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.

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.

Visualising mathematical knowledge and understanding

Contemporary mathematics education attempts to instil within learners the conceptualization of mathematics as a highly organized and inter-connected set of ideas. To support this, a means to graphically represent this organization of ideas is presented which reflects the cognitive mechanisms that shape a learner’s understanding. This organisation of information may then be analysed, with the view to informing the design of mathematics instruction in face-to-face and/or computer-mediated learning environments. However, this analysis requires significant work to develop both theory and practice.

Using a visually-based assignment to reinforce and assess design history knowledge and understanding

This paper presents a visual timeline-based assignment used in an undergraduate Industrial Design History, Theory and Critcism unit. The assignment was developed in order to find a better way of supporting design history learning than an exam or essay assessment. It was developed using constructive alignment and it allows design students to use their strong visual thinking skills to understand unfamiliar content, develop their visual literacy of design history, and think deeply about the links between the designs, styles, movements, events and people in their timeline. The task produced a variety of responses, from websites and electronic presentations to large paper timelines, scrolls and 3D models. These have been admired by peers and used for end of year shows and permanent displays. Questionnaires were issued to students to gain feedback about the assessment. Students stated that the visual nature of the assignment helped them to understand how different aspects of design history related to each other, assisted with retaining the information, and that it was more interesting and fun than a report or an exam. This paper explores the theories behind and the benefits of using such methods of assessment for design history courses.

Australian pre-service teachers' Knowledge and understanding of cyberbullying: Implications for school climate

The beliefs, attitudes and understandings of pre-service teachers towards bullying and more recently, cyberbullying remains unclear. Previous studies have found them to be generally lacking confidence to address bullying, which could impact negatively on school climate if, when they enter the profession, these beliefs undermine existing anti-bullying initiatives. This study explores Australian pre-service teachers' (N= 717) understanding and knowledge of traditional bullying and cyberbullying and their confidence and capacity to deal with it. Findings from self-report, anonymous questionnaires from students attending three universities in Australia indicated that two thirds (66%) of current pre-service teachers felt informed to very informed and capable to very capable (62%) of dealing with school bullying and 90% could discern cyber and traditional bullying behaviours from other online and offline aggressive acts. Gender and Year level differences were found. The potential impact of their knowledge and understanding of bullying and cyberbullying on school climate, and sustaining and maintaining anti-bullying interventions as they enter the profession is discussed.

Creating an eLearning Resource to Improve Knowledge and Understanding of Pregnancy in the Context of HIV Infection

Patient narratives have much to teach healthcare professionals about the experience of living with a chronic condition. While the biomedical narrative of HIV treatment is hugely encouraging, the narrative of living with HIV continues to be overshadowed by a persuasive perception of stigma. This paper presents how we sought to translate the evidence from a qualitative study of the perspectives of HIV affected pregnant women and expectant fathers on the care they received, from the pre conception to post natal period, into educational material for maternity care practice. Narrative scripts were written based on the original research interviews, with care taken to reflect the key themes from the research. We explore the way in which the qualitative findings bring to life patient and partner experiences and what it means for nurses, midwives and doctors to be prepared to care for couples affected by HIV. In so doing, we challenge the inequity between the dominance of biomedical knowledge over understanding the patient experience in the preparation of health professionals to care for HIV affected women and men who are having a baby or seeking to have a baby.

Primary English : knowledge and understanding.

Esta guía ayuda a los profesores en período de formación inicial a superar el Qualified Teacher Status (QTS) y a cumplir con la estrategia nacional sobre enseñanza primaria y el nuevo marco de alfabetización. Ofrece información sobre el lenguaje, la alfabetización, las tecnologías de la información y la comunicación (ICT) y los textos electrónicos.

Comprehension : Developing inference skills Applying knowledge and understanding.

Comprensión es un libro para Key Stage 2 y los primeros años de Key Stage 3. La serie anima a los chicos a pensar, requiere no solo que interpreten lo que ellos leen, sino que usen la información que ellos han reunido de una manera constructiva, aplicándolo por ejemplo a gráficos, mapas, diagramas, dibujos y tablas. Alternativamente, muchas de las actividades requieren que los chicos expliquen con palabras información que está contenida en diferentes representaciones visuales tales como gráficos, diagramas e ilustraciones. El libro desarrollará las capacidades evaluativas y de deducción de los chicos. Muchas de las actividades toman aspectos de la ciencia, historia y geografía por ejemplo. Otras actividades están centradas en los intereses de los chicos, y tópicos tales como magia, marcianos y dragones están incluidos.

Knowledge and understanding of choice diversity of residential care staff working with individuals with intellectual disabilities

The present study aimed to trial the effectiveness of 10 newly developed brief vignettes portraying typical interactions between staff and people with intellectual disabilities in residential care settings to assess the knowledge and understanding of staff about choice diversity, pre- and post-attendance at a staff training workshop. A total of 29 residential staff completed the Vignette Rating Scale and a knowledge questionnaire pre- and post-training. A t- test conducted on the vignettes revealed that respondents identified fewer choices in the post-test vignettes compared to the pre-test vignettes. Results showed no significant difference between the pre- and post-test data on the knowledge questionnaire. The questionnaire revealed a high level of knowledge about choice prior to and following training. The vignettes, however, proved effective in measuring changes in awareness of choice diversity among residential staff following participation in a staff training workshop.

A preliminary investigation of 4 to 11-year-old children's knowledge and understanding of stress

Objective. To examine children's knowledge, understanding and experience of stress from 4 to 11 years of age across four age groups (4–5, 6–7, 8–9, and 10–11 years old). Methods. A semi-structured interview format was used to elicit information from 50 children about their understanding and experience of stress. Results. Most children were able to define stress, with older children providing more complex responses. Many children had indirect and/or personal experience of stress. Younger children were more likely than older children to report that there was nothing people could do to stop stress; children reported using both adaptive and maladaptive coping strategies to deal with stress. Conclusion. Some young children have a basic understanding of stress and many have experience of stress; both understanding and experience develop with age. Practice Implications. The research has potential implications for provider-patient communication, particularly within preventative health education and clinically within the field of childhood post-traumatic stress disorder (PTSD).