Beliefs, Subject Conceptions, and Teaching Practice
This study explores the conceptions of mathematics held by exemplary secondary school mathematics teachers, their instructional practices, and the links between subject images and teaching. The background literature related to such a study is wide ranging and follows a number of themes: beliefs, teachers' personal knowledge, teacher thinking, and subject images, in particular as these relate to mathematics; teaching practices, and work examining relations between the foregoing. The following literature survey examines research related to these themes and is organized into two main sections addressing subject conceptions and teaching practice.
A teacher's subject conception resides in their belief system and comprises part of their personal knowledge. The translation of subject beliefs into instructional action is initiated by teacher thinking. Thus the literature review begins with a brief examination of research dealing with beliefs, personal knowledge and thinking in general. This is followed by a section addressing philosophies of mathematics and contrasting the various conceptions of the discipline found in the literature.
Those studies most closely related to the questions addressed by this thesis are surveyed in the third and fourth sections of the review. Research linking teacher conceptions of academic disciplines to instructional practices have been largely conducted with elementary school teachers or have involved subjects other than mathematics. When secondary school mathematics has been the object of study, the research has focussed on very specific topics within the overall curriculum. Thus there remains a need for further research in this field.
In addition, the studies reported in the literature have been able to address only one-half of the question concerning links between teaching practice and images of mathematics. The mathematics teachers participating in the small number of studies at the secondary school level have been found to employ traditional transmissive modes of instruction. This style of teaching has been linked to an instrumentalist conception of mathematics; an image of the discipline as a collection of fixed rules and procedures. Questions concerning the images of mathematics held by teachers employing non-traditional reform oriented instruction are still open.
The second unit of the chapter, addressing teaching practice, begins with an examination of the professional literature from the NCTM and OAME and describes the instruction advocated by the mathematics education reform movement. Adopting these reforms means a shift from teacher-centred to student-centred lessons and this in turn necessitates a change in methods for observational studies. Literature addressing the observation of mathematics instruction is surveyed and an emerging sociological and epistemological perspective, appropriate for the study of non-traditional classrooms, is identified.
The issue, then, is not, What is the best way to teach? but, What is mathematics really all about? (Hersh, 1979, p. 33)
Thompson (1992) defines a teacher's conceptions of the nature of mathematics as their "conscious or subconscious beliefs, concepts, meanings, rules, mental images, and preferences concerning the discipline", which, "constitute the rudiments of a philosophy of mathematics" (p. 132). As a component of teachers' belief systems, conceptions of subject are closely allied with knowledge and thought. Distinctions between teacher beliefs, knowledge and thinking are not clear in the literature with the three concepts being employed interchangeably under a variety of labels (Clandinin & Connelly, 1987). In this, authors are following the path set by Dewey (1910) who virtually equated the three ideas in noting that in one sense, "thought denotes belief resting upon some basis, that is, real or supposed knowledge" (p. 4). A precise and universally agreed upon separation between the three concepts is probably not possible, but for the purposes of this study there is a need to set out the relationships between the three ideas and identify the focus of the research. This is the task of the following section.
Beliefs, Personal Knowledge, and Teacher Thinking
"When I use a word," Humpty Dumpty said in a rather scornful tone, "it means just what I chose it to mean - neither more nor less."
"The question is," said Alice, "whether you can make words mean different things."
"The question is," said Humpty Dumpty, "which is to be master - that's all." (Carroll, 1954, p. 185)
Beliefs are personal principles, constructed from experience, that an individual employs, often unconsciously, to interpret new experiences and information and to guide action (Pajares, 1992). As personally held mental constructs, beliefs, unlike knowledge, do not require community consensus or agreement to establish their validity (Nespor, 1987). Knowledge is taken to be built up through intellectual activity: experimentation, debate and reasoning, and is stored in the form of propositions that are open to further evaluation and change. Beliefs on the other hand are not developed through rational thought, but are mental summaries of significant past episodes. An individual's beliefs may fail to exhibit logical consistency, conflicting with each other and with knowledge or the observed world. The power of beliefs to filter new information and colour one's comprehension of events means that once established they are not likely to change (Nespor, 1987) even in the presence of contradictory evidence. Despite their looser structure and potential inconsistencies, beliefs are much more influential than knowledge in defining individual behaviour.
The many dimensions and experiences of an individual teacher's life generate a vast number of beliefs which are linked in a belief system, a network of belief clusters each focused on a particular situation or facet of life. "When researchers speak of teachers' beliefs, however, they seldom refer to the teachers' broader belief system of which educational beliefs are but a part, but to teachers' educational beliefs" (Pajares, 1992, p. 316). Pajares goes on to note that the educational belief cluster, while only part of a teacher's broader belief system, is still too large and complex for research purposes. A further sub-division into clusters of "educational beliefs about" is required -
beliefs about confidence to affect students' performance (teacher efficacy), about the nature of knowledge (epistemological beliefs), about causes of teachers' or students' performance (attributions, locus of control, motivation, writing apprehension, math anxiety), about perceptions of self and feelings of self-worth (self-concept, self-esteem), about confidence to perform specific tasks (self-efficacy) (Pajares, 1992, p. 316).
Epistemological beliefs or beliefs about the nature of knowledge, in particular mathematical knowledge, are the focus of this study. Epistemological beliefs play a major role in the organization and interpretation of one's knowledge (Abelson, 1979; Kitchener, 1986). Thus it is possible for two teachers, possessing similar mathematical knowledge, to, in expressing different epistemological beliefs, present distinctly different interpretations of content through their teaching (Ernest, 1989a). Models of teachers' knowledge, such as those of Grossman, Wilson and Shulman (1989) for school subjects in general or Fennema and Franke (1992) for mathematics in particular, indicate this guiding role of disciplinary image by placing beliefs about the nature of subject above content knowledge.
Thus as a first stage, distinctions may be made between beliefs and knowledge by making reference to the evidence bases upon which they rest and their relative strengths. Knowledge must meet certain cannons of evidence, that is there are publicly recognized criteria for the acceptance or rejection of knowledge claims, while beliefs may exist without supporting, or in fact in the presence of contradictory, evidence. Beliefs often take precedence over knowledge, shaping the interpretation of presently held knowledge and selectively admitting or rejecting new knowledge claims. The introduction of the term "personal knowledge" into the research literature has complicated the belief-knowledge picture and made separation of the concepts more difficult. Connelly and Clandinin, in describing personal knowledge as, "that body of convictions and meanings, conscious or unconscious, which have arisen from experience" (1984, p. 137), significantly blur the knowledge-belief boundary.
In a similar manner Elbaz (1981) appears to extend the use of the term "knowledge" to include concepts that might by others be labelled as beliefs. Image, Elbaz's third level in the structure of practical knowledge, like belief, guides teacher action in an intuitive rather than logically reasoned manner. Subject matter image is a composite of the teacher's visions, metaphors, and values concerning the subject content to be taught.
In this study the terms employed by those researching personal practical knowledge: conviction, view, vision, and image are used interchangeably with Thompson's (1992) term, conception, as labels for the teacher's epistemological beliefs about mathematics. In each case these terms are taken to refer to beliefs rather than knowledge. That is, a teacher's vision of subject is an intuitive sense or belief built unconsciously through experience rather than a rationally constructed picture developed through the purposeful acquisition of knowledge in formal or informal study. Of course, a teacher may have a significant knowledge base for their epistemological beliefs and, in fact, Shulman (1986), presents such knowledge, knowledge of the substantive and syntactic structures of subject, as necessary for effective teaching.
Substantive structure refers to the organization of a discipline's basic concepts and syntactic structure is the set of principles and processes by which new knowledge is generated and proven within the subject (Schwab, 1964). For Shulman (1986) and Grossman, Wilson and Shulman (1989) substantive and syntactic knowledge is a necessary component of a teacher's content understandings. Substantive and syntactic knowledge, if present, are likely to contribute to a teacher's conception of their discipline, but such knowledge would not be necessary for the development of an image of what a subject is about and how its "facts" are developed. Schooling, professional preparation, teaching and general life, all provide considerable subject experience for mathematics teachers. Beliefs concerning the nature of mathematics develop from this experience without opportunities to explore more formally the subject's substantive or syntactic structures.
Perry (1981) provides a theory that describes an individual's progress in the development of beliefs concerning knowledge; their progress through epistemological positions. In the Perry scheme one ideally moves through four stages of development: Dualism, Multiplicity, Relativism, and Commitment; but may, in fact, become fixed at any one level. These stages correspond to attitudes or beliefs about knowledge that may be briefly described as follows:
DualismAll knowledge claims are either true or false and authorities in any discipline can determine validity.
MultiplicityDiversity of opinion is legitimate and any one view is as valid as another.
RelativismDiversity of opinion is legitimate, but some positions have more validity than others. There is a need to employ evidence and reasoning in the evaluation of positions.
CommitmentThe adopting of a position from among those seen to be potentially valid.
The stages of the Perry scheme and the transitions between them provide a framework within which teachers' images of subject may be analysed and this approach will be taken in the concluding chapter of this study.
Teaching may be viewed as transformation and reflection; transformation of knowledge and beliefs into pedagogical acts followed by reflection on the outcomes of instruction. Thinking is the process through which these complementary processes proceed.
As we have come to view teaching, it begins with an act of reason, continues with a process of reasoning, culminates in performances of imparting, eliciting, involving, or enticing, and is then thought about some more until the process can begin again. ... We ... emphasize teaching as comprehension and reasoning, as transformation and reflection. ... Sound reasoning requires both a process of thinking about what they are doing and an adequate base of facts, principles, and experiences from which to reason. (Shulman, 1987a, p. 13)
Thus thinking is the cognitive activity through which the mental structures of belief and knowledge are revealed in action and through which reflection on action in turn may modify cognitive and belief structures. Thought also links knowledge and belief. Thinking is involved when beliefs influence the interpretation of knowledge and reflective thought on knowledge can on occasion lead to modified beliefs.
As with belief and knowledge, some research usage of the concept of thinking expands its meaning and generates possible confusion. For Clandinin and Connelly (1987) teacher thought includes both prior experience stored as knowledge and belief and teacher action, along with the mental processes linking these end points. In this study to avoid the overlap in terms, teacher thinking is used in the narrower sense of the mental activity that links experience, beliefs, knowledge and practice.
Images of Mathematics
Philosophy [Nature] is written in that great book which ever lies before our eyes - I mean the universe - but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth. (Galileo, 1610)
Insofar as the propositions of mathematics give an account of reality they are not certain; and insofar as they are certain they do not describe reality. ... But it is, on the other hand , certain that mathematics in general and geometry in particular owe their existence to our need to learn something about the properties of real objects. (Einstein, 1921)
Research in philosophy of mathematics can be roughly divided into two streams (Van Bendegem, 1993). One branch, the study of the foundations of mathematics, has become formalized and absorbed into the discipline to become part of the mainstream of mathematical research (Rav, 1993). Being imbedded in the discipline, foundational studies can not fully address the question of, What is mathematics? A second, more recently initiated but growing (Rav, 1993) research direction is the analysis of mathematical practice. This field of enquiry is concerned with questions such as: What are the origins of mathematics?, What does it mean to do mathematics?, When and why are mathematical statements accepted as true?, Is it possible to gather evidence of the correctness of a mathematical statement?, and How is it possible for flaws and errors to arise in mathematics? Personal and collective answers to these questions have significance for mathematics education and it is this second orientation to the philosophy of mathematics that is of concern in this study. "If you wish to study problems related to the educational aspects of mathematics ... you will obviously need a theory or at least a model of what mathematical practice is about" (Van Bendegem, 1993, p. 22).
One central question concerning the nature of mathematics and the discipline's development can be simply stated as, Do humans "discover" or "construct" mathematics? Do we live in a world that is governed by fixed mathematical rules which, over the centuries, we have discovered and recorded as mathematical theorems, or is mathematics a human construct that we project onto our world whenever we find patterns of events that appear to support our creations?
Platonism takes the former view. For Platonists, mathematics is a body of facts existing independent of human knowledge. These facts, as part of the laws of nature, have held from the beginning of time and will not change in the future. For any mathematical question the universe's basic mathematical rules determine a definite answer. Knowledge in the discipline develops as the natural mathematical laws are identified and open puzzles persist only because we have not yet discovered the appropriate procedures for solving them. The Platonist image of mathematics as a static and unified body of truths is an example of what Ernest (1991) labels the "absolutist" view of mathematics, the belief that mathematical knowledge is certain and without flaw.
Deductive methods make it possible to maintain an absolutist position while mathematical knowledge expands with the discovery of new facts that lack direct reference to the physical world. Initial postulates, assumed to be true due to their roots in observations of the world, form the bases for mathematical systems and strict application of the laws of logic ensures that validity is preserved in any statements proven from these axioms. For centuries the geometry of Euclid, as expressed in the Elements (300 B.C./1956), was the model for this program designed to ensure mathematical truth. Geometry was also the source of the absolutist view's first major crisis when in the mid 1800's Lobachevsky and Riemann produced examples of geometries that revealed contradictions in the Euclidean axioms (Davis & Hersh, 1981). Platonists now needed to find a more secure footing for mathematics if the absolutist view of the discipline was to be preserved. Arithmetic as formalized in set theory and logic appeared to have potential as a solid starting point for mathematics, but here again contradictions were found and the search for concrete foundations lost favour. The image of mathematics as the embodiment of nature's truths holds strong attraction for mathematicians and thus Platonist views persist despite the failures to axiomatize the discipline.
The Realist [Platonist] position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism. (Cohen, 1971, p. 11)
Formalists, admitting that mathematics might not truly correspond to the experienced world, seek to preserve the consistency of the discipline by denying all claims of representation. The "theory is not properly a theory at all as we formerly understood the term, but a system of meaningless objects like the moves in chess" (Kleene, 1950, p. 62). With statements encoded in precisely defined symbols and explicit proof procedures, formal systems appear to allow mathematics to claim to be the model for truth and certitude, but even with formalism the absolutist position is not secure. Gödel, in 1930, established that formal systems lack completeness in that there still exist statements, the truth of which can not be determined (Penrose, 1989).
Formalism may describe research in some branches of mathematics, but in doing so it separates the discipline as an academic activity from mathematics in practice as a tool in commerce and industry. For users of mathematics and some teachers of the subject formalism translates into an instrumentalist (Ernest, 1989b) view of the subject. Instrumentalists, while not directly involved in the logical deduction of new mathematical knowledge, have faith in the formalists' program and are willing to employ the procedures developed to accomplish their everyday mathematical tasks. Thus mathematics becomes a collection of often unrelated facts, rules and skills to be used in the pursuit of solutions to problems external to the subject.
Despite the problems of Platonism and formalism in their attempts to provide firm foundations for mathematics, absolutist conceptions of the subject still find considerable favour with both mathematicians (Mura, 1993) and the general public. Witness the common use of the arithmetic statement, "1+1=2", as the prototypical example of absolute truth. With the contradictions existing on the periphery of the discipline, many everyday users of mathematics are not aware of, or concerned about the potential flaws in the subject. Their experience of mathematics, as a set of rules that if employed accurately produce single correct answers, supports instrumentalist and Platonists personal philosophies. "For many educated persons mathematics is a discipline characterized by accurate results and infallible procedures, whose basic elements are arithmetic operations, algebraic procedures, and geometric terms and theorems." (Thompson, 1992, August, p. 2).
Mura (1993) in a survey of Canadian university based research mathematicians identified a tendency towards the formalist emphasis on abstraction, logic and rigour. A second study of the views held by university teachers of mathematics education (Mura, 1995) showed only a slight reduction in the dominance of the formalist view, with mathematics educators also giving inductive processes and the study of patterns a place in mathematics. While Platonist and instrumentalist views were not prevalent among university faculty, Roulet (1995) found that such images were common among university students preparing to teach secondary school mathematics.
With a growing acceptance that "the concept of a universally accepted, infallible body of reasoning - the majestic mathematics of 1800 and the pride of man - is a grand illusion" (Kline, 1980, p. 6) alternative conceptions of the nature of mathematics are taking form. In contrast to the absolutist view of mathematics, Ernest (1991) proposes a fallibilist position, one that accepts that the statements of mathematics are potentially flawed and must be held open to revision and correction. But if the development of mathematics is not the discovery of universal truths, how is knowledge in the discipline generated and warranted? To address this issue a social constructivist (Bishop, 1985; Ernest, 1991, 1992) or problem-solving (Ernest, 1989b; Lerman, 1983) philosophy of mathematics is proposed. This conception of mathematics begins with "the assumption that the concepts, structures, methods, results and rules that make up mathematics are the inventions of humankind" (Ernest, 1992, p. 93). Social constructivism is concerned with the nature of mathematical knowledge and the process of its development both in individuals and societies. In locating mathematics in a larger social context this philosophical approach avoids the separation between the discipline of the research mathematician and mathematics as employed in the community.
Mathematics is seen as an extension of natural language and, as a language, is acquired and developed through social interaction. Individuals, through observations of patterns in the physical world and reflection on previously constructed concepts, develop new mathematical ideas. This individual or subjective knowledge is communicated to a wider audience where it undergoes critique, debate and modification. If, through this social interaction, the new concept is found to fit with previously accepted mathematical knowledge it becomes warranted by the social group and attains the status of objective knowledge. At any time the truth of a statement is a matter of its fit with other accepted mathematical concepts and human experience with the world.
Mathematics is a branch of knowledge which is indissolubly connected with other knowledge, through the web of language. Language functions by facilitating the formations of theories about social situations and physical reality. Dialogue with other persons and interactions with the physical world play a key role in refining these theories, which consequently are continually being revised to improve "fit". As a part of the web of language, mathematics thus maintains contact with the theories describing social and physical reality. (Ernest, 1992, p. 94)
Most often new concepts originate from analysis of previously accepted theories and as such, if accepted, result in incremental additions to the body of mathematical knowledge. More dramatically, it is possible for mathematicians, by linking previously separate theories or by drawing from knowledge beyond the discipline, to challenge existing truths and structures. Old ideas may be found to be flawed and require modification or be abandoned and mathematical knowledge may undergo restructuring. This image of the construction and refinement of mathematical knowledge builds on Lakatos' (1976) earlier quasi-empirical mechanism for the development of mathematical ideas and Popper's (1959) description of the growth of scientific knowledge.
Mathematics teachers, many of whom may not have formally studied philosophy of mathematics, might not describe their personal conceptions of the nature of the subject in terms of the Platonist, formalist (instrumentalist), or social constructivist (problem-solving) positions, but such categories may be employed in the analysis of teachers' less formally defined images of the subject. Through their experiences with mathematics, teachers construct images of the discipline, and this rudimentary philosophy is important for "controversies about high-school teaching cannot be resolved without confronting problems about the nature of mathematics" (Hersh, 1979, p. 33). In particular the social constructivist position in presenting "a view of mathematics as a way of knowing and as a social construct can powerfully affect the aims, content, teaching approaches, implicit values, and assessment of the mathematics curriculum" (Ernest, 1992, p. 89). The validity of the foregoing statement is evident in the social constructivist themes running through the recent mathematics education reform proposals such as those presented in the NCTM's (1989) Curriculum and Evaluation Standards for School Mathematics. In fact, Thomas Romberg (1992a), the chair of the Commission on Standards for School Mathematics, cites Ernest's (1991) work in arguing for the adoption of the school mathematics program projected by the standards. The instructional practices suggested by the mathematics education reform movement and their connection to the social constructivist philosophy will be explored in later sections focussing on new visions of teaching.
Conceptions of Subject and Teaching Practice: Mathematics
In fact, whether one wishes it or not, all mathematical pedagogy, even if scarcely coherent, rests on a philosophy of mathematics" (Thom, 1973, p. 204)
There is considerable agreement that beliefs influence action (Abelson, 1979) and in particular that teachers' educational beliefs (Pajares, 1992) guide teaching practice (Campbell, 1985; Clark, 1988; Crocker, 1983; Munby, 1982; Nespor, 1987). More specifically still, there is evidence that epistemological beliefs or subject images affect teachers' interpretations of content knowledge (Kitchener, 1986) and influence their instructional approaches (Elbaz, 1981; Pope & Scott, 1984).
Ernest (1989b, 1991) describes anticipated instructional styles associated with the three philosophical positions, instrumentalism, Platonism, and social constructivism. Teachers holding either of the two absolutist views of mathematics are expected to adopt teacher-centred transmissive modes of instruction but the content focus of lessons would vary with subject image. For instrumentalists pupil mastery of mathematical skills is the key objective and thus good instruction entails clear presentation of the steps in any procedure followed by extensive drill to ensure memorization. Platonists, taking mathematics to have an underlying structure, see a need to demonstrate the logical nature of the subject and explain the reasons for rules and procedures. In both cases mathematical authority in the classroom resides with the teacher and textbook who together determine the truth or correctness of any student answer or solution method. Teachers holding problem-solving or social constructivist views of mathematics would see their classroom role as a facilitator, providing stimulating problems for investigation and building an environment in which pupils may discuss their emerging understandings. Research studies have explored these potential links between epistemological beliefs and teaching methods. While constraints such as: fixed curricula, time pressures, external examinations, and school or departmental rules and standards, provide competing incentives for the guidance of practice, there is evidence that teachers' instructional practices do reflect their visions of subject.
Thompson (1984) studied the beliefs and teaching of three junior high school (grades 7 and 8) mathematics teachers. While Ernest's labels for the various conceptions of mathematics were not employed by Thompson, descriptions of the teachers' professed views allow her subjects' positions to be identified with the three positions: Platonism, instrumentalism, and a mild form of problem-solving. Jeanne (Thompson's assigned pseudonym) held that mathematics, originating from ideas present in the physical world, is fixed and predetermined. Mathematics is certain and consistent, with a logical and coherent underlying structure. Thompson's report of Jeanne's teaching shows the influence of her absolutist philosophy.
Although Jeanne conducted class in a question-and-answer fashion, there were no observable signs that she was making an effort to encourage discussions among the students or between them and herself. The students' participation typically was limited to elicit short, simple answers, and she had a tendency to disregard the students' suggestions and not to follow through with their ideas. (p. 112)
Thus we see the transmissive mode of instruction expected from a teacher holding Platonist views, but there is little evidence of an effort to build student understanding as predicted by Ernest.
Lynn, the teacher with an instrumentalist conception in Thompson's study, viewed mathematics as an exact discipline, producing procedures that guarantee correct answers. Her comparison of mathematical activity to "mental calisthenics" (p. 116) echoes the formalist position. As expected, Lynn's teaching followed a highly transmissive mode aimed exclusively at skill development.
She conducted class in a way that allowed as little interaction as possible. Her explanations were brief and aimed at demonstrating the procedures that students were to use in working out the day's assignment. The bulk of the remaining class time was given to independent seat work during which the students practiced the procedures taught. (p. 117)
For Thompson's third research participant (Kay) teaching practice follows from belief, but her conception of mathematics does not fit exclusively into any of the identified philosophical categories. Kay pictures the subject as continuously growing and changing in response to practical problem-solving needs, but she sees this expansion as proceeding by rigorous axiomatic methods and thus producing propositions of guaranteed validity. Kay's teaching, in which she frequently encourages students to guess, conjecture and reason out their own solutions to problems was compatible with a problem-solving image of mathematics, while her emphasis on formal geometric proofs reflects her formalist conceptions of the discipline.
McGalliard (1983), employing the Perry (1981) scheme of ethical and intellectual development to classify conceptions of geometry, found that the four senior high school mathematics teachers in his study held dualistic images of the subject. This absolutist view, that every question has a single correct answer that can be defended by reference to authorities, was consistent with the observed instructional practice which focused on the preparation of pupils for the next mathematics course by careful compliance with the syllabus and the teaching of rules without explanations. In emphasizing the taking of notes in class and the memorization of answers, the teachers communicated their beliefs in external authority as the source of mathematical truth.
The Perry (1981) scheme was also used by Kesler (1985) (as reported in Thompson, 1992) to analyse the conceptions of mathematics held by senior secondary school teachers. Here the two teachers with dualistic views were observed to teach in a transmissive authoritarian style consistent with their epistemological beliefs. On the other hand two participants with multiplistic views, acknowledging a variety of approaches and answers to questions, did not translate their subject images into any one consistent teaching method, using both inquiry and authoritarian modes of instruction.
Studies investigating links between teaching practices and instructors' conceptions of the nature of mathematics have been conducted in programs both following and preceding junior and senior high school. Ferrell (1995) looked at instruction in a community college level trigonometry course. Despite the college leadership's promotion of teaching that actively involved students, all three instructors in the study rejected the role of facilitator and presented themselves as dispensers of knowledge. This transmissive style of teaching was compatible with their instrumentalist images of mathematics as a body of useful facts and procedures. Raymond (1993) found similar agreement between conceptions of mathematics and instructional styles with six beginning elementary school teachers. Where practice was not entirely consistent with beliefs, teachers possessing problem-solving images explained deviations from their desired approaches as accommodations to time constraints and lack of resources. In a further analysis of her data, Raymond (1997) confirmed the strong influence of beliefs about the nature of mathematics on teaching practice. The six study participants expressed non-traditional images of learning and teaching mathematics, ideas that they had met in their recent teacher education program, but the practices implied by these views did not appear in their lessons. Their traditional absolutist conceptions of the discipline dominated thinking and encouraged a teacher-centred transmissive style of instruction.
In research with close parallels to the study conducted for this thesis, Philipp, Flores, Sowder, and Schappelle (1994) examined the conceptions of mathematics and its teaching and learning, classroom practices, and the mathematical preparation of four teachers working in grades 3 to 7. As in this present inquiry, these four individuals were invited to participate in the study due to their reputations as exemplary teachers. In previous contacts with these teachers, those conducting the research had observed classroom practices that reflected ideas expressed in the Standards, the guideline for reform issued by the National Council of Teachers of Mathematics (1989). These teachers did not appear to have fully developed social constructivist conceptions of mathematics but all rejected an image of the subject as a collection of disconnected algorithms. Mathematics as language and its development out of problem solving were central to their visions of the discipline. Classroom activities that regularly included explorations using manipulatives, student discussions demonstrating, explaining and justifying answers, and journal writing, reflected the teachers' problem-solving conceptions of mathematics. The researchers report that they did not observe any extensive teacher explanations of procedures or the use of exercise sets for drill.
The success of these teachers in putting their problem-solving images of mathematics into practice could be partially attributed to the professional support they had experienced in graduate level courses, national research conferences, and participation in local and state-level curriculum and assessment reform projects. Here the four teachers possessed images of mathematics compatible with teaching reform proposals and were heavily involved in collaborative efforts for educational change. In contrast to this situation, four extensive case studies (Heaton, 1992; Prawat, 1992; Putnam, 1992; Remillard, 1992) examined the results when fifth-grade teachers with instrumentalist views of mathematics were expected to implement a reformed mathematics program with only a new textbook for guidance.
The four participants in these studies, selected to be representative of the professional population, were found to have rather restricted images of mathematics.
The teachers in our cases believe that the computational algorithms that pervade the traditional elementary school curriculum constitute the core of mathematics. The teachers have differing views on what it means to understand those algorithms and how important that understanding is, but it is the algorithms of arithmetic that define their mathematics. (Putnam, Heaton, Prawat & Remillard, 1992, p. 223)
For these teachers mathematics is a collection of disjoint, precisely defined tools and techniques that prove useful, once mastered, in solving questions arising in daily life. When confronted with new textbooks giving increased emphasis to: estimation processes, the use of manipulatives for student exploration of concepts, and rich problem situations designed to encourage student reflection and discussion, these teachers made major adjustments to the intended program to maintain teaching practices compatible with their epistemological beliefs. Textbook units on problem solving and activities requiring the use of calculators were skipped while supplementary drills for the practice of computational skills were added. Even when textbook units were used for lessons, the teachers' interpretations of the content and instructional emphases portrayed their images of mathematics rather than that intended by the new program.
In the textbook lessons that she did follow fairly closely, Valerie highlighted procedural aspects of content and downplayed opportunities for students to reflect on and discuss mathematical ideas. We saw this procedural emphasis in the textbook-centred lesson on averages. Valerie omitted a number of questions in the student text that required reflection on the adequacy of averages for various purposes and the kinds of information lost with any average (e.g., change over time). Instead, she spent the lesson time carrying out the computational steps for getting an average. (Putnam, 1992, p. 176)
Dorgan (1994) noted similar interactions between teachers' conceptions of mathematics and the implementation of a revised fifth and sixth grade mathematics curriculum designed to reflect the NCTM Standards (1989). Teachers with instrumentalist and Platonist views maintained their traditional teaching styles with an emphasis on basic skills and computation. Lacking knowledge of how to enact a Standards oriented program on a daily basis, the teacher with an emerging problem solving conception of mathematics struggled with change and made little progress towards the desired instructional practice.
Ernest's (1989b) conceptual model of the relationships between a teacher's beliefs about the nature of mathematics and his or her instructional practice acknowledges the "constraints and opportunities provided by the social context of teaching" (p. 252). From the studies cited it would appear that the long standing, dominant tradition of teacher-directed transmissive instruction in school mathematics makes it easier for those holding absolutist conceptions to translate their subject images into teaching practice. Teachers possessing instrumentalist or Platonist conceptions of the subject continue to employ instructional styles compatible with these beliefs even in the presence of official curriculum reform programs based upon alternative problem solving images. Teachers with social constructivist views of mathematics must employ newer and less popular (Black & Atkin, 1996) styles of teaching if they are to put their epistemological beliefs into practice. In the face of time pressures that discourage the provision of opportunities for student discussion of new approaches to the subject and meeting opposition from parents, whose only experience is with traditional mathematics programs, these teachers compromise and employ a mix of traditional teacher-centred lessons and activities promoted by the mathematics education reform movement. Not surprisingly the connections of beliefs to intended or preferred teaching practices are stronger or more consistent than those to actual observed teaching. Teachers and teacher candidates can imagine employing less popular instructional methods free of the constraints provided by the general school milieu. Day (1993), working with preservice secondary mathematics teachers, found that three students who saw mathematics as a growing, flexible and changing discipline with problem solving as a core activity stressed the importance of their future pupils' active involvement in learning and favoured the use of guided-discovery activities and student communication. Lerman (1990) and Roulet (1995) have also connected student-teachers' beliefs about mathematics and preferred teaching practices.
Through the use of a questionnaire, Lerman (1990) identified two pairs of mathematics teacher candidates that represented the extremes of the absolutist and fallibilist perspectives. After the presentation, via video recording, of a short mathematics lesson, interviews were conducted to elicit these four students' assessments of the teaching observed. The observations "that the two student teachers who were the most 'absolutist' felt that the teacher in the extract was not directing the students enough and was too open, whereas the most 'fallibilist' thought she was not open enough, and was too directed" (p. 59) shows the link between views of mathematics and orientations to teaching practice.
In the initial days of a preservice mathematics methods course Roulet (1995) had students write position papers setting out their personal views of the nature of mathematics and their descriptions of "best" mathematics lessons (Roulet, 1995). Analysis of the student writing showed a predominant "toolkit" view of mathematics.
For two-thirds of the class "mathematics in its simplest form is only a set of rules", or "methodologies of how to work with numbers and symbols" which "demands from you patience and logical thinking to follow step by step procedures". These students find comfort in a system that "is rigorously precise" with "no grey areas or exceptions" and where one can "get the right answer just by following rules and methods". (p. 133)
This instrumentalist conception of mathematics was reflected in the students' descriptions of good teaching which involved the careful and organized presentation of clear examples for students to emulate followed by time for student practice of the modelled procedure.
For a smaller portion of the class, mathematics was a language for expressing patterns and relationships. A more fallibilist view of the discipline was shown in noting that there are often multiple ways to solve a problem and in some cases, starting from different assumptions, a variety of valid answers may be found. These teacher candidates saw good lessons as involving pupils in personal discovery of concepts and they suggested that investigations could effectively begin with a problem or application context. In this study, as in the previously cited research linking beliefs and observed practice, it was possible to identify definitive instrumentalist images of mathematics while social constructivist conceptions were less fully formed. Links between instrumentalist views and teacher-centred transmissive instruction were evident while teachers with emerging social constructivist images were not as adamant in their support for compatible teaching practices.
A number of published reports of programs designed to help teachers change their mathematics teaching practices have noted a link between instructional choices and epistemological beliefs. After working with more than 500 elementary school teachers at summer workshops exploring alternative ways for teaching mathematics, Schifter and Twomey Fosnot (1993) conclude that "constructing a practice consistent with the new paradigm entails a new conception of the nature of mathematics" (p. 193). Evidence of the necessity for epistemological change is given in the results of a study by Arsac, Balacheff and Mante (1992). Two grade 8 teachers were extensively prepared for implementing instructional scenarios designed from a social constructivist perspective. Students, grouped in pairs, were to develop and validate solutions to a geometric problem and then the expected multiple solution techniques would be debated by the whole class. Teachers were instructed to provide only organizational support and intervene only to manage the flow of debate. During actual implementation of the lessons it was found that, "precautions consisting of carefully presenting the situation, and the theoretical ideas behind it, before the class session, to the teacher were not enough to avoid difficulties" (p. 26). In each case, beliefs that mathematics consists of fixed true statements conflicted with the proposed instructional methods. The teachers' images of the discipline led them to intervene in the pupils' explorations and debates with the purpose of directing progress towards the "discovery" of an acceptable solution.
A similar failure to effect changes in teaching without altering beliefs was observed in a study involving a preservice secondary teacher (Wilson, 1994). Here a course presenting content knowledge and instructional methods related to the function concept did not have an effect on the teacher candidate's instrumentalist image of mathematics. Despite increased knowledge, of both content and alternative methods of instruction the student teacher's views of mathematics and mathematics teaching remained narrow and traditional.
Wood, Cobb, and Yackel (1991) describe work with a second-grade teacher during her efforts to move from a traditional transmissive mode of instruction to a style similar to that attempted in the previously described study by Arsac, Balacheff and Mante (1992). The target instruction envisioned pupils collaboratively constructing shared understandings through exploration of open-ended problem situations, first in pairs, and them in whole-class discussions. As the teacher struggled with change, attempting new practices, reflecting on lessons given and discussing issues with the research team, the authors noted that parallel to progress towards the intended new practice was a complimentary change in conceptions of mathematics. As teaching moved from transmission of information to the initiation and guidance of students' collaborative development of knowledge, images of mathematics changed from that of a collection of rules and procedures to a meaningful human activity. A similar shift in teachers beliefs about mathematics was observed (O'Brien, 1995) in a larger scale ten-year program designed to bring about change in the instructional styles employed by the teachers in one elementary school. Although growth was uneven among teachers, all experienced some movement away from an instrumentalist image as instructional roles changed from lecturer to facilitator.
Although the studies identified in the preceding survey provide evidence of the effects of conceptions of mathematics on teaching practice, in a number of ways the link has not been fully explored. Only two of the reports found in the literature dealt with secondary school mathematics teaching and one of these focused on a single content area, geometry. Other research addressing high school mathematics involved preservice teachers and their intended instructional methods. There are many constraints that can lie between envisioning certain practices and effectively employing them in the classroom. The translation is especially difficult for those teachers holding social constructivist views of mathematics.
The cited studies were able to identify definitive examples of Platonist and instrumentalist images of mathematics but clear and fully developed examples of social constructivist or problem solving conceptions were not located. The tentative nature of the problem solving images exhibited in the research and the strong legacy of opposing teaching traditions combine to make it difficult to establish clear links between beliefs and practices at this end of the scale. The study reported here was designed to address the need for research that explores for social constructivist views of mathematics at the high school level, and, if teachers possessing such images are found, investigates the extent to which their subject images are translated into practice and follows the struggles involved in efforts to make this translation.
Conceptions of Subject and Teaching Practice: Other Disciplines
A teacher is a member of a scholarly community. He or she must understand the structures of subject matter, the principles of conceptual organization, and the principles of inquiry that help answer two kinds of questions in each field: What are the important ideas and skills in this domain? and How are new ideas added and deficient ones dropped by those who produce knowledge in this area? That is, what are the rules and procedures of good scholarship or inquiry? (Shulman, 1987a, p. 9)
The impact of beliefs about subject matter on teachers' choices of content and instructional approaches is not restricted to mathematics. Ball and McDiarmid (1990) assert that generally, "teachers' conceptions of knowledge shape their practice - the kinds of questions they ask, the ideas they reinforce, the sorts of tasks they assign" (p. 438). At the secondary level, where schools are most often organized into subject based departments, "the nature of the parent discipline and features of the school subject, as well as teachers' beliefs regarding the subject, help create a conceptual context within which teachers work" (Grossman & Stodolsky, 1995, p. 5).
Pope and Scott (1984), in an study of the epistemologies of science held by science teachers, employed Kelly's (1955) categories of: "constructive alternativism", which takes knowledge to be constructed by individuals, and "accumulative fragmentalism", the vision of knowledge as a collection of substantiated facts. These terms closely parallel those employed by Ernest (1991), fallibilist and absolutist, to describe conceptions of the nature of mathematical knowledge. Pope and Scott noted that generally science teachers held accumulative fragmentalist or absolutist views of science knowledge, and in turn, the manner in which students were taught placed little value on pupils' own conceptions and did not encourage active student participation.
In a study similar to Roulet's (1995) research with mathematics teacher-candidates, Aquirre, Haggerty, and Linder (1990) analysed the written responses given by beginning science student-teachers to questions concerning: the nature of science, how scientific knowledge is produced, effective ways for teaching the subject, and the manner in which high school pupils learn science. The results of this study were also parallel to those of Roulet (1995), with the responses showing that "over the course of at least 16 years of formal education, students had grasped only some of the characteristics of the 'whys' and 'hows' of science." "The most crucial aspects of the ways in which scientific knowledge is produced (the most important of which is the concept that scientific knowledge is a creation of the human mind) are not being assimilated by undergraduate science students" (p. 388). Again, as in the mathematics study, these teacher-candidates adopted a transmissive approach to teaching. The teacher was seen as the source of knowledge and the teaching-learning process, as viewed by these students, involved the transfer of this knowledge through logically clear explanations and demonstrations.
Experienced (minimum of five years and average of 15.8 years of teaching) senior high school biology teachers were the subjects of a study by Lederman and Zeidler (1987). Here, completion of a survey form, the Nature of Scientific Knowledge Scale, was employed to measure the teachers' understandings of the nature of science. Data from subsequent observations of teaching were analyzed to determine differences in classroom atmosphere and instructional procedures used by the participating teachers. When the conceptions of science and teaching practices data were compared, the results of the study did, "not support the prevalent assumption that a teacher's classroom behavior is directly influenced by his/her conception of the nature of science" (p. 729). The fact that the outcomes of this project differ from those of the previously cited mathematics and science studies and from those of the research to be described in the following paragraphs, may rest on the fact that here, to generate quantitative data, the authors employed a questionnaire. Through courses in the history or philosophy of their disciplines or independent reading, teachers may have met academic presentations of the nature of their teaching subjects. Unless these formal images fit with teachers' more extensive experience learning subject matter content, this information is likely to be held strictly as surface knowledge and not be incorporated into belief systems. Such knowledge may show up on a survey while relatively unrestricted writing activities such as those employed by Roulet (1995) with mathematics teacher-candidates and Aquirre, Haggerty, and Linder(1990) with science student-teachers and also in this present study are more likely to accurately reveal beliefs.
Wilson and Wineburg (1988) studied the subject images and instructional approaches taken in secondary school American history courses by four teachers with a variety of university academic backgrounds: anthropology, political science, American Studies, and American history, and found that "their disciplinary backgrounds wielded a strong - and often decisive - influence on their instructional decision making" (p. 526). For the political science and American Studies graduates, history consisted of facts, while the teacher with a history background saw her subject as an exciting narrative. For the anthropologist, history involved collecting archeological evidence and from this identifying strings of discrete events. Both the history and American Studies graduates identified unifying themes that linked events and periods. These disparate images of history resulted in distinctly different subject presentations. In the classrooms led by the political science and American Studies graduates, the study of history became, essentially, debates around current events with references back to related facts concerning past human experience. The single historian among the four teachers incorporated music, literature, the study of old photographs, and debates into lessons to paint pictures of social, cultural, political and economic issues of past periods.
In a study similar to that of Wilson and Wineburg (1988), Grossman (1991) examined the disciplinary orientations and instructional practices of two teachers of English. At university Colleen majored in English literature with the intention of becoming a teacher while Martha followed an interdisciplinary route, examining literature from a number of language sources. Colleen's orientation towards literature centred on the text itself, while Martha took a more personal interpretive or reader-response theory approach, looking at overall themes and linking the story to personal experience. In turn these teachers held differing goals for instruction and planned and presented different lessons. Colleen's focus on the text translated into a goal of encouraging students to pay close attention to the author's words when interpreting literature and her questions to students, to a large extent, concerned eliciting definitions for unfamiliar words. Martha was much less concerned with students' understanding of particular texts and class time was consumed with pupils writing personal responses to the stories and poetry studied.
It would appear from the above studies, with the exception of Lederman and Zeidler (1987), that the subject orientations developed by teachers during their university years play a significant role in their subsequent instructional decisions made while teaching. It is also interesting that, while considerable research related to conceptions of subject and teaching practices has been conducted with secondary school teachers of subjects other than mathematics, most of the mathematics related studies have been situated in the primary and junior or middle school grades.
A number of studies (Grossman & Stodolsky, 1995; Siskin, 1994; Stodolsky, 1993) have looked at differing views of knowledge across high school subjects. Although teachers are somewhat removed from their past discipline based university studies, "the discipline's language and epistemology are interwoven in the ways teachers - as subject-matter specialists - conceptualize the world, their roles within it, and the nature of knowledge, teaching, and learning" (Siskin, 1994, p. 152). In these studies mathematics teachers described their subject as well defined, having clear boundaries, highly structured with a definite sequence of topics, and relatively static. Science and modern languages teachers also saw their subjects as structured, with required sequences of study, but did not see their disciples as being as static as did mathematics teachers. On the other hand, teachers of English and social studies saw their subjects as being less well defined and having less structure. As a result, members of English and social studies departments were relatively free to personally select course topics and instructional methods, since the outcomes of each individual course would have little direct impact on subsequent study. In mathematics departments there was more collegial knowledge of what instructors did in other courses, and more uniformity in terms of content coverage, teaching approaches, assessment standards, and curricular materials. Here there were significant departmental pressures to conform to the majority, traditional ways of organizing courses and lessons. "Subject subcultures may be characterized by both beliefs about subject matter that bind teachers together and by norms regarding teaching practice, curricular autonomy, and coordination" (Grossman & Stodolsky, 1995, p. 8).
I think a reading of both the Curriculum and Evaluation Standards and the Professional Teaching Standards will show that both documents were heavily influenced by contemporary thinking on students building meaning and constructing their own knowledge. (Black & Atkin, 1996, p. 81)
While the terms, constructivism and social constructivism, are not employed in the NCTM's (1989, 1991, 1995) mathematics education reform documents, the leaders of this project have, as in the quote above, and elsewhere (McLeod, Stake, Schappelle, Mellissinos & Gierl, 1996; Romberg, 1992a, 1992b), made it clear that such a philosophy of knowledge informed their efforts. "The term that we did not use in writing up the Standards (but we certainly talked about) is what might be called the social constructivist's notion of learning" (McLeod et al., 1996, p. 38). This conception of mathematics that underlies the reforms proposed by both the NCTM and OAME leads to a call for the implementation of specific standards (NCTM, 1989, 1991, 1995; OAME/OMCA, 1993, 1995) of practice. An expanded view of teaching is described by these documents, as the development of students' mathematical understandings is no longer seen as flowing exclusively from teacher explanations. If the classroom development of mathematics is to reflect the discipline's social constructive epistemology, then an environment in which students interact with each other and the teacher while pursuing meaningful and stimulating mathematical tasks (NCTM, 1991) is required. The following section of the literature review will first present an overview of the expected teaching practices described by the NCTM (1989, 1991, 1995) and OAME/OMCA (1993, 1995). This will be followed by a brief description of the research paradigms through which mathematics teaching practice has been examined and evaluated and a discussion of these methodologies in relation to the pedagogical practices encouraged by the present reforms. Finally an epistemological perspective (Koehler & Grouws, 1992) for the examination of teaching will be examined and identified as appropriate for the proposed study.
New Visions of Practice
Just as our vision of the twenty-first century differs dramatically from that of the twentieth century, so the vision of mathematics education described in this document [Focus on Renewal of Mathematics Education] differs significantly from the traditional vision of the subject. (OAME/OMCA, 1993, p. 2)
Calls for change in the teaching of mathematics are heard in many parts of the world (Black & Atkin, 1996) and proposals for reform share common themes across national boundaries (Romberg, 1992b). The National Council of Teachers of Mathematics [NCTM], through its publication and promotion of the Curriculum and Evaluation Standards for School Mathematics (1989) and related documents, has been a major player in this international movement. The NCTM publications and the parallel but less ambitious Focus on Renewal of Mathematics Education, produced by the Ontario Association for Mathematics Education [OAME] and Ontario Mathematics Coordinators Association [OMCA] (1993), are the major sources of guidance for Ontario teachers wishing to change their mathematics curricula and instruction. The observation and analysis of teaching practice conducted within this study were pursued from the perspective of the proposals made in these documents.
In 1986 the NCTM, in an effort to stimulate improvements in the quality of elementary and secondary school mathematics curricula and instruction, established the Commission on Standards for School Mathematics (Romberg, 1992b). The Commission and its various Working Groups, composed of a cross section of the mathematics education community: classroom teachers, supervisors, educational researchers, teacher educators, and research mathematicians, produced a draft set of standards by the fall of 1987. After addressing feedback gathered from the educational community during the 1987-88 academic year, the Commission and NCTM in 1989 issued the Curriculum and Evaluation Standards for School Mathematics. The work of the NCTM did not end with the publication of this report, for companion documents: Professional Standards for Teaching Mathematics (1991) and Assessment Standards for School Mathematics (1995), and grade specific support materials have subsequently been developed. In recent studies the NCTM (1996) and others have begun to refer collectively to the three "standards" documents (NCTM, 1989, 1991, 1995) as the Standards. This approach will be followed here with year of publication given when reference is to one of the three documents in particular.
The first four standards issued by the NCTM (1989) present goals for instructional practice that address the full curriculum and apply to the teaching of all topics. The proposals set out in these standards and their elaboration in subsequent NCTM documents have been employed in this study as a background when analyzing teaching practice. The images of mathematics teaching embodied in these standards will be described next.
Teaching in mathematics is expected to move beyond the traditional sequence of presentation of specific techniques followed by the solution of word problems to which the recently developed procedures apply. Exploration for problem solutions is to become the vehicle for the development of course content. "Problems and applications should be used to introduce new mathematical content, to help students develop both understanding of concepts and facility with procedures, and to apply and review processes they have already learned" (NCTM, 1989, p. 137). The problems presented by the teacher should serve to link the course content to the world outside the classroom and to make connections within mathematics itself. The use of calculator and computer technology, to support a shift in focus from computational details to the construction of mathematical models, is encouraged.
Students should be actively involved in their learning through whole-class and small-group explorations that provide opportunities for discussion, questioning, listening, and summarizing. Teacher questions should call for more than a report of an answer or a listing of solution steps. Students should describe in their own words how they arrived at a problem solution, elaborating their thinking and the difficulties they encountered. Classroom discussion should go beyond a focus on mathematical procedures and address social issues making the connection between the course content and society at large.
Both inductive and deductive reasoning should be employed in mathematics lessons. Through the examination of patterns students should be encouraged to make generalizations and formulate conjectures. In individual work or group debate the validity of these hypothesis may be explored, with students encouraged to develop deductive arguments supporting the conjecture or to construct counter examples revealing its flaws. Thus mathematics is created within the classroom rather than delivered by the teacher as a complete formal package.
Instruction should illustrate the connections between mathematics and other disciplines and within the subject itself. Students are to experience the process of constructing mathematical models for problem situations occurring in the natural and social sciences and arising in commercial and industrial contexts. The parallels between alternative representations of mathematical concepts, such as the numeric, algebraic and graphical views of functions, should be explored.
The United States has in fact lagged behind other jurisdictions in addressing mathematics education reform and the themes developed in the Standards (1989), as calls for policy change, are found in the official curricula of Australia, France, Germany, Japan, the Netherlands, Norway, Spain, the United Kingdom. Although content emphases and the terms employed differ, "the four standards for mathematics teaching and learning in the NCTM's Standards, problem solving, communication, reasoning, and connections, are reflected in all eight national curricula, not just in the reform curricula" (Romberg, 1992b, p. 229). Similarly these themes are part of the official policy for Ontario, the setting of this study, appearing in the guidelines issued in 1985 (Ontario Ministry of Education).
The NCTM's image of mathematical instruction can not be captured in a prescriptive list of distinct teacher behaviours, but must be seen as a set of complex interactions between teacher, students, mathematical tasks, and materials. The picture of this learning environment is further developed by the NCTM in the Professional Standards for Teaching Mathematics (1991). The presentation of appropriate and worthwhile mathematical tasks is the teacher's first step in building an environment for mathematical learning. "Teachers should choose and develop tasks that are likely to promote the development of students' understandings of concepts and procedures in a way that also fosters their ability to solve problems and to reason and communicate mathematically" (NCTM, 1991, p. 25). Problems that capture students' curiosity and invite them to pose conjectures for further exploration are required to address the Standards call for mathematical discourse and reasoning. Tasks that may be approached in multiple ways and those that may have more than one reasonable solution are valuable in generating the student debate that can lead to deeper understanding.
Having posed a mathematical task for exploration the teacher must encourage, support and orchestrate the discussion that ensues. Here the traditional teacher processes of presenting examples and explaining procedures must give way to careful listening and prompting of students to clarify and justify their ideas. The teacher must be constantly analyzing the ideas put forth and making decisions concerning productive leads to follow, the balance between free flowing discussion in common language and the use of formal mathematical language and notation, when to intervene to highlight an idea or clarify an issue and when to let students struggle towards their own solutions. The flow of discussion must be carefully monitored to ensure that all pupils are participating, both in making contributions and listening and responding to the ideas presented by others. Classroom norms of conversation, in which students critique each other's hypothesis and suggestions with civility and respect, must be encouraged.
The physical organization of the classroom and appropriate student groupings can encourage communication. The teacher must decide when they wish students to develop ideas individually, possibly writing journal entries for later sharing, and when it is appropriate for pupils to exchange ideas in pairs, small groups, and in a whole class format. Any one lesson is likely to employ a variety of such groupings and thus class time must be effectively sub-divided to provide the opportunities for students to complete significant explorations and fully exchange their thoughts arising out of the activity.
Concrete materials may serve as models for mathematical concepts and exploration with these can provide patterns for student observation and generalization. Calculator and computer supported explorations can similarly promote discussion and learning.
This expanded vision of what may occur in a mathematics classroom and what it means to teach and learn within the subject requires a parallel expansion of the activities subsumed under the term "teaching practice". Teaching is no-longer restricted to mean the activities of a teacher during a lesson but now includes the preparation tasks of designing or selecting and modifying the learning activities, arranging student groupings, and choosing and organizing the supporting resources that will be employed in the leaning environment.
Sharing the same concerns for mathematics education as the NCTM, the Ontario Association for Mathematics Education [OAME] and the Ontario Mathematics Coordinators Association [OMCA] in 1993 issued Focus on Renewal of Mathematics Education. As an Ontario interpretation of the Standards (NCTM, 1989) the first four teaching practices identified as key components of all mathematics programs: communication, reasoning, problem solving and connections, echo the NCTM vision described above. Reflecting the rapid advances in technology and its later publication date the Focus on Renewal of Mathematics Education (OAME/OMCA, 1993) identifies a fifth key component of effective mathematics teaching, the use of computers and calculators. "Teachers and students should use computers as tools to assist with the exploration and discovery of concepts, with the transition from concrete experiences to abstract mathematical ideas, with the practice of skills, and with the process of problem solving" (p. 5).
The vision of mathematics instruction expressed by the NCTM (1989, 1991) and OAME/OMCA (1993) was, in many ways, not a new one for the high school mathematics teachers of Ontario. Although the data from the provincial reviews of 1990 (Ministry of Education, 1992a, 1992b) suggests that, in Ontario, mathematics teaching practice was much different from that advocated in the reform documents, the basic themes outlined in the new approaches had been, to a great extent, presented to the province's teachers five years earlier in the introductory pages of new curriculum guidelines (Ontario Ministry of Education, 1985). Here, in a less complete and elaborate form, the curriculum developers presented a program that took problem solving as a major goal and encouraged the development of concepts and skills through the investigation of applications located in other disciplines. Student understandings were to be developed through the generalization of patterns observed in explorations employing manipulative materials, computers and calculators. Just as for the NCTM (1989), the development of mathematical reasoning through student communication was key.
Students should be consistently asked "Why?" and helped to use language that expresses their answer to such a question and conveys their meaning to others. Opportunities should be provided for students to engage in writing activities in which they explore their own perceptions of mathematical concepts and report on their attempts to apply mathematics to problem solving. Students should also be provided with opportunities to engage in speaking and listening activities in which they test their thinking and reasoning against those of their peers. It is through these types of activities that students modify, verify, and consolidate their learning. (Ontario Ministry of Education, 1985, p. 17)
Although the developers of this new curriculum wished to see mathematics instruction in Ontario move beyond its traditional teacher-centred transmissive style, the structure of the course guideline documents did not encourage change. Instructional issues in the program were given just five pages at the beginning of the guideline. The individual course descriptions that followed this severely limited discussion of process, consisted of lists of topics with no references back to the teaching ideas given in the introductory pages. Teachers could ignore the calls for change, turn to the content lists for their assigned courses, and continue their practice of past years. That this was in fact the route taken by many of Ontario's secondary school mathematics teachers is indicated by the predominance of teacher presentations followed by individual student practice recorded in the provincial reviews (Ontario Ministry of Education, 1991a, 1991b).
Recognizing that "testing drives curriculum", in 1995, both the NCTM and OAME/OMCA, to compliment their previous reform efforts, released documents addressing issues concerning student assessment. Here assessment is seen as more than the task of assigning grades and is presented as a tool to enhance learning. As such, assessment tasks should be linked to and expand upon instructional activities. While pencil-and-paper quizzes, tests, and examinations may still be employed, the reform movement sees this list of assessment instruments significantly expanded to include: projects, student writing activities, oral presentations, teacher observations of student action, pupil-teacher conferences exploring problem solving processes, and self, peer, and group evaluations. Such activities can provide data on student thought processes and attitudes as well as measure pupils' abilities with mathematical procedures. Such data may be used by: students to identify strengths and weaknesses and adjust efforts and educational plans accordingly, parents to help guide their support and encouragement efforts, teachers to help plan changes in instructional processes, and administrators to assess program effectiveness. Emphasis has shifted from measuring student performance on discrete mathematical skills to determining their understanding of mathematical processes and ability to combine concepts and procedures to tackle larger scale and un-rehearsed problems.
Mathematics teaching, learning, and assessment as described in the reform proposals of the professional associations and in the official Ontario curriculum documents is a complex interaction among task, resources to support exploration, students and teacher. Instruction is no longer just the teacher's performance at the front of the classroom. Research projects that wish to record practice in this new style must take a broad view of the classroom, recording the organization of the learning environment, materials used, problems posed, and the actions and conversations of the pupils and teacher.
Studying Teaching Practice: Research Paradigms
Even when researchers find reasonable variables such as 'lesson development', which are demonstrably effective, they are only effective within the traditional conception of teaching. They can only make current teaching more efficient or effective, but they cannot make it radically different. (Romberg & Carpenter, 1986, p. 865)
Instructional methods in mathematics have been objects of study for the past hundred years. As the 19th century closed, those participating in the emerging discipline of educational research began to adopt scientific methods for the study of teaching technique. "The result of the wish to employ scientific methods was a rather single-minded focus on quantification and 'experimentation' - the kind of experimentation that gave rise to data that could be analysed statistically" (Schoenfeld, 1994, p. 699). The need for larger student samples and a desire for control of variables made mathematics or more specifically arithmetic, being a dominant subject and having fairly consistent curricula across teachers and schools, a favourite topic for study. "No other school subject has even approached the level and frequency of studies conducted in the area of arithmetic" (Shulman, 1970, p. 25). Only in the past two decades has the dominance of the empirical-analytical tradition for research into mathematics teaching and learning been challenged (Kilpatrick, 1992).
The development of pencil-and-paper tests at the beginning of this century provided a means for researchers to measure in a standardized fashion the outcomes of instruction and thus gave birth to "process-product" studies of teaching. Here selected teachers, or often student-teachers, conduct lessons using specific contrasting styles of instruction and the success of each experimental group is measured by a common test. Alternatively, the practice of teachers in existing classrooms is observed and coded so that contrasting approaches may be identified. Again pupil success is determined by testing upon the completion of instruction. In either form of the research, correlation coefficients linking teaching practice (process) to student test scores (product) allow the identification of more and less effective instructional methods.
In many of these process-product studies the measured performance is of low-level procedural skills (Schoenfeld, 1994). Most of the mathematics studies cited in Brophy and Good's (1986) extensive survey of research linking teacher behaviour and student achievement, involved elementary school arithmetic. In general, process-product studies deal with discrete teacher actions or particular lesson structures and do not analyse in any depth the mathematics being addressed. Brophy and Good (1986) are able to provide lists of effective and ineffective processes that are independent, not just of mathematical topic, but also of school subject. The superficial measurement of outcomes and the traditional selection of topics addressed by standardized tests mean that these studies have little connection with the new curricula and deeper understandings called for by the NCTM Standards (1989, 1991, 1995).
Process-product research in identifying distinct instructional actions, such as, thinking level of questions posed, wait time for student responses, and the giving of praise, addresses the teacher-centred classroom. The image of instruction (Fennema & Peterson, 1986) is of the traditional Socratic lesson where concepts are transmitted from teacher to pupil via sequences of well crafted questions. In a study analysing the use of a reward system to encourage student accuracy in correcting mathematics homework, Miller, Duffy and Zane (1993) describe the introductory phase of the experimental lessons with,
The teacher then asked the students to take out their homework, and called on individual students to verbally provide the answer to each question and problem. When a student gave a correct answer, the teacher indicated so by repeating the answer one time. If a student responded incorrectly, the teacher either prompted the student or called on another student until the correct response was elicited (p. 185).
There is no sense here of the student to student communication of mathematical ideas called for by the NCTM (1989, 1991) and OAME/OMCA (1993) reform documents. The process-product research paradigm may be well suited to the study of the traditional transmissive mode of mathematics instruction but it can not capture the complexities of life in a classroom where the guiding principles for instruction are the NCTM (1989, 1991) standards.
Wishing to capture the complex cognitive processes of teaching, researchers have expanded the process-product research paradigm to studies of expertise in teaching. In this perspective, "one stands figuratively behind the shoulder of the teacher and watches as the teacher juggles the multiple goals of script completion, tactical information processing, decision making, problem solving, and planning" (Leinhardt, 1989, p. 52). These studies often also include observations of novice teachers, with the expert-novice contrasts being used to highlight the expert's instructional skills. In this research perspective, as in the process-product studies, the teacher is seen as the principal player in the teaching-learning process. Teaching mathematics is viewed as explaining (Leinhardt, 1989) and skill in elaboration of subject matter is taken as a measure of expertise.
Bromme and Steinbring (1994) selected expert mathematics teachers for their study using measures of skill in control of instructional flow, clarity of teacher statements, and clearness of presentations on blackboard and transparencies. Berliner (1986) describes an expert's lesson-opening homework review with,
The expert teacher was found to be brief, taking about one-third less time than a novice. She was able to pick up information about attendance, about who did or did not do homework, and to identify who was going to need help in the subsequent lesson. She was able to get all the homework corrected and elicited mostly correct answers throughout the activity. And she did so at a brisk pace and without ever losing control of the lesson (p. 5).
But is this "good" teaching in terms of the present reform proposals? Such "well-taught" lessons can be disasters when it comes to giving students some sense of mathematics as an evolving discipline (Schoenfeld, 1988). This model of teaching in no way reflects the spirit of the Standards (NCTM, 1989, 1991) and pupils leave such lessons believing that mathematics is just a collection of precisely defined procedures such as those that have been so carefully explained and illustrated by the expert pedagogue. Research methods that identify as expert teaching, the use of drills, games, and individual written work to address at least 40 problems per lesson (McKinney, 1986), can not be effectively employed to search for classrooms exhibiting the mathematics learning environments imagined by the present reform movement.
Capturing a Fuller Picture:
A Sociological and Epistemological Perspective
We believe that teaching involves reasoning as well as acting: it is an intellectual and imaginative process, not merely a behavioral one. (Shulman, 1987b, p. 41)
There has been a growing feeling in parts of the mathematics education research community that, while process-product and novice-expert studies have identified important teaching actions, "a much-needed emphasis on the central role of the students in their own learning is missing" (Anderson & Postlethwaite, 1989, p. 84). There is a need to examine the interactions among students as well as exchanges between teachers and pupils. The entire culture or environment of the classroom and the ways that children construct knowledge must become objects of study. In this way the full range of teacher responsibilities: setting mathematical tasks, structuring investigations, focussing pupil attention on critical points, managing student discourse, helping students identify and resolve contradictions, and encouraging metacognitive processes and generalization (Bishop, 1985; Hoyles, 1988), become decisions and actions to be recorded and analysed. Combining this wider view of teaching with a social constructivist epistemology of mathematics, Lampert (1990) examines instructional practice for its congruency "with ideas about what it means to do mathematics in the discipline" (p. 33).
The analysis of practice draws on both familiar social science approaches to educational research and the epistemological arguments that characterize the subject being taught. Mathematical notions about what constitutes knowledge are considered in concert with frameworks drawn from the sort of generic study of knowledge acquisition that has characterized research on teaching and learning. (p. 36)
Acknowledging the two components of Lampert's program, Koehler and Grouws (1992), in their survey of research on mathematics teaching practices, title this approach the "sociological and epistemological" view.
In Lampert's work the richness of the mathematical experience is conveyed in the form of detailed descriptions of the tasks posed, and narratives capturing the classroom interactions. These are then analysed, examining the teacher's planning and decision making and the children's construction of meaning, all against a social constructivist image of the discipline. In this work, access to the teacher's thinking is direct since, in her teaching experiments, Lampert acts as both teacher and researcher.
Using a similar research program, Simon (1995) presents and examines his teaching of mathematics at the college level in an experimental elementary school teacher preparation program. As in Lampert's research, "for each situation, a description is provided of the challenge that faced the teacher as construed by the teacher, the decision that he made to respond to that challenge, and subsequent classroom interaction that was constituted by the students and teacher" (p. 122). Again analysis is conducted against a social constructive perspective of teaching and learning.
While research reports have not always labelled the methodology employed as sociological and epistemological, a number of recent studies concerning classroom mathematics teaching and learning have taken this approach. Through narrative presentation of classroom episodes and sociological and epistemological analyses of the recorded vignettes, studies have examined children's construction of knowledge through explorations and mathematical discourse in: third, fifth, and seventh-grade classes working with fractions (Flores, Sowder, Philipp & Schappelle, 1995; Kieren, 1995), grade three pupils exploring addition of quantities greater than ten (Lo & Wheatley, 1994), a second grade class looking at number patterns (Wood, Cobb & Yackel, 1991), grade two pupils working in pairs on number problems (Yackel, Cobb & Wood, 1991), and high school students using computer support for exploration in geometry (Schwartz, 1989). In each case the focus is on the teachers' actions in establishing learning environments and students' making and exploring conjectures, resolving conflicts and negotiating meanings.
The establishment of taken-as-shared meanings for students' personal constructions enables mathematical ideas to be established that are accepted by members of the class. It is through this process of negotiation of meaning that students become acculturated to the mathematical truths established by society. (Wood, Cobb & Yackel, 1991, p. 594)
Cobb (1988) points out that translating a constructivist view of learning into a theory of instruction is a difficult task. Even when a teacher provides rich mathematical contexts, materials to support student experimentation, and an environment that encourages flowing discourse, miscommunication can occur and students fail to develop the intended understandings. "Because teachers and students each construct their own meanings for words and events in the context of ongoing interaction, it is readily apparent why communication often breaks down, why teachers and students frequently talk past each other" (p. 88). When working with secondary school pupils, who may have experienced eight or more years of transmissive mathematics instruction, the opportunities for miscommunication and tension increase. Thus, in the present study, the focus of classroom observations is on the participants' efforts to engage in teaching that implements features of the reform proposals and on the struggles involved in this exercise.
In the studies cited above, only those of Simon (1995) and Schwartz (1989) involved classes beyond the middle grades, and in Simon's research, although the students were adults, the topic, an introduction to the area concept, is from the middle school curriculum. While research employing a sociological and epistemological perspective to inquiry into social constructivist approaches to mathematics instruction appears to be increasing in popularity, studies have been concentrated at the primary and middle school levels. The study described in the following pages contributes to the expansion of this program to the secondary school level.
'Curiouser and curiouser!' cried Alice. (Carroll, 1954, p. 9)
This study was undertaken to explore the questions:
What are the conceptions of mathematics held by exemplary secondary school teachers, those who are attempting to implement the reform proposals?
How are these teachers' images of mathematics connected to their classroom practices?
or more specifically:
To what extent are these teachers' instructional practices expressions of their subject images?
What are the struggles involved in these teachers' efforts to translate subject images into classroom practice?
The research literature reviewed above provides some starting points for constructing answers to these questions, but there remain some important missing components. The conceptions of mathematics held by primary and middle school teachers have been quite extensively investigated. At the secondary level only teachers' images of selected topics have been explored and these in just a small number of studies. The research reported here builds a fuller picture by looking at the discipline of mathematics more generally. Teachers' translation of conceptions of mathematics into classroom practice has been reported in the published literature. Again this work has been mainly at the primary and middle school levels with secondary school based studies focussing on particular topic areas. As well, most of the study participants in published research have been teachers employing traditional transmissive modes of instruction and have been found to possess absolutist images of mathematics. Thus, at the secondary school level, connections between absolutist conceptions of mathematics and traditional transmissive modes of instruction have been made, but the practices employed by teachers possessing fallibilist philosophies have not been explored.
The present study involves two secondary school teachers who have been observed to employ teaching methods representative of those proposed by the mathematics education reform documents from the National Council of Teachers of Mathematics and the Ontario Association for Mathematics Education. While these teachers, as will be shown, do not both hold fully developed social constructivist views of mathematics, their conceptions of the nature of mathematics are definitely not absolutist. Thus they provide an opportunity to examine connections between subject images and instructional practices not yet reported.
The past research found in the literature provides an extensive background for this study but leaves the above questions open for exploration.
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