Mathematics and complex learning

There is unprecedented worldwide demand for mathematical solutions to complex problems. That demand has generated a further call to update mathematics education in a way that develops students’ abilities to deal with complex systems.

Professional growth through working together : a study of reciprocal benefits for teacher and education advisor through classroom based professional development

Teacher professional development provided by education advisors as one-off, centrally offered sessions does not always result in change in teacher knowledge, beliefs, attitudes or practice in the classroom. As the mathematics education advisor in this study, I set out to investigate a particular method of professional development so as to influence change in a practising classroom teacher’s knowledge and practices. The particular method of professional development utilised in this study was based on several principles of effective teacher professional development and saw me working regularly in a classroom with the classroom teacher as well as providing ongoing support for her for a full school year. The intention was to document the effects of this particular method of professional development in terms of the classroom teacher’s and my professional growth to provide insights for others working as education advisors. The professional development for the classroom teacher consisted of two components. The first was the co-operative development and implementation of a mental computation instructional program for the Year 3 class. The second component was the provision of ongoing support for the classroom teacher by the education advisor. The design of the professional development and the mental computation instructional program were progressively refined throughout the year. The education advisor fulfilled multiple roles in the study as teacher in the classroom, teacher educator working with the classroom teacher and researcher. Examples of the professional growth of the classroom teacher and the education advisor which occurred as sequences of changes (growth networks, Hollingsworth, 1999) in the domains of the professional world of the classroom teacher and education advisor were drawn from the large body of data collected through regular face-to-face and email communications between the classroom teacher and the education advisor as well as from transcripts of a structured interview. The Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999) was used to summarise and represent each example of the classroom teacher’s professional growth. A modified version of this model was used to summarise and represent the professional growth of the education advisor. This study confirmed that the method of professional development utilised could lead to significant teacher professional growth related directly to her work in the classroom. Using the Interconnected Model of Professional Growth to summarise and represent the classroom teacher’s professional growth and the modified version for my professional growth assisted with the recognition of examples of how we both changed. This model has potential to be used more widely by education advisors when preparing, implementing, evaluating and following-up on planned teacher professional development activities. The mental computation instructional program developed and trialled in the study was shown to be a successful way of sequencing and managing the teaching of mental computation strategies and related number sense understandings to Year 3 students. This study was conducted in one classroom, with one teacher in one school. The strength of this study was the depth of teacher support provided made possible by the particular method of the professional development, and the depth of analysis of the process. In another school, or with another teacher, this might not have been as successful. While I set out to change my practice as an education advisor I did not expect the depth of learning I experienced in terms of my knowledge, beliefs, attitudes and practices as an educator of teachers. This study has changed the way in which I plan to work as an education advisor in the future.

Get Into Vocational Education (GIVE) Gladstone Pilot Project : Final Report to Rio Tinto Alcan Community Fund

This report describes the Get Into Vocational Education (GIVE) pilot project run at Gladstone Central State School from September to December 2010. The report describes the aims, budget, and timeline of the project and its findings in relation to each of the three major objectives of the project, namely (a) build awareness of, interest in, and familiarity with trades as a future vocation and opportunity for advancement; (b) enhance literacy, numeracy and science knowledge and performance; and (c) provide motivation and engagement to stay on at school and build towards a productive future. The clear findings of the GIVE Gladstone Year 4 pilot project are that, for students at risk in terms of school attendance, engagement and learning: (1) awareness of trades, literacy, mathematics and science knowledge, and motivation and engagement all improve and, in most cases, dramatically improve, in the GIVE structure; (2) this improvement involves transfer to situations and concepts not directly addressed in the project; and (3) the crucial factor in the GIVE structure that gives the improvement is the integration of classroom work with trades experiences and not the classroom and trades experiences themselves (although it is better if these are good).

Get Into Vocational Education (GIVE) Rockhampton Region Pilot Project : Final Report to The John Villiers Trust

This report describes the Get Into Vocational Education (GIVE) pilot project run in the Rockhampton Region at two schools in 2011. The report includes a description of the project, including its aims, budget, and timeline; and the findings in relation to each of the three major objectives of the project, namely (a) build awareness of, interest in, and familiarity with, trades as a future vocation and opportunity for advancement; (b) enhance literacy, numeracy and science knowledge and performance; and (c) provide motivation and engagement to stay on at school and build towards a productive future. The clear findings of the GIVE Rockhampton Region pilot project are that, for students at risk in terms of school attendance, engagement and learning: (1) awareness of trade practices in horticulture, hospitality, retail, and design and engineering, literacy, mathematics and science knowledge, and motivation and engagement all improve and, in most cases, dramatically improve, in the GIVE structure; and (2) the crucial factor in the GIVE structure that gives the improvement is the integration of classroom work with trades experiences and not the classroom and trades experiences themselves (although it is better if these are good).

A framework for mathematics graphical tasks : The influence of the graphic element on student sense making

Graphical tasks have become a prominent aspect of mathematics assessment. From a conceptual stance, the purpose of this study was to better understand the composition of graphical tasks commonly used to assess students’ mathematics understandings. Through an iterative design, the investigation described the sense making of 11–12-year-olds as they decoded mathematics tasks which contained a graphic. An ongoing analysis of two phases of data collection was undertaken as we analysed the extent to which various elements of text, graphics, and symbols influenced student sense making. Specifically, the study outlined the changed behaviour (and performance) of the participants as they solved graphical tasks that had been modified with respect to these elements. We propose a theoretical framework for understanding the composition of a graphical task and identify three specific elements which are dependently and independently related to each other, namely: the graphic; the text; and the symbols. Results indicated that although changes to the graphical tasks were minimal, a change in student success and understanding was most evident when the graphic element was modified. Implications include the need for test designers to carefully consider the graphics embedded within mathematics tasks since the elements within graphical tasks greatly influence student understanding.

Get Into Vocational Education (GIVE) : motivating underperforming students

This study explores the effects of a vocational education-based program on academic motivation and engagement of primary school aged children. The Get Into Vocational Education (GIVE) program integrated ‘construction’ and the mathematics, English and science lessons of a Year 4 primary classroom. This paper focuses on investigating the components of the GIVE program that led to student changes in mathematical academic motivation and engagement resulting in outstanding gains in NAPLAN Numeracy results. The components proposed to have contributed to effectiveness of the GIVE program are: teacher and trainer expectations, task mastery and classroom relationships. These findings may be useful to researchers and educators who are interested in enhancing students’ mathematical academic motivation.

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.

Tracking structural development through data modelling in highly able Grade 1 students

A 3-year longitudinal study Transforming Children’s Mathematical and Scientific Development integrates, through data modelling, a pedagogical approach focused on mathematical patterns and structural relationships with learning in science. As part of this study, a purposive sample of 21 highly able Grade 1 students was engaged in an innovative data modelling program. In the majority of students, representational development was observed. Their complex graphs depicting categorical and continuous data revealed a high level of structure and enabled identification of structural features critical to this development.

Accelerating the Mathematics Learning of Low Socio-Economic Status Junior Secondary Students : an early report

The Accelerating the Mathematics Learning of Low Socio-Economic Status Junior Secondary Students project aims to address the issues faced by very underperforming mathematics students as they enter high school. Its aim is to accelerate learning of mathematics through a vertical curriculum to enable students to access Year 10 mathematics subjects, thus improving life chances. This paper reports upon the theory underpinning this project and illustrates it with examples of the curriculum that has been designed to achieve acceleration.

Innovation through action research in environmental education : from project to praxis

This thesis is a work-in-progress that articulates my research journey based on the development of a curriculum innovation in environmental education. This journey had two distinct, but intertwined phases: action research based fieldwork, conducted collaboratively, to create a whole school approach to environmental education curriculum planning; and a phase of analysis and reflection based on the emerging findings, as I sought to create personal "living educational theory" about change and innovation. A key stimulus for the study was the perceived theory-practice gap in environmental education, which is often presented in the literature as a criticism of teachers for failing to achieve the values and action objectives of critical environmental education. Hence, many programs and projects are considered to be superficial and inconsequential in terms of their ability to seriously address environmental issues. The intention of this study was to work with teachers in a project that would be an exemplar of critical environmental education. This would be in the form of a whole school "learnscaping" curriculum in a primary school whereby the schoolgrounds would be utilised for interdisciplinary critical environment education. Parallel with the three cycles of action research in this project, my research objectives were to identify and comment upon the factors that influence the generation of successful educational innovation. It was anticipated that the project would be a collaboration involving me, as researcher-facilitator, and many of the teachers in the school as active participants. As the project proceeded through its action cycles, however, it became obvious that the goal of developing a critical environmental education curriculum, and the use of highly participatory processes, were unrealistic. Institutional and organisational rigidities in education generally, teachers' day-to-day work demands, and the constant juggle of work, family and other responsibilities for all participants acted as significant constraints. Consequently, it became apparent that the learnscaping curriculum would not be the hoped-for exemplar. Progress was slow and, at times, the project was in danger of stalling permanently. While the curriculum had some elements of critical environmental education, these were minor and not well spread throughout the school. Overall, the outcome seemed best described as a "small win"; perhaps just another example of the theory-practice gap that I had hoped this project would bridge. Towards the project's end, however, my continuing reflection led to an exploration of chaos/complexity theory which gave new meaning to the concept of a "small win". According to this theory, change is not the product of linear processes applied methodically in purposeful and diligent ways, but emerges from serendipitous events that cannot be planned for, or forecast in advance. When this perspective of change is applied to human organisations - in this study, a busy school - the context for change is recognised not as a stable, predictable environment, but as a highly complex system where change happens all the time, cannot be controlled, and no one can be really sure where the impacts might lead. This so-called "butterfly effect" is a central idea of this theory where small changes or modifications are created - the effects of which are difficult to know, let alone determine - and which can have large-scale impacts. Allied with this effect is the belief that long term developments in an organisation that takes complexity into account, emerge by spontaneous self-organising evolution, requiring political interaction and learning in groups, rather than systematic progress towards predetermined goals or "visions". Hence, because change itself and the contexts of change are recognised as complex, chaos/complexity theory suggests that change is more likely to be slow and evolutionary - cultural change - rather than fast and revolutionary where the old is quickly ushered out by radical reforms and replaced by new structures and processes. Slow, small-scale changes are "normal", from a complexity viewpoint, while rapid, wholesale change is both unlikely and unrealistic. Therefore, the frustratingly slow, small-scale, imperfect educational changes that teachers create - including environmental education initiatives - should be seen for what they really are. They should be recognised as successful changes, the impacts of which cannot be known, but which have the potential to magnify into large-scale changes into the future. Rather than being regarded as failures for not meeting critical education criteria, "small wins" should be cause for celebration and support. The intertwined phases of collaborative action research and individual researcher reflection are mirrored in the thesis structure. The first three chapters, respectively, provide the thesis overview, the literature underpinning the study's central concern, and the research methodology. Chapters 4, 5, and 6 report on each of the three action research cycles of the study, namely Laying the Groundwork, Down to Work!, and The Never-ending Story. Each of these chapters presents a narrative of events, a literature review specific to developments in the cycle, and analysis and critique of the events, processes and outcomes of each cycle. Chapter 7 provides a synthesis of the whole of the study, outlining my interim propositions about facilitating curriculum change in schools through action research, and the implications of these for environmental education.

Modelling as a vehicle for philosophical inquiry in the mathematics curriculum

Philosophical inquiry in the teaching and learning of mathematics has received continued, albeit limited, attention over many years (e.g., Daniel, 2000; English, 1994; Lafortune, Daniel, Fallascio, & Schleider, 2000; Kennedy, 2012a). The rich contributions these communities can offer school mathematics, however, have not received the deserved recognition, especially from the mathematics education community. This is a perplexing situation given the close relationship between the two disciplines and their shared values for empowering students to solve a range of challenging problems, often unanticipated, and often requiring broadened reasoning. In this article, I first present my understanding of philosophical inquiry as it pertains to the mathematics classroom, taking into consideration the significant work that has been undertaken on socio-political contexts in mathematics education (e.g., Skovsmose & Greer, 2012). I then consider one approach to advancing philosophical inquiry in the mathematics classroom, namely, through modelling activities that require interpretation, questioning, and multiple approaches to solution. The design of these problem activities, set within life-based contexts, provides an ideal vehicle for stimulating philosophical inquiry.

An evaluation of the Australian ‘reconceptualising early mathematics learning’ project : key findings and implications

The Pattern and Structure Mathematics Awareness Project (PASMAP) has investigated the development of patterning and early algebraic reasoning among 4 to 8 year olds over a series of related studies. We assert that an awareness of mathematical pattern and structure (AMPS) enables mathematical thinking and simple forms of generalization from an early age. This paper provides an overview of key findings of the Reconceptualizing Early Mathematics Learning empirical evaluation study involving 316 Kindergarten students from 4 schools. The study found highly significant differences on PASA scores for PASMAP students. Analysis of structural development showed increased levels for the PASMAP students; those categorised as low ability developed improved structural responses over a short period of time.

Critical care nurses' perceptions of their educational needs

A survey of nurses working in critical care units in 89 Queensland hospitals was conducted to investigate their perceptions of critical care nurses' educational needs. Two thirds of the 62 respondents were from rural units and one third were from metropolitan units. Most respondents, irrespective of geographic location, wanted critical care education to be located in hospitals and to be accredited as a graduate diploma course. Rural and metropolitan nurses had similar educational needs and many worked for hospitals that were not offering adequate orientation or inservice critical care education. The findings that nursing staff turnover was a problem in metropolitan units and that the rural workforce was more stable have implications for the development of educational programs.