The most commonly used methods for teaching ... mathematics are presentation of information to the class by chalkboard or overhead projector and assignment of individual work (Ontario Ministry of Education, 1991a, 1991b, p. 1)
During April of 1990 the Ontario Ministry of Education conducted Provincial Reviews of two secondary school mathematics courses: Grade 12 (Advanced Level), and Grade 10 (General Level). The above statement, appearing in the public reports for both reviews, summarizes the data collected from questionnaires completed by a randomly selected set of school principals, teachers, and students. A more recent study by the Scarborough Board of Education (Colgan & Harrison, 1997), examining the teaching of Grade 12 mathematics at all academic levels, shows the continuing use of the rather limited range of activities described above. "On any given day in a Grade 12 mathematics classroom the routine is comprised of taking up the homework, teacher-centred delivery of a new concept, students' practice of that concept, and the completion of assigned work including homework" (p. 15).
This picture of mathematics teaching and learning in Ontario high schools contrasts with that presented in the guidelines under which these courses are offered (Ontario Ministry of Education, 1985). Here problem solving is presented as a major theme for the curriculum. The recommended program takes an experiential approach, with students employing manipulative materials and simulations to explore concepts and discussing their emerging understandings with each other. Mathematics is to be developed through applications and modelling supported by the use of calculators and computers.
Studies finding the rather unimaginative mathematics instruction described in the quotes above are not restricted to the province of Ontario. In fact, "the nature of classroom teaching is quite similar in all countries" (Anderson, 1987, p. 81), and is remarkably stable over time. Fey (1979), summarizing the results of three nationwide American studies of the mid-1970's, states, "the profile of mathematics classes emerging from the survey data is a pattern in which extensive teacher-directed explanation and questioning is followed by student seatwork on pencil-and-paper assignments" (p. 494). The US data from the 1980 Second International Mathematics Study [SIMS] confirmed that "students at the twelfth-grade level spent the major portion of their class time listening to teacher presentations. Doing seatwork and taking tests and quizzes accounted for other large blocks of time" (Crosswhite, 1987, pp. 75-76). Internationally, the teachers surveyed in the twenty-two countries (including Canada) that participated in SIMS reported that "most of their time was used in whole-class instruction" (McLean, Wolfe & Wahlstrom, 1987, p. 7). The student attitudinal data collected in the SIMS survey show the results of the dominant teacher-directed style of instruction. The majority of students saw mathematics as a set of rules rather than a discipline involving creativity, speculation and conjecture (Miwa, 1987; Rogers, 1990).
More recent international studies have replicated the SIMS findings. From 1981 to 83, the Classroom Environment Study conducted by the International Association for the Evaluation of Educational Achievement [IEA] made repeated observations of teaching in grades 5 and 8 mathematics classes in nine countries (Anderson, 1987). Again, teacher presentations and individual student work consumed the majority of class time. A decade later, in 1991, data from grade 8 classes in 20 countries, collected in an International Assessment of Educational Progress [IAEP] study, showed that "students in many countries regularly spent their instructional time listening to mathematics lessons" and that "another common classroom activity is to require students to work mathematics exercises on their own" (Lapointe, Mead & Askew, 1992, p. 48). The most recent international research, the Third International Mathematics and Science Study [TIMSS], found that, at the Grade 8 level, "the most frequent approaches used across countries involved students working individually with assistance from the teacher, and working as a class with the teacher leading" (Robitaille, Taylor & Orpwood, 1996, p. 5-6).
The United States, through its National Assessment of Educational Progress [NAEP] program, regularly samples mathematics instruction. NAEP surveys from 1973 to 1990 showed that teacher lectures were the predominant mode for high school mathematics lessons (Mullis, 1992) and the 1992 results were summarized with, "reports from students and their teachers show that no real progress has been made in shifting the instructional atmosphere in the nation's mathematics classrooms to one in which active learning and in depth problem solving are emphasized" (Lindquist, Dossey & Mullis, 1995, p. 49).
That effective teaching and learning, especially in mathematics, is more than simply the transfer of facts and must actively involve the pupil is not a new idea. In the third century B.C., Plato (1953) described a geometry lesson where the interaction between Socrates, the teacher, and a slave boy pupil is significantly greater than that suggested by the Provincial Review summary given at the beginning of this chapter. Recently, professional organizations for mathematics educators have produced documents (National Council of Teachers of Mathematics [NCTM], 1989; Ontario Association for Mathematics Education [OAME]/Ontario Mathematics Coordinators Association [OMCA], 1993) that call on teachers to develop mathematics curricula in which pupils actively construct their own personal mathematical understandings through investigating, conjecturing, testing hypothesis, and the sharing and discussing of ideas.
The curriculum initiatives taken by the professional associations have had an impact on government policies. In the US, a 1992 survey (Blank & Pechman, 1995) showed that 41 states had developed or were in the process of developing new mathematics frameworks modelled after the NCTM (1989) Curriculum and Evaluation Standards for School Mathematics (hereafter referred to as Standards). In Canada the Atlantic Provinces Education Foundation (1996) has adopted the NCTM's statements as the guiding principles for its new mathematics curriculum. The Ontario curriculum guide (Ontario Ministry of Education, 1985) for secondary school mathematics in many ways foreshadowed the present mathematics education reform movement, presenting in the introductory pages a vision of teaching and learning similar to that developed in the NCTM Standards.
Although the leaders of the teaching profession are calling for change in mathematics curricula and pedagogy and government policies are reflecting this thinking, "many high school teachers do not endorse the instructional strategies recommended" (Weiss, 1994, p. 5). Only a minority of mathematics classes are engaged in extended problem solving activities and mathematical exploration, conjecturing and communication. Why is progress towards reformed mathematics programs so slow? Why has implementation of the curriculum and teaching methodologies advocated by both government and the professional leadership been so limited?
Thomas Romberg (1992a) attributes the gulf between the program advocated by the NCTM Commission on Standards for School Mathematics, of which he was the chair, and the predominant school practices, to differences in subject conceptions and notes that "the single most compelling issue in improving school mathematics is to change the epistemology of mathematics in schools, the sense on the part of teachers and students of what the mathematical enterprise is all about" (p. 433). In this view, teachers' pedagogical choices are seen to flow from their personal conceptions of mathematics. Ernest (1989a, 1989b, 1991) has developed a theoretical scheme linking personally held philosophies of mathematics to instructional practices. Teaching styles that encourage students to personally and collaboratively construct mathematical concepts are linked to images of the discipline as a living, creative human endeavour, while teachers who see mathematics as a fixed set of rules and procedures are expected to adopt a predominantly transmissive mode of instruction. The slow pace of change in mathematics teaching and learning is linked, by those leading the present mathematics reform movement, to "the prevailing view of educators ... that mathematics consists of a set of procedures and that teaching means telling students how to perform those procedures" (Battista, 1994, p. 463). Thus the problem of teacher reluctance to embrace the mathematics education reform proposals is seen to originate in a conflict between disciplinary images.
While classroom implementation of the mathematics education reform proposals has generally been slow, some teachers are employing curricula and practices that capture the spirit of the NCTM (1989) Standards. The picture of mathematics instruction produced by the Ontario Provincial Reviews and quoted above is not universal. There exist classrooms where one can observe in action the teaching and learning processes described in the Ontario Ministry of Education (1985) guidelines. Are these exemplary teachers' instructional practices, as the theory suggests, expressions of conceptions of mathematics that are different from the predominant views?
In summary, a problem concerning curriculum implementation, that is, the slow paced and limited adoption of the process policies articulated by the Ontario Ministry of Education (1985) course guidelines and more recently expanded upon by professional associations for mathematics education (NCTM, 1989; OAME/OMCA, 1993), has been recast as a clash between differing conceptions of the nature of mathematics. Is this in fact the case? Looking at the general teacher reluctance to embrace the mathematics reform proposals as an opportunity to learn and noting that some teachers have adopted many of the instructional methods advocated by the curriculum guideline and the current reform movement, we can turn this problem into a set of questions.
What are the conceptions of mathematics held by teachers who are attempting to implement the reform proposals?
How are these teachers' images of their subject connected to their classroom practices?
What are the struggles involved in these teachers' efforts to translate subject images into classroom practice?
This study addresses the above questions, through an in-depth examination of the subject conceptions of two exemplary mathematics teachers, that is teachers who are working to implement mathematics education reform, detailed observation of their instructional practices, and exploration for connections between subject visions and pedagogy.
Different philosophies of mathematics have widely differing outcomes in terms of educational practice. However the link is not straightforward. (Ernest, 1991, p. 111)
The research for this thesis was conducted as separate case studies involving two purposely selected teachers, with invitations to participate based upon my personal observations that they were in fact attempting to incorporate aspects of the mathematics education reform proposals in their classroom activities. The case study approach permitted the extended, close contact with the participants necessary for an extensive examination of their teaching and conceptions of mathematics. Spending multiple full working days with each teacher allowed me to experience and record the contexts of their professional lives.
Collaborating teachers participated in three activities to help surface their views of the nature of mathematics: the writing of a brief personal description of the nature of mathematics and the discipline's historical development, the building of a repertory grid (Beail, 1985) comparing mathematics to other school subjects, and the drawing of a concept map (Novak & Gowin, 1984) presenting their image of the structure of mathematical knowledge. The products of these exercises were used, with each teacher separately, as the foci of subsequent interviews (Hewson & Hewson, 1989; Seidman, 1991) to collaboratively build a picture of their personal philosophy of mathematics. In addition, interviews were conducted to ascertain the teachers' motivations and intentions for observed teaching activities and to attempt to relate planning choices to the emerging pictures of their conceptions of mathematics.
Over the span of one school semester, a period of approximately four months, a minimum of 20 visits were made to each teacher's classroom. Observations over short blocks of consecutive days permitted the study to record the development of individual concepts. Visits were spread out and arranged to allow observations of lessons addressing different curriculum topics developed at a variety of grade levels.
A sociological and epistemological view (Koehler & Grouws, 1992) of mathematics teaching, one that examines and records intellectual processes as well as overt behaviours, was employed for the collection and analysis of data concerning the participating teachers' classroom practices. It was not the intention of the study to measure quantitatively the frequency of various teaching methods, but to build profiles of the practices of two exemplary teachers. As such, the observed teaching is reported in the form of narratives of illustrative classroom episodes.
The extended time spent with each participant, a minimum of 8 full teaching days, afforded many opportunities for observation of their professional environment and their interactions with fellow teachers, recording of resource materials employed in lesson planning and the products generated by students, and informal discussions concerning the nature of mathematics and its teaching and learning. All these were recorded in field notes.
Although the two teachers have in common the employment of classroom practices that incorporate some of the themes of the ongoing mathematics education reform, they are distinct, unique individuals, with differing personalities, life histories, and working contexts. To preserve this uniqueness, the data for each case was analysed separately with comparisons drawn only in the concluding chapter of this study. For each participant the data concerning visions of mathematics: writing on the nature of mathematics, school subjects repertory grid, concept map for the discipline, and transcripts of related interviews; were examined for recurring themes. With the dominant components of the subject conceptions identified, analysis efforts were focussed on the classroom observation record: field notes, audio tapes of lessons, materials employed, and student products. For each teacher, classroom practice was searched for incidents where mathematics was portrayed or presented in a style that captured one or more features of the participant's personal mathematical epistemology. In the case studies that follow, the analysed and reordered data is presented in sections alternating between mathematical image themes and classroom incidents putting these into practice.
As the full case studies that follow will show, the strengths and stabilities of the two teachers' subject images varied and in turn their successes in carrying their personal philosophy of mathematics into the classroom differed. During the school semester of the study, for both teachers, the use of non-traditional instruction was met with opposition from pupils, parents and school administrations. The individual responses to these challenges and the abilities to persevere with alternative teaching approaches was most revealing of the unity and strength of subject visions.
In the concluding chapter the struggles involved in these teachers' efforts to translate their visions of mathematics into classroom action are examined. The Perry (1981) theory of epistemological development is employed to analyse the two teachers' personal philosophies of mathematics and the stances taken in the face of conflict. Factors in teachers' professional environments that appear to play a role in the development of a strong mathematical epistemology are identified.
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