Understanding Professional Development Through the Analysis of the Mathematical Life Histories of Primary School Teachers

Jean Carroll

Concerns about the adequacy of primary teachers' knowledge of and attitudes towards mathematics have been ongoing over the last twenty years (Ball, 1988; Cockroft, 1982; Department of Employment, Education and Training, 1989). Such concerns are based on a belief that teachers' knowledge and feelings directly influence the achievement and attitudes of their students. Bishop and Nickson (1983) cited evidence to support this view. Relich (1995) also found that teachers' attitudes towards mathematics may have an influence on how their students perceive their own ability to do mathematics.

Teachers' attitudes towards mathematics were found by Ball (cited in Mayers, 1994) to be important because teachers who experience anxiety, or lack confidence in their own ability, are more likely to adopt instrumental teaching approaches. The ramification of negative attitudes of teachers was examined by McLeod (1994) who found that children come to school with positive attitudes towards mathematics but their attitudes become less positive as they progress through primary school. Often students' attitudes become negative during secondary school. As a consequence a considerable proportion of students entering primary teacher training has negative attitudes towards mathematics (Philippou and Christou, 1998).

The importance attributed to teachers' knowledge, beliefs and attitudes towards teaching mathematics is reflected by the number of studies that are focussed on measuring and changing preservice teachers' views of mathematics. A study conducted by Bobis and Cusworth (1997) found it was possible for teacher education programs to shift the attitudes and mathematics self concepts of preservice teachers to more positive ones. Other research asserted that many prospective primary teachers have negative attitudes toward the subject, however measurable positive changes were made during teacher training (Philippou & Christou, 1998; Mukhopadhyay, 1996). Although improvements have been shown to be made in student teachers' attitudes during their training, a number of studies maintain that teachers' attitude towards mathematics and mathematics teaching continue to be negative after they begin to teach (Pateman, 1989: Cockroft, 1982; Kanes & Nisbet, 1994). An Australian study found 52% of primary teachers had negative feelings about teaching mathematics (Carroll, 1997).

The importance of teachers' knowledge in the process of teaching and learning mathematics is generally accepted. Carpenter and Fennema (cited in Fennema and Franke, 1992) found that in the areas of mathematics in which the teacher was more knowledgeable, instruction and subsequent learning was richer. As well as mathematical content knowledge, primary school teachers require specialised knowledge of mathematical representations. Contrary to the beliefs of some prospective teachers (Ball 1988), that common sense and memories of their own mathematics education would supply the knowledge necessary to teach mathematics, the complex subject matter of mathematics must be translated into representations which can be understood by students.

A study of 100 primary teachers in suburban Melbourne schools reported that 43% of the teachers were negative regarding their knowledge of and feelings about doing and learning mathematics. In addition 47% felt that their knowledge of the methods and approaches for teaching mathematics was lacking (Carroll, 1997). Such high levels of negativity with respect to their own knowledge and feelings about teaching and learning mathematics in experienced teachers (the average experience in teaching for the group was 14 years) cannot be ignored given the influence of teachers' knowledge on student learning.

The Study

The research reported in this paper arose in response to the issues raised about teachers' attitudes and knowledge. This study was designed to investigate the mathematical histories of five primary school teachers, in an attempt to gain a greater understanding of the development of their views of mathematics and mathematics teaching and the significant influences on their professional development.

Five teachers were selected from 100 teachers who had responded to a Mathematics Attitude and Knowledge Survey (Carroll, 1998) which generated scores for each of three factors:

The teachers' factor scores indicated whether they viewed themselves positively or negatively with respect to the factors. (Positive and negative factor tendencies are described in Appendix 1.) Teachers were considered to have a negative factor tendency if their score was less than the mean on a factor and a positive factor tendency if their score was greater than the mean score on a factor. A profile of each teacher (the Teacher Type- see Appendix 2) was developed by combining the teachers' factor tendencies.

The teachers; Ann, Betty, Cathy, Dot and Ellen were selected because they each represented a different category of teachers in the Teacher Type Table (Appendix 2) and consequently represented diversity in the three factors, and their scores on one or more of the three factors were at a distance from the mean score on the factors and they displayed diversity in their factor tendencies. Selection of teachers with outlying scores was not always possible due to the limited number of case studies.

The teachers were asked to write a mathematical life history and given the following instructions:

The five life histories were analysed with the aim of identifying the, events, experiences and individuals that contributed to the professional development of the teachers.

The Teachers

Ann has been teaching in primary schools for 31 years. She is a classroom teacher, who studied mathematics at school up to Year 10 and completed a Trained Primary Teacher's Certificate in 1963. Her factor scores indicate Ann's teacher type to be F-M-K- (see Teacher Type Table in Appendix 2). Teachers in this category tended to have negative feelings about teaching mathematics, they indicated negativity regarding their knowledge and feelings about doing or studying mathematics and felt they were lacking in their knowledge of the methods and approaches for teaching mathematics to primary school children. Ann was selected for study because she had quite low scores on factors F and M and the lowest score on factor K.

Betty received her Diploma of Teaching in 1988 and has been teaching for 7 years. She is currently completing a Bachelor of Education through part time study. Her highest level of mathematics studied was year 12. Betty's teacher type was F-M+K- suggesting that she felt negative with respect to teaching mathematics, was confident in her ability to do mathematics, while expressing negativity about her knowledge of mathematics teaching. She scored quite low on factor F, just above average on factor M and just below average on factor K.

Cathy has been teaching primary school for 17 years. She obtained a two year teacher training certificate in 1972 and finished a Bachelor in Education in 1977. Year 11 mathematics at school was the highest level of mathematics she had studied. Cathy's teacher type was F-M-K+ with responses that showed negative feelings about teaching mathematics, was not confident in doing mathematics herself however rated her knowledge of the methods and approaches for teaching mathematics positively. Cathy was chosen for study because she scored below average on factor F, had the lowest score recorded on factor M and scored above average on factor K.

Dot was selected as the representative of the teacher type F+M+K-. Teachers of this type indicated that they had positive feelings about teaching and learning mathematics yet lacked confidence in their knowledge of current approaches for teaching it. Her scores on factors F and M were slightly above average and her factor K score was slightly below the mean. Dot has been teaching in primary school for 10 years. She studied mathematics at school until Year 12 and completed a diploma of teaching in 1984.

Ellen's teacher type was F+M+K+ relating to the fact that she was positive about all aspects of teaching and learning mathematics. Ellen has been teaching primary school for 30 years. She studied maths at high school at a level equivalent to Year 12 and completed her Trained Primary Teacher's Certificate in 1962. She was chosen for study because she had extremely high scores on factors F and K and quite a high score on Factor M. These scores indicate that she reported very positive feelings about teaching mathematics and felt knowledgeable of current approaches for teaching and that she is positive with respect to feelings about and knowledge of mathematics.

Discussion of Results

These case studies presented the mathematical life histories of five different teachers. The range of experience in primary teaching varied from 7 years to 31 years. Their mathematical histories varied in a many ways, however a number of themes could be identified. These themes; the teachers' school and teacher training experiences, their knowledge of and feelings about mathematics teaching and the significant influences on professional development are discussed below and analysed in order to better understand the professional development of the teachers during their careers.

Analysis of the case study data indicated that the teachers' own school experiences were important determinants of subsequent attitudes towards mathematics, their views of their knowledge of mathematics, and in determining their teacher types. The teachers who were M- (Ann and Cathy) described negative experiences of learning mathematics at school and described themselves as lacking in confidence in their knowledge of mathematics. While those who were "M+" (Betty, Dot and Ellen) recalled school mathematics learning as a more enjoyable experience and rated their mathematical knowledge more highly.

Ann's mathematical life history established that from her earliest days at school she believed that she was not mathematical and vividly recalled a traumatic experience from her school days. Cathy felt she did not understand mathematics at all and that her knowledge consisted of rote learning. She also recounted a vivid recollection of an upsetting school experience in learning mathematics. Betty, Dot and Ellen had positive feelings about learning maths and felt comfortable with their levels of mathematics knowledge as shown by their factor M scores. Dot had positive experiences of learning mathematics and stated that she generally had success during her primary education and enjoyed her secondary maths experiences. Betty had memories of "breezing through" mathematics at her local technical school and remembered the mathematics as challenging and enjoyable, although occasionally she experienced boredom while at primary school. She obviously did well in mathematics as she elected to study it at tertiary level. The love of learning mathematics that Ellen experienced when she was at school was very clear in her history. She stated that she was successful at school even though at times she didn't fully understand what she was doing.

Teacher training had varying effects on the five teachers. Ann had no significant recollections of her training, while Betty was disappointed by her experiences of teacher training. She had expected exciting and challenging experiences but found learning about mathematics teaching was boring and repetitive, however she admitted to learning some interesting teaching approaches. For Cathy, teacher training was the first time in her life that she found mathematics to be "really good". It was relevant and she was able to understand why the different processes were used. She passed the mathematics components at teachers' college well. Teacher training provided Dot with the opportunity to analyse her own mathematics learning for the first time. She critically evaluated the teaching that she had experienced at school and realised the importance of using concrete, relevant experiences to enable children to experience success in learning mathematics. Ellen's recollections of her teacher training include her shock in recognising to low level of mathematical understanding of her fellow student teachers and the lack of support they received in improving their mathematical knowledge

Ann admitted to beginning to think seriously about the teaching of mathematics later in her career. Ideas about how mathematics is learned by children became a focus for Ann after she had been teaching for about ten years. She began to focus on mathematics teaching and found colleagues particularly helpful in assisting her to find effective approaches. Developing the knowledge required for teaching mathematics seems to have been a struggle for her and there is some indication that some aspects were sometimes neglected in her teaching because they were difficult to organise. She indicated that she would feel very threatened by teaching mathematics to older primary children. Her life history gives the impression that her knowledge of mathematics teaching has been developed as a result of her experience of teaching and her interactions with colleagues.

Although she knew about the approaches for teaching mathematics that were currently thought to be most effective because she had been exposed to them during her teacher training, Betty's difficulties in implementing them meant that she developed feelings of inadequacy about her mathematics teaching. This lead her to use worksheets when she believed that other approaches would have been more appropriate. She seemed to have made a considerable effort to develop her knowledge through reading and professional development. Cathy developed the knowledge she needs for teaching, through experience. She expressed confidence in her ability to try new methods and incorporate them into her approaches. She still has doubts, stemming from her own school experiences of mathematics, about the adequacy of her knowledge if she had to teach at a higher level than the junior school.

Dot finds it challenging to develop interesting and exciting ways to teach essential mathematical skills. She said "there seems to be a direct relationship between the effort and preparation I put into a lesson to make it interesting and relevant to the students and the level of enjoyment and learning that takes place with my students". So, while she experiences success in her work, it comes as a result of considerable hard work. Dot felt that it took along time to develop her personal style of teaching mathematics and to successfully implement the approaches she learned in college. Like Betty she also found herself falling back on the more traditional approaches and types of lessons she had experienced at school, because they were familiar. However, at the same time she believed that they were not the most beneficial or effective methods.

Ellen learned tables and formulae easily at school and loved maths through secondary school. She recalled discovering as a teacher the reasons why formulae worked. Ellen described her knowledge of mathematics teaching as "above average" in the questionnaire and has continually upgraded it through extensive involvement in professional development activities.

The inadequacies experienced by Ann in learning mathematics at school, appear to continue to influence her current feelings about teaching mathematics. Her life history suggests that through experience she has developed the knowledge required to teach children at the junior level, but the thought that she might be required to teach mathematics to older children produces negative feelings. Betty stated that after overcoming some of the challenges of implementing her ideas about mathematics teaching, by attending inservices and trying out the ideas, she began to enjoy teaching mathematics. Initially, however, she had felt as though she didn't have enough knowledge to develop her teaching methods further.

Cathy feels good about mathematics teaching, having overcome the insecurities she felt a couple of years ago. She feels that there must be areas of mathematics that she is still overlooking and works closely with a curriculum document to ensure that this is not the case. She is still frustrated by the lack of support for her learning as she would like to be attend more inservices. Dot said that she enjoys mathematics and mathematics teaching. She went on to say that she finds mathematics teaching a challenge and is always seeking ways to make mathematics fun and meaningful for the children. She attributed her success in teaching to applying herself well and enjoyment of challenges. Ellen's responses indicate that following on from her own positive school experiences, she enjoys teaching mathematics and being involved in mathematics professional development programs and activities.

Personal relationships were very important in professional development for the teachers. They named people with whom they have had professional relationships that resulted in positive changes in their teaching and their feelings about teaching mathematics. Colleagues in their schools with whom they have worked closely on mathematics were important for Ann and Betty, while Ann and Cathy mentioned their interactions with other teachers in the EMIC (an ongoing professional development in mathematics) program. Ann attributes an awakening of interest in teaching mathematics and ongoing development to team teaching with an inspirational colleague. Dot remembered an enthusiastic lecturer at teachers' college as having the strongest influence on her and helped her focus on making her maths teaching successful through making maths fun, interesting, concrete, varied and meaningful for the children.

The role of experience was also acknowledged as a significant influence on professional development. This was evident in Ann's life history (31 years teaching) as she told of her struggle to teach mathematics over the years. Betty (7 years teaching) was influenced by the experiences she had as a learner of mathematics at teachers' college. While much of her training she found to be boring, she did discover some enjoyable ways of presenting mathematics. She realised that she required more knowledge of the approaches and attended inservices to increase her knowledge.

For Cathy (17 years teaching) her school, teacher training and experiences in working with children were important for her development as a teacher of mathematics. Remembrances of how bad learning mathematics at school was, have had a significant influence on Cathy's teaching of mathematics. She found her teacher training was influential because for the first time she began to develop her understanding of mathematics. The role of her experiences of teaching was also evident in Cathy's history as she described all the different approaches that she has tried over the years and how she incorporates new ideas that work into her approach. Like Cathy, Dot (10 years teaching) has found reflecting on her own experiences as a learner at school has influenced her teaching. Dot stated in her history that she often refers back to her own experiences of learning mathematics to help view her teaching from the students' point of view. Ellen (30 years teaching) has actively sought out professional development experiences in mathematics during her career, however it was to her experiences as a learner at school and the love of mathematics which she developed then, that she attributed most influence.

It was possible to identify personal philosophies held by some of the teachers which affected their teaching. Four of the teachers spoke of philosophies which guided their teaching. Betty had clear ideas about maths teaching when she began teaching but found them difficult to implement. She thought maths should be enjoyable and relevant for the children. At the end of her history she still had these priorities as well as the desire to ensure that all children participate and learn the concepts. Cathy expressed a clear personal philosophy of teaching mathematics stemming from her own unsatisfactory experiences as a learner. She seemed determined to have children avoid the negative experiences that she had. She encourages children to use trial and error in solving problems and makes mathematics relevant to their lives. Dot began to develop her ideas of what was important in teaching and learning mathematics during her teacher training, particularly as a result of the influence of one of her lecturers. Her statement of what is important seemed to reflect the influence of the lecturer rather than her own personal philosophy and she described the difficulties she had in trying to implement these ideas.

Ellen's philosophy is based upon the notion that children are born with a certain ability to understand mathematics and the teacher needs to work with that. Her life history is interspersed with what appears to be advice for teachers such as; "use hands on activities", "don't rely on text books", "rote learning is undesirable", and "teach concepts in different ways if children don't understand at first". It could be assumed that these strategies form part of her own philosophy of teaching. Ann's life history describes her struggle to manage teaching mathematics. Her philosophy was not evident in the history; perhaps the energy required for the struggle has precluded a reflective attitude.

Implications for Professional Development

The teachers' histories provide a "career-long" view of their professional development. While the views of these teachers do not generalise to a wider population of teachers, the understandings gained from this study may provide points for consideration when planning professional development in mathematics for primary school teachers.

These case studies suggest that there is a relationship between how teachers currently view their knowledge and feelings about mathematics and the experiences they had as learners at school. The teachers who recalled mathematics learning as a predominantly negative experience remember mathematics learning as memorising processes and procedures of which they had no understanding, and the frustration and lack of enjoyment that stemmed from it was apparent. When referring to their mathematical knowledge today, their lack of confidence is still evident. Those teachers whose recollections of learning mathematics at school were positive also inferred that their knowledge of mathematics is and has been adequate, or better.

Schuck (1997) identified different voices when teachers speak. She discussed "self as student" and "self as teacher" as two of the voices that teachers use. It is evident from these case studies that the "self as student" continues to speak many years after the teachers have ceased to be students the feelings about the "self as student" of mathematics appear to remain relatively unaffected by subsequent experiences of "self as teacher". Professional development for teachers like Ann and Cathy must recognise and allow them to acknowledge the "self as student" which continues to influence their views so strongly. They need to make explicit these prior experiences which exert such a powerful force upon subsequent constructions about themselves, so that they can reflect on and analyse them.

One of the most significant aspects of the teachers' views of professional development during their careers was that the development almost always occurred in environments in which significant personal relationships were established between the teacher and a more knowledgeable person. This study suggested that significant professional development is perceived by teachers to occur through personal interaction. The interactions were with particular teachers, lecturers, presenters, peers and colleagues and were mostly a result of ongoing relationships, which ranged in duration from several months (in the case of EMIC tutors and lecturers) to several years (relationships with peers and colleagues). Relationships are important when they value the teachers' experiences, include a climate of respect, and enable learning to be collaborative.

Ongoing personal relationships combined with learning that flows from the experience of teaching provides a powerful medium for professional development. It was apparent in this study, that teachers reflect upon their experiences of teaching and learning about mathematics. They described how growth flowed from their reflections on their experiences. An important aspect of their ability to reflect upon their experiences was the presence of a colleague, mentor or facilitator to discuss their reflections with.

The facilitator or mentor needs to be more knowledgeable about mathematics and mathematics teaching than the teachers he or she is working with and also needs to be aware of the attitudes and beliefs involved. The role of the facilitator or mentor is to encourage teachers to reflect upon their classroom experiences, provide catalysts for triggering new ways for teachers to think about their experiences and to facilitate growth in knowledge, improvement in attitudes and the development of systems of beliefs which will enhance the teaching of mathematics to young children.

The development of a personal philosophy appears to be linked to the development of the teachers' confidence in their knowledge of approaches for teaching and learning mathematics. The life histories of Cathy and Ellen contained clearly enunciated personal philosophies for teaching. These two teachers had positive views of their knowledge of the approaches for teaching mathematics. Ann, Betty and Dot's philosophies were less well developed and their teacher type indicated that they lacked confidence in their knowledge of mathematics pedagogy

The implications for professional development of the connection between the teachers' espoused views of mathematics teaching and learning and their confidence in their knowledge of the methods and approaches suggests that teachers need to be given assistance in developing their confidence in applying current methods and approaches at the same time as being encouraged to reflect upon their practise and build upon their successes. Such reflection presents a powerful means of professional development

Professional development for mathematics teaching must have its focus on providing opportunities for teachers to develop their own philosophy of mathematics teaching and learning. Many of the issues identified in this research can be addressed by assisting teachers to develop "personal philosophies" of mathematics teaching and learning. A "personal philosophy" provides them with a structure to determine the merit of new ideas for themselves.

The teachers must be provided with opportunities to analyse and critically evaluate their current knowledge, beliefs and attitudes and modify them to include new ideas. In the process of doing this they will develop confidence in their own ideas. There was evidence in this research that some teachers were so lacking in conviction of the importance of their own ideas that they adopt new ideas by rejecting their old ideas rather than evaluating both and arriving at a new personal view.

This research has identified the crucial components which interact to produce effective professional development for teachers. These are the:

Many teachers will require the help of a mentor or facilitator to assist in the development of the critical thinking skills that are required in developing their personal philosophies. However the goal is for teachers to be able to facilitate critical analysis of their own teaching.