The Peano Axioms: An IBL Unit Constructing the Natural Numbers

Abstract

A problem sequence is presented developing the basic properties of the set of natural numbers (including associativity and commutativity of addition and multiplication, among others) from the Peano axioms, with the last portion using von Neumann’s construction to provide a model satisfying these axioms. This sequence is appropriate for mathematically inclined students in their first or second years, and has regularly required about a week of class time. The sequence could be included as one unit within any of several appropriate courses, and could be an ideal “trial size” first attempt at guiding inquiry-based learning in the tradition of R.L. Moore.     

Accessing participatory research impact and legacy: developing the evidence base for participatory approaches in health research

Action research has been characterised as systematic enquiry into practice, undertaken by those involved, with the aim changing and improving that practice: an approach designed to have impact. Whilst much has been written about the process and practice of ‘researching’, historically ‘impact’ has been somewhat taken for granted. In recent years, however, the impact of all forms of research has become the focus of interest with many funding bodies now demanding that researchers not only articulate the prospective impact of their work, but what kinds of evidence will be proffered to demonstrate that impact. This has raised questions for action researchers, not about whether their work has an impact, but what form that impact takes, how it is recognised and by whom. This paper focuses on difficulties researchers find in both articulating the impact of participatory research and demonstrating links between such forms of research and impact. We draw on discussions about the notion of impact with authors that have self-reported and published their work as participatory. These discussions revealed that not only were there difficulties in clarifying the participatory dimension of their research but that whilst authors were able to discuss particular impacts of their work, articulating and evidencing that impact was often absent from their published papers. This paper offers insights into some of issues and barriers those who undertake participatory research face in explicating, for the external audience (and indeed sometimes for ourselves), the impact of this action-based form of enquiry.     

Development of reasoning associated with pictographs: Representing, interpreting, and predicting

A developmental model involving four response levels is proposed concerning how students arrange pictures to represent data in a pictograph, how they interpret these pictographs, and how they make predictions based on these pictographs. The model is exemplified by responses from three related interview-based studies. In Study 1, examples of each response level are provided from 48 preparatory- to tenth-grade students. Students from higher grades were more likely to respond at higher levels. In Study 2, 22 students were interviewed longitudinally after a three-year interval; many improved in response level over time, although a few responded at lower levels. In Study 3, 20 third-grade students were interviewed and then prompted with conflicting responses of other students on video; many improved their initial responses to higher levels after exposure to the conflicting prompts. Associations among levels of representing, interpreting, and predicting were explored. Educational implications are discussed concerning reasonable expectations of students and suggestions to develop these skills in students at different grades. This revised version was published online in July 2006 with corrections to the Cover Date.     

‘Educating rita’: Is an academic science course for prospective infant and primary teachers appropriate

Conclusion Our decision to choose the Open University S101 Science: A Foundation Course represents a direct attempt on our part to provide a background in science for our prospective infant and primary teachers. Our objective that this course achieve parity with first year Faculty of Science courses may appear to be an unattainable goal, given the methodological problems that have to be faced for such a general purpose course. It would certainly not be feasible for a lecture staff of four persons were it not for the printed readers, texts, assignments, examinations, and audio-visual software purchased, with approval, from the Open University in the United Kingdom. Our pilot study in 1983 gave us some encouragement and, most interestingly, 10 out of the 17 students continued with Science study into Year 2 in 1984, thus committing them to the major study in science in Years 3 and 4. Given that three students failed, 10 out of 14 is a far higher retention rate than anything we have experienced previously. Clearly, the evaluation of our efforts to influence the teaching of science in Tasmanian primary schools must wait, but these evaluative studies should make a contribution to the research data on curriculum issues in Australia. This course could play an important in-service function for present primary teachers. We are hopeful that such a content-based approach to science teacher education may help to convince teachers to take the extensive range of available curriculum materials seriously.     

Trends in Performance and Growth by Students With and Without Disabilities on Five State Summative Assessments

Indicators of student academic growth are desired in state accountability systems in order to approximate student learning over time and attribute observed growth to schooling inputs. Through an extant analysis of five states’ assessment data, this study offers evidence about whether longitudinal match rates and measures of growth differ at the state level for students with disabilities, relative to students without disabilities. There were three main findings: 1) In states in which a modified assessment was offered, students with disabilities were more likely to have missing prior year scores, and consequently missing growth scores; 2) Low scoring students, many of whom had a disability, were more likely to have missing prior scores on the state general assessment, and consequently missing growth scores; 3) Students with and without disabilities showed similar growth using transition and gain score definitions of growth, but students with disabilities had lower growth when estimated via a regression-based model. Measurement and policy considerations are discussed.     

Reassessing RCTs as the ‘gold standard’: synergy not separatism in evaluation designs

ABSTRACT

Randomised controlled trials (RCTs) are increasingly used to evaluate educational interventions in the UK. However, RCTs remain controversial for some elements of the research community. This paper argues that the widespread use of the term ‘gold standard’ to describe RCTs is problematic, as it implies that other research methods are inferior. The usefulness of RCTs can be greatly enhanced when used in conjunction with implementation-specific measures (e.g. observation tools, attitude/engagement surveys and interviews). The proposal is advanced through case studies of two evaluations. One relates to the development of science subject leader skills and expertise at primary school level and the other to co-operative learning of primary maths. Both evaluations randomised schools to the intervention or the business-as-usual control, and compared impact using subject knowledge tests. Integral to each study was a process evaluation which looked at evidence from classroom practice along with feedback from the teachers and pupils themselves. We contend that this enabled much more holistic and richly interpretative pieces of research. The paper concludes that privilege for particular paradigms should be set aside when designing effective evaluations of educational interventions, and that it is insufficient to ask ‘what works?’ without also asking ‘why?’, ‘where?’ and ‘how?’.     

Why do students drop out? Turning points and long-term experiences

Understanding the reasons that individuals drop out of high school is fundamental to improving intervention efforts to promote graduation. The authors present a mixed methods analysis of the reasons individuals drop out of high school before graduating. Using a survey methodology with young adults who left school before graduating (N?=?1,047, M age?=?21.5?years, SD?=?2.2 years, age range =18–25 years), the authors assessed ongoing life stressors, as well as the turning points individuals identified as the top reason for why they left school. Qualitative analysis provided seven categories of turning points: mobility, family, peers, school engagement and environment, health, crime, and multiple categories. More than half of respondents indicated that the reason they dropped out was related to school engagement and environment. Results showed differences across profiles of lived experiences, reported turning points, and sex.     

Fairness,trust, and school climate as foundational to growth mindset: A study among Brazilian children and adolescents

Recent research has established the importance of children and adolescents developing a growth mindset for future success and motivation. This research tests believes about fairness, adult trust, and school climate that are theoretically foundational for establishing a cognitive connection between effort and outcome. Regressions and MANOVAS were conducted to understand the direct and indirect relationships between perceptions of justice, adult trust, school climate and growth mindsets.

The first study included 363 children from Brazilian public schools and the findings supported our hypothesis that adult trust partially mediates the relationship between justice perceptions and growth mindset. The second study included an adolescent sample (n?=?497) from more diverse backgrounds and included a measure of institutional mindset. In this study, school fairness and solidarity mediated the relationship between perceptions of justice on institutional growth mindset. This research can help inform educators and researchers of other constructs necessary to foster a growth mindset.     

Computer-assisted reading intervention in a secondary school: an evaluation study

There is widespread concern over literacy standards in UK schools and growing evidence that problems become increasingly intractable as children grow older. Computer-based reading instruction may provide a 'fresh start' that helps a child to rediscover the path to literacy. Recent controlled studies demonstrated that infant and junior school children could be given very cost-effective support using RITA, a computer-based literacy support system that assists, rather than replaces, the teacher in providing support tailored to each child's profile of reading attainments. The present study evaluated the effectiveness of RITA in secondary school with 8 children (mean age 11.7 years) who were initially very seriously disadvantaged in terms of literacy skills (mean reading age 7.3 years). The RITA intervention over a 10 week period led to effective and cost-effective literacy gains. Significant overall improvements were made in the skills targeted, including reading standard scores, and reading speed, accuracy and comprehension. All children reacted positively to the RITA lessons, and most made good progress towards their Individual Education Plans. It is suggested that computer-assisted reading support can be effective in supporting the majority of children with reading failure, even in secondary school.