mathematical development think about its worthwhileness and practicality as ongoing professional development?
3. What do teachers who have had the opportunity to focus on a mathematical aspect of their teaching have to say about the importance of mathematical knowledge of a teacher with respect to their classroom practice and the mathematics learning of their students?

The project sought to understand how teachers can engage in the learning of mathematics to enhance their teaching as part of their professional lives. It gave a group of teachers the opportunity to undertake such learning in a supported fashion, and to reflect on and investigate its effect. A minor part of the study considered the students as learners, asking the teachers to reflect on the nature of mathematical knowledge that is required for effective classroom pedagogy.

This project is part of a much wider initiative being led from the Department of Mathematics at The University of Auckland. The university-based researchers are involved with the New Zealand Institute of Mathematics & its Applications (NZIMA), the national mathematics Centre of Research Excellence, to provide effective outreach to teachers and the promotion of mathematics. They are also involved in international research collaborations with Oxford University and The University of Michigan that are examining the nature of secondary teacher mathematics knowledge. This TLRI project became part of the research agenda of this collaboration.

A further aspect of this project was that the mathematics learning initiatives established were disseminated to the other teachers in the schools of the teacher-researchers. The teacherresearcher became a leader in their school community of practice and worked with their colleagues to set up mathematics learning environments for themselves.

Research design

The international literature on which this study is drawn concerns two main areas: first, the theorising led by Deborah Ball’s team at The University of Michigan about the type of mathematical knowledge that is indicated for teachers; second, the nature of professional development generally, including the potential of teacher research, particularly action research.

Literature: Mathematical knowledge

Researchers argue that it is important for professional development to have a general content base (e.g., Garet, Porter, Desimone, Birman, & Suk Yoon, 2001; Kennedy, 1999). Shulman’s (1987) work supports a trend away from the generic professional development offered in the 1980s towards programmes that recognise the importance of enabling teachers to learn how to teach particular content. Teachers need to understand the conceptions students are likely to hold of particular concepts, to know what representations and analogies are likely to be helpful and to be aware at what age students are developmentally ready to learn particular concepts (Kennedy, 1999). Garet et al. (2001) list content focus as one of the three core features of effective professional development activities.

The theoretical model developed by Ball and her colleagues (Ball, 2003) to describe content knowledge involves a picture with six components, from conventional subject learning on one side to Schulman’s (1986, 1987) pedagogic content knowledge on the other (see Appendix VIII (Delaney, Ball, Hill, Schilling, & Zopf, 2008, p. 179)). Their analysis suggests that knowing mathematics for teaching often involves making sense of methods other than one’s own and that teachers need opportunities to unpack mathematical ideas. The New Zealand best evidence synthesis on effective pedagogy in mathematics (Anthony & Walshaw, 2006) affirmed these conclusions in the New Zealand context.

A previous study conducted by one of the university-based researchers investigated the effect of creating a situation in which teachers could again be learners of mathematics (Paterson, 2007). These teachers showed clear evidence of being stimulated by the experience and, in a significant number of cases, re-energised for teaching and re-evaluating their practice. There was, however, variation in the levels of interest shown by teachers in different mathematical topics. Hence a design component of the present study was to invite teacher-researchers to choose their own mathematical area of study.

Literature: Professional development

The voluminous literature on professional development was drawn upon to develop key aspects of this study. Action research methods have a long history (see Mertler, 2006) and are used in many fields nationally (refer, for example, to the New Zealand Action Research Network at The University of Auckland, and the New Zealand Action Research and Review Centre at Unitec, Auckland). Its benefits include direct links between theory and practice, teacher empowerment and alternative ways of viewing (the “insider eye”). In practice it has the advantage of being immediate, community building and contributing to professional growth. Downsides include the unstructured and unconventional nature of the research, making it more, not less, difficult to implement effectively than standard methods.

Focusing on the context of this study, there is good evidence (e.g., Zeicher, 2003) that teacher action research into their own mathematical knowledge and their presentation of it in class will:

• highlight for teachers their own needs and motivate them to continue to work on developing specialist knowledge
• begin to develop a teacher community awareness of the issues surrounding specialist mathematical knowledge by reporting their experiences to colleagues in appropriate forums.

The literature on action research resonates with three important themes in general professional development literature: those of teacher control (e.g., in New Zealand, Begg, 1993); informed reflection (e.g., in New Zealand, Britt, Irwin, Ellis, & Ritchie, 1993); and professional communities (e.g., in New Zealand, Higgins, Tait-McCutcheon, Carman, & Yates, 2005; Timperley, Wilson, Barrar, & Fung, 2007).

Begg (1993) concluded that teacher control was of central importance to any professional development initiative. Britt et al. (1993), in the Teachers Raising Achievement in Mathematics project, concluded that reflection is a key mediating process by which teachers develop their knowledge and beliefs. Timperley et al. (2007) in their best evidence synthesis noted that opportunities to participate in a professional community of practice were more important than whether the professional development took place in school or externally. Effective communities provided teachers with opportunities to process new understandings and challenge problematic beliefs, with a focus on analysing the impact of teaching on student learning.

The action research model as adopted for this project was designed with teacher control, informed reflection and the establishment of a professional community in mind. The teacher-researchers controlled their own investigations, but were supported in their design, reflection and reporting by both the community (see below) and the university-based researchers. An intended additional outcome of this design was for the teacher-researchers to be inducted into research practice, and put in a position to become key players in ongoing research.

Methodology: Modified action research

The teacher-researchers used a modified action research model. Two cycles were attempted by each teacher-researcher, all completing at least one. In each cycle the classic action research stages (plan, act, observe, reflect) were undertaken in some form. Planning involved deciding on the area of study and means by which personal knowledge would be enhanced. Action was undertaking further study in the area and, in some cases, using that knowledge within the classroom. Observation was of each teacher’s own confidence, increased understanding and, in some cases, of student responses. Reflection took place within the wider research group discussions.

Each cycle began with teacher-researcher identification of the area to be addressed. Each teacher was asked to focus on either: a mathematical topic that they had found difficult to understand themselves; or one that they understood but felt they had not taught effectively; or one that they felt they had taught effectively, but which the students found difficult. Notes of their decisions were made at a group meeting by the university-based researchers.

The teacher-researchers each had control of the action they took to address the area identified. With a university-based researcher, each teacher-researcher planned a personal mathematical development scheme for this topic, including data collection and analysis concerning the influence on themselves or their teaching. This interaction was recorded in emails and notes by the university-based researcher.

The reporting phase was modified to share the process with the university-based researchers. At group meetings the teacher-researchers discussed their investigations, new learning, classroom consequences, observations and intended future actions. During the meetings the participants questioned and prompted each other to articulate plans for the next phase of the project. These meetings were recorded both on video and in meeting notes. One university-based researcher took notes and used the video to make a first draft of a written report for each teacher-researcher. Other sources, such as notes from individual meetings, school visits and emails, also contributed to the report. The draft was circulated between the writer, the other university-based researcher and the teacher-researcher involved, with amendments and extensions being made until a final version was agreed.

The group presentation at the New Zealand Association of Mathematics Teachers (NZAMT) conference became part of the action research cycle as preparing for it provided another chance to reflect on the manner in which the project had impacted on their practice.

We consider that three aspects of this modified action research model enhanced the outcome of the research. First, the sharing of the reporting load made it possible for teachers with busy professional lives to participate effectively. Second, having experienced researchers involved in the writing up of observations and reflections added a “researcher eye” to these reports. Third, the public nature of the reporting in group meetings and a joint conference presentation made the need for robust reporting more explicit and more immediate.

Methodology: Evaluating professional development

The collective part of the research (that is, the evaluation of the professional development overall) was naturalistic in its methodology. The term naturalistic refers to both the overall paradigm adopted and the data collection methods. The paradigm assumed that, although we set up a particular professional development situation, we recognise that there are unlikely to be direct deductive links between characteristics of the constructed situation and the outcomes (teacher attitudes, classroom practices, teacher knowledge). The teachers’ context is too complex for such claims, but a naturalistic approach enables the researchers to make evidence-based statements.

The data collection was naturalistic in that it was qualitative, and used participant observation, recorded group discussions and collaborative, evolving report writing.

The key components of the professional development model used in this study were:

• deliberate selection (for practical reasons) of teacher-researchers who had:
  • an existing relationship to the university-based researchers
  • shown prior interest in undertaking research
  • good communication skills y teacher-researcher self-selection of individual mathematical topics for study from mathematical topics that caused them concern
• conducting the study within a professional community of mathematics teachers, although each teacher had control of their own topic.

Four complementary sets of information were used in an iterative process to evaluate the impact of these components:

• The study involved extensive self-reporting by the teacher-researchers on all three aspects of the model. The reports took the form of both written reports and videotaped verbal reports in group meetings.
• The working of the community was recorded; that is, attendance at group meetings and participation in presentations. The videotaped discussions contain evidence of the collaborative and mutually supportive nature of discussions.
• The university-based researchers made observations of the teacher-researchers’ community and their participation in the professional development activities.
• The activities of participants after the study is also relevant information. This includes further involvement by the teacher-researchers in university study (both research and mathematical study), and activity in their own teacher communities as participants or leaders in professional development workshops.

How were the teachers’ experiences integrated into themes for the final report? Throughout the project the group continued to discuss the professional development experience as a whole as it impacted upon each teacher-researcher. Several themes evolved and were discussed explicitly at later meetings. Additional themes became apparent when the final teacher-researcher reports were read.

One of the university-based researchers compiled an initial draft of the collective report, collating the individual reports and identifying recurrent themes (several of which had already been identified by the group). The second university-based researcher then distilled the reports to create a report of suitable length and continued to develop the theme statements. Finally, two additional issues emerged through discussions with the Oxford University adviser, Dr Anne Watson, as a result of her viewing the videos. The report was shared with the teachers after each of these phases to ensure that their original reports were not misrepresented.

The project

First half-year

The objectives for each school term were documented in the milestone reports of April and July 2007. Progress at both times matched the original objectives.

The April report noted that the project had the strong support of principals of participating schools, and that the teacher-researchers reported that their developing mathematical understanding had begun to affect their approach to teaching.

By that time ethics approval had been gained for the project and each teacher-researcher had established their mathematical topic for development, as follows:

Anne Blundell	Proof and argument
Linda Crisford	Trigonometric functions
Margaret de Boer	Logarithms
Anna Dumnov	History of e and algebra
Jason Florence	Groups
Yoko Raike	Mathematical modelling
Peter Radonich	Mathematical modelling
Morgan Rangi	Probability at Year 13 (This participant subsequently withdrew from the project due to pressure of other work as he was made a Head of Department, Mathematics.)

The group met on 1 December 2006 to discuss the project and begin to establish each teacher’s area of interest. Another meeting was held in February 2007. Two further meetings were held in the second term (each attended by six teachers).

In July it was reported that each of the seven teacher-researchers had completed Cycle 1 and all teachers had been able to introduce aspects of their learning into their classes. Some had observed direct impacts on their students. Reports of Cycle 1 had been videotaped and draft write-ups completed.

Cycle 2 entered the planning stage and a joint presentation was developed for the NZAMT conference.

All teacher-researchers were visited in their schools during Term 1 or Term 2, and had been in contact with the university-based researchers. Meetings with the teachers focused on both their area of interest and how their developing understanding was impacting on their teaching. Teacherresearchers kept records of their own progress and thoughts, and notes of the meetings made by the university-based researchers were returned to the teacher-researchers for elaboration. A spinoff of the school visits was the establishment of informal contact with other members of each teacher-researcher’s department. Support for the teachers included books and articles relating to their field of interest, notes from lecturers on the topic (groups) and discussion of concepts in the topic.

This project is part of a shared collaboration with a team from Oxford University. During the first half-year we had two video-linked meetings. Video material from the project was viewed and discussed. (The Oxford team did the same with one of their projects.)

Second half-year

Most teachers continued to work on a second phase of the same topic that they had studied in the first two terms. Support for teachers continued in a similar way. The teacher-researchers shared their findings with their departments in a variety of ways—as presentations, shared units of work, talks for students or informal feedback to colleagues.

At a meeting on 22 November all the teacher-researchers’ reports were finalised and collected (see appendices).

NZAMT presentation

An extended meeting was held on 2 August at which we created a structure for the presentation and planned a practice presentation for the Mathematics Education Unit (Table 1). This was held at The University of Auckland on 7 September.

Table 1 Presentation to Mathematics Education Unit

	Anne	Anna	Linda	Jason	Peter	Margaret	Yoko	Time (minutes)
Introduction (Bill)								5
How I chose the subject					Yes			10
My learning process			Yes			Yes	Yes	25
Learning as a group				Yes				10
Impact on students now	Yes	Yes						20
In the future (Judy)								20

On 27 September we gave the final presentation at the NZAMT conference to an audience of 25. We received positive and interested verbal feedback after the presentation, two teachers asking directly to be able to be involved in further, similar work.

Project findings

The project was undertaken in two ways:

• Each teacher-researcher was supported individually in their exploration of a new mathematical topic and how possible this was while being a full-time teacher.
• A series of teacher meetings was held where all researchers shared their experiences and insights to form a group view of the efficacy of this mode of mathematical development and its pedagogical usefulness.

The research project is reported on three levels: first, the individual teacher-researchers’ cycles of experiences are briefly described (full accounts are in the appendices); then we consider the model of professional development, especially the role of the group as a professional learning community; and, finally, we discuss the role that increasing the depth and breadth of understanding of mathematical knowledge may play in promoting effective teaching of secondary mathematics.

Individual experiences

Anne Blundell (see Appendix A)

Anne began by reading around the ideas of proof. She reports that, during university education, she struggled with proofs and came to see the art of proof as “memorisation without insight or understanding”. She reported that, “It was with trepidation that I set out to correct a personal weakness this year, with the aim to increase my own confidence and . . . expose the art of proof to my own students.”

It took three months before she attempted her first proof and then it was nonmathematical. She then began discussing proofs with colleagues and tentatively introducing them into her teaching. Later in the year she introduced more proofs into her teaching, sometimes using an approach to scaffolding the teaching of proof described by Robyn Averill (McIntyre, 2006).

On a meta level she made it clear to her students that she had become a learner. She also actively involved her colleagues in her learning process. Her presentation at NZAMT began with her chronicling the ways she had changed—as she expressed it, “from proof-o-phobe to proof-ophile”. She reports that as the year progressed, “I found when teaching I would not settle for anything less than a full explanation, this being the fundamental reasoning behind proof.”

An interesting feature of Anne’s experience was that she found it linked in with other professional development undertaken during the year; for example with ICT training on Excel.

She considers that she has a long way to go in fully developing proofs in her mathematics teaching.

Linda Crisford (see Appendix B)

Linda’s mathematical content learning focused on developing a better understanding of the trigonometry that is taught at school. She freely acknowledged that this was an area in which her understanding was very weak—although it was not her first choice for a focus topic. Her exploration took her down two parallel courses. She began by “first asking teachers in my department what they felt about mathematics”, establishing that they believed that mathematical stimulus in schools for teachers is lacking. She then asked how they had felt about mathematics when they were younger, and their responses reflected the enthusiasm they had felt for learning and doing mathematics—“It was a turn-on.” They then discussed ways teachers could continue to feel enthused about their subject. In consequence, part of her focus also ended up being about the special nature of learning maths as teachers and as students.

Her ability to get other teachers in her school “on board” in her learning journey is probably a function of her very high energy and enthusiasm levels. However, her mode of engaging her colleagues in her learning may well be a key generalisable finding in this project. She summarises:

Because the teachers in my department are so busy I found that they were happy to assist me so long as I was fast; that is, ‘Can you just show me how to do this in four minutes? If I don’t get it in four, forget it—I’ll come back another time.’ Of course this meant I got seven to 10 minutes of personal one-to-one explanation. I also asked if I could be part of the class silently just for me to learn trig or anything they were teaching because they were so clever. This was marvellous for me as if I didn’t understand everything it didn’t matter because no one was going to ask me to answer a question (safe from embarrassment).

She reflects on the crucial nature of the input of other people into one’s learning process and of ways to incorporate this into busy teachers’ lives.

Margaret de Boer (see Appendix C)

Margaret’s decision to focus on logarithms was driven by both her students’ assessment needs and her own realisation that there are “a lot of applications of logs, and that I didn’t understand them well enough myself”. She was encouraged by other participants’ enthusiasm for history to begin to explore the history of logarithms. She originally refers to the log laws as, “these three or four little rules”, and was surprised that the students could not simply learn them. She reflects on ways her positive and negative energy may have affected her students’ performance in mathematics:

I thought how some of my enthusiasm may have had a positive effect on the students. Maybe my disappointment with many of the students not passing the topic had also been communicated to them without my realising it.

I spent a lot more time thinking about the way I taught than changing the lessons.

In developing her lessons on logarithms she introduced historical detail and planned them so that the students could see the links and processes involved in the development of the concept. She reports that while the students “weren’t really following the process in the same way I was, they did get something from it”. She continues:

The form of logarithm rules at least must have seemed familiar after the exercise, because they were very good at practising them after that. They even did some for homework. I wish I had found a better way for teaching them the indices laws, as that seemed to hold them back still, but they got straight into practising the questions, much more than they usually would. [In] previous work we had done, they would just do the bare minimum of work required, but this time they worked through whole exercises. With the reduced level of algebra skills in the class, this practice strengthened their algebra skills in other areas as well.

Margaret no longer thought there were “three or four little rules” and had become aware of the conceptual connections underlying an understanding of logarithms and of the way an understanding of the history of a concept can inform one’s pedagogy.

Anna Dumnov (see Appendix D)

From the outset Anna wanted to find ways to introduce more history of mathematics into her teaching. In 2005 she had studied complex numbers while on a Royal Society of New Zealand scholarship and found that her increased knowledge of the history of the development of these numbers had enriched the ways she discussed them with her students. In this project she turned her attention to algebra and to e, the base of the natural logarithms. In the first cycle of her studies she read around the topic, both generally and in the work of Ian Stewart whose Letters to a Young Mathematician (Stewart, 2006) both she and Anne have found seminal and inspirational.

She reported that her Year 13 class was more receptive to her historic input than her Year 11 class. She attributed this to the fact that “I taught most of this group the previous year and they are used to me going on small ‘excursions’ into history from time to time.”

In the second cycle of her engagement Anna worked alongside a student developing her understanding of the nature and history of e. This exploration began during a lesson on logarithms to her Year 13 class. She reports:

As this group is used to my incorporating historic developments into the lessons, it is only natural that they expected me to explain in detail why we use this strange number and the questions, What kind of number is this? and What is special about it?, as well as Who was the first to think about it?

The exploration enthused one student to make e the subject of her special presentation to a group of senior scholars at the school. She and Anna drew extensively on Maor’s book, e: The Story of a Number (Maor, 1994) as they worked together on her project.

Anna “wanted students to see that not only was I learning new historical information, but that this information leads me into the mathematical context, which was new for me as well”.

At the first meeting she expressed the opinion that in learning about the history of mathematics she might not learn a great deal of mathematics. Bill replied, “You may be surprised.” She has referred to this interchange on a number of occasions, acknowledging the accuracy of his response.

Jason Florence (see Appendix E)

Jason initially attempted to better understand concepts in university-level algebra and looked for ways to introduce some of the connecting ideas to his students. He began his search by working through a set of notes from a 300-level paper on integers, cryptology, groups, fields, polynomials and codes. After initially being enthusiastic, he found it a lonely process. Also, he could not really find a way to introduce the ideas to the students in a way that suited him and which would be meaningful.

The second phase of his learning was prompted by his studies in a Master’s-level statistics education paper. He found it very useful to be part of a group that met every week and he began to be fascinated by some of the concepts being discussed:

Had my second-to-last lecture and have been writing a paper about stats education. I think . . . statistical education is playing a significant role in my professional development at the moment and it seems silly to ignore the good learning I have done on this paper. After reading through my notes I have decided to go down the probability track. What I can gather is that people make bad calls due to a lack of understanding about probability.

He then developed an extensive unit of work for his department on the use of heuristics in probabilistic decision making. This was based on concepts identified by statistical education researchers following on from the work of Kahnemann, Slovic, and Tversky (1982). He presented this work at the Auckland Mathematics Association (AMA) statistics day in November.

Peter Radonich (see Appendix F)

Peter was interested in mathematical modelling and finding connections between doodling or drawing and working mathematically. His first explorations were inspired by Peter Hunter’s work on the heart, Len Lye’s creations and by the idea of building models using LOGO. He then focused on developing an approach to teaching algebra in which the students developed a “family” of creatures whose limbs increased according to a pattern. The first stage of the students’ creation was diagrammatic. They then produced tables, graphs and, finally, rules that described the connections between the family members. Validation of this approach was provided when his colleague reported how enthusiastically his students had responded to it.

He is now interested in the concept of “flow” as described by Mihaly Csikszentmihalyi (1990), how the geometry of the HIV virus features in how it operates, and the role 3D geometry software may play in learning mathematics.

Yoko Raike (see Appendix G)

Yoko’s work on this project appears as a series of small iterative cycles, focusing more and more closely until, in the end, she wrote, “I realise that at the end this project produced an answer to my fundamental question of why differential equations are useful. The question I did not originally regard as the fundamental.” Initially she said she was searching for a way to link big ideas across the material taught in the Cambridge curriculum, which she regarded as fragmented. She honed in on “rate” and then, after a meeting with Professor James Sneyd, a mathematical physiologist, looked more closely at differential equations that model the behaviour of HIV. She read around the subject in references he recommended to her.

She reported that rate became a more central concept in her teaching, with classes of all levels. She did four presentations based on the work in this project—at NZAMT, to her own department and two “lectures” for Years 12 and 13 students at Westlake Girls’ High School.

Yoko reflects that, “It turned out that the passage in Population Biology (Levin, 1984) gave me great joy in teaching this topic in spite of totally no progress in my skills in solving differential equations.” She continues:

I wondered why I did not develop this understanding of the usefulness of differential equations when I was a learner. My conclusion is that, as a learner, I focused only on how to solve equations. The rewards of learning at that time came from success in examinations.

She asks whether this joy she is experiencing makes a difference to her learners and, “Is it worth explaining the usefulness of differential equations in class?” She reaches the conclusion that, “I believe that it is worth explaining the usefulness of differential equations in class and a teacher having ‘fun’ with the topic must have an impact on his/her teaching.”

In 2008 she is going to continue her studies of mathematics and disease through short, contentbased courses at The University of Auckland.

Summary of individual experiences

Themes to emerge from the teachers’ very different experiences are the:

• importance of communication, openness, making connections between people and ideas and finding time to focus on one’s own mathematical development
• value of working in a domain the teacher identifies as potentially valuable to them and about which they were initially anxious
• value to students of seeing their teachers as learners y role of stimulation through reading, and talking to peers and experts
• power of teachers’ thirst for learning more mathematics.

In addition, we have realised again how important trust is in the learning situation, and how sensitively people who are exposing their vulnerabilities need to be handled.

We have questions about how to measure the change that the teachers report, and whether there is a negative side to increased enthusiasm—what Anne Watson refers to as “teacher lust”.

The professional development model

As noted above in the section on research design, the professional development model had three components: deliberate selection; teacher choice of area of study; and working as a professional community. These were evaluated using teacher-researcher reports, community activity records, university-based researcher notes and post-study activities:

• The teachers were enthusiastic in their self-reporting of the investigations and the impact on their teaching. It was clear that they had seen significant changes in their teaching practice. Their confidence had increased in relation to the topic or aspect they had chosen to examine— it had originally been an area in which they had an acknowledged weakness or lack of confidence.
• Meetings were well attended and retention was excellent. We only lost one participant very early on, owing to an increased workload on becoming a head of department. The teachers appeared to relate easily to one another and to grow in confidence as the year progressed. The final presentation at NZAMT demonstrated this publicly.
• Our observation as university-based researchers was that the participants grew in confidence during the project. We saw intense focus on self-identified areas of weakness and significant changes in their attitudes with respect to these areas over the course of the year. They were extremely proud of themselves and of the work that they had done.
• While we acknowledge that we selected teachers who had displayed an interest in research we have observed that they have continued to develop in this direction. One has a Post Primary Teachers Association (PPTA) study award, another a teacher fellowship, others are regular attenders at after-hours extension courses for teachers, two have made presentations at local teacher professional development days and one is actively involved in the local mathematics association. Three of four continue their Master’s study (the fourth has moved from the Auckland region to a new teaching position), and two began Master’s study immediately following the programme.

We concluded from the above that the model collectively was effective. We have no evidence to isolate any particular aspect of the model as being responsible for this, although the issue of a professional community of teachers came through strongly, and is reported in more detail below.

Role of the group as a professional learning community

In their reports and at the meetings, most of the teacher-researchers refer to the value they see in working together as a group. The collegial aspect of the project was evident from the first meeting:

Margaret leaves the first meeting at which Anna discussed her interest in history and, ‘Straight away I got out my old history of mathematics book, which I had fondly carried with me for the last 10 years, but rarely opened. There were always too many things that were more urgent and I couldn’t afford the time.’

Jason reflects: ‘It has been a long journey as well and at times I thought I may have taken on too much. My one regret is that I most likely will not get to work with such a diverse and great bunch of teachers [again]. Each of them brought something to the group, from Linda’s wacky and ‘out there’ antics to Peter’s steady and professional attitude.’

Linda reports: ‘For me, my colleagues have proved invaluable as they have helped me take more frequent academic risks and develop greater self-confidence, which has led to an upward spiral of not only my learning but that of the students I teach. This again stressed the importance of shared learning experiences. This has brought me to the realisation that it is important to build a mathematical community whether it is with our friends, colleagues or between schools—primary, intermediate and secondary—and university.’

Jason’s change of direction from algebra to probability was a result of the interaction between this project and his Master’s studies. If he had not been doing this project he may well not have had the impetus to develop the unit of work in the area he had become interested in while completing the statistical education course.

Anne refers directly to the effect the project has made to her approach to learning in other areas: ‘For example, in ICT training on Excel I developed a self-marking worksheet for the generalisation of the quadratic formula. This came about from students’ desire and requirement to understand quadratic patterning . . .’

Linda’s mode of working with her colleagues has led to her understanding more of the mathematics that underpins ideas in physics and how her colleagues teach trigonometry.

Yoko reported that she worked individually. She notes, ‘However, even if this is true it does not devalue having the project as a group. What has been generated through the group was the energy for carrying out the project. The contribution of Judy was huge but if I were working alone with her, I do not think I would have produced the outcome at the same level as I did with the group. [My] uncertainty and concern was eased by the sense of belonging to a group of people who were aiming to do the same thing.’

The reports of the teacher-researchers attribute much of the mathematical learning success of the project to collegiality. Not only is there a transfer of knowledge between teachers, but the sense of learning in a group and sharing the importance of mathematical learning also made it possible for teachers to progress. Although it had always been possible for the teachers to pick up their enthusiasm for mathematics at any time, it was only when they joined together and shared it that they actually engaged.

The reports contain several references to initial insecurity about approaching mathematical topics that the teacher-researchers felt they understood only poorly. In all cases, the group was a significant means of overcoming this insecurity.

The conclusions go further than this. Several of the teachers report their desire to continue working in such an environment, one expressing disgust when it was realised it would not continue. They also wished to involve other teachers, and other teachers who knew about the project wished to be included. There were no negative responses at all—the one teacher who withdrew doing so with disappointment because of time constraints.

We regard a strong outcome of this study to be further confirmation of the need for a professional community to be at the basis of professional development, not only for it to proceed effectively but also for its products to be enduring. To the extent that teacher accounts of the experience are valid, doubt is cast on what will happen to the gains from this project once the contact is lost.

In some cases, the effects were transferred to the professional group of teachers in the school, but this happened less often than expected, and more research needs to be done to understand why this is so. While the NZAMT conference presentation was a success, there was no noticeable effect on the overall participation of teacher-researchers in the wider professional community. We can hypothesise that this community is too big and meets too infrequently to provide the support needed.

We also conclude that this particular model of professional development was highly successful. It is difficult to know how far this conclusion can be generalised. Was it just this particular group (and if so, what was it about this group) that was successful? Was the timing right for these teachers on a personal or school level? Did the fact that it was an official research project make a difference?

We hypothesise some key elements, but emphasise that further research would be needed to fully establish the links:

• All the group knew each other prior to the start of this project and, in particular, all the teacher-researchers knew (and trusted) the university-based researchers. We believe that this level of collegiality is important, and takes time to be established.
• The link with university-level mathematicians appeared to be important for many of the teacher-researchers. We believe that maintaining links with mathematicians is important to gain the sense of authenticity about mathematics that teachers feel is needed for effective teaching.
• All teacher-researchers made their own decisions about the topics, and the direction of study— to the extent that the initial parameters of the study were sometimes not adhered to. We believe that this is further confirmation of the issue of teacher control over professional development noted in existing literature (Begg, 1993; Robinson, 1989).
• The university-based researchers in this study took significant responsibility for contact, steering teacher-researchers to resources or providing them directly, and writing up the results of meetings and reports. We believe that teachers are primarily focused on their teaching and classroom responsibilities, and that professional development needs to occur in a way that does not intrude on their time. This applies to the administrative issues. On the other hand, we also believe that there is good evidence that teachers who are inspired by some new ideas will find and invest large amounts of time in developing the ideas for themselves and their practice.

Role of teachers’ mathematical understanding in effective teaching

The teacher-researcher accounts are full of examples of their learning in the project directly affecting their teaching practices. Not only can this be read implicitly in their accounts, but they are also explicit about this effect. The project was designed on the assumption (from previous research) that teachers’ mathematical knowledge impacted upon their teaching. We can see in the data presented direct impacts of new mathematics learning while the learning is going on.

We have identified three aspects to the effects on teaching. They are manifested most obviously in the increased variety and richness of mathematical learning opportunities the teachers offered their students. Secondly, there is the effect on the students’ learning of seeing their teachers as learners. Further, there is evidence that the deep questions the teachers have been asking themselves have become part of their classroom practice. Below are examples of evidence from the teacher-researcher accounts under these three headings.

Increased variety and richness of mathematical learning opportunities

Anne no longer ignores the ‘grey boxes’ in the Barton textbooks that contain proofs. She says, ‘I found when teaching I would not settle for anything less than a full explanation, this being the fundamental reasoning behind proof!’

Peter presented his class an opportunity to develop their understanding of functions based on a ‘function family’ of their own construction, harnessing their interest in being creative and providing the opportunity to represent the functions diagrammatically, in table form, graphically and, finally, symbolically.

Jason presented the class with a hospitals-and-babies scenario. They began with an intuitive guess, and then ran a simulation. This was a hit with the students.

Yoko has found examples of differential equations that her students can understand and that describe a phenomenon in which they are interested, thus better helping the students comprehend the power of differential equations.

Effect on the students’ learning of seeing their teachers as learners

Anne told her students that she ‘had chosen an area of personal weakness to focus on as part of a research project’. We have no data on the effect this had on her students but would hypothesise that it would be positive.

Linda sat in on a number of colleagues’ lessons, in mathematics and in physics. The students will have observed her behaving as a learner, modelling how teachers can learn from each other, and not being scared to show that they need to learn.

Yoko says: ‘Does the statement I made earlier, ‘the passage in Population Biology gave me great joy in teaching this topic’, have any value to learners? . . . I believe that . . . a teacher having ‘fun’ with the topic must have an impact on his/her teaching.’

Changing classroom practice

Margaret began the project with the idea that the log laws were simple and that she only needed to tell them to the students for them to be able to learn to apply them. They were simply rules to be learnt and used. She did not make connections between them and the structure of the number system, nor did she use their historical development. Her report shows that she began to engage the students in trying to create logarithms of their own, introducing the process of development of concepts into the discussion.

Anna’s students became used to her raising questions about the origins of mathematical concepts, and now raise questions of their own. For example, on the introduction of e they asked, ‘What kind of number is this?’, ‘What is special about it?’ and, ‘Who was the first to think about it?’

Jason: ‘As I am Year 9 co-ordinator, I will set this up for the Year 9 course in Term 3. I am confident I have got something really good here . . . It has already had a huge impact on my way of thinking about how to teach probability and it will have a massive impact on what I do in the classroom. I wonder if I should do this just with my class or for the whole department?’

What lasting impact do any of these have on student learning? The project was not designed to assess any student impacts, and we believe that this is almost impossible in anything other than a very large-scale long-term study. However, our experience as teachers leads us to believe that it is likely to be possible to identify effects of the second of these characteristics (teachers indulging in learning and being enthusiastic for learning) from accounts from learners of their experiences as students of these teachers.

As mathematics educators we believe the potential for significant improvement in classroom learning is most likely to emerge from the third characteristic (deep changes to the way teachers think about particular aspects to teaching). The issue here is one of ensuring that the change in perspective is permanent or, better, continues to develop. We hypothesise that maintaining a professional community over a long period is likely to help in this respect.

The first characteristic (providing new, particular learning opportunities for students) is, we believe, likely to be the most short term and unstable of the characteristics, and therefore likely to have the least effect on learning. However, it should be noted that the evidence from the literature is clear that experiences that are richer mathematically are more likely to produce better learning (Ball, 2003; McGowen & Davis, 2001).

The future

The teachers have expressed an interest in continuing to study mathematics for themselves, making use of the methods they developed during the project—reading around their subjects, exploring areas they currently find problematic, continuing their exploration of their established areas of interest, working with colleagues, studying Master’s-level papers in mathematics education and attending short, content-based courses.

It is clear that learning communities were set up—both within the project group and in some of the teachers’ departments. We did not see evidence of increased participation in existing professional bodies. Whether the group will be sustained without regular meetings and outside input remains to be seen. The university-based researchers’ experience of groups of teachers from different schools maintaining a relationship outside personal friendships is that this does not happen without external input. Three of the teacher-researchers in this project will be full-time at university on study awards or fellowships in 2008, and this may be sufficient to keep the group together for another year. A follow-up project examining the long-term effects of this intervention could yield useful findings.

Our collective experience of school-based communities is that they also change and move on in response to local needs and priorities. Hence, continued professional community around mathematics learning is unlikely without external input. As one teacher-researcher noted:

We were also very fortunate to have two very focused and helpful mentors who gave the project direction, commitment and rewards. Sadly we heard that funding has not been approved to continue this project on and I think ultimately someone has made a dumb decision there. Nevertheless, it has been a great ride and one that I have really enjoyed despite the occasional stressful moments.

One of the researchers has set in motion a project that will continue to develop the model and to work with new groups of teachers in the future.

Limitations of the project

This was a small-scale study with good secondary teachers in supportive and mid- to high-decile schools who were familiar with the university-based researchers. This raises questions about whether the results would be similar with a larger group, less competent teachers or primary or tertiary teachers, in low-decile schools or with teachers with whom trust had not previously been established.

We believe that a larger group of similar teachers would have similar results—the issue would be whether the project could be replicated in terms of resources and support from outside. We believe that primary and tertiary teachers would also benefit from this, but have no evidence either way on this question.

The issue of teacher competence and trust with respect to the university-based researchers appears to us to be primarily concerned with insecurity about “not knowing”. Nearly all the teacherresearchers expressed some form of insecurity and discuss it explicitly. Whether a teacher who was struggling in class, or a teacher who did not know the external support staff well, would be as open with their weaknesses and as willing to address them is doubtful.

Our collective experience of low-decile schools makes us conclude that it is unlikely the results in such schools would be similar to this study (although one teacher was from a decile 2 school). We know from previous research that most teachers in high-stress environments with little collegial support on a day-to-day basis (features of low-decile schools) do not have the personal conditions necessary to undertake the type of activity that was evident in this project.

A final cause for questioning might be the observation that none of the results of this study were particularly surprising. We regard them as confirming previous work. The significance of this study was the unique way this particular group was managed, rather than the effect of mathematical learning for teachers, or the importance of a professional community itself. Both the university-based researchers and the teacher-researchers were keen to take part and enthusiastic from the beginning. So did the expectations of the researchers prejudice the results of the study? Possibly. On the other hand, research results that are in line with other work internationally build a larger picture of reliability with respect to those results.

Capacity building

A number of levels of capacity building resulted from this project:

• The teacher-researchers will continue developing their mathematics content knowledge for the classroom. The potential for them to actually do this was evident by their keenness to continue the project or be involved in similar projects. In addition, they will be more able to access the expertise of mathematicians for themselves, and to help their colleagues do this.
• The teacher-researchers are better potential leaders. All took a role in their school departments with respect to this project. Several actively worked within their departments to foster contentbased discussions. All were part of a conference presentation to their peers. Thus they have developed both confidence and skills in presenting and taking a leadership role.
• The teacher-researchers are better able to undertake future research. As a result of this study all the teacher-researchers are now part of a Master’s programme in mathematics education. They have been involved in research at an active level and are keen to be involved in more. To quote the most experienced teacher-researcher:

This is the fourth research project I have been involved in. Projects such as this are the reason I will finish my Master’s in Mathematics Education in 2008 with the aid of a PPTA study award and also a reason I will continue to teach. I look forward to being involved in many more.

• The university-based researchers are better able to continue research in this area. Discussion of the project with overseas colleagues has led to a better theoretical understanding of what has been happening and helped put the work in context. This study has been quoted overseas (most recently in a Cambridge University seminar of mathematical knowledge for teaching) as giving particular insights into professional communities in mathematics. The university-based researchers are motivated to continue to explore the model, to work with new groups of teachers and to challenge the state of their own knowledge.
• The practical model of professional development exemplified in this study has been seen to be effective. At the NZAMT presentation, for example, interest was expressed by a mathematics education researcher based in Wellington who wanted to replicate it.

Research outputs and disseminations

There were three components to the dissemination plan:

• Teachers present their own studies and findings to teachers in their departments. This output was exceeded. Not only did teachers present to their colleagues in both formal and informal sessions, but they also communicated their work to both students and teachers in other schools. Two of these teachers have now been invited to present to regional teacher meetings on topics associated with their learning on this project. In addition, the project provided the opportunity for sessions with other teachers in collaboration with the universitybased researchers.
• A joint presentation was made at the NZAMT conference, NZAMT-10, in Auckland, September 2007.
• The university-based researchers are preparing a collectively authored scholarly paper to be presented at the conference of the New Zealand Association for Research in Education in December 2008, and at the Mathematics Education Research Group Australasia (MERGA) conference in July 2009.

Recommendations

We have three recommendations:

1. The establishment of a professional community of teachers needs to be a vital component of all professional development interventions.

This recommendation is not a direct conclusion of this project, in that we do not have evidence that without the community the same successful results would not have occurred. Nevertheless, the project provides strong evidence to support it; in particular, it is the views of the teacher-researchers that this is true. Even the teacher-researcher who claims to “work on their own” comments explicitly on the value of the community. Both our previous research and the literature in mathematics education also lead to this conclusion, and we believe that it is sufficiently established to be implemented.

2. Teachers should be provided with opportunities to gain mathematical stimulation themselves if they are to provide mathematically rich and stimulating learning environments.

Again, this recommendation is a result of a combination of the study, existing literature and our own previous research. We know from this study that mathematical stimulation will lead to richer learning environments; we do not necessarily know that this is the only way to get there. However, the study confirms a further aspect (also present in the PhD work of one of the university-based researchers), namely, that learning mathematics oneself can lead a teacher to reconsider their own teaching practice. Thus, the act of learning mathematics can enhance not just the mathematical aspects of teaching but general classroom practice as well.

3. Programmes should be established to link teachers in schools with university (or workplace)-based practitioners of their subject matter.

The model of this study was effective. We are convinced that a significant component of this is the linking of the mathematics teachers with mathematicians, or their resources— and such a conclusion makes sense in the context of both this research and other writing.

Research Team Members

Principal researchers
Dr Judy Paterson and Associate-Professor Bill Barton, The University of Auckland

Co-researchers and contributing authors
Dr Barbara Kensington-Miller and Dr Hannah Bartholomew

Participating teacher-researchers
Anne Blundell, Auckland Girls’ Grammar School
Peter Radonich, Northcote College
Jason Florence, Otahuhu College
Anna Dumnov, Senior College of New Zealand
Margaret de Boer, Tamaki College
Linda Crisford, Westlake Boys’ High School
Yoko Raike, Westlake Girls’ High School

International advisers
Professor Anne Watson, Oxford University
Professor Deborah Ball, The University of Michigan

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The appendices for this online version of the report have been removed. However, you can access them here