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.

Evaluation of the "reconceptualising early mathematics learning" project

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 enables mathematical thinking and simple forms of generalisation from an early age. The project aims to promote a strong foundation for mathematical development by focusing on critical, underlying features of mathematics learning. This paper provides an overview of key aspects of the assessment and intervention, and analyses of the impact of PASMAP on students’ representation, abstraction and generalisation of mathematical ideas. A purposive sample of four large primary schools, two in Sydney and two in Brisbane, representing 316 students from diverse socio-economic and cultural contexts, participated in the evaluation throughout the 2009 school year and a follow-up assessment in 2010. Two different mathematics programs were implemented: in each school, two Kindergarten teachers implemented the PASMAP and another two implemented their regular program. The study shows that both groups of students made substantial gains on the ‘I Can Do Maths’ assessment and a Pattern and Structure Assessment (PASA) interview, but highly significant differences were found on the latter with PASMAP students outperforming the regular group on PASA scores. Qualitative analysis of students’ responses for structural development showed increased levels for the PASMAP students; those categorised as low ability developed improved structural responses over a relatively short period of time.

On the reliability of classifying programming tasks using a Neo-Piagetian theory of cognitive development

Recent research has proposed Neo-Piagetian theory as a useful way of describing the cognitive development of novice programmers. Neo-Piagetian theory may also be a useful way to classify materials used in learning and assessment. If Neo-Piagetian coding of learning resources is to be useful then it is important that practitioners can learn it and apply it reliably. We describe the design of an interactive web-based tutorial for Neo-Piagetian categorization of assessment tasks. We also report an evaluation of the tutorial's effectiveness, in which twenty computer science educators participated. The average classification accuracy of the participants on each of the three Neo-Piagetian stages were 85%, 71% and 78%. Participants also rated their agreement with the expert classifications, and indicated high agreement (91%, 83% and 91% across the three Neo-Piagetian stages). Self-rated confidence in applying Neo-Piagetian theory to classifying programming questions before and after the tutorial were 29% and 75% respectively. Our key contribution is the demonstration of the feasibility of the Neo-Piagetian approach to classifying assessment materials, by demonstrating that it is learnable and can be applied reliably by a group of educators. Our tutorial is freely available as a community resource.

Emerging from student to practitioner : a cumulative development tool for students

PURPOSE: This pilot project’s aim was to trial a tool and process for developing students’ ability to engage in self-assessment using reflection on their clinical experiences, including feedback from workplace learning, in order to aid them in linking theory to practice and develop strategies to improve performance. BACKGROUND: In nursing education, students can experience a mismatch in performance compared to theoretical learning, this is referred to as the ‘theory practice gap’ (Scully 2011, Chan Chan & Liu 2011). One specific contributing factor seems to be students’ inability to engage in meaningful reflection and self-correcting behaviours. A self-assessment strategy was implemented within a third year clinical unit to ameliorate this mismatch with encouraging results, as students developed self-direction in addressing learning needs. In this pilot project the above strategy was adapted for implementation between different clinical units, to create a whole of course approach to integrating workplace learning. METHOD: The methodology underpinning this project is a scaffolded, supported reflective practice process. Improved self-assessment skills is achieved by students reflecting on and engaging with feedback, then mapping this to learning outcomes to identify where performance can be improved. Evaluation of this project includes: collation of student feedback identifying successful strategies along with barriers encountered in implementation; feedback from students and teachers via above processes and tools; and comparison of the number of learning contracts issued in clinical nursing units with similar cohorts. RESULTS: Results will be complete by May 2012 and include analysis of the data collected via the above evaluation methods. Other outcomes will include the refined process and tool, plus resources that should improve cost effectiveness without reducing student support. CONCLUSION: Implementing these tools and processes over the entire student’s learning package, will assist them to demonstrate progressive development through the course. Students will have learnt to understand feedback and integrate these skills for life-long learning.

Collaborative research into the affordances of place for primary school children’s literacy learning

In the context of culturally diverse high poverty areas of Australia, we have conducted collaborative research with teachers and students in a primary school for more than a decade. Teachers have been exploring the affordances of place‐based pedagogies (Gruenewald & Smith, 2008) for the development of students’ spatial literacies and their understandings of the politics of places and built environments (Comber, Nixon, Ashmore, Loo & Cook, 2006; Comber, Thomson and Wells, 2001). This paper reports on a project in which the affordances of placedbased pedagogy are being explored through teacher inquiries and classroom‐based design experiments (Cobb, Confrey, di Sessa, Lehrer & Schauble, 2003). Located within a large‐scale urban renewal project in which houses are being demolished and families relocated, the original school has been replaced by a larger school that serves a population from a wider area. In this paper we draw on the study to consider the challenges of working with teachers and primary school students to study innovative ideas and practices in educational research. Specifically we consider issues raised by collaborative studies of the affordances of cross curricular projects focusing on social and environmental change to engage students in academic learning and expand their literate repertoires in a changing policy climate.

Mentoring pre-service teachers on school students’ differentiated learning

This Australian study explores the mentoring of pre-service teachers in selecting and implementing teaching strategies to meet students‟ learning needs. Two case studies involving 28 mentor teachers in a professional development program and a mentor-mentee partnership during a four week practicum provided data about mentoring teaching strategies for differentiated learning. Findings showed that contexts for learning about differentiation occurred at the pre action, in-action, and post-action stages. Central to each stage were pedagogical knowledge practices such as planning, preparation, classroom management, assessment, and problem solving (reflection-in-action to present solutions to problems) as key to in-action strategising and the mentoring processes. Mentoring pre-service teachers on how to devise teaching strategies for differentiated learning needs to be researched with a wider range of mentors and pre-service teachers, including those at different stages of development.

Widening participation in university learning

This paper reports how one Australian university and the Queensland Department of Education and Training (DET) are working together to increase the number of school students from low socio-economic backgrounds enrolling in undergraduate university degrees. This innovative program involves university lecturers and school teachers working together in the delivery and assessment of four Bachelor of Education units (or subjects) to a cohort of Year eleven and twelve students at a secondary school. Focus group interviews collected data from 26 students, 7 parents, 4 school and 3 university staff to assess the effectiveness of the program. All stakeholders viewed the program as a highly valuable opportunity to experience university learning with 31 high school graduating students being made offers to enter full-time university in the 2010 and 2011. This positive result has particular significance in the current drive in Australia and elsewhere to increase the participation in higher education of young people from under-represented groups.

Adaptive teachers embracing new ways of learning with robotics in Chinese schools

In July 2010, China announced the “National Plan for Medium and Long-term Education Reform and Development(2010-2020)” (PRC 2010). The Plan calls for an education system that: • promotes an integrated development which harnesses everyone’s talent; • combines learning and thinking; unifies knowledge and practice; • allows teachers to teach according to individuals’ needs; and • reforms education quality evaluation and personnel evaluation systems focusing on performance including character, knowledge, ability and other factors. This paper discusses the design and implementation of a Professional Learning Program (PLP) undertaken by 432 primary, middle and high school teachers in China. The aim of this initiative was to develop adaptive expertise in using technology that facilitated innovative science and technology teaching and learning as envisaged by the Chinese Ministry of Education’s (2010-2020) education reforms. Key principles derived from literature about professional learning and scaffolding of learning informed the design of the PLP. The analysis of data revealed that the participants had made substantial progress towards the development of adaptive expertise. This was manifested not only by advances in the participants’ repertoires of Subject Matter Knowledge and Pedagogical Content Knowledge but also in changes to their levels of confidence and identities as teachers. It was found that through time the participants had coalesced into a professional learning community that readily engaged in the sharing, peer review, reuse and adaption, and collaborative design of innovative science and technology learning and assessment activities.

Examining the Assessment Opportunities for Cultural Connectedness for Student Learning : A Sociocultural Analysis

The disparity that exists between the highest and lowest achievers together with deficit approaches to teaching, learning and assessment raise serious equity issues related to fairness, validity, culture and access, which were analysed in a recent Australian Research Council funded project. This chapter explores the potential that exists for teachers to work with Indigenous Teacher Assistants (ITAs) to secure cultural connectedness in teaching, learning and assessment of Indigenous students. The study was a design experiment, which was conducted in seven Catholic and Independent primary schools in northern Queensland and involved semi-structured focus group interviews with Year 4 and 6 Indigenous students, principals, teachers and Indigenous Teacher Assistants. Classroom observations and document analyses were also conducted. This corpus of data was analysed using a sociocultural theoretical lens. The use of a sociocultural analysis helped to identify cultural influences, Indigenous students’ funds of knowledge and values. The information from this analysis was made explicit to teachers to demonstrate how they could enhance their pedagogic and assessment practices by embracing and extending the cultural spaces for learning and teaching of Indigenous students. The way in which teachers construct their interactions for greater cultural connectedness and enhanced learning would appear to rely on relationship building with Indigenous staff, Indigenous students’ cultural knowledge, and improved understanding of assessment and related equity issues.

Epistemic beliefs and beliefs about teaching practices for moral learning in the early years of school : relationships and complexities

While investment in young children is recognised as important for the development of moral values for a cohesive society, little is known about early years teaching practices that promote learning of moral values. This paper reports on observations and interviews with 11 Australian teachers, focusing on their epistemic beliefs and beliefs about teaching practices for moral education with children aged 5 to 8 years. The analysis revealed three main patterns of thinking about moral education: following others, reflecting on points of view, and informing reflection for action. These patterns suggest a relationship between epistemic beliefs and beliefs about teaching practices for moral learning which have implications for teacher professional development concerning experiences in moral education.

Teaching and learning family and consumer sciences through K-W-L charts

While much of the control and many of the activities found in today’s classrooms have been placed in the hands of the learners and learning has become inquiry-based, there remains a need for teachers to use teaching tools that would facilitate this student-centered teaching process. This article identifies the K-W-L Chart as one such tool and follows a case study of four Kuwaiti ‘Family and Consumer Sciences’ teaching / learning events to evaluate their ability to enhance the learning outcomes of eight students. The research was designed from a qualitative, multi-tiered design approach and was assessed through a constant comparative method of data analysis of interview responses, classroom observations and worksheet-assessments. The results showed that the use of K-W-L Charts influenced the teachers and learners toward a more inquiry-based approach and facilitated a more student-centered and collaborative learning environment, raising the level of interest and the amount of personal input given by the students.

Work-integrated learning courses : an essential component for preparing students to work in statutory child protection?

Undergraduate programs can play an important role in the development of individuals wanting professional employment within statutory child protection agencies: both the coursework and the work-integrated learning (WIL) components of degrees have a role in this process. This paper uses a collective case study methodology to examine the perceptions and experiences of first year practitioners within a specific statutory child protection agency in order to identify if they felt prepared for their current role. The sample of 20 participants came from a range of discipline backgrounds with just over half of the sample (55 per cent) completing a WIL placement as part of their undergraduate studies. The results indicate that while some participants were able to identify and articulate specific benefits from their undergraduate coursework studies all participants who had undertaken a WIL placement as part of their degree believed the WIL placement was beneficial for their current work.

The OO files : providing online formative feedback on Contract Law learning in a narrative setting

Feedback, both formative and summative, enables students to reflect on their understandings and to restructure their thinking to develop their capabilities. It can also encourage positive motivation and help boost self-esteem. Online multiple choice questions can be an efficient and effective means of providing timely formative feedback. At the same time, locating learning in a narrative environment can facilitate engaging and effective learning experiences. Narratives can help learners to navigate through information and support cognitive and imaginative engagement. This article will discuss The 00 Files, an online suite of modules containing multiple choice questions situated in the narrative of a fictional law firm. It notes student responses to the program and discusses lessons that may be learnt from its development which may be of assistance to academics considering the development of similar programs for their courses.

Assessing the potential of mathematics textbooks to promote deep learning

Curriculum documents for mathematics emphasise the importance of promoting depth of knowledge rather than shallow coverage of the curriculum. In this paper, we report on a study that explored the analysis of junior secondary mathematics textbooks to assess their potential to assist in teaching and learning aimed at building and applying deep mathematical knowledge. The method of analysis involved the establishment of a set of specific curriculum goals and associated indicators, based on research into the teaching and learning of a particular field within the mathematics curriculum, namely proportion and proportional reasoning. Topic selection was due to its pervasive nature throughout the school mathematics curriculum at this level. As a result of this study, it was found that the five textbook series examined provided limited support for the development of multiplicative structures required for proportional reasoning, and hence would not serve well the development of deep learning of mathematics. The study demonstrated a method that could be applied to the analysis of junior secondary mathematics in many parts of the world.