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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).

*Objective*

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.
*Context*
`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.
*Methodology*
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).