Co-constructing decimal number knowledge

This mathematics education research provides significant insights for the teaching of decimals to children. It is well known that decimals is one of the most difficult topics to learn and teach. Annette’s research is unique in that it focuses not only on the cognitive, but also on the affective and conative aspects of learning and teaching of decimals. The study is innovative as it includes the students as co-constructors and co-researchers. The findings open new ways of thinking for educators about how students cognitively process decimal knowledge, as well as how students might develop a sense of self as a learner, teacher and researcher in mathematics

Connecting the Points: An Investigation into Student Learning about Decimal Numbers

The purpose of this research project was to investigate the effects of a short-term teaching experiment on the learning of decimal numbers by primary students. The literature describes this area of mathematics as highly problematic for students. The content first covered student understanding of decimal symbols, and how this impacted upon their ability to order decimal numbers and carry out additive operations. It was then extended to cover the density of number property, and the application of multiplicative operations to situations involving decimals. In doing so, three areas of cognitive conflict were encountered by students, the belief that longer decimal numbers are larger than shorter ones (irrespective of the actual digits), that multiplication always makes numbers bigger, and that division always makes numbers smaller. The use of a microgenetic approach yielded data was able to be presented that provides details of the environment surrounding the moments where new learning was constructed. The characteristics of this environment include the use of physical artifacts and situational contexts involving measurement that precipitate student discussion and reflection. The methodology allowed for the collection of evidence regarding the highly complex nature of the learning, with evidence of 'folding back' to earlier schema and the co-existence of competing schema. The discussion presents reasons as to why the pedagogical approach that was employed facilitated learning. One of the main findings was that the use of challenging problems situated in measurement contexts that involved direct student participation promoted the extension and/or re-organization of student schema with regard to decimal numbers. The study has important implications for teachers at the upper primary level wanting to support student learning about the decimal numbers system

Growing Prior Knowings of Zero: Growth of Mathematical Understanding of Learners with Difficulties in Mathematics

Currently, there is a large proportion of learners experiencing difficulties in mathematics. Much of the intervention research for children with great difficulty learning mathematics has focused on accommodations to the peripheral supports of mathematics, like creating step by step plans, and not on strategies to help children conceptualize mathematics or enable them to mathematize. We know very little about the conceptual development and how to effect change in conceptual understanding of mathematics for children who have great difficulty learning mathematics. At the same time, in mathematics education research, zero is a known area of difficulty for many students and misconceptions regarding zero can persist into university and adulthood. This dissertation explores growth in understanding with three learners experiencing difficulties in mathematics and their growing conceptions of zero. Utilizing the Pirie Kieren Theory for the Dynamical Growth of Mathematical Understanding and its model for tracking growth on a small scale, I ask the questions, (i) What is the process of change, the growth of understanding, that each child passes through? and (ii) What are the images and prior knowings that children experiencing difficulties in mathematics have about zero, and how do they thicken? The analysis presented here is mainly of the task-based clinical interviews in which each learner participated. Data from parental surveys, task-based interventions and classroom observations are used to support this analysis. Results of my research indicate how learners may be thickening and revisiting their prior knowings. Thickening occurs either as a foundation to anchor growth, or as a comparative for new growth. Results of my research also indicate that on the small-scale of tracking growth there is a juncture between expectation and result where growth has the potential to occur or not occur. This research provides descriptive evidence of intervention specifically for growth in understanding that takes into account the juncture between expectation and result. Finally, because zero is a paradox, understandings in Primitive Knowing around zero require multiple revisitings

The Feedback on Alignment and Support for Teachers (FAST) Program

Over the last decade, most states have adopted new college- and career-readiness standards in math and English language arts (ELA), standards that call for the mastery of ambitious content and raise expectations for student success and classroom instruction. To support teachers in the implementation of these new, challenging standards, the Center on Standards, Alignment, Instruction, and Learning (C-SAIL) has developed the Feedback on Alignment and Support for Teachers (FAST) program. The FAST program is a virtual coaching program designed to support 4th grade math and 5th grade English language arts (ELA) teachers in fully understanding the college- and career-readiness standards in their states and implementing instruction aligned with these standards to foster learning for all students, including English language learners (ELLs) and students with disabilities (SWDs). The program includes three key program components: personalized instructional coaching, tools to support reflection, and an online library of resources

Toward a Model of Intercultural Warrant: A Case of the Korean Decimal Classification\u27s Cross-cultural Adaptation of the Dewey Decimal Classification

I examined the Korean Decimal Classification (KDC)\u27s adaptation of the Dewey Decimal Classification (DDC) by comparing the two systems. This case manifests the sociocultural influences on KOSs in a cross-cultural context. I focused my analysis on the changes resulting from the meeting of the two cultures, answering the main research question: “How does KDC adapt DDC in terms of underlying sociocultural perspectives in a classificatory form?” I took a comparative approach and address the main research question in two phases. In Phase 1, quantities of class numbers were analyzed by edition and discipline. The main class with the most consistently high number of class numbers in DDC was the social sciences, while the main class with the most consistently high number of class numbers in KDC was technology. The two main classes are expected to differ in semantic contents or specificities. In Phase 2, patterns of adaptations were analyzed by examining the class numbers, captions, and hierarchical relations within the developed adaptation taxonomy. Implementing the taxonomy as a coding scheme brings two comparative features of classifications: 1) semantic contents determined by captions and quantity of subordinate numbers; and 2) structural arrangement determined by ranks, the broader category, presence and the order of subordinate numbers. Surveying proper forms of adaptation resulted in the development of an adaptation taxonomy that will serve as a framework to account for the conflicts between and harmonization of multiple social and cultural influences in knowledge structures. This study has ramifications in theoretical and empirical foundations for the development of “intercultural warrant” in KOSs

An application of brain-based education principles with ICT as a cognitive tool: a case study of grade 6 decimal instruction at Sunlands Primary School

The larger population of South African learners do not learn effectively and struggle with low academic achievements currently. This can be attributed to various factors such as frequent changes in the curriculum, underqualified educators, ineffective teaching methods and barriers to learning existing in classrooms today. Learners need extra support, including cognitive support, but in reality the heavy workload of educators may prevent them from giving learners the needed support. If support is given, it is minimal or not effective enough. Computer technologies may afford both educators and learners such opportunities when used as a cognitive tool in activities that provide the needed support. This research is concerned with the use of computer technology as a cognitive tool to activate learners' cognitive processes, thus enhancing learning, based on Brain Based Education principles. The objective is to lay the foundation in using computer technologies as cognitive tools in educators' teaching practice and instructional design to make teaching and learning more effective, interactive, real world based, giving meaning to what is learnt and to enhance understanding

A primary numeracy : a mapping review and analysis of Australian research in numeracy learning at the primary school level : report

The outcome from this project produces a database of over 185 projects and 726 publications relating to numeracy research to systematically &lsquo;mapped&rsquo; Australian research on primary school numeracy over the last decade. The database incorporates research summaries and findings that are easily accessible to teachers and teacher educators, and act as a valuable tool for determining further research directions. The project report examines the available research and organises the discussion of the research findings under a set of themes and sub-themes. Some summarised examples from the report reveals that: * Effective teachers of numeracy: - have high expectations of their students; - focus on children&rsquo;s mathematical learning, rather than on providing pleasant classroom experiences; - provide a challenging curriculum; - use higher-order questioning; - make connections both within mathematics and between mathematics in different contexts; and - use highly interactive teaching involvement with students in class discussion. * Effective professional development programmes: - provide teachers with the time and appropriate resources to enable them to reflect on their teaching; - provide continuing support and encouragement while teachers explore possibilities and trial new strategies in their classrooms; - involve teachers in school-based and wider networks; - are of sufficient duration to allow significant changes to habitual beliefs and practices; and - create opportunities for the exploration of theory-practice relationships.<br /

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