Understanding is not a matter of appreciating the "real" causal links prescribed by Nature but of imposing a purposive structure that emerges from the interaction between investigator and phenomenon....What emerges from the interaction is a constructed reality that is shaped in equal proportion by the investigator's purpose and the phenomenon's presentational aspect [emphasis in original]. (Lincoln & Guba, 1985, p. 152)
The study reported here has three central questions:
What are the conceptions of mathematics held by exemplary secondary school teachers, those who are attempting to implement the reform proposals?To what extent are these teachers' instructional practices expressions of their subject images?
and
What are the struggles involved in these teachers' efforts to translate subject images into classroom practice?
To a large extent the first two questions have been addressed within the case studies related in the two preceding chapters. Through the purposeful selection, as participants, of exemplary secondary school mathematics teachers, this study has been able to fill-in some gaps left by previous research concerning teachers' conceptions of mathematics and the links between these beliefs and instructional practices.
In the studies cited in the literature review, participants were generally teachers who employed traditional teacher-centred transmissive lessons; the kind of instruction that provincial
(Colgan & Harrison, 1997; Ontario Ministry of Education, 1992a, 1992b), and international (Crosswhite, 1987; Lapointe, Mead & Askew, 1992; McLean, Wolfe & Wahlstrom, 1987; Robitaille, Taylor & Orpwood, 1996) surveys have shown to be predominant at all levels of mathematics teaching. Most of these teachers were also found to hold absolutist conceptions of mathematics; images that matched their teaching styles. On the other hand, teachers holding distinctly problem solving conceptions of mathematics were found only in studies (Philipp, Flores, Sowder & Schappelle, 1994; Wood, Cobb & Yackel, 1991) at the elementary level and where there was extensive coaching and support provided by the researchers. In these cases the teachers' developing fallibilist conceptions of mathematics were compatible with their new styles of instruction. The present study with Jonathan and Randy extends these results in a number of directions.
Jonathan, in Chapter 4, through his writing, repertory grid, and concept map, and the interviews analysing these products displays a social constructivist conception of mathematics. The classroom episodes related in Jonathan's case show the main themes of his mathematical philosophy at work in his planning and instruction. Thus in this case we find a teacher, exemplary in his use of instruction advocated in the reform documents, possessing a compatible social constructivist subject conception.
Randy's instruction does not stray as far from tradition, but links may still be drawn between his subject conception and teaching. In Chapter 5, Randy presents a mixed image of mathematics; one that has both absolutist and fallibilist dimensions. In a similar fashion his teaching is mixed, being quite traditional when addressing standard school mathematics topics and becoming more adventurous whenever newer content such as fractal geometry or probability is the focus. We see parallels in Randy's emerging fallibilist philosophy of mathematics and his changing instructional practice. The development of Jonathan's social constructivist subject image and non-traditional teaching style, and the emergence of Randy's new fallibilist conceptions of mathematics and class use of mathematical experiments have occurred, unlike the situation in previous studies, in the absence of officially sanctioned reform programs.
The research did not set out to establish a causal link between subject conceptions and instructional practices, and the two case studies do not suggest that one exists. Although Jonathan possesses a social constructivist image of mathematics and his observed lessons predominantly display this philosophy, he is still capable, on occasion, of employing a transmissive style of instruction (JO-LES-N-06). Randy's vision of fractal geometry as an experimental discipline suggests that learning should progress through student investigations, but, in forcing his lessons, building to the Mandelbrot set (RW-LES-N-06, 09, 12a), to fit within the very limited available time, he uses a highly teacher-centred style. What can be said is, that these teachers' conceptions of mathematics appear to encourage them to make particular choices for instruction and, in the absence of significant competing factors, practice is compatible with beliefs. The translation from beliefs to instruction is not easy or smooth and involves considerable effort. My extended contact with Jonathan and Randy, observing their full teaching days, provided opportunities to record the obstacles placed in their way and the struggles to bring their visions of mathematics to the classroom. The visions into practice translation process and the struggles involved are the focus of the following section.
Teachers do enter into dialogue with innovation. The new practices and the old interact in complex ways. We can picture the new and the old overlapping to create a zone of turbulence and challenge. (Black & Atkin, 1996, p. 148)
Jonathan Ode and Randy Walker are unique individuals with different personal histories and professional contexts. Their conceptions of mathematics and teaching practices, while having some common themes, are quite different. Jonathan's philosophy of mathematics can be described as social constructivist while Randy presents a mixture of absolutist and fallibilist views. In their teaching practices, Jonathan more consistently than Randy displays features of the mathematics education reform ideas presented by the NCTM (1989, 1991, 1995) and the OAME/OMCA (1993). When facing opposition to their use of non-traditional instruction, Jonathan, after a short detour, continues along the reform path while Randy retreats.
Nevertheless, there are some parallels between the two cases. Both Jonathan and Randy experience challenge and turbulence in their efforts to put problem solving visions of mathematics into classroom practice. This challenge is partly external, in the opposition from students, parents, and school administrators, but it is also internal, as Jonathan and Randy work to construct arguments for instructional choices; arguments not for presentation to those in opposition but for themselves. In examining these teachers' thinking and action within the zone of turbulence, questions may be asked and partially answered. What is the nature of the struggles experienced by teachers as they work to put subject visions into practice? What characteristics of a philosophy of mathematics appear to influence the extent to which a teacher translates vision into action? What features in a teacher's professional environment contribute to the construction of a subject conception that promotes and supports changed instructional practices? In the following sub-sections these questions will be explored, using the data from the two cases previously developed and field notes recording features of the teachers' professional lives.
Visions into Practice: The Struggle
To open up one's class so that students are pursuing problems whose outcomes cannot be easily foreseen is a hazardous business....What happens when students refuse to work independently? When they balk at taking risks in their learning? What if one's colleagues, the students' parents, or the school administrators decline to take their own gamble of supporting...'adventurous teaching'? (Black & Atkin, 1996, p. 132)
Ernest (1991) acknowledges that the impact on practice of a teacher's beliefs concerning the nature of mathematics, "is mediated by the constraints and opportunities provided by the social context of teaching" (p. 290). Teachers may be supported or hindered in the classroom expression of their images of mathematics, by the expectations of students, parents, school administrations, and colleagues. The results of studies (Cooney, 1985; Dorgan, 1994; Ferrell, 1995; Heaton, 1992; Kesler, 1985; McGalliard, 1983; Philipp, Flores, Sowder & Schappelle, 1994; Prawat, 1992; Putnam, 1992; Raymond, 1993; Remillard, 1992; Thompson, 1984) examining the link between teachers' views of mathematics and their instructional practices reveals the constraints provided by the teaching context. In all cases, those teachers holding an absolutist philosophy of mathematics were also observed to employ a compatible transmissive style of teaching. Such agreement should not be surprising since in both their images of mathematics and their teaching practice these teachers fit the predominant mode. Neither their thoughts about mathematics nor their instructional methods would meet much opposition or challenge. On the other hand, of those teachers with a distinctly problem solving image of mathematics, only the elementary school teachers in the studies by Wood, Cobb and Yackel (1991) and Philipp, Flores, Sowder and Schappelle (1994) translated their views into practice. Here the teachers had received extensive support through participation in graduate level courses and curriculum reform projects. In all other situations, teachers were found to modify their teaching practices to bring them into line with the teacher centred norm.
In the present study, the teaching contexts of both Jonathan and Randy posed constraints to their translation of subject conception into teaching practice. They met opposition from pupils which, through the intervention of parents, resulted in challenges from their schools' administrations. In the face of these difficulties both teachers adjusted their teaching, but for Jonathan this was only a short detour. A month after being cautioned about student and parent complaints, Jonathan was observed taking a problem solving approach that portrayed mathematics as an inductively developed human construct (JO-LES-N-12). After his brush with authority, Randy's problem solving approach was consigned to the edges of his program, appearing in: optional bonus assignments, lesson extensions whenever a few minutes could be spared, a loosely organized experimental course, and in extra-curricular explorations with committed pupils. Randy realizes that, "there are always more questions to ask and more avenues to explore" (RW-INT-D-03), but notes that "it's very difficult, within the constraints of both the system and our own experience, to actually accomplish anything more than just the regular course of study" (RW-INT-D-05b). The two teachers of this study found that there are risks, conflicts and challenges that teachers seeking change must struggle against.
As noted above by Black and Atkin (1996), there are intellectual and administrative risks in setting open-ended tasks for students. Jonathan has experienced such situations. Reflecting on the Grade 9 periodic decimal art project (JO-LES-N-07), he reports,
That study actually can construct multiple 'what-ifs' and students are likely to ask why. There are some funny patterns. I predict, you can predict, and I mean it's natural that the students are going to then want to know why, which I think is great. But for some teachers that's threatening. It's, I guess, because they just haven't done it themselves. I mean it's tough. (JO-INT-D-15c)
Teachers generally are not ready to take such risks and thus stick to the "binomial theorem because they know how to do that and it's so much more difficult to do something else" (JO-INT-D-15c). Randy also recognizes teachers' reluctance to tackle difficult topics. Discussing open questions in probability, he notes, "There are difficult concepts there. There are things that you sort of don't want to touch, might not be able to answer, or complete usually. So I think they've kind of just been filtered out of the curriculum" (RW-INT-D-02). Jonathan sees teachers' adjustment to such risk situations as a growth process and notes that time and experience are required.
I think what has to happen is these teachers have to be exposed to that [open-ended problem solving] and at first they're going to get scared. After a while they'll get used to it and they'll begin to learn how to respond and their scope will widen. I think that the end result is we're going to have better teachers and better students. (JO-INT-D-15c)
The intellectual demands that come from a problems centred program do not all fall on the teacher. Open-ended investigations, reasoning, and mathematical discussions make intellectual demands of students. Secondary school pupils have experienced eight or more years of study in mathematics. This background defines the 'rules' of the discipline, what it consists of, how it is taught, and what the expectations of students will be. Schoenfeld's (1989) surveys of high school students' attitudes towards mathematics show that all were unwilling to commit any more than 20 minutes to the solution of any one problem. Colgan and Harrison (1997) found that Grade 12 students had come to accept teacher centred mathematics instruction with 81% reporting that they "liked to learn mathematics in school by listening to the teacher" (p. 7). Randy has observed such lack of intellectual commitment from students and notes that marks and not knowledge appear to be their goal.
The students just take the courses to get marks without any objective beyond that as to how they'd apply any of their knowledge. They just need a course that's a prerequisite for another course that's a prerequisite for a piece of paper that allows them into a profession. (RW-INT-D-14)
Changes in teaching style, especially those that create ambiguity and call for increased pupil effort, are likely to lead to student opposition. Such conflict reduces the range of possible classroom activities and effectively, pupils become covert curriculum decision makers.
The lesson does not simply belong to the teacher, children can and do make it their own. They put so much on the agenda of the lesson, to a point where, they are the curriculum decision-makers. They make a major contribution to the social construction of classroom knowledge. (Riseborough, 1985, p. 214)
Pupils' curriculum defining roles have been recorded in studies involving mathematics teaching. Cooney (1985) working with a beginning high school mathematics teacher, holding the view that problem solving was the core of mathematics, observed that "the means by which he had decided to teach mathematics conflicted with the expectations of many students, particularly the less able ones, about what constituted mathematics and how it should be taught" (p. 333). Within ten weeks of beginning teaching, while still maintaining that problem solving was important, the teacher had separated this from the content material of the course, the mathematical procedures that pupils were to learn. Problem solving was no longer the core of mathematics but an add-on to be addressed if time and student motivation would permit. Similarly Brown and Borko (1992) report how a beginning secondary school mathematics teacher's instructional intentions were adjusted in the face of what he believed his students were capable of or willing to do.
In this study Randy adjusted his demands of students to fit the efforts they were willing to give and converted his open-ended investigation projects into bonus assignments. He came to accept that only a few students are risk takers and willing to make the effort to come up with original ideas (RW-INT-D-03). Both Jonathan and Randy value intellectual commitment and are willing to put in considerable time with students who wish to explore mathematics beyond the confines of the curriculum. Randy worked one-on-one with Gary, a Grade 10 student, encouraging his computer supported explorations of fractal geometry. Jonathan founded a school club, the "Math Senior Scholars", which met in his classroom after school to look at problems that extended their course curricula (JO-INT-N-06).
Students often enlist the aid of parents in mounting opposition to changed teaching practices and increased work requirements. For most parents the image of mathematics is one of fixed rules and procedures that they developed while experiencing the traditional mathematics instruction that teachers such as Jonathan and Randy are working to change. Parents in middle class neighbourhoods, such as the locations of Western Secondary School and Northern High School, may have experience of mathematics study beyond the high school level, but that in many cases does not promote a more open view of the discipline. "Many educated persons, especially scientists and engineers, harbor an image of mathematics as akin to a tree of knowledge: formulas, theorems, and results hang like ripe fruits to be plucked by passing scientists to nourish their theories" (Steen, 1988, p. 611).
Parents may bring their concerns directly to teachers but more often they make their representations to school, board or government officials where they may find individuals with equally traditional views of mathematics teaching. During Jonathan's most highly teacher centred lesson, the school vice-principal visited his classroom to collect some student data for a board survey. On exiting the room, as an aside to Jonathan, he commented, "I'm very impressed, everyone is very quiet and working diligently" (JO-LES-N-19). When such definitions of "good" teaching and learning are held by school administrators it is not surprising that adventurous teachers receive little support.
In fact, often times it's, it's just the other way around. The teacher who is creative and wants to try something new and gets in there and takes a chance and has a student who reacts negatively to that, maybe sees the guidance counsellor or something like that, that's the teacher who's going to get reprimanded because they're doing something out of the ordinary. They might be cautioned about, you know, you should keep everything a little bit more in line with what's done traditionally. So it's almost there in a negative way to hold back the good teachers as opposed to help along the others. (JO-INT-D-10b)
In this study both Jonathan and Randy experienced the results of parental complaints. Jonathan was cautioned by the vice-principal about concerns that his problem solving approach had raised (JO-DIS-N-03) and Randy was reminded that assessing his Grade 12 students' work on extended open-ended investigations was against school policy (RW-INT-D-03). They discovered that "being a teacher is not merely a matter of subject knowledge and methodological skills. Teaching also implies operating as a member of an organization, that is 'loosely coupled' and highly political" (Kelchtermans & Vandenberghe, 1995, August, p. 9). Neither Randy nor Jonathan appear to have developed or be interested in developing the skills needed to operate in the political domain. Neither directly challenged the school authorities. Jonathan ignored the criticism and warning while Randy adjusted his practice to fit within the school's rules.
In secondary schools the strong departmental subcultures define what teaching approaches are acceptable and unacceptable (Grossman & Stodolsky, 1995; Siskin, 1991) thus reducing dissent and opportunities for intellectual development. "They constitute social communities of common thinking, feeling, and belief in relation to the nature of knowledge and learning, desirable and undesirable approaches to pedagogy, attitudes to student grouping, student discipline, and so on" (Hargreaves, Wignall & Macmillan, 1992, p. 7). Adopting innovative pedagogical methods can bring a teacher into conflict with colleagues. When techniques that violate the unwritten rules are employed, those following traditional patterns are threatened. Those opposed to change may organize covert interference, discontinue cordial social interactions, or indulge in openly hostile actions (Beynon, 1985).
Conversations in the Mathematics Department office of Northern High School (RW-DIS-N-07, 22) focused on issues of student discipline and how to get pupils to learn the rules and procedures of mathematics. All department members voiced opposition to alternative teaching practices such as collaborative group work. They "knew" it did not work and were not interested in research on the issue or experimenting with it themselves. While Randy accepted his colleagues' highly traditional views of teaching (RW-INT-D-05b, RW-INT-D-13), he felt very isolated by their collective lack of interest in mathematics beyond the school curriculum (RW-DIS-N-32). At Golden District Secondary School, his former workplace, he had found one "kindred spirit" with whom he was able to discuss his explorations in fractal geometry, but at Northern High School, without anyone with which to share ideas he was not sure that he could continue his work (RW-DIS-N-32).
Jonathan was not as isolated at Western Secondary School, for there he had colleagues who to some extent shared his interests in mathematics and also experimented with alternative teaching styles. With their desks clustered together in one corner of the departmental workroom these three were able to participate in supportive conversations concerning mathematics and its teaching (JO-DIS-N-08,09,12). The rest of Western Secondary's mathematics department appeared to employ quite traditional styles of teaching and Jonathan was frustrated in his efforts to bring about change. In preparation for an official school board-wide "Math Day", a day set aside for alternative exciting mathematics activities, Jonathan provided his fellow Grade 9 teachers with copies of collaborative problem solving activities from Get It Together (Erickson, 1989, pp. 132-137, JO-LES-D-17b, see Appendix G). Later, while using these activities in class (JO-LES-N-17), he confided to me that, despite the aims and plans for Math Day, he suspected that in most classes lessons were proceeding in their regular direct instruction manner (JO-DIS-N-13).
Recent studies (Barnes, 1995; Edwards, 1994; McGlamery, 1994; Secada & Byrd, 1993) examining the processes of change in secondary school mathematics teaching have highlighted the need for administrative support and department-wide commitment; features that were missing in both Jonathan's and Randy's school settings. Both participants in this study talked regularly of a lack of support, but put this in terms of the teaching population in general rather than relating it directly to their particular situations.
I see huge problems in education today. Ones that need to be solved before teachers are going to be able to go on and do what they need to do. It's too often the case that all of these other problems are at the root of what's not happening. What really happens is everybody comes in and says well what's wrong with the teachers? You're not teaching today's mathematics you're teaching a hundred years old mathematics. And it's not necessarily their fault. (JO-INT-D-15c)
"What we're learning in the teaching profession is that if change is to come about, then it has to be more than just policy. We don't have the kind of support that goes with a commitment to the change" (RW-INT-D-14).
For Jonathan and Randy the major problems lie in a lack of time; time within the classroom to address topics in depth:
If we were ever to set out to take any effective approach to improve mathematics education, I think we'd do less and we'd do it better. So many curriculum attempts have involved just adding more and taking nothing away. And then we're forced to try and be efficient in our presentation and we end up with the lecture formats that we have now, rather than much of a discovery and exploration approach. (RW-INT-D-14)
and preparation time to develop interesting and meaningful student activities:
I don't have time to go to Hydro and ask them, can you give me some unique applications that I can use in my classroom. Neither do any other math teachers. It happens because once in a while a teacher does something which goes above and beyond, not just the call of duty, but above and beyond what would normally be expected. That happens. But it happens at a cost to that individual too and sometimes their classroom. (JO-INT-D-15c)
Not all problems or impediments to change originate out in the teachers environment; some come from within. Randy's image of mathematics as a technical language (RW-INT-D-01a, RW-DOC-D-07, RW-DIS-N-07) that his students will someday employ in their future careers, leads him to provide lessons from which students can learn the details of that language: the terminology, definitions, and algorithms of use. Jonathan, although holding a social constructivist philosophy, still on occasions expressed concerns for students' learning that echoed an instrumentalist 'back-to-the-basics' view. While waiting for his OAC-Finite Mathematics class to complete a quiz (JO-LES-N-12), Jonathan remarked to me, "It's taking longer and longer each year for students to do simple work. They lack so many basic skills. This is making me think, maybe we do need a database of knowledge; things such as arithmetic skills" (JO-DIS-N-10).
Kilbourn (1992), in relating the struggles of a history teacher, who like Jonathan and Randy was attempting to change to an inquiry mode of instruction that reflected his new understandings of subject, notes, "In some teaching situations part of the skill is to honour the demands of one agenda without sacrificing the integrity of another. How can interaction be promoted while maintaining a degree of rigour with regard to the subject?" (p. 83). Teachers seeking change must work in a zone of turbulence that involves their classroom, teaching milieu and personal thoughts.
Visions into Practice: The Commitment
The fundamental issue from which mathematics teachers cannot escape is that a commitment to a theory of mathematical knowledge logically implies a particular choice of syllabus content and teaching style. (Lerman, 1983, p. 65)
It can be argued that there is a logical connection between a teacher's conception of mathematics and their choices for instructional practice, but, as discussed in the previous section, serious impediments can lie in the way of this logical progression. Teachers face considerable struggle in the classroom realization of images that motivate teaching in non-traditional styles. Studies (Cooney, 1985; Dorgan, 1994; Kesler, 1985; Raymond, 1993) have reported cases where the connection was not made and teachers employed methods that did not match their professed subject images, but were compatible with school norms and fit student expectations. The teachers had been socialized into adopting the school and department patterns. But socialization to the norm is not inevitable. The process involves "a constant interplay between choice and constraint, between individual and institutional factors. Individual teachers are not merely passive recipients in the process of socialization" (Brown & Borko, 1992, p. 221). They may resist normative forces and make their own independent instructional choices.
Both Jonathan and Randy do not accept their school and department patterns. They struggle against the norm to bring different views of mathematics into practice. They possess conceptions of mathematics that are strong enough to provide the motivation for efforts against the constraints provided by their school contexts. In particular, Jonathan chooses to silently ignore the school's administration and continue to make a problem solving approach the core of his teaching.
Senior teachers such as Jonathan and Randy are in no professional danger from the mild rebukes that they received. Their competence has not been and can not justifiably be questioned. Still, for experienced teachers, criticism of classroom practice generates a sense of vulnerability (Kelchtermans & Vandenberghe, 1995, August).
Conceptions of subject are just one cluster of a teacher's educational beliefs (Pajares, 1992). Teachers also possess perceptions of self and their effectiveness as a teacher. A criticism of teaching approach is first and foremost a negative comment on teacher efficacy and a challenge to self-esteem. There is a need for the teacher to rebuild self-confidence and self-concept through a reasoned defence of their actions. The arguing of such a defence is essentially internal and the teacher, as in this study, may not bother presenting the case to the challenging school officials, students or parents. Thus the argument must be satisfying to the individual and follow their rules for logic and proof.
For Jonathan and Randy the models for reasoning, logic, and proof lie in their knowledge of and experience with mathematics and physics. They project the requirements of mathematical logic onto their own thinking and thus an argument is not valid if it violates the rules of the predicate calculus. Basic postulates must be assembled and a deductive sequence developed from these to desired consequences. Consistency in the defence is of prime importance, for in logic the existence of one contradiction means that all statements may be taken as true. In turn, these high standards mean that the base from which action is argued must be strong.
Jonathan presents a complex, integrated and consistent set of beliefs about mathematics. There is a problem solving theme that runs throughout the writing, repertory grid and concept map generated for this study and the interviews that further explored these products. He acknowledges that for some users of mathematics, the discipline may be a set of algorithms, but argues that these procedures were originally constructed by humans through investigations in the pursuit of solutions to problems. Formal research mathematics, in Jonathan's view, is also built on a constructivist platform. Although mathematicians may work with systems that do not model the physical world, their actions continue the original experimental processes for building mathematics. They develop their understandings inductively, building from patterns observed in abstract symbol systems.
Jonathan's strong conception of mathematics provides him with a solid base for personally arguing for his classroom use of group work, student discussions, inductive reasoning from patterns, collaborative problem solving, and open-ended creative investigations. In the interviews and discussions exploring the reasons for his pedagogical choices (JO-INT-D-01,02,11b,15d; JO-DIS-N-06,07,08,11,14) Jonathan linked his decisions back to his image of mathematics. When his teaching approach was questioned, Jonathan was able to construct a personally satisfying argument that allowed him, with reason, to persist in his efforts to reform mathematics education.
Randy's conception of mathematics, although complex, is less integrated and complete than Jonathan's. His philosophy is a mixture of absolutist and fallibilist beliefs. On a number of points, such as the origins of mathematical ideas and the nature of proof, Randy reports that he has not yet made up his mind (RW-INT-D-05b). Randy's teaching practices are also mixed. The episodes related in the case study show both examples of teacher centred direct instruction and situations where pupils are involved in open-ended investigations. Of Randy's variety of teaching approaches, only those that require students to participate in significant problem solving activities are challenged and thus require a defence. With his more fractured image of the discipline, Randy has difficulties completing an argument that would support pushing ahead with his plans and he adjusts his program to fit the school rules and student wishes.
Randy does hold one strong, tightly linked cluster of mathematical beliefs; those related to his recent work in fractal geometry. He consistently presents this sub-discipline as experimental, inductive, open, growing, and creative. His teaching efforts in this area, although reduced to extras in his curriculum, carry his image into practice. As Randy explores the subject of fractal geometry and its teaching, new ideas emerge that challenge both his more general conceptions of mathematics and his traditional teaching style.
The relative success of Jonathan and Randy in putting their subject conceptions into practice is similar to the results of Thompson's (1984) research. Of her three research participants, the teacher with the most integrated belief system was most successful in translating subject image into teaching action. Moreover when practice did not match subject beliefs this participant was aware of the conflict and reflected upon it. Shealy (1995) found a similar pattern in a recent study examining the translation of beliefs about mathematics teaching into practice. This study involved two beginning secondary school teachers, recently graduated from an education program that emphasized open-ended investigations. Both were excited about such student activities and had intentions of bringing them to their classrooms. The participant whose beliefs about teaching were well-developed and integrated was able to translate these into practice while the other who held isolated and competing beliefs retreated from the use of open-ended investigations.
The Perry (1981) scheme of cognitive growth can be employed to compare Jonathan's and Randy's positions concerning mathematical epistemologies and teaching. Their statements in multiple interviews indicate that both are at least at the third stage in the scheme, Relativism. They realize that there exist multiple images of mathematics and a variety of approaches to teaching. Moreover, as Jonathan indicates, they understand that one can establish reasons for each position.
This teacher approached everything from an algorithmic point of view because that's where the teacher obtained the security to teach. Because they, the teacher, would essentially learn an algorithm and then do it in class. And the students were successful in that teacher's class because they always knew that the teacher was going to test them on something just like they had been shown how to do. (JO-INT-D-10b)
Acknowledging other teachers' reasons for traditional approaches, Jonathan and Randy are not judgemental, but they do believe that their positions have greater validity. Randy, speaking of his fellow department members makes the argument,
I could hardly criticize them in some way. I really believe that they intend to do a very good job at what they do, but their view of their job is just different from mine. It is a little bit narrower. They stick to doing things exactly the same way all the time. They're not the slightest bit interested in mathematical problems or anything like that. It is just getting the job done every day. (RW-INT-D-13)
Jonathan has in fact progressed beyond Relativism and has reached the stage of Commitment. He has made a choice of epistemologies and through reasoned thought come to a social constructivist position. From this stage of commitment he is able to take action and adopt practices that embody his philosophy. Randy appears to be in transition between Relativism and Commitment, and has not yet settled on an epistemological stance. Perry notes that periods of transition between stages, when one is trying to find a path, are unsettling. Arguments are not clear and there are choices to be made. It is difficult to select definitive actions and one may act in contradictory manners. Randy would appear to be at this point in forming his philosophy of mathematics. During periods of cognitive turmoil there may be a temporary return to Dualism and a desire for authorities to make decisions and select a course of action. At times Randy takes this approach to mathematics education reform, calling for new official guidelines explicitly mandating the change.
It has to come down from the top that certain changes are necessary, and maybe the teachers are given some flexibility to learn how to interpret those changes. But if there aren't prescribed changes in the curriculum and support materials to go with them, nothing happens....There won't be positive results until teachers really know in a day-to-day sort of way what the changes mean. (RW-INT-D-14b)
Looking Above and Beyond the Curriculum
Each project began with the premise that developing collegiality among professional mathematicians and teachers can reduce teachers' sense of isolation, foster their professional enthusiasm, expose them to a vast array of new developments and trends in mathematics, and encourage innovation in classroom teaching. (Black & Atkin, 1996, pp. 145-146)
Studies (Underhill, 1988) have found that mathematics teachers' subject conceptions show little change with years of professional experience. Randy and Jonathan are exceptions to this pattern. In interviews Randy talks of his changing image of mathematics: that experimentation can be a source of mathematical ideas (RW-INT-D-01b), that mathematics may not be a perfect representation of the universe (RW-INT-D-05b), and that there may be methods of proof beyond the Euclidean deductive model (RW-INT-D-05b). In the case study we witness Randy working on these new ideas and fitting them with his former conceptions of mathematics. Jonathan, with his graduate degrees in mathematics and philosophy, entered the teaching profession with an expanded conception of the discipline. Still he reports that, although he has retained his core philosophy, there have been changes in details, and he sees that his classroom practice has "changed considerably since [his] first year of teaching" (JO-INT-D-14b).
There are features in these two teachers' professional lives that appear to have contributed to expanding subject images; conditions that may be missing from the experience of others. Jonathan and Randy have not directly sought out alternative visions of mathematics, but they have pursued mathematical knowledge. In recent years Randy has spent much time studying fractal geometry and for Jonathan a "primary motivation has always been to learn" (JO-INT-D-14b).
"Distinguishing knowledge from belief is a daunting undertaking" (Pajares, 1992, p. 309) and in fact beliefs are often taken to have a cognitive component. Ernest (1989a), when examining teachers' knowledge and beliefs about mathematics, links the two with the view that teacher thought has a cognitive outcome, knowledge, and an affective outcome, belief. Thus as teachers pursue mathematical knowledge they also meet opportunities for change in their beliefs about the discipline. In particular, when teachers work on their own personal mathematical investigations and become "creators" of mathematical knowledge they gain firsthand experience with the substantive and syntactic structures of the discipline, the critical elements in Shulman's (1986, 1987a) subject knowledge base for teaching. In the search for such knowledge, teachers must look outside their classrooms and schools.
Increasing classroom experience brings greater instructional mastery and comfort, but generally this is due to developing instructional craft knowledge and does not involve intellectual growth within the teacher's subject. In fact, for many teachers, as the years since formal subject study pass, interest in their discipline fades (Sikes, 1985). Similarly department and school collegial relationships, although important, may not support intellectual development. Department based collaborative activities involve exchanging "tricks of the trade" rather than discussions concerning the nature of disciplines or underlying principles of instruction (Hargreaves, 1992). Staffroom conversations focus on classroom anecdotes, comments on individual pupils, and issues on which participants are unlikely to have professional disagreements.
Educational theory, long-term plans, discussions about basic purposes and underlying assumptions are virtually absent features of teacher talk. Sharing is confined to stories, tips and news - to things that will not intrude upon or challenge the autonomous judgement of the classroom-isolated teacher. (Hargreaves, 1992, p. 221)
Thus relying on classroom experience and relationships that arise by chance within a school placement may not provide opportunities for growth. There is a need to seek opportunities beyond the immediate professional environment.
The study by Philipp, Flores, Sowder, and Schappelle (1994), the one case where teachers were found to translate problem solving conceptions of mathematics into classroom practice, shows the power of extended collegial networks. In this case the four elementary school teachers involved had national and state level opportunities to meet with other teachers interested in mathematics education reform and they had developed ongoing collaborative links with university faculty. Similarly a US study comparing the professional interactions of presidential award-winning mathematics teachers to those of other mathematics teachers (Noddings, 1992) showed that awardees had more collegial contacts outside their schools, not only with other teachers but also with university professors and district and state administrators.
Furthermore there is a need for such extended professional interaction to address more than just issues of teaching and learning mathematics. Raymond's (1997) study with beginning elementary school teachers shows the dominance of absolutist conceptions of the nature of mathematics over non-traditional pedagogical beliefs. Here, although the teachers had been introduced to the mathematics teaching reform messages of the NCTM (1989, 1991, 1995) and professed agreement with such non-traditional instructional methods, in their classrooms they employed teacher-centred transmissive lessons. In the struggle between deeply held instrumentalist images of mathematics and the goals of the NCTM, the conceptions of discipline won out, suggesting that "beliefs about the nature of mathematics are more strongly linked to actual teacher practice than are pedagogical beliefs" (Raymond, 1997, p. 573). Teachers require experiences that encourage change and growth in beliefs about the nature of mathematics.
Jonathan and Randy have sought out opportunities to interact with university faculty and fellow teachers and in doing so have looked at issues that lie above and beyond their course curricula and related teaching methods. Randy has established a link with a professor at a local university; one who has published extensively in the field of fractal geometries. It is here that he has received guidance and encouragement in his study of the subject and his development of related classroom activities (RW-INT-D-13). Randy's changing conception of mathematics and the investigations that he provides for his pupils are evidence of the impact of this liaison.
Jonathan also has contacts with university faculty, but in his case the links are more numerous and eclectic. Through these relationships he has explored: new interesting problems in calculus and differential equations, fractal geometry, chaos theory, and the history and philosophy of mathematics (JO-DIS-N-03,14,20,29). These activities and curriculum development projects have put Jonathan in contact with other like-minded teachers who are also seeking new mathematical knowledge. Jonathan values this network of colleagues and the intellectual stimulation their interchange provides.
There's a large number of people that I've met and have established an intellectual bond with. They always affect me when I listen to them, and I'm sure it's reciprocal when I talk to them about things. I mean that's what the intellectual community is all about as far as I'm concerned. You stimulate each other with new thoughts. (JO-INT-D-14b)
In this intellectual community, conversations are not about school mathematics or possible methods of instruction, but focus on philosophical issues and interesting mathematical investigations that lie above and beyond the school curriculum. The opportunities provided by these conversations for Jonathan to analyse, re-work, strengthen, and extend his conception of mathematics is evidenced in the complex integrated subject image that he presented through his writings, repertory grid, concept map, and interviews.
Studies of open classrooms, free schools, or other radical educational innovations are often conducted using case studies or ethnographic methods. In these studies the researcher is attempting to portray the workings of circumstances that differ dramatically from what typically presents itself in the "natural" functioning of our society and our educational systems. It is as if the researcher is attempting to document with vivid characterizations that nature need not be the way it typically is. (Shulman, 1988, p. 14)
This study has developed pictures of two unique teachers' conceptions of mathematics and classroom practices. Links between the participants' subject beliefs and their instructional methods have been made, and in particular their struggles in translating subject images into action have been related. The use of a case study approach has permitted the development of vivid characterizations, but also raises questions of internal and external validity (Merriam, 1988). There is a need to ask, "How faithfully do the descriptions and stories related here capture the realities of the participants' beliefs about mathematics and their teaching practices?", and "To what extent are the findings concerning Jonathan and Randy applicable to other teachers and educational settings?". The two issues of "trustworthiness" (Lincoln & Guba, 1985, p. 281) and "transferability" (Lincoln & Guba, 1985, p. 124) are addressed in this section.
Thompson (1992), in a survey of the research concerning teachers' beliefs about mathematics, notes that:
Any serious attempt to characterize a teacher's conception of the discipline he or she teaches should not be limited to an analysis of the teacher's professed views. It should also include an examination of the instructional setting, the practices characteristic of that teacher, and the relationship between the teacher's professed views and actual practice. (p. 134)
The present study goes beyond Thompson's call for "triangulation" (Lincoln & Guba, 1985, p. 283). Along with addressing the issues that Thompson identifies, the study also employed "methodological triangulation" (Mathison, 1988, p. 14) when exploring the participants' conceptions of mathematics. Writing in response to questions concerning the nature, history and foundations of mathematics, repertory grids comparing mathematics to other school subjects, and concept maps displaying the teachers' structures of the discipline were all used to provide data on the study participants' beliefs. Each of these techniques was followed by an interview in which the teacher and I explored ideas that surfaced in their earlier work. Through this use of multiple methods, the study went beyond an examination of the teachers' "professed views" of mathematics.
In exploring teachers' subject conceptions through a variety of techniques, this study has gone beyond previous research investigating beliefs about mathematics. All but two of the studies found in a search of the literature and cited in Chapter 2 report using only structured interviews or surveys or a combination of these two research instruments. McGalliard (1983) and Raymond (1993, 1997) employed, as did the present study, written responses to open-ended questions concerning the nature of mathematics, but only Raymond and the research reported here used the teachers' responses as the focus of subsequent interviews. No study reported using instruments such as repertory grid or concept mapping; techniques that require participants to reflect on and reveal deeply held images of mathematics. The application of three research instruments and follow-up interviews illuminated the participants' conceptions of mathematics from a variety of angles and allowed the study to explore the depth and complexity of Jonathan's and Randy's subject beliefs. It is this complexity and connectedness that supported Jonathan's efforts to carry visions into everyday instructional practice. Previous studies (Cooney, 1985; Dorgan, 1994; Kesler, 1985; Thompson, 1984) found that teachers with a problem-solving orientation to mathematics failed to fully translate their subject images into classroom practice. It is possible that these research projects' methods only explored the surface level of the teachers' belief systems and failed to locate underlying inconsistencies and instrumentalist views. Raymond's (1997) close analysis of her data showed conflicts between deeply held subject beliefs and more surface opinions about how a subject should be taught. The teachers' instrumentalist discipline images were dominant and encouraged transmissive styles of instruction.
The classroom observation schedule employed in this study was also more extensive than that of most reported research. Only Thompson (1984), Wood, Cobb, and Yackel (1991), and Ferrell (1995) watched 20 or more individual lessons by single teachers, the minimum number of observations in this project. This study, in a style similar to that employed by Wood, Cobb and Yackel, spread classroom visits out over an extended time period and thus observed teaching of a variety of topics. This comprehensive classroom data, more extensive than that recorded in other secondary school based studies (Kesler, 1985; McGalliard, 1983; Wilson, 1994), permitted the identification of links between beliefs and practice.
In addition, the full teaching days spent with the study participants and observation of regular school routines allowed me to describe their teaching contexts. Observation of the struggles involved in bringing beliefs to practice and identification of features in these teachers' professional environments that aided or hindered the translation process came from this extended contact and opportunities to indulge in spontaneous conversations.
A study wishing to develop understanding of individuals' actions must go beyond objective observation and ask for the "facts of the case" as perceived by the research participants. "The case study worker constantly attempts to capture and portray the world as it appears to the people in it. In a sense for the case study worker what seems true is more important than what is true [italic in original]" (Walker, 1980, p. 45). For a study to be trustworthy the report must first be credible to the participants (Lincoln & Guba, 1985) and this necessitates negotiation of data interpretation. This approach was taken in the present study. Interviews focussing on the writing, repertory grids, and concept maps provided the teachers with opportunities to flesh out the ideas emerging through these research instruments and to interpret them in their own words. It is these words that comprise the core of those sections presenting the participants' images of mathematics. In interviews and less formal discussions, Jonathan and Randy provided their reasons for selecting lesson approaches and their understandings of the resulting action. They were provided with the lesson descriptions selected for this report and invited to give their interpretations. The problems and struggles encountered in translating subject visions into teaching practice are reported here as related by Jonathan and Randy.
Although it can be argued that the conceptions of mathematics and narratives of classroom experience presented in previous chapters faithfully capture what was said and happened, there remains the question of whether these beliefs and practices were altered by my presence. That is, what was the "observer effect" (Bogdan & Biklen, 1992, p. 47) in this study?
I believe that my presence had little impact on Randy's and Jonathan's teaching. Visits to Western Secondary and Northern High School did not follow any fixed schedule and the only days avoided were those that involved extensive paper and pencil testing. Thus I frequently arrived while Randy and Jonathan were in the middle of lesson sequences and were not in a position to adjust activities or styles. Both teachers planned their lessons without consulting me and did not make adjustments as a result of our discussions. This is in contrast to those projects (Philipp, Flores, Sowder & Schappelle, 1994; Wood, Cobb & Yackel, 1991) where, in the presence of considerable coaching, teachers' emerging social constructivist subject images did translate into compatible instruction. In the research reported here, Randy and Jonathan independently selected their approaches to lessons.
On the other hand, this study, in encouraging focussed reflection on personal mathematical philosophies, may have had some impact on the participants' subject images. Clarke (1997), in a project with two Grade 6 teachers implementing a nonroutine problems unit, studied the participants' instructional practices and conceptions of their teaching roles. He noted that the one teacher, with whom he had numerous opportunities to meet and collaboratively reflect upon lessons and roles, developed expanded beliefs concerning mathematics teaching and learning. A similar effect on subject conceptions may have occurred in the present study. In fact, Jonathan, in describing his intellectual community and its effects on his thinking about mathematics, included myself and the recent research activities (JO-INT-D-14b). I believe that the project activities did not suggest alternative subject conceptions, but they likely provided opportunities for Jonathan and Randy to strengthen their images of mathematics.
The research reported here employed "purposeful sampling" (Bogdan & Biklen, 1992, p. 71; Lincoln & Guba, 1985, p. 199) in the selection of participants. That is, sampling "based on the assumption that one wants to discover, understand, gain insight; therefore one needs to select a sample from which one can learn the most" (Merriam, 1988, p. 48).
Previous studies examining teachers' beliefs about mathematics and instructional practices had, except for two cases (Philipp, Flores, Sowder & Schappelle, 1994; Thompson, 1984), involved teachers exhibiting traditional transmissive instructional styles. The present study, in an effort to expand our knowledge of the interplay between subject beliefs and practice, took the opposite tack and involved two teachers who were moving beyond teacher centred traditional instruction. Jonathan, and Randy to a somewhat lesser degree, are atypical in their teaching practices. Surveys, both in Ontario (Colgan & Harrison, 1997; Ontario Ministry of Education, 1992a, 1992b), and internationally (Crosswhite, 1987; Lapointe, Mead & Askew, 1992; McLean, Wolfe & Wahlstrom, 1987; Robitaille, Taylor & Orpwood, 1996), show that engaging students in mathematical investigations, conjecturing and discussion are practices that are absent in most mathematics classrooms.
Although focusing on teachers who are not representative of the typical case produces problems of generalizability, for this research as an example of "studying what could be" (Schofield, 1990, p. 203) such "uniqueness is an asset rather than a liability" (Donmoyer, 1990, p. 194). It allows us to look for exactly those conditions that make the case exceptional; in this study, the conceptions of mathematics held by teachers who are pursuing mathematics education reform. For such research, the aim should not be universal generalizability, but to provide thick description to permit others to assess the potential for transfer to their sites of interest (Schofield, 1990).
Lincoln and Guba (1985) argue that, in qualitative research, we should look for transferability rather than generalizability. The degree to which the findings of any particular study apply to another case is a function of the similarity between contexts. Thus the chief research requirement is not to provide a general case, but to describe the particular cases studied in detail sufficient to allow others to assess the degree of fit. The two cases developed in this report contain extensive descriptions of Jonathan's and Randy's conceptions of mathematics and teaching practices. The teachers' reports of the struggles to translate beliefs into action give information concerning their teaching environments. It is left to the reader "to ask, what is there in this study that I can apply to my situation, and what clearly does not apply?" (Walker, 1980, p. 34).
On the other hand, this study when combined with the parallel research cited earlier in the literature survey, does support the development of a "working hypothesis" (Cronbach, 1975; Lincoln & Guba, 1985). Instrumentalist views of mathematics appear to translate easily into the predominant transmissive modes of instruction. Teachers with less absolutist discipline images, unless provided with considerable coaching and support, also adopt traditional teaching practices. With this present study, we see that social constructivist conceptions of mathematics encourage alternative styles of teaching, and a well developed social constructivist philosophy can, in the face of considerable opposition, support a teacher's efforts to bring mathematics education reform to their classroom.
Understanding how teachers, individually and collectively, think, act, develop professionally and change during their careers might provide new insights as to how one might approach the reform, change and improvements in education that are necessary to equip our students for a desirable future within a context that is rapidly altering the nature of teachers' work. (Butt, Raymond, McCue & Yamagishi, 1992, p. 51)
The extensive time spent with Jonathan and Randy, studying their images of mathematics and classroom practices and observing their professional lives, has raised, for me, a number of questions not directly addressed by this present study. In addition, the working hypothesis that flows from this and related research, that a strong social constructivist image of mathematics can support a teachers' struggle to adopt and implement non-traditional instruction, has implications for those who promote mathematics education reform.
This study and parallel research suggest that teachers' images of mathematics influence their choices of instructional methods. Thus teachers' conceptions of mathematics are important, and a question that flows from this is:
What factors, events and experiences contribute to the development of a teacher's conception of mathematics? In particular, what experiences encourage the development of a social constructivist image of the discipline?
Discussions with Jonathan and Randy have identified some recent experiences that have influenced their mathematical beliefs. Interactions with university faculty and fellow professionals that encourage the exploration of mathematics beyond the school curriculum and raise philosophical questions appear to be productive. But, teachers construct their personal knowledge and meanings from a wide range of experiences (Kelly, 1955; Polanyi, 1958) including those prior to and beyond their professional lives. Research (Butt, Raymond, McCue & Yamagishi, 1992) has shown the influence of childhood experiences on teachers' general philosophies of education and drawn links between family life and teacher-candidates' images of mathematics (Roulet, 1995). Elementary and secondary schooling provide opportunities for future teachers to develop subject matter beliefs (Ball & McDiarmid, 1990; Roulet, 1995; Schoenfeld, 1988, 1989). We might thus ask a more focussed question of Jonathan and Randy, and of teachers with similar views of mathematics.
Are there childhood, elementary school, and secondary school experiences that encourage the development of social constructivist images of mathematics?
Educational systems are not in a position to influence teachers' childhood and adolescent experiences, but they can, to some extent, specify the formal university programs of those students who plan to enter the teaching profession. There is evidence that the present policy of mandating numbers of course credits has little positive effect on prospective teachers' understandings of the nature of individual disciplines (Ball & McDiarmid, 1990; Roulet, 1995). In fact, undergraduate study in mathematics may promote a narrowing of discipline conceptions and a reduction in enjoyment of the subject (Galbraith, 1984). On the other hand, research has shown that graduate level study in mathematics increases a teacher's syntactic knowledge of the discipline and inclination to lead school lessons that present a larger picture of mathematics (Grossman, Wilson & Shulman, 1989). There is a need to ask:
What university level mathematics courses or topics and what styles of university teaching promote prospective teachers' understanding of the syntactic and substantive knowledge of the discipline?
The present work with Jonathan and Randy provides some hints as to processes by which teachers may alter and expand their conceptions of mathematics. Following these leads might prove to be fruitful research.
This study and related research indicate that subject beliefs can have an effect on practice. The reverse may also be true. Clarke (1997) and Cobb, Wood and Yackel (1990), in projects with elementary school teachers who were moving to non-traditional teaching styles, found that reflection on teaching resulted in a shift in beliefs about mathematics. Randy's and Jonathan's cases also suggest that "beliefs and practices are dialectically related" (Clarke, 1997, p. 279). Randy's observations that many students can not follow and really do not believe formal proofs, has led him to explore more visual presentations and suggested an extended image of proof, one based upon the idea of "convincibility" (RW-INT-D-05b). Jonathan has found the seeds for some of his mathematical investigations within the school curriculum. Early in his career he became interested in the topic of geometric transformations, and over a five year period explored higher dimensional extensions of the high school work (JO-INT-D-14b).
Are there secondary school topics and classroom activities through which teachers may reflect on and alter mathematical beliefs? Could such activities be employed for in-service work with teachers?
There is considerable agreement that collaborative activities and supporting professional networks contribute to teacher growth (Black & Atkin, 1996; Butt, Raymond & Townsend, 1990, April; Hargreaves, Davis, Fullan, Wignall, Stager & Macmillan, 1992). This professional development is usually pictured in terms of changed or expanded classroom practice, but the experiences of Jonathan and Randy show that altered subject images are also possible outcomes.
What types of professional networks and activities are supportive of change in subject beliefs? What roles could university based mathematicians play in these networks? Can effective networks be provided for large numbers of teachers?
The collaborative activities experienced by Randy and Jonathan were available to other teachers in their schools, but in general these people did not make the efforts to seek them out.
How can teachers be encouraged to take advantage of professional networks and collaborative activities?
This study also raises some research and educational policy questions that involve participants or audiences beyond mathematics teachers. The opposition to Jonathan's and Randy's teaching approaches expressed by parents and the school administrations appears to be rooted in differing conceptions of mathematics. The instrumentalist image of mathematics, prevalent within the mathematics education community, has even greater popularity in the general population. Reform oriented teachers will continue to experience conflicts unless major efforts are made to alter public images of what it means to do mathematics.
What strategies might the professional organizations and allies from the university based mathematics community employ to counteract the dominant instrumentalist public conception of mathematics?
Both Jonathan and Randy felt constrained by the official mandated curriculum. It is obvious that these teachers, if freed from curricular restrictions, would provide stimulating and challenging mathematics courses; programs that would be unique. Such diversity would be likely to clash with the government's perceived need for uniformity.
What would be the results of giving adventurous reform minded mathematics teachers freedom to design their own curricula? What is the appropriate balance between freedom for individual teachers and Ministry control of curriculum? Can guidelines be designed to provide directions for the majority of teachers while providing sufficient options for reform minded educators?
The focus of this study was teachers; their beliefs about mathematics and teaching practices. Although students appear as characters in the narratives of classroom experience, they are there primarily as foils for the teacher. There is a need for studies that shift this focus and examine the learning of students in classrooms lead by teachers such as Jonathan.
What are the conceptions of mathematics held by students who have experienced secondary school courses designed around a social constructivist philosophy of the discipline? Do students receive and internalize the messages concerning the nature of mathematics delivered by such courses and teaching? Are students who have experienced social constructivist oriented mathematics instruction more or less inclined than the general population to pursue tertiary education or careers that involve mathematics? What is the experience of such students in post secondary mathematics courses?
Such questions are important, for in the final analysis the goal of mathematics education reform is increased student knowledge and appreciation of mathematics.
In this study innovative instructional practice, activities that put into place ideas expressed in the mathematics education reform literature (NCTM, 1989, 1991, 1995; OAME/OMCA, 1993), were found to be reflections of the conceptions of mathematics held by the teachers. The teachers struggled against considerable opposition in their efforts to express their subject visions in classroom practice. Jonathan, who possessed a well developed social constructivist philosophy, persevered and managed to regularly bring to his classroom lessons that captured his view of mathematics. Randy's mixed and changing conception of mathematics did not support continuing reform efforts within the core mathematics program, but did motivate the provision of non-traditional activities in a style that would not invite student complaints or official notice. Randy found no intellectual companions within the staff of his school and received no support in his efforts to alter mathematics education. Jonathan had the company of two department members for conversations concerning mathematics and its teaching, but generally the school was not welcoming of his style of mathematics teaching. Both teachers sought out and found the intellectual stimulation that contributed to their images of mathematics in collegial relationships that went beyond the school. Both found stimulation in their contacts with university faculty, and Jonathan was able to build a supportive intellectual network within his professional community.
Intellectually alive mathematics teachers, those who are natural knowledge seekers, can, with effort, find collegial links that support their interests in furthering their mathematical understanding. The resulting explorations can help develop conceptions of mathematics that support changed instructional practices. The potential for such sequences exist, but the teacher effort involved is considerable. In the present environment of little or no support for intellectually adventurous teachers, the population of mathematics educators such as Jonathan and Randy is not likely to rapidly grow.