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.

The continuing decline of science and mathematics enrolments in Australian high schools

Is there a crisis in Australian science and mathematics education? Declining enrolments in upper secondary Science and Mathematics courses have gained much attention from the media, politicians and high-profile scientists over the last few years, yet there is no consensus amongst stakeholders about either the nature or the magnitude of the changes. We have collected raw enrolment data from the education departments of each of the Australian states and territories from 1992 to 2012 and analysed the trends for Biology, Chemistry, Physics, two composite subject groups (Earth Sciences and Multidisciplinary Sciences), as well as entry, intermediate and advanced Mathematics. The results of these analyses are discussed in terms of participation rates, raw enrolments and gender balance. We have found that the total number of students in Year 12 increased by around 16% from 1992 to 2012 while the participation rates for most Science and Mathematics subjects, as a proportion of the total Year 12 cohort, fell (Biology (-10%), Chemistry (-5%), Physics (-7%), Multidisciplinary Science (-5%), intermediate Mathematics (-11%), advanced Mathematics (-7%) in the same period. There were increased participation rates in Earth Sciences (+0.3%) and entry Mathematics (+11%). In each case the greatest rates of change occurred prior to 2001 and have been slower and steadier since. We propose that the broadening of curriculum offerings, further driven by students' self-perception of ability and perceptions of subject difficulty and usefulness, are the most likely cause of the changes in participation. While these continuing declines may not amount to a crisis, there is undoubtedly serious cause for concern.

Accelerating junior-secondary mathematics

Unfortunately, in Australia there is a prevalence of mathematically underperforming junior-secondary students in low-socioeconomic status schools. This requires targeted intervention to develop the affected students’ requisite understanding in preparation for post-compulsory study and employment and, ultimately, to increase their life chances. To address this, the ongoing action research project presented in this paper is developing a curriculum of accelerated learning, informed by a lineage of cognitivist-based structural sequence theory building activity (e.g., Cooper & Warren, 2011). The project’s conceptual framework features three pillars: the vertically structured sequencing of concepts; pedagogy grounded in students’ reality and culture; and professional learning to support teachers’ implementation of the curriculum (Cooper, Nutchey, & Grant, 2013). Quantitative and qualitative data informs the ongoing refinement of the theory, the curriculum, and the teacher support.

Problem solving in a 21st century mathematics curriculum

Research on problem solving in the mathematics curriculum has spanned many decades, yielding pendulum-like swings in recommendations on various issues. Ongoing debates concern the effectiveness of teaching general strategies and heuristics, the role of mathematical content (as the means versus the learning goal of problem solving), the role of context, and the proper emphasis on the social and affective dimensions of problem solving (e.g., Lesh & Zawojewski, 2007; Lester, 2013; Lester & Kehle, 2003; Schoenfeld, 1985, 2008; Silver, 1985). Various scholarly perspectives—including cognitive and behavioral science, neuroscience, the discipline of mathematics, educational philosophy, and sociocultural stances—have informed these debates, often generating divergent resolutions. Perhaps due to this uncertainty, educators’ efforts over the years to improve students’ mathematical problem-solving skills have had disappointing results. Qualitative and quantitative studies consistently reveal mathematics students’ struggles to solve problems more significant than routine exercises (OECD, 2014; Boaler, 2009)...

Mathematics, programming, and STEM

Learning mathematics is a complex and dynamic process. In this paper, the authors adopt a semiotic framework (Yeh & Nason, 2004) and highlight programming as one of the main aspects of the semiosis or meaning-making for the learning of mathematics. During a 10-week teaching experiment, mathematical meaning-making was enriched when primary students wrote Logo programs to create 3D virtual worlds. The analysis of results found deep learning in mathematics, as well as in technology and engineering areas. This prompted a rethinking about the nature of learning mathematics and a need to employ and examine a more holistic learning approach for the learning in science, technology, engineering, and mathematics (STEM) areas.

A model eliciting framework for integrating mathematics and robotics learning

Robotics is taught in many Australian ICT classrooms, in both primary and secondary schools. Robotics activities, including those developed using the LEGO Mindstorms NXT technology, are mathematics-rich and provide a fertile round for learners to develop and extend their mathematical thinking. However, this context for learning mathematics is often under-exploited. In this paper a variant of the model construction sequence (Lesh, Cramer, Doerr, Post, & Zawojewski, 2003) is proposed, with the purpose of explicitly integrating robotics and mathematics teaching and learning. Lesh et al.’s model construction sequence and the model eliciting activities it embeds were initially researched in primary mathematics classrooms and more recently in university engineering courses. The model construction sequence involves learners working collaboratively upon product-focussed tasks, through which they develop and expose their conceptual understanding. The integrating model proposed in this paper has been used to design and analyse a sequence of activities in an Australian Year 4 classroom. In that sequence more traditional classroom learning was complemented by the programming of LEGO-based robots to ‘act out’ the addition and subtraction of simple fractions (tenths) on a number-line. The framework was found to be useful for planning the sequence of learning and, more importantly, provided the participating teacher with the ability to critically reflect upon robotics technology as a tool to scaffold the learning of mathematics.