Ann Anderson

This research investigates African immigrant families' experiences and perspectives of their children's (10-15-year-old) mathematics learning in the home, community and school settings within the Greater Vancouver area. The study employs the concept of cultural capital framework and Afrocentric worldviews to understand how African immigrant parents support their children’s mathematics learning at the upper elementary and middle grade levels (Grades 4 - 8) in Canada. Specifically, the study addressed the following questions: i) which socio-cultural strategies and understandings do African immigrant families living in a large metropolis draw upon to support their children’s mathematics learning? ii) what are African immigrant families’ experiences and perspectives with respect to their (10–15-year olds) children’s mathematics learning in the home, community and school settings, within the Greater Vancouver Area? The study’s findings indicate that, African immigrant parents 1) were aware of the gatekeeping function of mathematics and its role in reproducing or disrupting race or class hierarchies and 2) perceived their support for their children’s mathematics fluency as a sure way for securing a status for counterbalancing their social exclusion and positioning themselves as people of education and intellect. The study’s findings also indicate that immigrant families leveraged various forms of capital in support of their children’s mathematics learning: Cultural, linguistic, aspirational, technological/digital and navigational capital. These forms of capital extend the narrowly conceived definitions of capital as accumulated wealth, resources, and experiences of white middle-class families. The results of the study are significant in that they provide much- needed insights into ways to build culturally responsive mathematics education informed by African immigrant students’ and families’ funds of knowledge.

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Emphasis on the importance of subject matter expertise in teaching secondary mathematics is found in the research literature and in policy. In the United States, for instance, the No Child Left Behind Act, calls for secondary teachers to be certified in a subject specialization. In Canada, admission to secondary teacher education programs requires extensive subject-specific university coursework. However, it is unclear if or how extensive subject matter expertise impacts the practices of teachers in a secondary classroom.This study aims to explore how advanced coursework in mathematics, beyond the scope of the high school curriculum, impacts the ways prospective teachers understand and teach secondary content. Using a qualitative case study methodology, five prospective secondary mathematics teachers participated, with data obtained through document analysis and semi-structured task-based interviews. Participants engaged with classroom-relevant tasks and were explicitly asked how they could draw upon advanced mathematics to inform their teaching. Participants also detailed their perceptions of the role advanced mathematics plays in their development as teachers.Results from this study reveal that participants saw little value in the content of advanced mathematics to their teaching, but expressed value towards the beliefs and values gained through advanced mathematics, such as problem solving and rigour. Some participants demonstrated misconceptions at the secondary level, which had direct connections to content from their post-secondary mathematics coursework. For example, all participants made the false claim that a real-valued polynomial can be factored if and only if it has a root.Results extend the literature through rich empirical data which illuminates how prospective secondary mathematics teachers perceive and use advanced mathematics in understanding the secondary curriculum. While participants held content knowledge beyond the secondary curriculum, this knowledge was not integrated in a way that impacted their understanding of secondary mathematics. An understanding of post-secondary mathematics has the potential to be of value to secondary teachers in the classroom, but this potential needs a space to be unlocked. I argue that mathematicians and teacher educators need to work together to build opportunities for prospective teachers to build connections between the mathematics they know and the mathematics they need to teach.

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Even though previous research points to the significance of kindergarten teachers’ practices, that take into consideration the nature of children and how they learn, there is limited research regarding developmentally appropriate practices in various socio-cultural contexts. To address this gap in the literature, a qualitative multi-case study into the perceptions and classroom practices of four kindergarten teachers in two Ghanaian schools, Tata and Kariba, was carried out over a six-month period. Four research questions guided the study, namely: How do teachers interpret and apply DAP in kindergarten classrooms within the Ghanaian sociocultural context?; With what kind of learning activities do teachers engage kindergarten children?; Which instructional strategies do teachers use in a kindergarten classroom?; and What factors and beliefs influence teachers’ instructional decision-making in a kindergarten classroom? Cognitive constructivist theory (Piaget, 1951) and sociocultural theory (Vygotsky,1978) informed the research. Data used were semi-structured individual interviews and pair-based interviews and fieldnotes of classroom observations. Both within and across case interpretative analysis, as outlined by Stake (2006), was used.The findings of this study revealed these teachers’ practices were developmentally appropriate and they interpreted DAP within the Ghanaian socio-cultural context through contextually relevant language of instruction (English language, Nfante language), age- and culturally- appropriate learning materials, and the use of storytelling, traditional songs, and traditional rhymes. Moreover, teachers in both the urban and rural setting, described a variety of learning activities they believed impacted children’s development in different ways; they pointed to play-based instruction and integration as well as specific strategies such as picture-walk and think-pair-share that they believed promoted effective DAP; and discussed their explicit and implicit theories of teaching involved in their instructional decision-making processes. These findings are discussed in light of current research in early childhood education to provide insights into how DAP, as interpreted and applied in the Ghanaian socio-cultural context, can inform teaching and learning in kindergarten classrooms, globally. Implications for future research and practice both within Ghana and elsewhere are established.

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This dissertation uses a discursive psychological perspective to investigate how young children use the structure of spoken language to support their participation in mathematical discussion. There is a general consensus that participation in the practice of collective argumentation not only promotes, but actually constitutes the learning of mathematics (Cobb, Yackel & Wood, 1992a; Krummheuer, 2007). However, given the wide variety of forms that mathematical communication might take (Barwell, Leung, Morgan & Street, 2005), how to fruitfully define and document the practices of communication involved, especially for young learners, remains an open question. Through a series of related discourse analyses (three linked studies carried out on a common data set), I explore some of the discursive practices 5- to 7-year-olds use as they negotiate what will become a taken-as-shared understanding regarding mathematical validity: a sociomathematical norm regarding what it means to know in mathematics. Discursive psychology (Edwards, 1997) affords an examination of how participants treat notions of knowing and understanding during interaction. Using corpus linguistic analysis in the first study here allows me to elaborate features of this group’s culture of negotiation by illuminating patterns in the interactional sequences involving doing knowing. Conversation analysis in the second study affords an examination of the practices by which young children incorporate mathematical content within the social act of negotiation: doing mathematical understanding. Further analysis attending to multimodal aspects of communication in the third study shows how the participants used those previously noted discursive practices to develop and sustain a six week long investigation into the meaning of the square root symbol: doing algebraic reasoning. These analyses show how the children are able to draw upon a range of sociomathematical norms as resources that enable them to participate in ways that co-ordinate with each other. The findings suggest that expanding our expectations for what mathematical knowing looks like with young children affords the development of learning environments that support all children’s sustained, successful engagement with mathematics.

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This dissertation investigates the problem of student engagement in elementary mathematics through the particular theoretical framework of imaginative education (IE) (Egan, 1997, 2005). The question at the centre of this study is what the use of IE and imaginative lesson planning frameworks means to children and for their engagement in elementary mathematics. For this study, five intermediate-aged elementary students were tracked through a unit of shape and space (geometry). The unit, framed with the binary opposites of vision and blindness, asked students how they might come to understand shape and space as a sighted and as a visually impaired person. Thus a humanized perspective was brought to the learning of mathematics. After the unit the five focus students took part in an individual and a whole-group semi-structured interview with the teacher/researcher. The study used qualitative instrumental case study methods; data sources included students’ mathematics journals, activity pages, transcripts of audio- and videotaped semi-structured individual and group interviews, a teacher/researcher diary, and a detailed unit overview and lesson plans. The study gathered rich descriptive data focused on bringing out the students’ perspective of their experience. Results indicate that the students demonstrated positive engagement with mathematics and that the IE theory, which utilized the students’ imaginations and affective responses, allowed multiple access points through which to connect with the mathematical concepts. Three conclusions of the study were that the students expanded their mathematical awareness through making a variety of connections, they were able to develop self-confidence in their learning of mathematics by using emotions and imagination, and they were able to use cognitive tools, particularly a sense of wonder, to engage with mathematics. The dissertation concludes with a discussion of implications and recommendations in four areas, including the need for further research in different educational contexts, in the interaction of imagination and affective responses, and into characteristics of mathematical engagement such as self-confidence. Recommendations for how future pedagogical practice might use the IE theory and embrace the expansion of students’ perspectives in classroom practice bring the dissertation to a close.

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This dissertation investigates the problem posing patterns that emerge as small groups of students work collectively on a mathematics task, and describes the characteristics of problem posing that result. This case study is a naturalistic inquiry about four small groups of Grade 8 students in the Lower Mainland of British Columbia who are working in a classroom setting, with the researcher acting as participant/observer and videographer. The concept of author/ity is used to highlight human agency in mathematics. Small groups, as learning systems, are being considered to be “authors” of their discourse, and the improvisational nature of authoring is discussed. A parallel is drawn between the storyline of a literary work and the storyline that emerges as a group poses problems in order to work its way through a mathematical task. The metaphor of a tapestry is used as a way of describing how the threads of group discourse weave together. To address the challenge of documenting collective behavior at the group level, a method of data analysis is introduced that “blurs” the data in order to capture patterns that emerge over time – transcripts are color-coded and then shrunk to create tapestries that provide visual evidence of collective problem posing patterns. This dissertation finds that collective problem posing is an emergent process. Each group poses its own set of problems, and the number of problems posed and their frequency also vary, resulting in individual tapestries for each group. The tapestry patterns are then used to compare characteristics of the groups’ discussions. Problem posing appears to be an activity that these groups are able to do without receiving formal instruction or direction. The reposing of problems helps to structure each group’s discussion, with the role that each problem plays in the conversation evolving as it reemerges. The concept of groups working as bricoleurs is also explored, with bricolage in mathematics being characterized as a creative and generative process. The dissertation concludes with a discussion of expertise in school mathematics and what implications an “aesthetic of imperfection” might have in the mathematics classroom.

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