Theories of learning mathematics

According to Karl Popper, widely regarded as one of the greatest philosophers of science in the 20th century, falsifiability is the primary characteristic that distinguishes scientific theories from ideologies – or dogma. For example, for people who argue that schools should treat creationism as a scientific theory, comparable to modern theories of evolution, advocates of creationism would need to become engaged in the generation of falsifiable hypothesis, and would need to abandon the practice of discouraging questioning and inquiry. Ironically, scientific theories themselves are accepted or rejected based on a principle that might be called survival of the fittest. So, for healthy theories on development to occur, four Darwinian functions should function: (a) variation – avoid orthodoxy and encourage divergent thinking, (b) selection – submit all assumptions and innovations to rigorous testing, (c) diffusion – encourage the shareability of new and/or viable ways of thinking, and (d) accumulation – encourage the reuseability of viable aspects of productive innovations.

Discovering, applying and integrating: The process of learning in coaching

Professional coaching is a rapidly expanding field with interdisciplinary roots and broad application. However, despite abundant prescriptive literature, research into the process of coaching is minimal. Similarly, although learning is inherently recognised in the process of coaching, the process of learning in coaching is little understood and learning theory makes up only a small part of the evidence-based coaching literature. In this grounded theory study of coaches and their clients, the process of learning in coaching across a range of coaching models is examined and discussed. The findings demonstrate how learning in coaching emerged as a process of discovering, applying and integrating new knowledge, which culminated in a process of developing. This process occurred through eight key coaching processes shared between coaches and clients and combined a multitude of learning theories.

A constraints-led perspective to understanding skill acquisition and game play: A basis for integration of motor learning theory and physical education praxis?

Background: In order to design appropriate environments for performance and learning of movement skills, physical educators need a sound theoretical model of the learner and of processes of learning. In physical education, this type of modelling informs the organization of learning environments and effective and efficient use of practice time. An emerging theoretical framework in motor learning, relevant to physical education, advocates a constraints-led perspective for acquisition of movement skills and game play knowledge. This framework shows how physical educators could use task, performer and environmental constraints to channel acquisition of movement skills and decision making behaviours in learners. From this viewpoint, learners generate specific movement solutions to satisfy the unique combination of constraints imposed on them, a process which can be harnessed during physical education lessons. Purpose: In this paper the aim is to provide an overview of the motor learning approach emanating from the constraints-led perspective, and examine how it can substantiate a platform for a new pedagogical framework in physical education: nonlinear pedagogy. We aim to demonstrate that it is only through theoretically valid and objective empirical work of an applied nature that a conceptually sound nonlinear pedagogy model can continue to evolve and support research in physical education. We present some important implications for designing practices in games lessons, showing how a constraints-led perspective on motor learning could assist physical educators in understanding how to structure learning experiences for learners at different stages, with specific focus on understanding the design of games teaching programmes in physical education, using exemplars from Rugby Union and Cricket. Findings: Research evidence from recent studies examining movement models demonstrates that physical education teachers need a strong understanding of sport performance so that task constraints can be manipulated so that information-movement couplings are maintained in a learning environment that is representative of real performance situations. Physical educators should also understand that movement variability may not necessarily be detrimental to learning and could be an important phenomenon prior to the acquisition of a stable and functional movement pattern. We highlight how the nonlinear pedagogical approach is student-centred and empowers individuals to become active learners via a more hands-off approach to learning. Summary: A constraints-based perspective has the potential to provide physical educators with a framework for understanding how performer, task and environmental constraints shape each individual‟s physical education. Understanding the underlying neurobiological processes present in a constraints-led perspective to skill acquisition and game play can raise awareness of physical educators that teaching is a dynamic 'art' interwoven with the 'science' of motor learning theories.

Revisiting 'how we learn' in academia : practice-based learning exchanges in three Australian universities

Ideas of 'how we learn' in formal academic settings have changed markedly in recent decades. The primary position that universities once held on shaping what constitutes learning has come into question from a range of experience-led and situated learning models. Drawing on findings from a study conducted across three Australian universities, the article focuses on the multifarious learning experiences indicative of practice-based learning exchanges such as student placements. Building on both experiential and situated learning theories, the authors found that students can experience transformative and emotional elucidations of learning, that can challenge tacit assumptions and transform the ways they understand the world. It was found that all participants (hosts, students, academics) both teach and learn in these educative scenarios and that, contrary to common (mis)perceptions that academics live in 'ivory towers', they play a crucial role in contributing to learning that takes place in the so-called 'real world'.

Understanding the optimal learning environment in palliative care

The learning experiences of student nurses undertaking clinical placement are reported widely, however little is known about the learning experiences of health professionals undertaking continuing professional development (CPD) in a clinical setting, especially in palliative care. The aim of this study, which was conducted as part of the national evaluation of a professional development program involving clinical attachments with palliative care services (The Program of Experience in the Palliative Approach [PEPA]), was to explore factors influencing the learning experiences of participants over time. Thirteen semi-structured, one-to-one telephone interviews were conducted with five participants throughout their PEPA experience. The analysis was informed by the traditions of adult, social and psychological learning theories and relevant literature. The participants' learning was enhanced by engaging interactively with host site staff and patients, and by the validation of their personal and professional life experiences together with the reciprocation of their knowledge with host site staff. Self-directed learning strategies maximised the participants' learning outcomes. Inclusion in team activities aided the participants to feel accepted within the host site. Personal interactions with host site staff and patients shaped this social/cultural environment of the host site. Optimal learning was promoted when participants were actively engaged, felt accepted and supported by, and experienced positive interpersonal interactions with, the host site staff.

Designing Assessment for Quality Learning

- Considers broad-scale assessment approaches and how they impact on educational opportunity and outcomes. - Brings together internationally recognised scholars providing new insights into assessment for learning improvement and accountability. - Presents different theoretical and methodological perspectives influential in the field of assessment, learning and social change. - Contributes to the theorising of assessment in contexts characterised by heightened accountability requirements and constant change. This book brings together internationally recognised scholars with an interest in how to use the power of assessment to improve student learning and to engage with accountability priorities at both national and global levels. It includes distinguished writers who have worked together for some two decades to shift the assessment paradigm from a dominant focus on assessment as measurement towards assessment as central to efforts to improve learning. These writers have worked with the teaching profession and, in so doing, have researched and generated key insights into different ways of understanding assessment and its relationship to learning. The volume contributes to the theorising of assessment in contexts characterised by heightened accountability requirements and constant change. The book’s structure and content reflect already significant and growing international interest in assessment as contextualised practice, as well as theories of learning and teaching that underpin and drive particular assessment approaches. Learning theories and practices, assessment literacies, teachers’ responsibilities in assessment, the role of leadership, and assessment futures are the organisers within the book’s structure and content. The contributors to this book have in common the view that quality assessment, and quality learning and teaching are integrally related. Another shared view is that the alignment of assessment with curriculum, teaching and learning is linchpin to efforts to improve both learning opportunities and outcomes for all. Essentially, the book presents new perspectives on the enabling power of assessment. In so doing, the writers recognise that validity and reliability - the traditional canons of assessment – remain foundational and therefore necessary. However, they are not of themselves sufficient for quality education. The book argues that assessment needs to be radically reconsidered in the context of unprecedented societal change. Increasingly, communities are segregating more by wealth, with clear signs of social, political, economic and environmental instability. These changes raise important issues relating to ethics and equity, taken to be core dimensions in enabling the power of assessment to contribute to quality learning for all. This book offers readers new knowledge about how assessment can be used to re/engage learners across all phases of education.

Informed systems approach : new directions for organizational learning

The Informed Systems Approach offers models for advancing workplace learning within collaboratively designed systems that promote using information to learn through collegial exchange and reflective dialogue. This systemic approach integrates theoretical antecedents and process models, including the learning theories of Peter Checkland (Soft Systems Methodology), which advance systems design and informed action, and Christine Bruce (informed learning), which generate information experiences and professional practices. Ikujiro Nonaka’s systems ideas (SECI model) and Mary Crossan’s learning framework (4i framework) further animate workplace knowledge creation through learning relationships engaging individuals with ideas.

Implementation of e-learning: A case study of three schools

As Western Australian schools move to implement technology into the classroom, there appears to be prevalence in combining e-learning with face to face traditional classroom practice. This has been accompanied by a shift toward a digital curriculum that incorporates re-usable learning objects. Essential to any teacher contemplating the use of a digital curriculum resource is not only the knowledge of learning theories but models of best practice to create online curriculum for students to use in every day classrooms. This paper explores the e-learning practices in three case study schools (n=3) in Western Australia. Data were collected by observation and interviews (n=11) conducted with the teachers and the ICT co-ordinators, to ascertain their perceptions and experiences with regard to the e-learning environment. There were challenges associated with the implementation of e-learning by teachers into their classroom such as skill development, changes in their role and the pedagogies they employ. The case study schools were pilot schools breaking new ground in order to test a new portal technology. Findings indicated that successful implementation of the e-learning environment was dependent on the four key factors of ICT infrastructure, ICT leadership, support and training initiatives and the teachers’ ICT capacity.

How current pedagogy methods in games teaching in the UK, Australia and the US have been shaped by historical, socio cultural, environmental and political constraints

Crucial to enhancing the status and quality of games teaching in schools is a developed understanding of the teaching strategies adopted by practitioners. In this paper, we will demonstrate that contemporary games‟ teaching is a product of individual, task and environmental constraints (Newell, 1986). More specifically, we will show that current pedagogy in the U.K., Australia and the United States is strongly influenced by historical, socio-cultural environmental and political constraints. In summary, we will aim to answer the question „why do teachers teach games the way they do.‟ In answering this question, we conclude that teacher educators, who are trying to influence pedagogical practice, must understand these potential constraints and provide appropriate pre-service experiences to give future physical education teachers the knowledge, confidence and ability to adopt a range of teaching styles when they become fully fledged teachers. Essential to this process is the need to enable future practitioners to base their pedagogical practice on a sound understanding of contemporary learning theories of skill acquisition.

Integrating information literacy into higher education curricula : an IL curricular integration model

This study investigates a way to systematically integrate information literacy (IL) into an undergraduate academic programme and develops a model for integrating information literacy across higher education curricula. Curricular integration of information literacy in this study means weaving information literacy into an academic curriculum. In the associated literature, it is also referred to as the information literacy embedding approach or the intra-curricular approach. The key findings identified from this study are presented in 4 categories: the characteristics of IL integration; the key stakeholders in IL integration; IL curricular design strategies; and the process of IL curricular integration. Three key characteristics of the curricular integration of IL are identified: collaboration and negotiation, contextualisation and ongoing interaction with information. The key stakeholders in the curricular integration of IL are recognised as the librarians, the course coordinators and lecturers, the heads of faculties or departments, and the students. Some strategies for IL curricular design include: the use of IL policies and standards in IL curricular design; the combination of face to face and online teaching as an emerging trend; the use of IL assessment tools which play an important role in IL integration. IL can be integrated into the intended curriculum (what an institution expects its students to learn), the offered curriculum (what the teachers teach) and the received curriculum (what students actually learn). IL integration is a process of negotiation, collaboration and the implementation of the intended curriculum. IL can be integrated at different levels of curricula such as: institutional, faculty, departmental, course and class curriculum levels. Based on these key findings, an IL curricular integration model is developed. The model integrates curriculum, pedagogy and learning theories, IL theories, IL guidelines and the collaboration of multiple partners. The model provides a practical approach to integrating IL into multiple courses across an academic degree. The development of the model was based on the IL integration experiences of various disciplines in three universities and the implementation experience of an engineering programme at another university; thus it may be of interest to other disciplines. The model has the potential to enhance IL teaching and learning, curricular development and to implement graduate attributes in higher education. Sociocultural theories are applied to the research process and IL curricular design of this study. Sociocultural theories describe learning as being embedded within social events and occurring as learners interact with other people, objects, and events in a collaborative environment. Sociocultural theories are applied to explore how academic staff and librarians experience the curricular integration of IL; they also support collaboration in the curricular integration of IL and the development of an IL integration model. This study consists of two phases. Phase I (2007) was the interview phase where both academic staff and librarians at three IL active universities were interviewed. During this phase, attention was paid specifically to the practical process of curricular integration of IL and IL activity design. Phase II, the development phase (2007-2008), was conducted at a fourth university. This phase explores the systematic integration of IL into an engineering degree from Year 1 to Year 4. Learning theories such as sociocultural theories, Bloom’s Taxonomy and IL theories are used in IL curricular development. Based on the findings from both phases, an IL integration model was developed. The findings and the model contribute to IL education, research and curricular development in higher education. The sociocultural approach adopted in this study also extends the application of sociocultural theories to the IL integration process and curricular design in higher education.

Towards a model for the analysis and description of mathematical knowledge and understanding

Mathematics education literature has called for an abandonment of ontological and epistemological ideologies that have often divided theory-based practice. Instead, a consilience of theories has been sought which would leverage the strengths of each learning theory and so positively impact upon contemporary educational practice. This research activity is based upon Popper’s notion of three knowledge worlds which differentiates the knowledge shared in a community from the personal knowledge of the individual, and Bereiter’s characterisation of understanding as the individual’s relationship to tool-like knowledge. Using these notions, a re-conceptualisation of knowledge and understanding and a subsequent re-consideration of learning theories are proposed as a way to address the challenge set by literature. Referred to as the alternative theoretical framework, the proposed theory accounts for the scaffolded transformation of each individual’s unique understanding, whilst acknowledging the existence of a body of domain knowledge shared amongst participants in a scientific community of practice. The alternative theoretical framework is embodied within an operational model that is accompanied by a visual nomenclature with which to describe consensually developed shared knowledge and personal understanding. This research activity has sought to iteratively evaluate this proposed theory through the practical application of the operational model and visual nomenclature to the domain of early-number counting, addition and subtraction. This domain of mathematical knowledge has been comprehensively analysed and described. Through this process, the viability of the proposed theory as a tool with which to discuss and thus improve the knowledge and understanding with the domain of mathematics has been validated. Putting of the proposed theory into practice has lead to the theory’s refinement and the subsequent achievement of a solid theoretical base for the future development of educational tools to support teaching and learning practice, including computer-mediated learning environments. Such future activity, using the proposed theory, will advance contemporary mathematics educational practice by bringing together the strengths of cognitivist, constructivist and post-constructivist learning theories.

Objectifying early-number : a visual nomenclature to express mathematical domain knowledge

To address issues of divisive ideologies in the Mathematics Education community and to subsequently advance educational practice, an alternative theoretical framework and operational model is proposed which represents a consilience of constructivist learning theories whilst acknowledging the objective but improvable nature of domain knowledge. Based upon Popper’s three-world model of knowledge, the proposed theory supports the differentiation and explicit modelling of both shared domain knowledge and idiosyncratic personal understanding using a visual nomenclature. The visual nomenclature embodies Piaget’s notion of reflective abstraction and so may support an individual’s experience-based transformation of personal understanding with regards to shared domain knowledge. Using the operational model and visual nomenclature, seminal literature regarding early-number counting and addition was analysed and described. Exemplars of the resultant visual artefacts demonstrate the proposed theory’s viability as a tool with which to characterise the reflective abstraction-based organisation of a domain’s shared knowledge. Utilising such a description of knowledge, future research needs to consider the refinement of the operational model and visual nomenclature to include the analysis, description and scaffolded transformation of personal understanding. A detailed model of knowledge and understanding may then underpin the future development of educational software tools such as computer-mediated teaching and learning environments.

A Popperian consilience : modelling mathematical knowledge and understanding

Goldin (2003) and McDonald, Yanchar, and Osguthorpe (2005) have called for mathematics learning theory that reconciles the chasm between ideologies, and which may advance mathematics teaching and learning practice. This paper discusses the theoretical underpinnings of a recently completed PhD study that draws upon Popper’s (1978) three-world model of knowledge as a lens through which to reconsider a variety of learning theories, including Piaget’s reflective abstraction. Based upon this consideration of theories, an alternative theoretical framework and complementary operational model was synthesised, the viability of which was demonstrated by its use to analyse the domain of early-number counting, addition and subtraction.

An information literacy integration model and its application in higher education

Purpose - The purpose of this paper is to present a model for curricular integration of information literacy for undergraduate programs in higher education. Design/methodology/approach - Data are drawn from individual interviews at three universities in Australia and curricular integration working experience at a New Zealand university. Sociocultural theories are adopted in the research process and in the development of the model, Findings - Key characteristics of the curriculum integration of information literacy were identified and an information literacy integration model was developed. The S2J2 key behaviours for campus-wide multi-partner collaboration in information literacy integration were also identified. Research limitations/implications - The model was developed without including the employer needs. Through the process of further research, the point of view of the employer on how to provide information literacy education needs to be explored in order to strengthen the model in curricular design. Practical implications - The information literacy integration model was developed based on practical experience in higher education and has been applied in different undergraduate curricular programs. The model could be used or adapted by both librarians and academics when they integrate information literacy into an undergraduate curriculum from a lower level to a higher level. Originality/value - The information literacy integration model was developed based on recent PhD research. The model integrates curriculum, pedagogy and learning theories, information literacy theories, information literacy guidelines, people and collaborative together. The model provides a framework of how information literacy can be integrated into multiple courses across an undergraduate academic degree in higher education.

The effect of consumer education on unit price usage : an early exploratory study

Consumer awareness and usage of Unit Price (UP) information continues to hold academic interest. Originally designed as a device to enable shoppers to make comparisons between grocery products, it is argued consumers still lack a sufficient understanding of the device. Previous research has tended to focus on product choice, effect of time, and structural changes to price presentation. No studies have tested the effect of UP consumer education on grocery shopping expenditure. Supported by distributed learning theories, this is the first study to condition participants over a twenty week period, to comprehend and employ UP information while shopping. A 3x5 mixed factorial design was employed to collect data from 357 shoppers. A 3 (Control, Massed, Spaced) x 5 (Time Point: Week 0, 5, 10, 15 and 20) mixed factorial analysis of variance (ANOVA) was performed to analyse the data. Preliminary results revealed that the three groups differed in their average expenditure over the twenty weeks. The Control group remained stable across the five time points. Results indicated that both intensive (Massed) and less intensive (Spaced) exposure to UP information achieved similar results, with both group reducing average expenditure similarly by Week 5. These patterns held for twenty weeks, with conditioned groups reducing their grocery expenditure by over 10%. This research has academic value as a test of applied learning theories. We argue, retailers can attain considerable market advantages as efforts to enhance customers’ knowledge, through consumer education campaigns, can have a positive and strong impact on customer trust and goodwill toward the organisation. Hence, major practical implications for both regulators and retailers exist.