Learning to ‘deal’: A microgenetic case study of a struggling student’s representational strategies for partitive division

International audienceThis paper focuses on the arithmetical understandings and behaviours of one fifteen-year old student with very low attainment in mathematics, as she worked on a sequence of scenario-based partitive division (sharing) tasks with individually-tailored verbal and visuospatial support. The student’s independent and co-created visuospatial representations of arithmetical structures, along with her verbal comments, were analysed qualitatively using a multimodal microgenetic approach. This paper focuses on three particular excerpts which illustrate the fundamentally componential nature of the concept and practice of division, some difficulties that may be experienced when modelling ‘sharing’ tasks, and the pedagogical importance of spatial structuring when a learner is moving between different kinds of representation

An investigation of how visual arts can be used to teach mathematical concepts of space and shape in Grade R

The impetus for this study came from the findings of an evaluation of a Maths and Science through Arts and Culture (MStAC) Curriculum Intervention undertaken with Grade R teachers registered for a BEd(in-service) qualification at Rhodes University, South Africa. The intervention aimed to enrich Grade R teachers’ teaching of mathematics. Post-intervention classroom observations showed that, in spite of the intervention, teachers’ classroom practices did not change, and they were not using visual arts to teach mathematical concepts. This, together with the lack of research in the field of mathematics in early childhood, particularly in South Africa, motivated this research, a case study, which investigates how visual arts can be used to teach space and shape conceptualization in Grade R. I designed a research intervention underpinned by a constructivist model of teacher professional development located in reflective practice (Borko & Putman, 1995; Zeichner & Liston, 1996; Wilmot, 2005). Guided by Stacey’s (2009) notion of an emergent curriculum, I designed a three phase research intervention which involved selected Grade R teachers undertaking classroom-based research. Phase I built awareness around the notion of creativity; Phase II focused on making meaning of children’s behaviour and interests; and Phase III applied the knowledge and ideas from the Phases I and II to the teaching of space and shape. As an interpretive research study, it closely examines the participating teachers’ perceptions, experiences and reflections which were articulated in reflective reports and assignments. Following action research processes, the participant teachers engaged in the process of an emergent curriculum. They observed the behaviour interests of Grade R children, interpreted and made meaning of the evident behaviours, made decisions regarding extension activities, and planned accordingly. The findings of the study illuminate a model of teacher professional development that can support and enhance teachers’ practice. Understanding the notion of creativity and the ability to create a classroom conducive to creativity, are necessary components for teaching space and shape through visual arts activities. An emergent curriculum approach is proposed as an appropriate pedagogy for teaching children about space and shape through visual arts activities

An investigation of how visual arts can be used to teach mathematical concepts of space and shape in Grade R

The impetus for this study came from the findings of an evaluation of a Maths and Science through Arts and Culture (MStAC) Curriculum Intervention undertaken with Grade R teachers registered for a BEd(in-service) qualification at Rhodes University, South Africa. The intervention aimed to enrich Grade R teachers’ teaching of mathematics. Post-intervention classroom observations showed that, in spite of the intervention, teachers’ classroom practices did not change, and they were not using visual arts to teach mathematical concepts. This, together with the lack of research in the field of mathematics in early childhood, particularly in South Africa, motivated this research, a case study, which investigates how visual arts can be used to teach space and shape conceptualization in Grade R. I designed a research intervention underpinned by a constructivist model of teacher professional development located in reflective practice (Borko & Putman, 1995; Zeichner & Liston, 1996; Wilmot, 2005). Guided by Stacey’s (2009) notion of an emergent curriculum, I designed a three phase research intervention which involved selected Grade R teachers undertaking classroom-based research. Phase I built awareness around the notion of creativity; Phase II focused on making meaning of children’s behaviour and interests; and Phase III applied the knowledge and ideas from the Phases I and II to the teaching of space and shape. As an interpretive research study, it closely examines the participating teachers’ perceptions, experiences and reflections which were articulated in reflective reports and assignments. Following action research processes, the participant teachers engaged in the process of an emergent curriculum. They observed the behaviour interests of Grade R children, interpreted and made meaning of the evident behaviours, made decisions regarding extension activities, and planned accordingly. The findings of the study illuminate a model of teacher professional development that can support and enhance teachers’ practice. Understanding the notion of creativity and the ability to create a classroom conducive to creativity, are necessary components for teaching space and shape through visual arts activities. An emergent curriculum approach is proposed as an appropriate pedagogy for teaching children about space and shape through visual arts activities

Visualising young children's play: exploring multimodal transcription of video-recorded interaction

This thesis considers child-initiated play from a multimodal social semiotic perspective, giving close attention to the ways in which children collaboratively make meaning in play in a multitude of ways. Such a perspective resists instrumental, developmental perspectives on play, and comes at a time when play-based approaches are in tension with increasingly formalised learning agendas and changes to early years assessment. In order to explore the multimodality of child-initiated play, apt theories and research methods are necessary for attending to the ways children make meaning in multiple modes. The study consists of video-based observations of child-initiated play collected through an ethnographic, teacher-research case study carried out in a nursery school in England. A particular challenge in multimodal research is developing forms of transcription which account for multiple modes in fine-grained detail, with the conventions developed for transcribing language proving insufficient. This thesis presents four multimodal transcript designs as analytic devices that bring multimodal aspects of play to the fore, and critically discusses the gains and losses of each multimodal transcript. The multimodal transcripts highlight the richness and complexity of child-initiated play as learning, making visible ways in which children’s play is complex, layered, transformative, creative and agentive meaning-making. This thesis proposes that multimodal transcription not only ‘visualises’ play by making it visible and sharable, but also offers a new lens through which we might understand the semiotic complexity of play. Through interwoven substantive and methodological strands, this thesis therefore offers a contribution towards the tools and dispositions necessary for recognising and valuing meaning-making in play, in early years research methodology, educational theory and practice

Drawing division : emerging and developing multiplicative structure in low-attaining students representational strategies

This thesis examines the particular difficulties with multiplicative thinking experienced by students with very low attainment in school mathematics, and the representational strategies they use for multiplication and division-based tasks.\ud Selected students in two mainstream secondary schools, all performing significantly below age-related expectations in mathematics, placed in ‘bottom sets’, and described by their teachers as having particularly weak numeracy, received a series of tuition sessions (individual or paired). These involved ongoing qualitative diagnosis of their arithmetical strengths and weaknesses, and personalised, flexible learning support, delivered by the author. Students engaged mainly in division-based scenario tasks designed to encourage their engagement in multiplicative thinking, and explored various visuospatial representational strategies tailored to their specific areas of conceptual and procedural difficulty.\ud Multimodal audiovisual data collected from tuition sessions was analysed qualitatively across multiple analytic dimensions using a microgenetic approach. This led to the development of an adaptable framework for the analysis of nonstandard visuospatial representations of arithmetical structures and relationships. Analysis of changes in individual students’ strategies provided insight into some possible learning trajectories for multiplicative thinking. Parallel comparison of students’ varied representational strategies resulted in evidence for the psychological power of certain fundamental representation types, such as unit arrays and containers.\ud The main findings of this thesis concern: the fundamentally componential nature of the concept and practice of division, the potential difficulties this causes in understanding, and the importance of modelling and manipulating unitary multiplicative structures; and the relationship between representational strategies, economy and efficiency in carrying out multiplication and division-based tasks.\ud Conclusions are drawn on the relationship between the development of representational strategies and multiplicative thinking. Recommendations are given regarding learning and teaching practice for students with severe and milder difficulties in mathematics, and particularly the nature of 1:1 support provision for those considered to have Special Educational Needs

The development of pattern-related abilities through play activities in young children

This thesis investigates nursery children's knowledge, understanding and skills in patternmaking as an aspect of early mathematical development. It presents two discrete but closely related studies, a cross-sectional and a longitudinal study. The methodology includes use of structured assessment activities using familiar play materials. Assessment focuses firstly on different aspects of pattern-making; secondly, on pattern perception; and thirdly, on wider aspects of developing cognition. The methodology includes collection of case study data in the naturalistic setting of the nursery class. The cross-sectional study, focused on knowledge, understanding and skil1s in pattern-making at 3½ and 4½ years, finds an increasing minority of children successful in repeated pattern-making and 2D spatial pattern-making but not linear symmetrical pattern-making. Few children evidence pattern perception at either age but an increasing minority evidences an emergent understanding of the word 'pattern.' The longitudinal study tracks children's development towards and within pattern-making from 3½ to 4½ years. It details development in two key aspects of pattern-making, colour and spatial organisation, through case study data. An examination of commonalities in development leads to hypothesised developmental pathways in both aspects of pattern-making. A single pathway leads towards complex colour organisation. Distinct pathways lead to basic and complex spatial organisation and to the basic elements of pattern. Pathways to 2D spatial pattern-making are more varied than pathways to repeated pattern-making. Quantitative analysis confirms key features of the pathways although some findings remain tentative. Differences in the detail of individual pathways are highlighted, as are wide differences in children's rates of development and in their interests and motivation. There are no findings of significant gender-related differences. Children's competencies in the colour organisation strand of pattern-making are significantly associated with abilities across key areas of developing cognition. Spatial organisation competencies are at first associated with a narrow range of primarily spatial abilities but this extends to include number and rhythmic abilities at 4½ years. The study confirms and extends some earlier findings, and also presents new findings. Findings lead to questioning of guidance on curriculum goals and pedagogy for the age-group. They highlight a need to acknowledge the creativity of many young children in this area of mathematics

An Exploration of young children’s engagements in research behaviour

Natural research behaviours may present in children younger than eight years but tend to be overlooked by professional researchers, with the result that young children are rarely recognised as agents in enquiry concerning matters affecting them. This exclusion amounts to social injustice as it underestimates children’s capabilities and denies them particular rights. The thesis proposes that young children engage in research activity congruent with professional adult researchers’ behaviours, as part of their daily lives. Furthermore, the inequity caused by excluding children from recognition as researchers may be addressed if professional researchers were to find ways to recognise and value the children’s contributions as researchers. The empirical study that is the focus for the thesis secured a taxonomy of research behaviours from professional adult researchers which was then applied to naturalistic observations conducted with - and by - children aged 4-8 years in their settings and homes. A ‘jigsaw’ methodology was adopted, featuring constructivist grounded theory and critical ethnography, among other methodologies. Throughout, the project was committed to participatory, emancipatory and inductive principles, though challenges were encountered along the way. Alongside observations, multiple other methods and analysis were employed in the co-construction of data with children and their practitioners in three English early childhood settings and children and their parents in five homes. Professional adult researchers also contributed to primary and meta-data. Results indicated that problem solving, exploring, conceptualising and basing decisions on evidence were regarded by professional researchers as the ‘most important’ research behaviours. Children engaged in these behaviours of their own volition, alongside other research behaviours. Their activities included exploring materials to create novel artefacts in art work, rolling in giant cylinders, cooking and ordering objects systematically. While undertaking these activities, children often revealed higher order cognitive processes such as trial and error elimination, causality, analogy and a posteriori conceptualisation. The study produced a ‘plausible account’ suggesting that children aged 4-8 years do engage in research activity naturalistically as part of their daily lives and that this activity is congruent with professional adult researchers’ behaviours

An investigation into Grade R teachers' experiences of implementing numeracy in Grade R

This research study investigates how selected Grade R teachers implement numeracy in their classrooms, based on their experiences, personal beliefs and perceptions. This study adopts the view that a teacher's practice is not only informed on what she believes about her learners, but also on how she organises her classroom environment and her own teaching practice. However, the teacher cannot be divorced from the political and historical background of Early Childhood Development (ECD), as these factors have an impact on her numeracy implementation. Integrated in this research study is thus not only a comprehensive historical analysis of the historical and political background of ECD, but also an in-depth look at the complex curriculum road the Grade R teacher had to travel over the past twelve years. The aim of this case study is to understand and describe what is happening in Grade R classrooms when numeracy is implemented, as well as identifying possible barriers which Grade R teachers may experience in implementing numeracy. The author utilized an adapted Interactive Qualitative Analysis Framework to explore Grade R teachers' views and experiences. Open-ended focus group interviews were used to develop a framework for individual interviews. This interview framework guided the data collection of nine audio-video tapes of classroom activities and nine semi-structured individual interviews. Analysis of the individual interviews revealed each teacher's understanding as well as her perceptions and needs regarding the implementation of numeracy in her classroom. The transcriptions of the individual interviews were compared to the audio-video tapes of what happened in each classroom. This was done in order to determine whether the teachers' practices were consistent with what they said in the interviews. Paradoxes and inconsistencies were documented. The historical and political analysis of ECD illustrates the complex development of ECD in South Africa over the last few decades. It therefore also highlights and brings to the fore the complex journey that Grade R teachers had to navigate in adapting their practice to ever changing curriculum requirements. The empirical results show that there are inconsistencies between what selected teachers know and believe, and what they are implementing. However, "silent themes" which did not come to the fore in the focus group or individual interviews, were identified in the audiovideo tapes of classroom activities. Also, many "silent themes" indicate that there are numerous concerns, such as lack of proficiency in bridging the gap between theory and practice, regarding the implementation of numeracy in Grade R. It was found that many of the selected Grade R teachers struggle to implement numeracy in their classrooms, as they lack the skills and support to teach numeracy in a Grade R context

An investigation into Grade R teachers' experiences of implementing numeracy in Grade R

This research study investigates how selected Grade R teachers implement numeracy in their classrooms, based on their experiences, personal beliefs and perceptions. This study adopts the view that a teacher's practice is not only informed on what she believes about her learners, but also on how she organises her classroom environment and her own teaching practice. However, the teacher cannot be divorced from the political and historical background of Early Childhood Development (ECD), as these factors have an impact on her numeracy implementation. Integrated in this research study is thus not only a comprehensive historical analysis of the historical and political background of ECD, but also an in-depth look at the complex curriculum road the Grade R teacher had to travel over the past twelve years. The aim of this case study is to understand and describe what is happening in Grade R classrooms when numeracy is implemented, as well as identifying possible barriers which Grade R teachers may experience in implementing numeracy. The author utilized an adapted Interactive Qualitative Analysis Framework to explore Grade R teachers' views and experiences. Open-ended focus group interviews were used to develop a framework for individual interviews. This interview framework guided the data collection of nine audio-video tapes of classroom activities and nine semi-structured individual interviews. Analysis of the individual interviews revealed each teacher's understanding as well as her perceptions and needs regarding the implementation of numeracy in her classroom. The transcriptions of the individual interviews were compared to the audio-video tapes of what happened in each classroom. This was done in order to determine whether the teachers' practices were consistent with what they said in the interviews. Paradoxes and inconsistencies were documented. The historical and political analysis of ECD illustrates the complex development of ECD in South Africa over the last few decades. It therefore also highlights and brings to the fore the complex journey that Grade R teachers had to navigate in adapting their practice to ever changing curriculum requirements. The empirical results show that there are inconsistencies between what selected teachers know and believe, and what they are implementing. However, "silent themes" which did not come to the fore in the focus group or individual interviews, were identified in the audiovideo tapes of classroom activities. Also, many "silent themes" indicate that there are numerous concerns, such as lack of proficiency in bridging the gap between theory and practice, regarding the implementation of numeracy in Grade R. It was found that many of the selected Grade R teachers struggle to implement numeracy in their classrooms, as they lack the skills and support to teach numeracy in a Grade R context