Author: Prof Robert Perry

Faculty of Education, University of Western Sydney

Keywords: primary mathematics, mathematics education, positive attitudes

Article style and source: Committee for the Advancement of University Teaching
Final Report for a 1994 National Teaching Development Grant

Contents

• 1 Project summary
• 2 Major objectives of the project
• 3 Major achievements of the project
• 4 The teaching development
• 5 Output
• 6 Personnel Outcomes
• 7 Networks
• 8 CAUT Activities

1 Project summary

The project involved preservice primary (K-6) teacher education students, hereinafter called students, in the development of their mathematics knowledge through interaction with peers while seeking solutions to specially developed tasks. The students worked in pairs to produce initial solution attempts, shared these attempts with the rest of the class, discussed them and reached agreement on viable solutions. The teacher stimulated discussion, maintained the established class norms but did not indicate "correctness" of solution attempts. The mathematics was constructed by the learners, not transmitted by the teacher. The teaching process in this project has application to teaching mathematics and other content beyond teacher education. It would seem to provide a viable approach to teaching a wide variety of material at university level.

2 Major objectives of the project

The objectives for the project were to:

• establish an approach to teaching the subject Mathematics for K-6 Teachers which relies on the learning cycle:

• measure the preservice teachers' beliefs values and attitudes to mathematics;
• establish, within classes of the subject Mathematics for K-6 Teachers, interactively constituted social norms which determine acceptable classroom behaviour;
• develop preservice teachers' knowledge of and attitudes to mathematics;
• evaluate the processes and products of the learning in terms of changes in mathematics knowledge, beliefs, attitudes and values of the preservice teachers involved;
• help satisfy employment requirements of primary school teachers in the NSW Department of School Education;
• test the approach in various contexts including;

- different teachers
- different student groups (both Indigenous and non-Indigenous)
- different modes of delivery (weekly classes and concentrated blocks of time)

3 Major achievements of the project

The original application for this grant expected that the results of the project would include:

• the development of a set of mathematical tasks which will be applicable in a variety of university teaching contexts;
• a significant development of mathematics knowledge and skillsamong the preservice teachers;
• the development of a more positive attitude to mathematics and mathematics teachers among the preservice teachers;
• changes in the preservice teachers' beliefs and values about mathematics;
• the development of the preservice teachers as reflective practitioners, not only in mathematics but in all of their learning and teaching.

The project team believes that all of these results have been achieved.

• A set of mathematical tasks has been developed which are specific to the Mathematics in Society syllabus from the NSW Higher School Certificate. The approach taken in the development of these tasks is applicable in many different contexts.
• There have been significant advances in the students' mathematical knowledge. This can be judged by their results in assignments and examinations, as well as from interview data.
• Pre- and post-surveys of attitudes to and beliefs about mathematics and mathematics teaching and learning indicate substantial improvements. Anecdotal and interview data suggest that the approach taken in this project has had a great deal of positive impact on these measures.
• The student teachers have continued in their development as reflective practitioners through this subject. Inbuilt to the approach is the analysis of the students' own learning. It must be said that not all students, at least initially, felt that the approach was conducive to their learning. Nevertheless, they did develop their awareness of an alternative approach to the teaching and their learning of mathematics.

There were some difficulties encountered with the project implementation.

• The technology used to record all classes was intrusive initially but became less so as time went on.
• Individual differences among students required careful selection of activities and adaptations to these as they were undertaken.
• Teachers need to be aware of the need to display total commitment to the approach and to use it consistently.
• Students need to develop a commitment which includes a willingness to pursue the approach outside the class times.
• The concentrated AREP mode of teaching required major modifications of the approach which worked with some students better than for others.

Evaluation of the project was carried out using analysis of videorecords of all classes, analysis of the students and teachers' reflective journals, attitude and belief surveys, assignments and examinations, reflective interviews and anecdotal comments.

The project team believes that the objectives of the project as listed were achieved.

4 The teaching development

The Mathematics Education Team at the Faculty of Education in UWS Macarthur is very grateful for the opportunity provided by the Committee on the Advancement of University Teaching through its grant. This enabled the team to refine and implement much more widely an approach which had been trialled during 1993 to help preservice teachers who were reluctant mathematics learners to develop their skills in mathematics.

The approach developed was consciously quite different from approaches likely to have been undertaken in previous learning by these students. It was an approach based upon that used by the Purdue Problem Based Mathematics Project. This approach has been used in schools, particularly the early years of school, in the United States. It has also been used in schools in Australia by Dr Bob Wright from Southern Cross University. The adaptations required for the current project included the development of a series of tasks designed to cover much of the Mathematics in Society syllabus from the Higher School Certificate in NSW. As well, the teachers (lecturers) involved in the team needed to develop their own skills in providing a problem based approach for their students.

The approach was trialled in 1993 by the project leader with a group of students undertaking the Aboriginal Rural Education Program within the Bachelor of Teaching (Primary) course at UWS Macarthur. It was this trial which was reported briefly in the application for the CAUT grant.

The objectives for the project were to:

• establish an approach to teaching the subject Mathematics for K-6 Teachers which relies on the learning cycle:

• measure the preservice teachers' beliefs values and attitudes to mathematics;
• establish, within classes of the subject Mathematics for K-6 Teachers, interactively constituted social norms which determine acceptable classroom behaviour;
• develop preservice teachers' knowledge of and attitudes to mathematics;
• evaluate the processes and products of the learning in terms of changes in mathematics knowledge, beliefs, attitudes and values of the preservice teachers involved;
• help satisfy employment requirements of primary school teachers in the NSW Department of School Education;
• test the approach in various contexts including;

- different teachers
- different student groups (both Indigenous and non-Indigenous)
- different modes of delivery (weekly classes and concentrated blocks of time).

The project team feels that all of these objectives were met, although some created more difficulties than others. It is instructive, perhaps, to report on what actually happened within each of the classrooms using this approach.

1. In every case, students worked in small groups, usually pairs, to undertake activities which were presented to them on worksheets. At any one time, all the students in the class were working on the same activity. A period was set aside to introduce the activity, to ensure that students knew and understood the language used in the activity and to encourage them in the approach.
2. Students, in their groups, worked on the activities for a period. The length of this period was flexible and depended on the level of interest of the students, the speed which they achieved solutions to the problems and the time available in the overall class.
3. Students were invited to present their solutions or attempted solutions to the class. It was insisted that this be done with the entire group of students (usually a pair) coming to the front of the class and presenting their materials via the whiteboard and overhead projector. Students were encouraged to present not only completed solutions but attempts at a solution and to justify their methods in whatever ways they could.
4. Students who were not presenting were encouraged to discuss the solutions presented, to argue about the validity of these solutions and to make comparisons between the presented solutions and any that they may have devised. In general, the teacher observed these sharing sessions, only intervening if there were breaches of the social norms established within the classroom or to provide information concerning vocabulary or notation. At no stage, did the lecturer comment on the validity of the solutions or the correctness of the results.
5. After several groups of students had presented their results, the class was challenged to come up with an acceptable approach and result for all of them. This generalisation phase of the overall approach requires a great deal of discussion and final agreement on what will be "taken-as-shared" by all the members of the class.
6. It was sometimes the case that the taken-as-shared agreement reached by the class would not match that which the teacher knew was "correct" in the outside world. However, an important part of the approach is for that to be left until one of the students obtains further information which might challenge the taken-as-shared agreement.

EVALUATION

The project was evaluated using numerous separate techniques.

1. Video Recording
2. Video records were made of all classes undertaken in 1994. These were analysed and have resulted in a series of critical points for student's learning being identified. These critical points have provided teachers with substantial markers by which they can judge their students learning. The video records also provide a great deal of information concerning changes in attitude and beliefs towards mathematics, mathematics teaching and mathematics learning.
3. Reflective Journals
4. All students and teachers kept detailed reflective journals as they moved through the subject. These journals display much thought concerning the approach and the development of the students' mathematical knowledge and attitudes.
5. Surveys
6. Surveys of attitudes and beliefs about mathematics, mathematics teaching and mathematics learning were conducted before and after treatment.
7. Assignments and Examination
8. Students were required to undertake a number of assignments and a final examination in order to meet university requirements in this subject. Included were pre- and post- treatment knowledge tests. These pieces of work showed a development in the students' knowledge and understanding of mathematics at Higher School Certificate level and displayed their achievements in this field.
9. Reflective Interviews
10. Each pair of students was asked to be involved in a reflective interview with one of the mathematics team in order that they might put forward their own ideas about their approach and their own mathematical development. These interviews were audio taped and analysed.

DIFFICULTIES WITH THE APPROACH

There were numerous difficulties encountered in the approach which should be noted by anyone wishing to undertake a similar approach in their university teaching. The difficulties are:

• The technology used to record all classes was intrusive initially but became less so as time went on.
• Individual differences among students required careful selection of activities and adaptations to these as they were undertaken.

• Teachers need to be aware of the need to display total commitment to the approach and to use it consistently.
• Students need to develop a commitment which includes a willingness to pursue the approach outside the class times.
• The concentrated AREP mode of teaching required major modifications of the approach which worked with some students better than for others.

In encountering and working through these difficulties, the mathematics team at UWS Macarthur has become much more coherent and has developed a strategy which is taken-as-shared. The overall effect of this is that future projects are being planned, some of which evolved directly from the CAUT project and others which move into other areas of mathematics teacher education. We believe that it is important for the Committee on the Advancement of University Teaching to acknowledge the professional development aspects of their grants, not only in terms of the development of innovative approaches to teaching but also in the development of collaborative teams of university teachers.

REFERENCE GROUP

In the initial stages of the project, the Reference Group played a major role in developing the approach. Special thanks should go to Dr Bob Wright from Southern Cross University and Debra Evans who was a student in the original pilot group for the project. As the project was implemented, individual discussions were held with members of the Project Reference Group, with the Sydney based members, Richard Parker and Peter Howard playing major roles. Professor Erna Yackel has been kept informed of the progress of the project and its publications.

The Project Reference Group has not met together formally but has provided a great deal of individual support for the project team. We are very grateful for this support.

RESULTS

The project has been evaluated extensively and the results are encouraging. We believe that all the objectives have been reached and that the approach has proved itself to be useful in the development of mathematical knowledge and attitudes among students who have not been terribly successful in their mathematics in the past. We feel that the project has shown sufficient flexibility to enable it to be adapted in other settings, including High Schools and TAFE Colleges, as well as in other Faculties of Universities. There has been a deal of interest shown by the other Faculties of the University of Western Sydney Macarthur and by some members of other Universities.

The University of Western Sydney Macarthur supplemented the CAUT grant to this project and has indicated its willingness to implement the approach, where appropriate, across the University. Certainly, the mathematics team within the Faculty of Education at UWS Macarthur will continue to implement the approach in the future.

5 Output

Various publications and conference presentations have arisen from the project. Details are given below.

Publications

*IR - Internationally refereed; R - refereed; AR - abstract refereed; PR - peer reviewed

Geogeghan, N., Howe, P., Owens, K., Perry, B. (1995). Interactive collaboration and affective processes in the construction of mathematical understanding. Proceedings of PME19, International Group for Psychology of Mathematics Education, Recife, Brazil. (IR)

Howe, P., Owens, K., Perry, B., Geoghegan, N. (1995). Teaching adults high school mathematics using an approach compatible with constructivist theories of knowing. Reflection 20 (4), 120-124. (PR)

Owens, K., Perry, B., Conroy, J., Geoghegan, N., Howe, P. (1994). Affective processes in interactive construction of understanding in mathematics. In R. Killen (Ed.) Educational research: Innovation and practice. Swinburne: Australian Association for Research in Education. (AR)

Perry, B. & Conroy, J. (1994). Early childhood and primary mathematics: A participative text for teachers. Sydney: Harcourt Brace.

Perry, B., Geoghegan, N., Howe, P., Owens, K. (1994). Interactive constitution of mathematics by teacher education students. In G. Bell, B. Wright, N. Leeson, J. Geake (Eds.) Merga 17, Proceedings of the 17th annual conference: Challenges in mathematics education: Constraints on construction, (pp. 487-496). Lismore: Mathematics Education Research Group of Australasia. (R)

Perry, B., Geoghegan, N., Howe, P., Owens, K. (1995). An approach for developing the mathematics backgrounds of primary school teachers. In R. Hunting, G. Fitzsimons, P. Clarkson, & A . Bishop, Regional collaboration in mathematics education (pp. 583-592). Melbourne: Monash. (R)

It is planned to publish a summary article on the project in the Teaching Review, the internal University of Western Sydney Macarthur journal devoted to innovative university teaching.

Conference presentations

(Other than those published above)

Geoghegan, N., Howe, P., Owens, K., Perry, B. (1994). Reflecting on the interactive construction of mathematics by primary teachers education students. ICMI - China Conference on Mathematics Education, Shanghai.

Geoghegan, N., Owens, K., Perry, B., Howe, P. (1995). Cooperative learning and social constructivism in mathematics education. Annual Conference of the Mathematics Education Research Group of Australasia, Darwin, NT.

Perry, B. (1995). The use of participative texts in mathematics teacher education. University of Technology, Sydney.

Informal discussions have occurred among colleagues from the Faculties of Education in the University of Western Sydney and other faculties within UWS Macarthur.

6 Personnel Outcomes

The project was undertaken by four mathematics lecturers within the Faculty of Education at UWS Macarthur. These people are Noel Geoghegan, Peter Howe, Kay Owens and Bob Perry. As well, two other mathematics educators who worked part time within the Faculty, Diane Baraclough and John Conroy, were involved in the teahcing and/or analysis of video records of the teaching. As has already been mentioned, the project provided the catalyst for the development of a strong team spirit among these mathematics educators.

The Reference Group for the project contained five members, four in Australia and one from the United States of America. The project team did not meet with the Reference Group as a whole at any time but did take advice from them individually. There were infrequent ongoing discussions with all members of the Reference team.

In all, 64 students were involved in classes undertaken by the project. As the approach involved small group work, each class needed to be restricted in size to less than 20.

It was necessary to use the assistance of a number of technical officers employed by the Audio-Visual Services section of UWS Macarthur and two research assistants. Because of the complexity of the technical set up required to record both individual and group responses to the activities within the project, it was necessary to undertake substantial training of the technical officers. This requirement was met through numerous meetings between the mathematics lecturers involved and various technical officers and their managers from the Audio-Visual Services section. In this way, the project also provided the catalyst for quite extensive training of these technical officers.

7 Networks

The project has excited much interest in numerous other institutions and among many colleagues. For example, reports on the project findings and approaches have been very well accepted by conferences of the Australian Teacher Education Association, Mathematics Education Research Group of Australasia and the Mathematics Association of NSW.

One of the team members had the pleasure of attending the UTS November Conference considering CAUT projects and was struck by the wide ranging acceptance of the approach used in our project by colleagues at the conference. It is important to note that this project is one of the few sponsored by CAUT in the 1994 round which does not involve the extensive use of computer or distance learning technologies. While there can be no doubt that such issues are important and will lead to a great deal of innovative teaching in universities, there is still room for improving the face-to-face presentation of university classes.

Interest in the project from other institutions includes discussions which have been held with Curtin University, the National Institute of Education in Singapore, the Australian International School in Singapore, the University of Technology, Sydney and Southern Cross University.

The project has been supported substantially by the University of Western Sydney Macarthur. On a formal level, a teaching incentive grant of $1000.00 was won by the project team to supplement the funding for the project. On a less formal basis, the Faculty of Education at UWS Macarthur has provided support in kind in the form of office space, infrastructure such as photocopying, computing facilities and some support staff assistance.

8 CAUT Activities

As we have stated earlier, the mathematics education team at the Faculty of Education in UWS Macarthur is grateful for the opportunity provided by the CAUT grant. We would want to encourage the Committee to continue offering such grants, while realising that this is a matter of Government policy and funding. We believe that the Committee for the Advancement of University Teaching has in fact advanced university teaching through its activities. However, this would indeed need to be tested through an independent evaluation of these activities.

We believe that the Committee needs to address two major issues.

1. The continued encouragement for innovative teaching projectswhich are not solely based on the development of approaches using technology.
2. There needs to be further emphasis on students improving their own learning as well as university lecturers improving their teaching. That is, an emphasis needs to be placed on the importance of independence in university learning and it is hoped that the Committee might encourage student groups to develop innovative practices in this area.

About the author

Prof Robert Perry
Faculty of Education
University of Western Sydney, Macarthur
P.O. Box 555
Campbelltown 2560
New South Wales
Australia

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