The original objective of the PRISM project was to continue the development of a model for describing students' reasoning that includes needs. This model is intended to be applicable to the description of mathematical reasoning at every level of schooling. It distinguishes among individuals' degrees of formulation of deductive reasoning, styles of reasoning, and needs related to reasoning (especially explaining, exploring and verifying). "Needs" in this model describes the student's goals or purposes in reasoning in a particular way and context. Three questions guided the project:

- In what ways can the mathematical reasoning of primary school students and the needs associated with their reasoning differ from those of secondary school students?
- How can teaching interventions and other features of learning mathematics in schools foster emotional orientations toward reasoning and create occasions for deductive reasoning and the formulating of that reasoning?
- To what extent do the models of needs to reason developed in previous research describe students' reasoning at various school levels, and how might they be improved?

Question one was addressed through gathering data on the reasoning of students in grade 2 and grade 5 classrooms, as well as in grade 8 and grade 10 classrooms. Examples of reasoning at each level were described using a model for reasoning developed for this purpose and the ways in which the model was appropriate or inappropriate gave insight into the differences between the reasoning of primary school students and secondary school students and differences in the needs associated with their reasoning. A steady increase in the degree of formulation (that is, students' awareness of their own reasoning) was observed as the grade level of students increased. On the other hand the complexity of students' reasoning seemed to be as much dependent on teaching approaches and the importance placed on explanation as it did on the grade level of the students. For example the reasoning observed in the grade 8 class was less complex than the reasoning observed in the grade 5 class. Analysis of the data with regard to this question is ongoing.

Question two was addressed though categorization of students' reasoning and comparison with teaching activities occurring at that time. In the grade 2 classroom it was observed that teacher interventions were very important in provoking a need to explain and the formulation of thinking. This result has been elaborated in more detail in a research report (Reid, Dowden, Jeans, & d'Entremont 2000) a refereed conference presentation (Reid 2002) and a refereed scholarly article (Reid, 2002). In the grade 5 classroom a strong need to explain was observed among some students, and deductive reasoning was often used to address this need. This seems to be associated with the emphasis on problem solving and communication between students in that classroom. More details of the results from the grade five classroom have been published in refereed journals (Reid 2001, 2002) and presented at conferences (Zack and Reid 2000). The observations in the grade 8 classroom occurred at the end of the research project and data analysis is ongoing. Preliminary results suggest that in this classroom the need to explain arose primarily in the context of students' explaining to the teacher, and rarely in students' discussions with each other. This may be related to the short periods of time allotted to student discussion, or the nature of the questions the students were asked to investigate.

Question three was addressed in the course of the data analysis from all the sites. As noted above the original model described the degree of formulation of reasoning (more or less formulated), the style or type of reasoning (deductive, inductive, by analogy), and the need the reasoning is intended to satisfy (explaining, exploring, verifying). As result of work on the PRISM project this model is now much more detailed, including six dimensions (Need, Actor, Audience, Type, Formulation and Formality), and the types of reasoning are now divided into subcategories. The current state of the model has been presented at an invited colloquium (Reid 2002) and elements of it are described in several journal articles (e.g., Reid 2001, 2002) and conference presentations (e.g., Reid 2002). In addition, patterns of reasoning involving chains of reasoning acts described with the model have been identified and described in a refereed journal article (Reid 2002)

The PRISM project has contributed to the advancement of knowledge in mathematics education in two ways.

First by providing detailed descriptions of students' reasoning at a range of levels the project has offered researchers interested in reasoning opportunities to compare reasoning they have observed in different circumstances to the reasoning described by the model developed through the PRISM project. Such descriptions have been published in internationally known journals (e.g., JRME, JMB, MTL), presented at international conferences (especially PME, but also AERA, CMESG and PME-NA), and made available on the web (see http://www.acadiau.ca/~dreid/publications/home.htm).

Second the model itself allows a more detailed description of some aspects of students reasoning than is possible using the terminology already available in the mathematics education literature. In fact, because elements of other models have been incorporated into the PRISM model whenever they have been useful, the PRISM model offers the most complete language for describing reasoning available to this point. The application of the revised model to the description of reasoning in other contexts (e.g., other countries, informal mathematical activity, other ages of students,) will be the focus of future research.

In the course of the PRISM project four graduate students and eight undergraduate students were involved in preparing materials, data collection and analysis. At minimum this exposed these students to the activities involved in a large scale qualitative research project, including analysis of video and audio tapes, classroom observations, and planning. In some cases students took a major role in data collection and analysis, especially at the grade 2 site. The research report describing that aspect of the PRISM project (Reid et al 2000) was truly a collaborative effort that could not have been completed without the involvement of students at all stages in the process.

The research team included one collaborator (Dr. Vicki Zack), and numerous student research assistants in significant roles. Without the involvement of the collaborator the grade five site, which is in many ways unique, would not have been available, and the subsequent discussions, collaborative data analysis, shared presentations and ongoing writing would not have occurred. The involvement of the collaborator was very important to the success of the project and much of the elaboration of the model came about as a direct result of this collaboration.

The methodology of the project required that multiple researchers operating from multiple perspective be involved. These perspectives were provided by the collaborator, the student research assistants, and colleagues in mathematics education. Without them the methodological approach, which proved to be very fruitful, would have been impossible.

The data collection for the PRISM project involved work in three Canadian provinces, but it was also guided by the involvement of colleagues in mathematics education from other countries. Most importantly, Laurinda Brown of the University of Bristol, UK participated in classroom observations, data analysis and conference presentations (at AERA in 1999 and PME in 2001). The agenda of the PRISM project in inviting multiple perspectives coincided with interest internationally in data collection methods and theoretical perspectives. This led to a discussion group at the 2001 annual meeting of the International Group for the Psychology of Education which made use of data from the PRISM project. Very recently a collaboration has begun with a researcher from Germany (Christine Knipping) whose own work complements in many ways the PRISM project. A symposium at the 2002 meeting on Mathematics Education and Society (Knipping & Reid 2002) built on material from the PRISM project as well as other projects and experiences concerning reasoning and proof.

At present all video tapes, audio tapes, transcriptions, etc., are being maintained either by me, or by Dr. Zack. This will be the case as long as our ongoing analysis of the data continues. We are handling requests for access to this data directly, as long as suitable allowances for the preservation of confidentiality and anonymity can be ensured. At the conclusion of the data analysis the tapes themselves will be destroyed, in accordance to the arrangements made with research participants to ensure their privacy, but all transcripts, etc. will remain available electronically.

The only issue that arose for me was my move from Memorial University to Acadia University partway through the project. The automatic provision of an extra year in which to complete the work as planned meant that this was not a serious issue. An extra year allowed me to establish myself in a new location, locate suitable students to employ as research assistants, and develop links with local schools to provide research sites.

Supported by a research grant from the Social Sciences and Humanities Research Council of Canada