The research examined in the previous chapter clearly shows the complexity of the interactions between beliefs and practice. Teachers possessing specific sets of beliefs may be found to usually employ specific sets of practices, but these apparently coincident occurrences of events and qualities are not the result of deterministic laws. These are examples of "mutual simultaneous shaping" (Lincoln & Guba, 1985)
Everything influences everything else, in the here and now. Many elements are implicated in any given action, and each element interacts with all of the others in ways that change them all while simultaneously resulting in something that we, as outside observers, label as outcomes or effects. (p. 151)
A study examining phenomena in such an environment and wishing to make any significant headway in addressing the questions posed at the close of the previous chapter needs to focus on particular cases and explore these in detail. Thus a case study approach has been employed for the research reported here.
The case study offers a means of investigating complex social units consisting of multiple variables of potential importance in understanding the phenomenon. Anchored in real-life situations, the case study results in a rich and holistic account of a phenomenon. (Merriam, 1988, p. 32)
The methodologies employed within the two cases of this study, like the preceding literature review, come from the two domains of research into: teachers' beliefs or thinking and classroom practice.
Reason ... must approach nature in order to be taught by it. It must not, however, do so in the character of a pupil who listens to everything that the teacher chooses to say, but of an appointed judge who compels the witnesses to answer questions which he has himself formulated ... It is thus that the study of nature has entered on the secure path of science, after having for so many centuries been nothing but a process of merely random groping. (Kant, 1787)
The present mathematics education reform efforts, designed to emphasize the inductive aspects of mathematical development and to reduce the subject's portrayal as a pure deductive science, and teachers' struggles to come to grips with this change in philosophical orientation, are parallelled in the selection of methods for this inquiry. I, like the two teachers who agreed to participate in this study, am a product of a mathematics education that focused upon deduction. Textbook chapters and mathematics lessons began with a statement of a mathematical 'truth', the theorem to be proved or the algorithm to be illustrated in the following pages of text or the teacher's blackboard presentation. The result or theory was already set. All that remained to do, was to provide a proof or explanation. In a similar fashion, from my own experiences as a mathematics student, mathematician, and mathematics teacher and from the reading of the literature previously cited, I had, prior to beginning my research, developed an image of what the results of this study might say. There was thus a temptation to approach the research with a predetermined thesis and to employ the study as a proof of this theory. This I did not want to do. It was intended that the summary thesis of this study, if one was in fact found, emerge from the data. To hopefully avoid the selective observation and interpretation that may follow from a previously determined theory, or at least to identify and acknowledge such selectivity if and when it occurs, a brief overview of the research perspective is provided.
The research literature previously cited suggests that, within the constraints provided by context, teachers employ practices that are warranted by their beliefs. In particular, studies have found that teachers holding absolutist beliefs concerning mathematics adopt transmissive modes of instruction in the subject. Turning this around, it might be postulated that teachers in whose classrooms pupils are encouraged to investigate mathematical situations, to make and test hypotheses, and to discuss and defend conjectures hold fallibilist images of the subject, possibly social constructivist views. Is this the case? For one who has come to take a social constructivist perspective on mathematics and who supports the reform efforts of the professional associations, a positive answer to this question would be desirable.
While efforts can be made to reduce the effects of researcher bias, it is probably undesirable and likely impossible for a researcher to initiate a study without some prior view of the potential results. As Kant asserts in the quote above, to enter the research field (nature) without prior questions, constructed from some initial tentative theory, is likely to lead to random groping. The researcher can not focus his powers of observation and details that might answer his questions are likely to be lost in a overwhelming mass of data. "Anticipatory data reduction" (Miles & Huberman, 1984a, p. 24) guided by an initial conceptual framework can help focus a study. Thus the present study acknowledges the above "hypotheses" or speculations and looks for evidence of these while also being open to other interpretations of the data.
The fact that lawfulness and individuality are considered antitheses has two sorts of effect on actual research. It signifies in the first place a limitation of research. It makes it appear hopeless to try to understand the real, unique, course of an emotion or the actual structure of a particular individual's personality. (Lewin, 1935, p. 18).
Lewin argues that knowledge and theory might be advanced more rapidly by abandoning the common search for repetition and frequency in our world and by examining fortuitous individual special cases (p. 13). This study takes this approach, employing "purposeful sampling" (Bogdan & Biklen, 1992, p. 71; Lincoln & Guba, 1985, p. 199) in the selection of participants.
The literature reviewed in the previous chapter presents examples where teachers' absolutist conceptions of mathematics were enacted in the classroom through traditional transmissive modes of instruction. The prevalence of such situations in research reports and survey data describing the most common mathematics instructional processes suggest that small scale random sampling of secondary school mathematics teachers would be likely to generate case studies that again featured teachers with limited views of mathematics and traditional teaching styles. Such a study would not contribute much that was new to our understandings of mathematical beliefs and teaching. Case studies that focus on less representative special examples from a population can be enlightening. These purposefully selected cases, by their contrast to the more general, may cast light on the larger situation and "reveal the properties of the class to which the instance being studied belongs" (Guba & Lincoln, 1981, p. 371). "The importance of a case, and its validity as proof, cannot be evaluated by the frequency of its occurrence" (Lewin, 1935, p. 42).
This research consists of two case studies involving exemplary teachers of secondary school mathematics. Here the adjective "exemplary" is employed with the meaning given in The Random House Dictionary (Stein et al., 1967, p. 498), "serving as an illustration or specimen". These teachers are examples of those professionals who are attempting to implement classroom practices consistent with the reforms outlined in the documents from the NCTM and OAME. Following the hypothesized connection between teachers' conceptions of mathematics and instructional practice (Ernest, 1989; Lerman, 1983) it was anticipated that, if teachers holding fallibilist images of mathematics are to be found at the secondary school level, they are most likely to be among those employing non-traditional teaching methods. Working with exemplary mathematics teachers, as the study results will show, does not guarantee the finding of social constructivist subject images, but the chances are likely to be increased.
These teachers' purposeful selection for this research was based upon their reputations as users of non-traditional instructional methods. Limited prior contact and observation had revealed that in their classrooms students developed concepts through investigations, and that mathematical reasoning was encouraged through student discussion. The competence of both teachers had been recognized by their school and board administrations and peers. Both at times had acted as mentor teachers for pre-service teacher education programs.
These two teachers were approached and after extensive discussions concerning the purposes and nature of the project agreed to participate. Since part of this study involved observing classes taught by the participating teachers, permission to conduct such research was obtained from the administrations of the boards and schools in which the participants were employed.
Jonathan Ode (all names are pseudonyms) is the more senior of the two teachers, having considerable work and educational experience prior to beginning secondary school teaching. Jonathan began university studies in engineering but left after two years to enter the military where he received training in electronics technology. After five years of military service, Jonathan returned to university completing a B.Sc. degree in mathematics and physics and continuing with four years of graduate school, receiving masters degrees in both philosophy and mathematics. Jonathan completed education courses through summer programs during his first three years of teaching. He has taught mathematics for 23 years and held the position of Assistant Department Head in large secondary schools (1500 plus students) in a suburban region of a large metropolitan centre. During his teaching career Jonathan has written articles in professional journals, made presentations at teachers' conferences, and participated in curriculum development activities.
Randy Walker has taught mathematics, physics and general science for 17 years and been the Assistant Head of the mathematics department in small (200 pupils) and medium sized (1000 pupils) secondary schools located in an industrial city with a population of 160,000. Randy began university studies in engineering but after two years switched to science, completing a degree with a major in mathematics. This was followed by five years employment in a variety of jobs, the longest lasting (3 years) as a management trainee in banking. Deciding that he wanted to be a teacher, Randy returned to university for a year and received a B.Ed. degree in mathematics and physics teaching. Recent personally directed exploration and study has lead to the development of materials for the high school teaching of the mathematical topics of fractal geometry and chaos. Using his teaching materials, Randy has conducted teacher workshops and made conference presentations in a variety of Canadian and US cities.
Qualitative research is essentially an investigative process, not unlike detective work. (Miles & Huberman, 1984b, p. 37)
Explorations in the two strands of the research: beliefs and teaching practice, required use of the full range of data collection techniques associated with qualitative research. These included: semi-structured interviews to explore beliefs, with the structure provided by participants' previously developed writings, repertory grids and concept maps, 'participant observation' of lessons, and the collection of documents employed in teaching and the products of student work. The materials collected or generated during these activities were stored in the form of: document files, audio tapes of interviews and lessons, and field notes recorded in log books. Extensive contact with the teachers involved also provided opportunities for informal discussions of issues related to the teaching of mathematics and schooling in general. Field notes recording the substance of such discussions were kept. Involvement in these teachers' daily school routines provided opportunities to experience their interactions with other professionals during departmental meetings and in less formal social gatherings in staffrooms and lunch rooms. While intrusive data collection techniques, such as audio taping, were not employed during the participants' contacts with other teachers, field notes of such occasions were kept.
To keep track of the data, all contacts with participants and materials collected or generated were assigned codes (Miles & Huberman, 1984b) and recorded in ongoing data logs (see Appendix A for data coding scheme and data logs).
The research consumed considerable amounts of the participating teachers' time and was rather invasive of their lives. To avoid creating undue pressures that could lead to hurried casual treatment of the research activities, or possibly create participant resentment of the project, visits to the schools were arranged at the teachers' convenience. Classroom practice data was collected over a four month period while the activities related to gathering information of the participants' images of mathematics were spread over 10 months.
Until the concluding data analysis the two cases were addressed separately but data collection in both proceeded simultaneously. Thus, while breathing spaces were provided for the teachers involved, data collection and recording required the full-time attention of the researcher. While the two studies proceeded in parallel, there was no need to have activities follow a common timetable. Scheduling decisions were made with the primary purpose of generating maximum participant comfort.
The extended data collection period had benefits beyond participant convenience. Prolonged engagement at a research site and the opportunity to repeatedly sample over an extended time increased the internal validity and reliability of the study (Lincoln & Guba, 1985).
The following sections provide descriptions of the procedures employed in the data collection within the study's two strands: conceptions of mathematics and teaching practice.
Conceptions of the Nature of Mathematics
There is no clear window into the inner life of a person, for any window is always filtered through the glaze of language, signs, and the process of signification. And language, in both its written and spoken forms, is always inherently unstable, in flux, and made up of the traces of other signs and symbolic statements. Hence there can never be clear, unambiguous statement of anything, including an intention or meaning. (Denzin, 1989, p. 14)
Conceptions of the nature of a discipline, being a component of one's belief system, are subjective and difficult to access. The language potentially employed by research participants to articulate their personal subject images may use common words, but with differences in personal meanings, the researcher may misinterpret the communication (Rose, 1962). In addition, participants themselves may not be fully aware of their personal points of view or may not have readily at hand the words or 'significant symbols' (Mead, 1934) to present their meanings. Thus they may require assistance in exploring and articulating their personal knowledge, but any aids provided have the potential to impose external meanings. While the possibility of misinterpreting research participants' words and symbols can never be totally eliminated, there exist procedures to support the exploration of personal meanings that can reduce this danger. Three of these: reflective writing, repertory grid technique, and concept mapping were employed in this study.
Researchers (Aguirre, Haggerty & Linder, 1990; Beyerbach, 1988; Easterby-Smith, 1981; Morine-Dershimer, 1991) in other studies have employed these techniques in a single step manner, taking the participants' meanings directly from their writings, repertory grids, or concept maps. Such an approach substitutes new possibilities for misinterpretation in place of potential confusion over spoken words. By combining the interpretation of personal writings, repertory grids, and concept maps with interviews, the danger of false interpretations can be reduced. In this study, interviews were scheduled with each participant after they had completed each task: writing, constructing a repertory grid, and drawing a subject concept map. Their products from the respective tasks were used as the focal points of these interviews where I and the teacher collaboratively analysed their writing, grid, or map. In this manner the interviews were given structure, but this structure was one provided by participant rather than researcher. Following the advice of Seidman (1991) all interviews were kept to less than 90 minutes in length and, in fact, no session exceeded an hour (55 minutes maximum). Interviews were audio taped and later transcribed.
The use of four data sources: personal writing, repertory grid, concept map, and the accompanying interviews, provided triangulation in this study: triangulation, not to just provide verification but in the sense of Mathison (1988) as a means of lighting a phenomenon from multiple angles and thus revealing more and better detailed evidence.
Personal statements: Images of mathematics.
During the first meeting, each participant was given a request to write a brief personal statement concerning the nature of mathematics, the sources of mathematical knowledge, and the measures of mathematical truth (see Appendix B for a copy of the instructions to participants). To avoid putting pressure on the participating teachers no deadline for completion of this task was set. Once a participant's response to this task had been received he and I set a time for an interview to examine his statement. The intent of this interview was not to test or challenge the teacher, but to collaboratively explore the meanings of his words to ensure that my interpretation was essentially correct and to give the participant an opportunity to expand upon any of the ideas expressed in his paper.
Personal writing provided an opportunity for each teacher to set out his views on mathematics, but without probes to initiate deeper investigation it was possible that underlying meanings would be missed. Repertory grid technique can serve as such a probe to, "elicit, systematise and exhibit personal meanings" (Thomas & Harri-Augstein, 1985, p. 260). In this study participants constructed grids that focused on their conceptions of school subjects. By having each participant compare and contrast subjects, including mathematics, features of their image of mathematics were revealed.
The development of a grid begins with the identification of a set of elements, in this case 8 school subjects, including: mathematics, English, physical education, a science, a modern language, an art, a subject from the social sciences or humanities, and a commercial or technical subject (see Appendix C for instructions to participants concerning the selection of subjects and examples of displays of repertory grids).
Elicitation of constructs proceeded by presenting to the teacher random selections of three elements and having him describe some way in which two are similar and different from the third. This description and its opposite were used to generate a scale upon which the elements were arranged. Elicitation continued until the constructs revealed began to overlap and became redundant.
The computer program, RepGrid (Shaw, 1990), was employed to provide an analysis of the arrangements of elements and constructs. Of the various displays generated by this software (see Appendix C), that of the principle components (PrinCom) is the easiest to read and was used as the focus of subsequent interviews.
On the PrinCom display the elements (subjects) are arranged according to the descriptors provided by the constructs. Subjects that have descriptions that are similar are plotted in close proximity. The constructs appear as lines with the degree of collinearity between two lines being a measure of the constructs' dependence. Thus perpendicular constructs may be regarded as independent. In addition, the length of a line is an indication of the polarity of the construct. Those constructs for which subjects were clustered at the extremes of the scale are displayed with extended lines. The PrinCom display identifies those subjects that carry similar descriptions and records the qualities attributed to each discipline.
In the subsequent interviews, participants were asked to react to their grids, and to elaborate on any information with which they agreed or disagreed. In this conversation new and hidden features of their images of mathematics surfaced.
In constructing concept maps one externalizes personal frameworks of knowledge. In this process, developed by Novak and Gowin (1984), one begins with the title of a knowledge area (mathematics) set on the mapping surface and around this are placed the labels of concepts that are components of this area. A network of lines is drawn between these concept labels to indicate the relationships between them and their hierarchy (see Appendix D for an example of a map and the instructions given to the study participants).
This study employed concept mapping in its least restricted form, asking participants to generate both the concepts to be mapped and the connections to be shown. Novak and Gowin (1984) note that users often have difficulties finding appropriate labels for the connecting lines. Anticipating that such a problem might arise in this study, participants were encouraged to label connections, but this detail was not considered essential. Mapping a large knowledge domain such as mathematics, especially when one has considerable experience with the subject, is likely to be a time consuming task. Participants were encouraged to tackle the problem in parts and to take as long as required. No deadline was set for the completion of the map.
Once a participant's map was, in their terms, complete, an interview, in which he and I would explore its content and organization, was arranged. The maps and subsequent interviews provided pictures of what within the discipline was important to each teacher and give some sense of how each saw the subject growing.
Mathematics Teaching Practice
The task of accounting for successful instruction is not one of explaining how students take in and process information transmitted by the teacher. Instead, it is to explain how students actively construct knowledge in ways that satisfy constraints inherent in instruction. (Cobb, 1988, p. 87)
A sociological and epistemological perspective, as described in Chapter 2, guided the gathering of data on the teachers' instructional practices. That is, the focus was on the processes by which pupils constructed mathematical knowledge and the teacher's role in this, in: planning, creating supportive environments, setting tasks, grouping students, and orchestrating discourse.
During whole class activity the interactions between students and between the teacher and students are observable from a position at the back of the room. This is not true when students are engaged in individual or small group work. In these situations it is necessary for a researcher to move into the class to closely observe student and teacher activity and to potentially interact with pupils. This has the potential to disrupt the regular lesson flow unless pupils see the researcher's presence as natural. Efforts were made to ensure that this was the case.
I was introduced to each class by the participating teachers, as a high school mathematics teacher who now works at a university, one who is interested in how students learn mathematics in this course, and as a person who at times may provide assistance in the class. As such, my role approached that of the participant observer (Spradley, 1980) common to much qualitative research. I had previously visited the classrooms of both participants and had discussed my potential actions with them. Both teachers appeared to be comfortable with the planned approach.
A minimum of 20 lessons led by each teacher were observed during the course of the semester. This provided opportunities to participate in classes addressing a variety of topics and grade levels. Whole class instruction portions of lessons were unobtrusively audio taped and illustrative segments later transcribed. Extensive field notes recording: classroom arrangements, questions posed, responses obtained, locations of interacting pupils, and interval times for various activities were kept. In addition, handouts used during the lesson and copies of representative student products were gathered.
Two further research activities served to link images of mathematics to practice. As often as possible, within the busy schedules of the participating teachers, short interviews were conducted to obtain the reasons behind their instructional decisions: Why did they organize the lesson in this way? Why was a particular task employed? Why were certain materials and resources used?. In addition, during the time I spent with the teachers, observations were made of the resource materials they consulted and conversations they had with fellow professionals and students outside regular class time.
'It seems very pretty,' she said when she had finished it, 'but it's rather hard to understand!' (You see she didn't like to confess even to herself, that she couldn't make it out at all.) 'Somehow it seems to fill my head with ideas - only I don't exactly know what they are! (Carroll, 1954, p. 130)
Although Jonathan and Randy have in common their reputations as teachers contributing to mathematics education reform, they are still distinctly unique individuals. Anticipating that their conceptions of mathematics and the instructional practices they employed would be equally unique, the initial data analysis was conducted on the two cases separately.
Once the cases were developed, a second level analysis was conducted to search for common themes. These are reported in a concluding chapter of the thesis along with implications derived from common trends.
Most conclusion-drawing tactics amount to doing two things: reducing the bulk of data and bringing a pattern to them. Such tactics are sometimes rationally trackable, sometimes not. (Miles & Huberman, 1984a, p. 27)
A "rationally trackable" tactic for data analysis, one following the methods described by Bogdan and Biklen (1992) and Miles and Huberman (1984b) was employed. Initial data analysis was conducted as materials were collected.
Beginning with the data connected to a single participant's image of mathematics, all items were examined for the existence of frequently occurring ideas or themes. Theme labels were developed and attached to the various 'conceptions of mathematics' articles in the data. Once the themes apparent in the participant's conceptions of the discipline had been identified, the data related to practice was examined for incidents of the same ideas. Although analysis began with beliefs, if a new re-occurring idea was located in the practices material a new label was created and the conceptions of mathematics data re-examined for incidents of the same theme.
Through repeated cycles of the analysis process it was possible to locate themes that existed in both data strands: subject image and instructional practice. Themes that did not fully find development in both strands were also noted since they marked cases where beliefs were not appearing in practice.
One starting point of almost all research on teacher thinking has been the concern for the tacit aspect of teachers' knowledge and for the paradox implied by this quality; but while knowledge must be made explicit if the teacher's voice is to be heard, we thereby risk turning teachers' knowledge into researchers' knowledge, colonizing it, and thus silencing the voice of the teacher. (Elbaz, 1991, p. 11)
The results of the data analysis are reported in two chapters, each dealing with just one case. For each case, the report consists of sections alternately presenting a theme from the teacher's conception of mathematics and a narrative of illustrating classroom practice. In each case this pair is related but the link may be either an example of belief in action or of the teacher's struggles and incomplete transfer of image to pedagogy. As much as possible the participants' own words are used to present their visions of mathematics. In the concluding chapter themes that cross both cases are addressed.
We can argue for negotiated outcomes on the ground that such negotiation is essential if the criteria of trustworthiness are to be met adequately. We shall see that a major trustworthiness criterion is credibility in the eyes of the information sources, for without such credibility the findings and conclusions as a whole cannot be found credible by the consumer of the inquiry report. (Lincoln & Guba, 1985, p. 213)
Within the limits of the participants' schedules, efforts were made to collaboratively develop, meanings and interpretations of the data. This process, by the research design, took place within the investigation of subject image with interviews to explore and elaborate on the participant's writing, repertory grid and concept map. In these interviews there were opportunities for the teacher to point out problems with my interpretations and to provide alternative meanings. Vignettes illustrating classroom interactions were provided to each teacher for his feedback. Such participant commentary not only helped verify the account, but also addressed issues of confidentiality.
Walker (1980) notes that in case studies, confidentiality can not be guaranteed. In delivering 'thick' descriptions of participants and locations, the study makes cases identifiable even when pseudonyms are employed. Walker suggests that this issue be addressed, not by promising the impossible, but by admitting that anonymity can not be ensured. An alternative form of protection was provided for the participants by continually involving them in negotiations of meaning and data release.
Expatiate free o'er all this scene of man;
A mighty maze! but not without a plan;(Pope, 1733/1846, p. 10)
Having completed, as a minimum, university honours degree programs in mathematics and physics, both Jonathan and Randy have participated in significant formal studies of mathematics. This focussed academic experience with the discipline has been, for both, augmented by periods of employment in military, industrial or business settings requiring the use of mathematical skills and knowledge. These opportunities, coupled with self-initiated reading, personal independent study, and reflection on teaching during at least 17 years in the classroom, have contributed to the development of multidimensional conceptions of mathematics. Similarly, reading of the mathematics education professional literature and regular attempts to put new ideas into practice mean that both teachers have developed multifaceted styles of teaching. The structure of the cases set out in the following two chapters was developed to show this complexity and highlight both connections and missing connections between conceptions and practice. This section is designed as an orientation for a reading of the two cases and provides an outline of their organization.
Multiple readings of the data related to the teachers' images of mathematics led in both cases to the identification of a number of re-occurring themes. With these themes in mind the field notes, gathered materials, and audio tapes of the observed lessons were examined and teaching incidents that conveyed compatible or contrary messages were noted.
The two case chapters are organized in alternating subject conception-teaching practice units. That is, each case, after opening with an introductory view of classroom practice, follows the pattern of a section presenting one aspect of the teacher's image of mathematics followed by a unit giving a picture of instruction that appears to either support or disagree with the identified belief.
Composing a Picture of Subject Vision
The participants, through their writing, subjects repertory grid, concept map for mathematics and interviews focussed on these, articulated their images of mathematics. Although the sequence of particular statements has been altered to permit the thematic grouping of evidence, it is the teacher's voice, their words and products, that is used to paint a picture of their conception of mathematics.
Individual image themes arose repeatedly in interviews, informal discussions, and in the teacher products: writing, repertory grid, and concept map. These are gathered together to develop a particular aspect of the teacher's vision of mathematics, but in each case the original source is given by a data code appearing at the end of the quote or description (see Appendix A for the coding scheme and data logs).
Telling the Story of Teaching Practice
Teaching practice is described through narratives of selected classroom events, those most closely related, either positively or negatively, to subject images. To give a sense of "being there" in the classroom, these stories are told in the present tense. In this way the reader can see the event unfolding and gain a sense of the mathematical epistemology being displayed by the teacher's actions and words. Images and text displayed on the classroom blackboards or overhead projector, or appearing in student handouts are reproduced in the narratives whenever essential to the understanding of the story. In the class dialogues the teachers are identified as Mr. Ode and Mr. Walker, while the generic terms "pupil", "student" and "class" are employed to identify other speakers. Student names (pseudonyms) are used only when there is a continuing conversation with or about a particular pupil.
Teaching involves much more than conducting lessons. It is the choices made during course and lesson planning that determine the activities and thus the epistemology revealed through individual classroom events. In the cases that follow, resource materials consulted by the teachers are identified and their thoughts in planning for and reflecting on lessons, as revealed in interviews and discussions, are given. In activities and conversations outside the classroom the participants are identified by their given names, Jonathan and Randy.
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