## PROGRAM OF RESEARCH: DETAILED DESCRIPTION

### 1. Objectives

School mathematics achievement varies widely across Canada (HRSDC, CMEC & SC, 2007). In international comparisons, some provinces of Canada are ranked among the highest achieving, while others compare much less favourably. The proposed research seeks to account for some of these disparities through a focus on pedagogy. The objectives of the research are to describe regional differences in mathematics pedagogy across Canada, and to relate these differences to regional variation in student achievement in mathematics. The guiding questions for this research are:

- How does middle school mathematics pedagogy differ across regions of Canada?
- How can differences in pedagogy establish the basis for further research into regional differences in mathematics achievement?

To meet these objectives the research team, composed of researchers from across Canada, will establish focus groups of middle school mathematics teachers in five regions of Canada. Each group will observe and discuss mathematics teaching from other regions. We will analyse these discussions in order to characterise regional differences in mathematics pedagogy. This work extends comparative studies of mathematics teaching which have previously only been conducted internationally. The findings will provide a basis for teacher education, professional development and educational policy across Canada, with important implications for student achievement in mathematics.

### 2. Context

#### 2.1 Teaching and pedagogy

We make a distinction between teaching and pedagogy. Teaching refers to the observable practices of teachers and their interactions with learners. Pedagogy refers to what Tobin et al. (2009) call the “‘implicit cultural practices’ of teachers […] practices that though not taught explicitly in schools of education or written down in textbooks reflect an implicit cultural logic” (p. 19). As Tobin et al. note, these implicit practices are related to teachers’ “knowledge in practice” (Anderson-Levitt, 2002, p. 109) and “embodied knowledge” (Anderson-Levitt, 2002, p. 8). Such knowledge is related to Bruner’s (1996) concept of folk pedagogy, the “taken-for-granted practices that emerge from embedded cultural beliefs about how children learn and how teachers should ‘teach’” (p. 46). Folk pedagogy clearly varies from one place to another. Underlying these various characterisations are two related points: that pedagogy is implicit and that it guides practice.

#### 2.2 Large-scale comparisons in mathematics achievement

Large-scale international and national assessments have revealed a considerable range of student achievement in mathematics across Canada (HRSDC, CMEC & SC, 2007). When compared to international results, some Canadian provinces, notably Québec, rank among the top countries, while other provinces, especially in the Atlantic region, are significantly below the Canadian average. A number of factors have been suggested to explain these differences including curriculum, gender, attitudes, beliefs, aspirations, time spent working outside school, parents’ education, involvement and socio-economic status and school resources (see, e.g., Anderson et al., 2006; Beaton & O’Dwyer, 2002; Schmidt et al., 2001; Wilkins, Zembylas, & Travers, 2002). Teaching, which might be expected to have the most direct effect on student achievement, is considered less often.

While most international comparative studies have not considered teaching or pedagogy, some studies at international and regional levels have at least considered easily observable features of teaching, including:

- OECD (2010), which examined students’ answers to questions like “The teacher helps students with their learning” on the 2003 Programme for International Student Assessment (PISA) study to analyse features like teacher support, student-teacher relations, and disciplinary climate. They conclude that no single factor was consistently related to student achievement and speculate that a combination of factors might account for achievement differences.
- Mullis et al.’s (2000) study of teaching and student achievement in the 38 countries that participated in the 1999 repeat of the Third International Mathematics and Science Study (TIMSS-1999). Again, however, they only examine easily quantifiable aspects of teaching, such as teachers’ academic backgrounds, their professed beliefs about mathematics and learning, and their self-confidence. They conclude that higher achievement is related to higher levels of teachers’ confidence in their preparation.
- The TIMSS video studies (e.g. Hiebert et al., 2003), which compared video recordings of mathematics lessons in 8 countries according to a number of observable and quantifiable variables. Differences were observed in various aspects of teaching, including the amount of time spent on different aspects of teaching (practising new content, introducing new content, or reviewing), the topics studied, the complexity of the problems solved, and the type of problem solved (making connections, stating concepts or using procedures). Hiebert et al. conclude that there are few clear patterns in the teaching observed in different high achieving countries.
- Simmt et al.’s (1999) comparison of teaching practices in schools whose students had a history of performing well on provincial assessments in mathematics in Alberta. The study identified teaching practices that were common to all the teachers studied: “making mathematics real and relevant for students; making connections within mathematics and between mathematics and the physical world; caring about students’ mathematics knowing; demanding instruction […]; challenging content; highly interactive instruction; being well prepared for teaching; reflecting on one’s own teaching practices and assessing one’s own instruction; taking ownership of the curriculum” (pp.1-2). Simmt et al. note, however, that there is no identifiable ‘recipe’ for good teaching.

Studies like the ones summarised above have generally failed to establish clear patterns relating observable, often quantifiable, features of mathematics teaching and mathematics achievement. They have not, however, attempted to examine more implicit cultural practices of teaching i.e. pedagogy. Furthermore, no comparison of mathematics teaching between regions of Canada has been made, in spite of the noticeable differences in student achievement between regions. The proposed research will, then, fill two gaps left by the studies mentioned above. First, it will explore both externally observable teaching practices and the pedagogy underlying the teaching observed. Second, it will compare mathematics teaching and pedagogy in different regions of Canada.

#### 2.3 Connection to previous research

Since the proposed research will be conducted by a large research team, the connections to our previous research are varied. Nevertheless, most members of the team have been involved in comparative classroom research of some kind, whether focusing on mathematics teaching, learning or teachers’ professional development.

For Reid, this work builds on enactivist
methodological considerations (in Reid, 1996), research on somatic
markers as a basis of teacher decision making (Brown & Reid, 2004,
2006) and current comparative work on the emergence of disparity in
perceived mathematical ability in middle school mathematics classrooms^{*}.
Barwell is currently PI for a SSHRC-funded
project (2008-2011) that brings a comparative approach to the analysis
of mathematics learning involving second language learners. This work
applies discourse theories and methods to examine the construction of
mathematical thinking in mathematics classrooms in different settings
(Barwell, 2003, 2005). The proposed research will extend this work to
examine the construction of mathematical learning and thinking by
mathematics teachers. For Knipping, this
research builds on her comparative research on proving in mathematics
teaching in France and Germany (Knipping, 2001, 2002, 2003ab) and
current work on the emergence of disparity in middle school
mathematics classrooms^{*}.
For Mason, the research provides an ideal
opportunity to investigate further the dialectic between teachers and
students when changes in the instruction that students experience call
for changes in their approaches to learning (Mason, 2009; Mason &
McFeetors, 2007, 2009). The research will extend Savard’s
work on teaching practices in middle school mathematics classroom
(Theis & Savard, 2010). She has previously focussed on teachers’
practices and understanding of teaching tasks in the area of
probability using technology. The proposed study extends the work Simmt
has done to explore teaching practices in secondary school
mathematics (Simmt et al., 1999) and the ethical implications of high
activity mathematics classes (Towers & Simmt, 2007; Simmt, Gordon,
& Towers, 2002). Suurtamm is the
Director of the Pi Lab, a CFI infrastructure for the analysis of video
data from mathematics classrooms. Her research examines the complexity
of mathematics teacher practice, particularly as teachers facilitate
mathematical inquiry and engage in formative assessment. The proposed
research will build on her recent work on teachers’ enactment of an
inquiry-oriented mathematics curriculum in Grades 7 – 10, which was
based on video-recorded classroom data. All of the researchers look
forward to bringing their own extensive experiences researching
classroom practices within the umbrella of a national collaboration.

#### 2.4 Significance

Significant differences between regions of Canada occur in mathematics achievement. Social factors such as gender, aspirations, work outside school, parents’ education, and socio-economic status, account for some of these differences, but much remains unaccounted for. Mathematics teaching and the underlying pedagogy have an effect on students’ achievement, but this has not been studied from a pan-Canadian perspective. Furthermore, finding that social factors are associated with student achievement is not directly useful in improving student achievement. Changing curriculum, students’ attitudes, parents’ education and occupation, or socio-economic status are long-term processes. Studies that focus on these factors do little to inform policy makers and educators of things that can be done in the near-term to positively affect student achievement. Changes to mathematics teaching and pedagogy can occur more quickly. Hence the main impact of this research will derive from an enhanced understanding of differences in regional mathematics pedagogies in Canada. The results will provide a basis for guiding teacher education and professional development in ways that have the potential to increase student achievement across Canada. Ministries of education, school boards, and university faculties of education will gain insight into similarities and differences in mathematics teaching in their region compare with others, and into how changes to regional approaches might affect achievement. As a result of this study, teachers will be able to learn about teaching in other parts of Canada, and to think about their own teaching differently. Outside of Canada, educators in other federal systems (e.g., USA, Germany, Australia, South Africa) will be provided with an example of the ways in which regional pedagogies can differ and with a methodology for investigating such differences in their own contexts.

Theoretically, the research will contribute to the ongoing debate on the degree to which teaching in different contexts is unified by a common “culture of schooling” or differs according to local circumstances (Anderson-Levitt, 2003) . A starting hypothesis for this research is that within a region there are sufficient points of agreement in pedagogy to allow for a regional pedagogy of the region to be recognised, and that there are differences between regions in their regional pedagogies. This hypothesis may be refuted by this research; however, such an outcome would still result in useful information about teaching in Canada concerning either its regional variability or its national uniformity.

Finally, our proposed methodology is singularly appropriate to the topic, but has not previously been applied to the investigation of teaching in schools.

#### 2.5 Theoretical framework

The theoretical framework for this research is enactivist, based on the work of Maturana & Varela (1992; Varela, Thompson & Rosch, 1991). Enactivism is primarily a theory of cognition; however, as cognition occurs in any system capable of sustaining itself, enactivism can be applied to social systems (see Luhmann, 1995) and to the interactions between individuals embedded in social systems. From an enactivist perspective, we can characterise more precisely what we mean by “pedagogy” and account for the emergence of regional pedagogies in the Canadian context. Later, under “methodology” we will describe links between this theoretical framework and the methodology adopted.

The nature of pedagogy

Pedagogy has two particular characteristics: it is implicit and guides practice. Pedagogy, the basis for teachers’ behaviour in classrooms, must be implicit because of the complexity of the environment in which teachers teach. Maturana (1988) uses his concept of “emotional orientation” to explain science as a domain of explanation, that is, as a domain in which characteristic implicit criteria for making and accepting explanations apply. The same idea applies to teaching as a domain of practice. The criteria for accepting an explanation in science must be implicit at some level, in order to escape an infinite regress, since explicit criteria themselves require an explanation that must be based on more fundamental criteria. Basing one’s behaviour on implicit criteria and hence participating in a community of practice is, according to Maturana, ultimately a choice based on emotions: “Finally, whether an observer operates in one domain of explanations or in another depends on his or her preference (emotion of acceptance) for the basic premises that constitute the domain in which he or she operates” (p. 33). For this reason, Maturana uses the phrase “emotional orientation” to refer to that which defines a domain of explanation and determines membership in the domain. Similarly, pedagogy defines a domain of teaching practice and determines membership in it. Many participants in a domain of practice, for example mathematicians and teachers, agree on what counts as acceptable behaviour for a member of that community. Different communities, however, accept different kinds of behaviours. Being recognised as a teacher by other teachers is a matter of behaving in ways other teachers expect teachers to behave. What we are calling pedagogy could also be called a “teaching emotional orientation”.

Brown and Reid (2004, 2006) make a connection between teaching emotional orientations and the explanatory construct “somatic markers”. Teachers (and other people) continually act in complex situations, often without time for reflection. Damasio’s (1996) “somatic marker hypothesis” accounts for how people manage to accomplish this. A “somatic marker” refers to the juxtaposition of image, emotion and bodily feeling that informs decision-making. Somatic markers are established through experience, as emotions become associated with circumstances, meaning similar circumstances will trigger similar emotions in advance, guiding decision making:

Somatic markers are ... acquired through experience, under the control of an internal preference system and under the influence of an external set of circumstances which include not only entities and events with which the organism must interact, but also social conventions and ethical rules. (Damasio, 1996, p. 179).

Damasio’s explanation of the emergence of somatic markers fits with an enactivist view of cognition where cognition is seen as arising from:

two interrelated points: (1) that perception consists of perceptually guided action; and (2) that cognitive structures emerge from the recurrent sensorimotor patterns that enable action to be perceptually guided. (Varela, 1999, p. 12).

An individual teacher’s pedagogy can be seen as a constellation of somatic markers that emerge in the course of a history of teaching, interactions with other teachers, and other experiences. In the next section, we consider why, from an enactivist perspective, it makes sense to refer to a regional pedagogy.

The emergence of regional pedagogies: structural coupling

Mathematics teachers in different places teach mathematics differently. This phenomenon has been observed by researchers (e.g., Knipping, 2002; Hiebert et al., 2003) and fits our own experiences as observers of mathematics teaching. Some of these differences are differences between individuals, as each individual has a different history of experiences and hence a different pedagogy. However, mathematics teachers also form communities, in which pedagogies co-emerge. For two individuals to interact they must be able to remain in the interaction (they must be structurally coupled, Maturana & Varela, 1992). Characteristics of one individual must be related to the characteristics of the other individual in such a way that what one individual says or does must trigger a compatible act in the other. Over a history of interactions, each individual’s structures change in ways that facilitate future interactions. The individuals can be said to co-emerge in the interaction.

Which characteristics are relevant to maintaining an interaction depends on context. Two individuals may be capable of interacting about cooking, but not about biological evolution. Two teachers in a school will interact in the context of that school and characteristics of their structures related to the school as a community will be most strongly affected by their interactions.

In what contexts do mathematics teachers interact with other mathematics teachers? To some extent, in their schools; however, unless there are explicit mechanisms encouraging subject-specific interactions, mathematics teachers are as likely to interact with teacher of other subjects in their school as they are to interact with the other mathematics teachers, especially at the middle school level. At the regional level, mathematics teachers are likely to attend professional development sessions and conferences specific to mathematics, and hence to interact with other mathematics teachers primarily at the regional or provincial level. Because the community of mathematics teachers in Canada is primarily a regional one, individual teachers’ pedagogies co-emerge primarily through interactions limited to other teachers within a region. Hence it makes sense to hypothesise a regional pedagogy of mathematics teaching.

### 3. Methodology and methods

Consistent with our theoretical approach, the methodology for this research is an enactivist one (Reid, 1996). Recalling Maturana’s (1987) statement that “everything said is said by an observer”, the study will seek to research teaching by examining teachers’ observations of teaching. The research will involve a multivocal ethnography approach similar to that described by Tobin (1999; Tobin, Hsueh & Karasawa, 2009; Tobin, Wu & Davidson, 1989).

The effects of regional pedagogy will not be evident from a single case or many similar cases. To explore how different regional pedagogies have different effects, comparisons must be made. It is through comparing that observers notice similarities and differences between their own practices and those they are observing, so that some of the implicit aspects of their practice are brought to the surface. Therefore, this project will involve comparing regional pedagogies in middle school mathematics in regions of Canada that show significant differences in student achievement. Tobin et al. (1989) describe a layered process of documenting insiders’ implicit criteria. This process involves working with insiders to construct a visual ethnography, an auto-ethnography and an ethno-ethnography. At each stage insiders observe either their own or others’ practices, first by creating a video record of their own practice, then by commenting on video recordings of classroom teaching within their region, and finally discussing video recordings of classroom teaching from other regions.

Within each region classrooms where the language of instruction is English and classrooms where the language of instruction is French will be studied, as large scale assessments have revealed that there are differences of achievement along linguistic lines in some regions of Canada (HRSDC, CMEC & SC, 2007). The regions to be compared are listed below, along with student achievement in mathematics compared to the Canadian average, linguistic differences and the gap between those who do well and those who do poorly (as reported in HRSDC, CMEC & SC, 2007):

- Atlantic Canada (represented by Nova Scotia): Below Canadian average, English performed better than French, smallest gap between high and low achievement in Canada.
- Quebec: Above Canadian and OECD average, French performed better than English, largest gap between high and low achieving in Canada.
- Ontario: At Canadian average, English performed better than French, gap between high and low smaller than Canadian average.
- Alberta: Slightly above Canadian average, no linguistic differences, gap between high and low smaller than Canadian average.
- Manitoba: At Canadian average, no linguistic differences, gap between high and low at Canadian average.

Visual ethnographies: Ten focus groups will be established, one for each region/language group combination. Four teachers (the “participants”) will participate in each focus group. In the first year, each participant will be asked to choose three lessons to be video recorded: a lesson that the participant judges to be a “typical” lesson in her/his classroom; a lesson the participant considers “exemplary”; and a lesson in which specified content (e.g., the proof of the Pythagorean theorem) is introduced. The researchers will collaboratively choose a content area that is closely parallel in the curricula of each region.

Each teacher will review the videos from their classroom with a researcher and collaboratively describe the structure of the lesson in a format similar to that used in the TIMSS video study (Hiebert, et al., 2003). They will then select segments to be included in an edited video. An edited video of 20 minutes or less will be produced for each lesson recorded by each teacher. The videos will present the lesson in an accessible format, including key episodes and omitting sequences (e.g., individual paper and pencil tasks) that are not usefully portrayed visually. Omitted sequences will be marked with descriptive titles. Students’ faces will be blurred unless permission to include them has been obtained. The final edited video will be shown to the teacher to check that it portrays the lesson adequately. These edited videos provide the visual ethnography of the teacher’s teaching.

Auto-ethnographies: At the end of the first year or the start of the second year, the four teachers in each focus group will view the twelve edited videos from their classrooms and attempt to identify three that they feel show “typical” teaching in their region. The recordings of these focus group discussions will form the first data set: as responses of regionally and linguistically internal observers they will provide an auto-ethnography of mathematics teaching in each region. If the participants can agree on three representative videos, these will be used as stimuli for the other groups. If the participants cannot agree then three videos that demonstrate the diversity of pedagogies in the region will be selected instead.

Ethno-ethnographies: In the second year each focus group will view and discuss the three videos from the same language group from the four other regions. Guiding questions will be provided by the researchers, including questions related to accounting for regional differences in achievement based on what the teachers observe. The recordings of these focus group discussions will form the second data set and will be analysed to identify points of difference and similarity observed by the participants. While the emphasis will be on pedagogy, it may be that regional curriculum differences and socio-economic differences will also be discussed if their effects are evident in classroom practices. Recordings of these discussions will provide the ethno-ethnography of the pedagogy revealed in the videos. Encounters with other pedagogies will offer the participants a way to reflect on their own familiar beliefs and practices, by comparison with others.

The final year will be spent analysing the focus group discussions and preparing publications.

The proposed research differs from many other studies in that it seeks to explore both teaching practices and the underlying pedagogy. This is impossible through approaches that involve external observers, as they have no access to what is implicit to the participants themselves. However, by making the participants the observers, one gains insight through what they observe and how they observe it into the implicit criteria that guide their observations.

### 4. Communication of Results

Research results will be reported at national and international conferences and in research journal articles at the two stages of auto-ethnography and ethno-ethnography for each region. Article will be submitted to journals such as Educational Studies in Mathematics, the Journal of Research in Mathematics Education, the Journal of Mathematics Teacher Education, and the Canadian Journal of Science, Mathematics and Technology Education. The reports will be gathered together in an edited book at the conclusion of the research process. The book will be written with an audience of government policy makers, teachers and teacher educators in mind. In each region researchers and participating teachers will also present at professional teaching conferences (such as National Council of Teachers of Mathematics conferences and regional mathematics teachers’ association conferences) to report the results of the research at each stage.

* see http://www.acadiau.ca/~cknippin/sd/

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