The development of multiplicative thinking and proportional reasoning: Models of conceptual learning and transfer

This thesis considers the development of multiplicative thinking and proportional reasoning from two perspectives. Firstly, it examines the research literature on progressions in conceptual understanding to create a Hypothetical Learning Trajectory (HLT). Secondly, it surveys modern views of how transfer by learners occurs in and between situations, contrasting object views of abstraction with knowledge in pieces views. Case studies of six students aged 11-13 years illustrate conceptual changes that occur during the course of a school year. The students are involved in a design experiment in which I (the researcher) co-teach with the classroom teacher. The students represent a mix of gender, ethnicity and level of achievement. Comparison of the HLT with the actual learning trajectory for each student establishes its validity as a generic growth path. Examination of the data suggests that two models of learning and by inference, transfer, describe the conceptual development of the students. There is consideration of students’ use of anticipated actions on physical and imaged embodiments as objects of thought with a focus on the significance of object creation for conceptual growth, and the encapsulation, completeness and contextual detachment of objects. There is broad consistency in students’ progress through the phases of the HTL within each sub-construct though the developmental patterns of individual are variable and temporal alignment across the sub-constructs does not uniformly hold. Some consistency of order effect in concept development is noted. Discussion on the limitations of the HTL includes the difference between knowledge types from a pedagogical perspective, absence of significant model-representation-situation transfer, and order relations in conceptual development. Considerable situational variation occurs as students solve problems that involve applications of the same concepts. Partial construction of concepts is common. This was true of all learners, irrespective of level of achievement. High-achieving students more readily anticipate actions and trust these anticipations as objects of thought than middle and low achievers. The data supports knowledge in pieces views of conceptual development. Complexity for learners in observing affordances in situations, and in co-ordinating the fine-grained knowledge required, explains the difficulty of transfer. While supporting the anticipation of action as significant from a learning perspective the research suggests that expertise in applying concepts involves a process of noticing similarity across contextually bound situations and cueing appropriate knowledge resources

Fractions speak louder than words: Investigating preservice primary teachers’ knowledge and understanding for teaching fractions with representations

Mathematics presents specific challenges for primary preservice teachers and fractions is among the most problematic of topics. This thesis investigates preservice primary teachers’ understanding and use of fractions and fraction representations. Preservice teachers have particular difficulty explaining the rationale behind fraction operations, often only demonstrating superficial knowledge of symbolic procedures. This level of knowledge is insufficient for teaching and, thus, initial teacher education presents a crucial opportunity to deepen teachers’ knowledge before they begin their teaching careers. The study addresses the crucial need for further research into the initial teacher education of preservice teachers at a time where there is a national agenda for improving education in Australia. However, despite the potential to redress preservice teachers’ knowledge of fractions, there is a dearth of studies elucidating how fraction knowledge develops over a program in initial teacher education, particularly in an Australian context. To address this gap, the current study aimed to investigate the development of preservice primary teachers’ knowledge about teaching fractions during a Graduate Diploma of Education (GradDipEd) program with a focus on their understanding and use of fraction representations. To focus the study, the following research questions were posed: RQ1. How do preservice teachers’ understandings of fractions and fraction representations develop over a teacher education program? RQ2. How and why do preservice teachers use fraction representations for learning and teaching tasks over the course of a teacher education program

Reconfigurable-Hardware Accelerated Stream Aggregation

High throughput and low latency stream aggregation is essential for many applications that analyze massive volumes of data in real-time. Incoming data need to be stored in a single sliding-window before processing, in cases where incremental aggregations are wasteful or not possible at all. However, storing all incoming values in a single-window puts tremendous pressure on the memory bandwidth and capacity. GPU and CPU memory management is inefficient for this task as it introduces unnecessary data movement that wastes bandwidth. FPGAs can make more efficient use of their memory but existing approaches employ only on-chip memory and therefore, can only support small problem sizes (i.e. small sliding windows and number of keys) due to the limited capacity. This thesis addresses the above limitations of stream processing systems by proposing techniques for accelerating single sliding-window stream aggregation using FPGAs to achieve line-rate processing throughput and ultra low latency. It does so first by building accelerators using FPGAs and second, by alleviating the memory pressure posed by single-window stream aggregation. The initial part of this thesis presents the accelerators for both windowing policies, namely, tuple- and time-\ua0based, using Maxeler\u27s DataFlow Engines\ua0(DFEs) which have a direct feed of incoming data from the network as well as direct access to off-chip DRAM. Compared to state-of-the-art stream processing software system, the DFEs offer 1-2 orders of magnitude higher processing throughput and 4 orders of magnitude lower latency. The later part of this thesis focuses on alleviating the memory pressure due to the various steps in single-window stream aggregation. Updating the window with new incoming values and reading it to feed the aggregation functions are the two primary steps in stream aggregation. The high on-chip SRAM bandwidth enables line-rate processing, but only for small problem sizes due to the limited capacity. The larger off-chip DRAM size supports larger problems, but falls short on performance due to lower bandwidth. In order to bridge this gap, this thesis introduces a specialized memory hierarchy for stream aggregation. It employs Multi-Level Queues (MLQs) spanning across multiple memory levels with different characteristics to offer both high bandwidth and capacity. In doing so, larger stream aggregation problems can be supported at line-rate performance, outperforming existing competing solutions. Compared to designs with only on-chip memory, our approach supports 4 orders of magnitude larger problems. Compared to designs that use only DRAM, our design achieves up to 8x higher throughput. Finally, this thesis aims to alleviate the memory pressure due to the window-aggregation step. Although window-updates can be supported efficiently using MLQs, frequent window-aggregations remain a performance bottleneck. This thesis addresses this problem by introducing StreamZip, a dataflow stream aggregation engine that is able to compress the sliding-windows. StreamZip deals with a number of data and control dependency challenges to integrate a compressor in the stream aggregation pipeline and alleviate the memory pressure posed by frequent aggregations. In doing so, StreamZip offers higher throughput as well as larger effective window capacity to support larger problems. StreamZip supports diverse compression algorithms offering both lossless and lossy compression to fixed- as well as floating- point numbers. Compared to designs using MLQs, StreamZip lossless and lossy designs achieve up to 7.5x and 22x higher throughput, while improving the effective memory capacity by up to 5x and 23x, respectively

Role of an artefact of Dynamic algebra in the conceptualisation of the algebraic equality

In this contribution, we explore the impact of Alnuset, an artefact of dynamic algebra, on the conceptualisation of algebraic equality. Many research works report about obstacles to conceptualise this notion due to interference of the previous arithmetic knowledge. New meanings need to be assigned to the equal sign and to letters used in algebraic expressions. Based on the hypothesis that Alnuset can be effectively used to mediate the conceptual development necessary to master the algebraic equality notion, two experiments have been designed and implemented in Italy and in France. They are reported in the second part of this pape

Classification-based phrase structure grammar: an extended revised version of HPSG

This thesis is concerned with a presentation of Classification -based Phrase Structure Grammar (or cPSG), a grammatical theory that has grown out of extensive revisions of, and extensions to, HPSG. The fundamental difference between this theory and HPSG concerns the central role that classification plays in the grammar: the grammar classifies strings, according to their feature structure descriptions, as being of various types. Apart from the role of classification, the theory bears a close resemblance to HPSG, though it is by no means a direct translation, including numerous revisions and extensions. A central goal in the development of the theory has been its computational implementation, which is included in the thesis.The presentation may be divided into four parts. In the first, chapters 1 and 2, we present the grammatical formalism within which the theory is stated. This consists of a development of the notion of a classificatory system (chapter 1), and the incorporation of hierarchality into that notion (chapter 2).The second part concerns syntactic issues. Chapter 3 revises the HPSG treatment of specifiers, complements and adjuncts, incorporating ideas that specifiers and complements should be distinguished and presenting a treatment of adjuncts whereby the head is selected for by the adjunct. Chapter 4 presents several options for an account of unbounded dependencies. The accounts are based loosely on that of GPSG, and a reconstruction of GPSG's Foot Feature Principle is presented which does not involve a notion of default. Chapter 5 discusses coordination, employing an extension of Rounds- Kasper logic to allow a treatment of cross -categorial coordination.In the third part, chapters 6, 7 and 8, we turn to semantic issues. We begin (Chapter 6) with a discussion of Situation Theory, the background semantic theory, attempting to establish a precise and coherent version of the theory within which to work. Chapter 7 presents the bulk of the treatment of semantics, and can be seen as an extensive revision of the HPSG treatment of semantics. The aim is to provide a semantic treatment which is faithful to the version of Situation Theory presented in Chapter 6. Chapter 8 deals with quantification, discussing the nature of quantification in Situation Theory before presenting a treatment of quantification in CPSG. Some residual questions about the semantics of coordinated noun phrases are also addressed in this chapter.The final part, Chapter 9, concerns the actual computational implementation of the theory. A parsing algorithm based on hierarchical classification is presented, along with four strategies that might be adopted given that algorithm. Also discussed are some implementation details. A concluding chapter summarises the arguments of the thesis and outlines some avenues for future research

Developing early algebraic reasoning in a mathematical community of inquiry

This study explores the development of early algebraic reasoning in mathematical communities of inquiry. Under consideration is the different pathways teachers take as they develop their own understanding of early algebra and then enact changes in their classroom to facilitate algebraic reasoning opportunities. Teachers participated in a professional development intervention which focused on understanding of early algebraic concepts, task development, modification, and enactment, and classroom and mathematical practices. Design research was employed to investigate both teaching and learning in the naturalistic setting of the schools and classrooms. The design approach supported the development of a model of professional development and the framework of teacher actions to facilitate algebraic reasoning. Data collection over the school year included participant observations, video recorded observations, documents, teacher interviews, and photo elicitation interviews with students. Retrospective data analysis drew the results together to be presented as cases of two teachers, their classrooms, and students. The findings show that the integration of algebraic reasoning into classroom mathematical activity is a gradual process. It requires teachers to develop their own understanding of algebraic concepts which includes understanding of student reasoning, progression, and potential misconceptions. Task implementation and design, shifts in pedagogical actions, and the facilitation of new classroom and mathematical practices were also key elements of change. The important role which students have in the development of classrooms where algebraic reasoning is a focus was also highlighted. These findings have significant implications for how teachers can be supported to develop their understanding of early algebra and use this understanding in their own classrooms to facilitate early algebraic reasoning