A12 Detailed Description

The Psychology of Reasoning In School Mathematics (PRISM) Research Project 
The western culture to which we modern scientists belong depreciates emotions, or at least considers them a source of arbitrary actions that are unreliable because they do not arise from reasons.This attitude blinds us about the participation of our emotions in all that we do as the background of bodyhood that makes possible all our actions and specifies the domains in which they take place (Maturana, 1988, p. 48).


Descriptions of mathematical reasoning rarely refer to the emotional basis for reasoning, what Maturana (1988) calls “emotional orientations.” In mathematics the use of deductive reasoning to satisfy needs for explanation and verification is one example of an emotional orientation. The PRISM project aims to investigate the role of emotions, especially needs to explain, explore, and verify, in the mathematical reasoning of students in mathematics classrooms at both the primary and secondary levels.
The main objective of the PRISM project is 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. Deductive reasoning has been identified by the National Council of Teachers of Mathematics (NCTM, 1989), the Cockcroft Report (1982) and other calls for reform in mathematics teaching as an area of central importance to mathematics instruction. Research in this area contributes directly to the psychology of mathematics education, and has implications for the current reform of mathematics teaching and learning in both primary and secondary schools.
Three specific questions will be investigated:

1.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?
2.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? 
3.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? 
The first question addresses the need for research that bridges the divide between research done with young students and research at the secondary and post-secondary level. Work at the primary level has shown that young children can reason in sophisticated ways (English, 1996; Zack, 1995, 1997; Graves & Zack, 1996; Maher & Martino, 1996; Lampert 1990 ). Research on deductive reasoning at the secondary level, however, suggests that students are not able to apply their reasoning when called to do so in formal mathematical contexts (Fischbein, 1982; Senk, 1985). In the PRISM project the same model for reasoning will be used to describe the reasoning of students at a wide range of grade levels allowing comparisons to be made that may provide insight into the development of mathematical reasoning from primary to secondary school.
The second question addresses the extent to which it is possible to create occasions for deductive reasoning while working within the constraints of existing provincial curricula and the aims of reform documents, such as the Curriculum and Evaluation Standards and Professional Standards for Teaching Mathematics (NCTM 1989, 1991). This question is ultimately tied to a long term goal of offering researchers and teachers examples of teaching in which deductive reasoning was encouraged, and providing materials that will assist them in fostering such occasions in their own classrooms.
The third question is concerned with the feasibility of understanding the deductive reasoning of younger students using existing models. Can characteristics of the development of deductive reasoning in primary school students be described using models for describing the reasoning of secondary school students? Investigations of this question will produce improved models for examining mathematical reasoning at all levels.

Previous research on deductive reasoning.
Research into deductive reasoning in mathematics has been conducted from several different perspectives. The considerable body of research into logic and mathematical proof, conducted especially in connection with the New Math reforms, contains much that is relevant. A smaller body of research (Reid, 1995a; de Villiers, 1992; Bell, 1976) exists that is concerned directly with the nature of deductive reasoning in mathematics, and the needs that reasoning can satisfy. There also exist several research projects directly concerned with mathematical reasoning in school contexts.
The main conclusion of research into mathematical proof and logic is that most students do not understand the nature of proof (e.g., Fischbein, 1982; Senk, 1985) and do not reason logically in many situations (Henle, 1962; Wason, 1966). Large scale studies in the United States have found that most students (e.g., 70% in Senk, 1985) do not understand the proofs they study in high school geometry. Other studies have indicated that students are confused about the relationship between examples and proofs (Alibert, 1988; Bell, 1976; Chazan, 1993; de Villiers, 1992; Finlow-Bates, 1994; Fischbein, 1982). Other researchers (Balacheff, 1991; Braconne & Dionne, 1987; Schoenfeld, 1988; Shanny & Erlich, 1992) have found that both students and teachers concentrate much more on the appearance of proofs than on their logical content. This focus contributes to problems in understanding proof (Schoenfeld, 1988).
Most of the research on proof mentioned above does not consider students’ emotional orientations to be important to their reasoning. There is an implicit assumption that the need of many mathematicians to use deductive reasoning to verify mathematical statements is shared by secondary school students. My own research (Reid, 1995a), building on the psychological studies of de Villiers (1991) and Bell (1976), indicates that students often use deductive reasoning to explore and explain mathematical phenomena, suggesting that an emotional orientation to satisfy a need to explain or explore through deductive reasoning is already present in some secondary students. The assumption that students have an emotional orientation toward using deductive reasoning to satisfy a need to verify may be at the root of students’ poor understanding of proofs and other topics related to deductive reasoning. Work in the philosophy and sociology of mathematics by Hanna (1983), Tymoczko (1986), Lakatos (1976), and others, shows that the use of deduction to explore and explain is also widespread among professional mathematicians, suggesting that the concept of mathematical proof at the post-secondary level could be developed from the emotional orientations secondary school students already have.
The PRISM project will be a continuation of a program of research into students’ reasoning processes that I began in 1990. Initial research into university students’ understanding of proof by mathematical induction (Reid, 1992) revealed the importance of formulation in students’ reasoning, especially in the development of informal or everyday reasoning into mathematical proving (Reid, 1993). It also suggested that the needs students had for reasoning deductively had significant implications for the development of mathematical reasoning. More recently a study of the needs deductive reasoning satisfies for high school and university students and the contexts in which these needs arise has revealed the importance of exploring and explaining as needs for reasoning deductively (Reid 1994a, b, 1995a, b, c). This research into the psychology of mathematical reasoning has provided a basis for developing the PRISM project, which seeks to investigate whether these results are applicable in schools.
Preliminary work related directly to the PRISM project involved the development and testing of materials for Grade 10 Coordinate and Euclidean Geometry that create opportunities for students to reason deductively. These materials will be refined further as the PRISM project progresses. Results of a pilot study (Blackmore, Cluett, & Reid, 1996) and additional work at the grade 10 site have suggested that the existing model will require adjustment to take into consideration systemic constraints in classrooms (Reid 1997a). Preliminary analysis of students’ reasoning in this context is underway, in preparation for comparison with the data from other contexts. Initial results will be reported in Reid (1997b).
The model developed in earlier research addresses four aspects of reasoning: needs; types; formulation; and formality. In the PRISM project, special attention will be paid to elaborating on needs and formulation.

Needs are the purposes or goals of reasoning.Explaining, exploring and verification are the main needs identified in the current model.
Types of reasoning covered are: deductive (reasoning from general principles); inductive (the generalization of specific cases); and analogical (reasoning by similarities).These three types are suggested by Polya (1968).Two other types, abductive reasoning and generalizing, have been added as a result of research done since the model was first developed.
Formulation applies chiefly to deductive reasoning.It describes the extent to which the reasoner is aware of her or his own reasoning.It also includes reasoning that is automatic (e.g., the use of algebra) or superficial (i.e., it has the appropriate form, but the underlying understanding is absent).
Formality applies only to written deductive proofs, and categorizes them according to their adherence to mathematical norms for the presentation of reasoning.
Classroom research
While some research has been done in classrooms on the development of students’ reasoning, none has specifically addressed the issue of the development of deductive reasoning at both the primary and secondary levels or the role of students’ needs in this development. Furthermore, much of the research that has focused on deductive reasoning is not directly applicable to contemporary Canadian schools as it has been done in contexts far removed from contemporary Canadian classrooms, and has employed models of deductive reasoning that neglect needs for reasoning. 
One example is Fawcett’s (1938) research which analyzed the development of students’ reasoning in geometry and in real world situations during an experimental full year course in Euclidean geometry. While many of Fawcett’s experimental methods are still useful, the curricular shift away from Euclidean geometry means that his findings are no longer directly applicable. Balacheff, Arsac, Mante, and others (Arsac, Balacheff, & Mante, 1992; Arsac, Chapiron, Colonna, Germain, Guichard, & Mante, 1992) engaged in classroom research in France in which the development of mathematical reasoning was observed in the course of a unit on mathematical proof that is first introduced in the equivalent of grade 7. As Canadian curricula do not provide such opportunities to focus directly on mathematical reasoning, this research is also not directly applicable, however some of the methods used by these researchers are. 
Some research employing methods similar to those of the PRISM project has been done in primary schools. Teacher/researchers have created classroom situations in which deductive reasoning might be expected to flourish, but their research focus has been on other issues. Lampert (1990) and Maher and Martino (1996) have conducted classroom research with grade 5 students on the development of mathematical communication and community that offers occasional hints at the reasoning of the students involved. Zack (1995, 1997; Graves & Zack, 1996) is also engaged in research at the grade 5 level, focused on children’s construction of mathematical knowledge through problem solving and joint activity. She is a collaborator in the PRISM project.

Theoretical and methodological background.
The PRISM project derives both its psychological perspective and its general methodological structure from enactivist theories of cognition, as described by Maturana and Varela (Maturana & Varela, 1992; Varela, Thompson & Rosch, 1991). Enactivism as a theory of cognition acknowledges the importance of the individual in the construction of a lived world, but emphasizes that the structure of the individual coemerges with this world in the course of, and as a requirement for, the continuing inter-action of the individual and the situation. From this perspective the role of reasoning in mathematics is seen not as part of mathematics, nor as a result of the students’ needs in a situation, but as a part of the inter-action between them. Similarly the data and interpretations of enactivist research are seen as coming out of the inter-actions between students, teachers, teaching materials, and the research context. This approach captures much of the richness of classroom activity, a richness that is difficult to capture in other ways.
This enactivist style of inquiry (described further in Kieren, Gordon Calvert, Reid & Simmt, 1996, 1995) has also been described as “brico-logical” (Reid 1996, 1995a). Brico-logical research combines the flexibility and creativity of bricolage with an underlying logic of inquiry. Bricolage, as it is used in conceptualizing brico-logical research, favors the production of complex structures, theories, models, etc. appropriate to research on complex systems such as human learners. The logic of the brico-logical methodology comes from the questions chosen for research, and the theories and models with which the research begins. In practice this involves a continuous and continuing reinterpretation of data from multiple perspectives that are themselves being continually reinterpreted by the data. 
In the PRISM project these multiple perspectives will be provided both through the active involvement of a number of researchers, and through the use of multiple sites for research. The cooperating teachers, research assistants, collaborator and principal investigator will each bring to the research unique perspectives and goals. The backgrounds of the teachers and researchers who will be involved suggest that their goals will include developing better teaching methods, exploring classroom social structures, and investigating learning of particular concepts. The interaction of investigations with different goals and perspectives allows for the creation of richer research contexts, encouraging the development of deeper understandings of all the questions brought to the situation. The presence of multiple perspectives at each research site corresponds to the presence of the principal investigator’s perspective at multiple sites. The same complexity is offered by the multiple data sources in different classrooms, at different grade levels, over several years.

The PRISM project is concerned with the development of mathematical reasoning in students at several grade levels. Teachers and researchers at four research sites will be involved over a period of three years. The research sites will offer a range of student ages and contexts in which to test models of reasoning. The project will be cumulative in that each year of the project will expand on the previous year’s work, with modifications based on the experiences from the previous year. The first year’s research involves classrooms observations at grades 2, 5, and 10. In the second year a grade 7 classroom will be added.  In the third year the grade 2, grade 5 and grade 7 sites will be revisited. 
The classes involved in the research project will be employing patterns of instruction involving students’ active participation in the discourse of the classroom. This pattern of instruction is a common thread in the research of Zack, Lampert, Fawcett, and Balacheff (described above). Instruction is structured through a sequence of activities. A problem or task is proposed to the class, and small groups discuss it, attempting to arrive at a solution acceptable to all. Then the class as a whole engages in discussion, with competing solutions being evaluated for validity, clarity and utility. At this point the teacher’s role becomes more active, and more formal styles of verification and explanation come into play.
The first year of the project will build from ongoing studies in a grade 10 classroom (Blackmore, Cluett, & Reid, 1996; Reid 1997a ,b) and on data from ongoing research in a grade 5 classroom (Zack, 1995, 1997; Graves & Zack, 1996). 
At the high school site work in the grade 10 class will continue, and data from prior work will be analyzed in conjunction with new data as it is gathered. The research will involve observations and video and audio tape recordings of students’ discussions and group work. Observations in the grade 10 class will concentrate on a unit in Euclidean geometry. This unit has been chosen as it offers the greatest scope for deductive reasoning within the constraints of the curriculum objectives. Previous work in the grade 10 classroom included observations during the teaching of the Coordinate Geometry unit as well, which yielded some useful data, but for purposes of testing the descriptive capacity of the model for reasoning the Euclidean geometry unit offers greater scope. The cooperating teacher of the grade 10 class will make extensive use of group work, problem solving and class discussions, based on the recommendations of the NCTM (1991) and research results (Fawcett, 1938; Lampert, 1991; Arsac, Balacheff, & Mante, 1992). Research assistants and I will observe individuals working in class, and interpret the classroom activity through the revised model for describing reasoning (see above) by noting occurrences of deductive reasoning, explaining, exploring and verifying, as suggested by verbal cues. One group of students will also be videotaped, and other groups audio taped, for later analysis. Throughout the period of classroom observation there will be regular meetings with the cooperating teacher to discuss the interplay between the teaching methods and the development of students’ reasoning. This will allow the cooperating teacher to make use of the research data in her teaching. At the end of the unit semi-structured interviews will be held with selected students to supplement the classroom observations.
Following the period of classroom observation the collected notes and recordings will be used to develop a description of the mathematical activity of individual students and groups of students in the course of the unit, what these activities reveal about the students’ reasoning, and the influence of teaching on the development of their reasoning. These descriptions will then be interpreted within currently existing models. Examples of previously identified categories will be identified, as well as cases that do not fit the existing categories. Cases that are not easily categorized will be analyzed in more detail, and descriptions of students reasoning in these cases will be made.  Within the set of descriptions produced in this way, commonalities will be sought that might provide additional features for a revised model of students’ reasoning. Anomalous cases will be kept for comparison with data from other sites and subsequent years of the project.
At the grade 5 site a large body of data has been assembled over the past five years, as part on an ongoing research project investigating the construction of mathematical knowledge thorough problem solving and joint activity (Zack, 1995, 1997; Graves & Zack, 1996). This body of data includes video tapes of students solving problems in groups analogous to those at the high school site. As a part of the PRISM project these tapes will be re-analyzed by the principal investigator in collaboration with Zack, using the existing model for reasoning, and modified models will be produced. This will take place concurrently with the analysis of tapes from the high school classes to allow for connections to be made and contrasts observed.
At the grade 2 site preliminary work will be done with three groups of students working independently at learning stations. Several learning stations will be developed to provide opportunities for students to reason deductively and in other ways in playing games and solving problems. Data will be collected primarily as a basis for developing additional materials and observation protocols for Year Two, but also for comparison with the existing data from the grade 5 site.
In the second year new data will be generated from the grade 5 site and the grade 2 site. A pilot study will be done at a grade 7 site in preparation for collection of data in Year Three specifically related to developmental aspects of the model under revision. Collection of data at the grade 10 site will cease to allow additional time for analysis of new and existing data. The revised model produced in the previous year will be used to analyze data from all three sites, and structures will be added to the model to better describe a possible path for the development of mathematical reasoning over time. While three sites are too few to provide a basis for generalizations across the wide range of teaching contexts in Canada, the various sites will allow for the discovery of important constraints on students’ reasoning in schools. As well as the differences in students’ ages, the sites also offer differences in socio-cultural background that will further test the applicability of the model for describing needs and reasoning as it is being developed.
 In the third year observations will continue at the grade 2, grade 5 and grade 7 sites, and the data analysis will focus the developmental aspects of the modified model. In addition to the refinement of the model, another focus of the third year will be describing the interactions between teaching and reasoning, making use of case studies from the project and the revised model, in order to provide research results in a form useful to teachers interested in the development of reasoning in their own students.

Communication of Results
Results of the research project will be communicated to mathematics educators both within the research community and the teaching community. This will be achieved through conference presentations and workshops, publication in refereed journals of wide distribution, and by making materials, research reports, and multimedia depictions available on the Internet. As the study moves to new research sites many opportunities for interaction with teachers in the area will occur, providing another means of communicating research results.
Results of the PRISM project will be reported in international journals, such as the Journal for Research in Mathematics Education, the Journal for Mathematical Behavior, Educational Studies in Mathematics, and For the Learning of Mathematics. The research community will also be reached through presentations at international conferences such as the International Congress on Mathematical Education (ICME), the International Group for the Psychology of Mathematics Education (PME) and its North American affiliate (PME-NA), the American Educational Research Association (AERA), as well as Canadian organizations such as the Canadian Society for the Study of Education (CSSE) and the Canadian Mathematics Education Study Group (CMESG). 
The research data produced in the course of this project will provide useful case studies for teachers interested in mathematical reasoning and in student involvement in their classes. Conference presentations and workshops at the conferences of the National Council of Teachers of Mathematics and publication in NCTM journals (Mathematics Teacher, and Mathematics Teaching in the Middle School) will be an important avenue for communicating research results to practicing teachers. At the local level teacher in-service courses and workshops will provide a further avenue of communication. The announcement of a special issue of the Mathematics Teacher on mathematical proof (Fall 1998) and an NCTM yearbook on mathematical reasoning (1999) illustrate the interest of practicing teachers in results of research like the PRISM project.
A large amount of organized data and research results will be made available on the Internet. Materials will be included of interest both to the research community and to teachers. Research reports providing more detail than is possible in journal publications, detailed unit plans, and copies of text materials from classes will be available, as well as students’ work and lessons taught at the research sites. Direct interaction with researchers and willing teachers involved in the project will also be possible via electronic mail. Some data will also be used as part of a set of teacher education materials currently under development. This will provide a resource for teacher educators similar to that described by Mousley & Sullivan (1995, 1997).