An investigation of post-primary students' images of mathematics

This research study investigates the image of mathematics held by 5th-year post-primary students in Ireland. For this study, “image of mathematics” is conceptualized as a mental representation or view of mathematics, presumably constructed as a result of past experiences, mediated through school, parents, peers or society. It is also understood to include attitudes, beliefs, emotions, self-concept and motivation in relation to mathematics. This study explores the image of mathematics held by a sample of 356 5th-year students studying ordinary level mathematics. Students were aged between 15 and 18 years. In addition, this study examines the factors influencing students‟ images of mathematics and the possible reasons for students choosing not to study higher level mathematics for the Leaving Certificate. The design for this study is chiefly explorative. A questionnaire survey was created containing both quantitative and qualitative methods to investigate the research interest. The quantitative aspect incorporated eight pre-established scales to examine students‟ attitudes, beliefs, emotions, self-concept and motivation regarding mathematics. The qualitative element explored students‟ past experiences of mathematics, their causal attributions for success or failure in mathematics and their influences in mathematics. The quantitative and qualitative data was analysed for all students and also for students grouped by gender, prior achievement, type of post-primary school attending, co-educational status of the post-primary school and the attendance of a Project Maths pilot school. Students‟ images of mathematics were seen to be strongly indicated by their attitudes (enjoyment and value), beliefs, motivation, self-concept and anxiety, with each of these elements strongly correlated with each other, particularly self-concept and anxiety. Students‟ current images of mathematics were found to be influenced by their past experiences of mathematics, by their mathematics teachers, parents and peers, and by their prior mathematical achievement. Gender differences occur for students in their images of mathematics, with males having more positive images of mathematics than females and this is most noticeable with regards to anxiety about mathematics. Mathematics anxiety was identified as a possible reason for the low number of students continuing with higher level mathematics for the Leaving Certificate. Some students also expressed low mathematical self-concept with regards to higher level mathematics specifically. Students with low prior achievement in mathematics tended to believe that mathematics requires a natural ability which they do not possess. Rote-learning was found to be common among many students in the sample. The most positive image of mathematics held by students was the “problem-solving image”, with resulting implications for the new Project Maths syllabus in post-primary education. Findings from this research study provide important insights into the image of mathematics held by the sample of Irish post-primary students and make an innovative contribution to mathematics education research. In particular, findings contribute to the current national interest in Ireland in post-primary mathematics education, highlighting issues regarding the low uptake of higher level mathematics for the Leaving Certificate and also making a preliminary comparison between students who took part in the piloting of Project Maths and students who were more recently introduced to the new syllabus. This research study also holds implications for mathematics teachers, parents and the mathematics education community in Ireland, with some suggestions made on improving students‟ images of mathematics.

Iterative reconstruction as a novel method of radiation dose reduction at computed tomography in patients with Crohn's disease

Prior work of our research group, that quantified the alarming levels of radiation dose to patients with Crohn’s disease from medical imaging and the notable shift towards CT imaging making these patients an at risk group, provided context for this work. CT delivers some of the highest doses of ionising radiation in diagnostic radiology. Once a medical imaging examination is deemed justified, there is an onus on the imaging team to endeavour to produce diagnostic quality CT images at the lowest possible radiation dose to that patient. The fundamental limitation with conventional CT raw data reconstruction was the inherent coupling of administered radiation dose with observed image noise – the lower the radiation dose, the noisier the image. The renaissance, rediscovery and refinement of iterative reconstruction removes this limitation allowing either an improvement in image quality without increasing radiation dose or maintenance of image quality at a lower radiation dose compared with traditional image reconstruction. This thesis is fundamentally an exercise in optimisation in clinical CT practice with the objectives of assessment of iterative reconstruction as a method for improvement of image quality in CT, exploration of the associated potential for radiation dose reduction, and development of a new split dose CT protocol with the aim of achieving and validating diagnostic quality submillisiever t CT imaging in patients with Crohn’s disease. In this study, we investigated the interplay of user-selected parameters on radiation dose and image quality in phantoms and cadavers, comparing traditional filtered back projection (FBP) with iterative reconstruction algorithms. This resulted in the development of an optimised, refined and appropriate split dose protocol for CT of the abdomen and pelvis in clinical patients with Crohn’s disease allowing contemporaneous acquisition of both modified and conventional dose CT studies. This novel algorithm was then applied to 50 patients with a suspected acute complication of known Crohn’s disease and the raw data reconstructed with FBP, adaptive statistical iterative reconstruction (ASiR) and model based iterative reconstruction (MBIR). Conventional dose CT images with FBP reconstruction were used as the reference standard with which the modified dose CT images were compared in terms of radiation dose, diagnostic findings and image quality indices. As there are multiple possible user-selected strengths of ASiR available, these were compared in terms of image quality to determine the optimal strength for this modified dose CT protocol. Modified dose CT images with MBIR were also compared with contemporaneous abdominal radiograph, where performed, in terms of diagnostic yield and radiation dose. Finally, attenuation measurements in organs, tissues, etc. with each reconstruction algorithm were compared to assess for preservation of tissue characterisation capabilities. In the phantom and cadaveric models, both forms of iterative reconstruction examined (ASiR and MBIR) were superior to FBP across a wide variety of imaging protocols, with MBIR superior to ASiR in all areas other than reconstruction speed. We established that ASiR appears to work to a target percentage noise reduction whilst MBIR works to a target residual level of absolute noise in the image. Modified dose CT images reconstructed with both ASiR and MBIR were non-inferior to conventional dose CT with FBP in terms of diagnostic findings, despite reduced subjective and objective indices of image quality. Mean dose reductions of 72.9-73.5% were achieved with the modified dose protocol with a mean effective dose of 1.26mSv. MBIR was again demonstrated superior to ASiR in terms of image quality. The overall optimal ASiR strength for the modified dose protocol used in this work is ASiR 80%, as this provides the most favourable balance of peak subjective image quality indices with less objective image noise than the corresponding conventional dose CT images reconstructed with FBP. Despite guidelines to the contrary, abdominal radiographs are still often used in the initial imaging of patients with a suspected complication of Crohn’s disease. We confirmed the superiority of modified dose CT with MBIR over abdominal radiographs at comparable doses in detection of Crohn’s disease and non-Crohn’s disease related findings. Finally, we demonstrated (in phantoms, cadavers and in vivo) that attenuation values do not change significantly across reconstruction algorithms meaning preserved tissue characterisation capabilities with iterative reconstruction. Both adaptive statistical and model based iterative reconstruction algorithms represent feasible methods of facilitating acquisition diagnostic quality CT images of the abdomen and pelvis in patients with Crohn’s disease at markedly reduced radiation doses. Our modified dose CT protocol allows dose savings of up to 73.5% compared with conventional dose CT, meaning submillisievert imaging is possible in many of these patients.

A Jaworskian analysis of four senior class primary teachers endeavouring to teach mathematics from a constructivist-compatible perspective

A constructivist philosophy underlies the Irish primary mathematics curriculum. As constructivism is a theory of learning its implications for teaching need to be addressed. This study explores the experiences of four senior class primary teachers as they endeavour to teach mathematics from a constructivist-compatible perspective with primary school children in Ireland over a school-year period. Such a perspective implies that children should take ownership of their learning while working in groups on tasks which challenge them at their zone of proximal development. The key question on which the research is based is: to what extent will an exposure to constructivism and its implications for the classroom impact on teaching practices within the senior primary mathematics classroom in both the short and longer term? Although several perspectives on constructivism have evolved (von Glaserfeld (1995), Cobb and Yackel (1996), Ernest (1991,1998)), it is the synthesis of the emergent perspective which becomes pivotal to the Irish primary mathematics curriculum. Tracking the development of four primary teachers in a professional learning initiative involving constructivist-compatible approaches necessitated the use of Borko’s (2004) Phase 1 research methodology to account for the evolution in teachers’ understanding of constructivism. Teachers’ and pupils’ viewpoints were recorded using both audio and video technology. Teachers were interviewed at the beginning and end of the project and also one year on to ascertain how their views had evolved. Pupils were interviewed at the end of the project only. The data were analysed from a Jaworskian perspective i.e. using the categories of her Teaching Triad of management of learning, mathematical challenge and sensitivity to students. Management of learning concerns how the teacher organises her classroom to maximise learning opportunities for pupils. Mathematical challenge is reminiscent of the Vygotskian (1978) construct of the zone of proximal development. Sensitivity to students involves a consciousness on the part of the teacher as to how pupils are progressing with a mathematical task and whether or not to intervene to scaffold their learning. Through this analysis a synthesis of the teachers’ interpretations of constructivist philosophy with concomitant implications for theory, policy and practice emerges. The study identifies strategies for teachers wishing to adopt a constructivist-compatible approach to their work. Like O’Shea (2009) it also highlights the likely difficulties to be experienced by such teachers as they move from utilising teacher-dominated methods of teaching mathematics to ones in which pupils have more ownership over their learning.