Beyond questionnaires – Exploring adult education teachers' mathematical beliefs with pictures and interviews

Because of the impact that mathematical beliefs have on an individual’s behaviour, they are generally well researched. However, little mathematical belief research has taken place in the field of adult education. This paper presents preliminary results from a study conducted in this field in Switzerland. It is based on Ernest’s (1989) description of mathematics as an instrumental, Platonist or problem solving construct. The analysis uses pictures drawn by the participants and interviews conducted with them as data. Using a categorising scheme developed by Rolka and Halverscheid (2011), the author argues that adults’ mathematical beliefs are complex and especially personal aspects are difficult to capture with said scheme. Particularly the analysis of visual data requires a more refined method of analysis.

Experience of problem-based learning (PBL) in virtual space : a phenomenographical study

This thesis reports the outcomes of an investigation into students’ experience of Problem-based learning (PBL) in virtual space. PBL is increasingly being used in many fields including engineering education. At the same time many engineering education providers are turning to online distance education. Unfortunately there is a dearth of research into what constitutes an effective learning experience for adult learners who undertake PBL instruction through online distance education. Research was therefore focussed on discovering the qualitatively different ways that students experience PBL in virtual space. Data was collected in an electronic environment from a course, which adopted the PBL strategy and was delivered entirely in virtual space. Students in this course were asked to respond to open-ended questions designed to elicit their learning experience in the course. Data was analysed using the phenomenographical approach. This interpretative research method concentrated on mapping the qualitative differences in students’ interpretations of their experience in the course. Five qualitatively different ways of experiencing were discovered: Conception 1: ‘A necessary evil for program progression’; Conception 2: ‘Developing skills to understand, evaluate, and solve technical Engineering and Surveying problems’; Conception 3: ‘Developing skills to work effectively in teams in virtual space’; Conception 4: ‘A unique approach to learning how to learn’; Conception 5: ‘Enhancing personal growth’. Each conception reveals variation in how students attend to learning by PBL in virtual space. Results indicate that the design of students’ online learning experience was responsible for making students aware of deeper ways of experiencing PBL in virtual space. Results also suggest that the quality and quantity of interaction with the team facilitator may have a significant impact on the student experience in virtual PBL courses. The outcomes imply pedagogical strategies can be devised for shifting students’ focus as they engage in the virtual PBL experience to effectively manage the student learning experience and thereby ensure that they gain maximum benefit. The results from this research hold important ramifications for graduates with respect to their ease of transition into professional work as well as their later professional competence in terms of problem solving, ability to transfer basic knowledge to real-life engineering scenarios, ability to adapt to changes and apply knowledge in unusual situations, ability to think critically and creatively, and a commitment to continuous life-long learning and self-improvement.

A soft approach to solving hard problems in construction project management

Construction projects are faced with a challenge that must not be underestimated. These projects are increasingly becoming highly competitive, more complex, and difficult to manage. They become ‘wicked problems’, which are difficult to solve using traditional approaches. Soft Systems Methodology (SSM) is a systems approach that is used for analysis and problem solving in such complex and messy situations. SSM uses “systems thinking” in a cycle of action research, learning and reflection to help understand the various perceptions that exist in the minds of the different people involved in the situation. This paper examines the benefits of applying SSM to wicked problems in construction project management, especially those situations that are challenging to understand and difficult to act upon. It includes relevant examples of its use in dealing with the confusing situations that incorporate human, organizational and technical aspects.

Trailblazers

A recent study in the United Kingdom (Ofsted Report 2008) provides strong evidence that well-organized activities outside the classroom contribute significantly to the quality and depth of children's learning, including their personal, social, and emotional development. Outdoor math trails supply further evidence of such enhanced learning: They are meaningful, stimulating, challenging, and exciting for children. Most important, these trails invite all students, irrespective of their classroom achievement level, to participate successfully in the problem activities and gain a sense of pride in the mathematics they create. Additionally, Math trails empower lifelong learning. Integrating "outside" mathematics with "inside" classroom mathematics can sow the seeds to develop flexible, creative, future-oriented mathematical thinkers and problem solvers. Here, English et al discuss how to design and implement math trails to promote active, meaningful, real-world mathematical learning beyond the classroom walls.

Individual differences in learning from worked examples by senior secondary mathematics students

The primary purpose of this research was to examine individual differences in learning from worked examples. By integrating cognitive style theory and cognitive load theory, it was hypothesised that an interaction existed between individual cognitive style and the structure and presentation of worked examples in their effect upon subsequent student problem solving. In particular, it was hypothesised that Analytic-Verbalisers, Analytic-Imagers, and Wholist-lmagers would perform better on a posttest after learning from structured-pictorial worked examples than after learning from unstructured worked examples. For Analytic-Verbalisers it was reasoned that the cognitive effort required to impose structure on unstructured worked examples would hinder learning. Alternatively, it was expected that Wholist-Verbalisers would display superior performances after learning from unstructured worked examples than after learning from structured-pictorial worked examples. The images of the structured-pictorial format, incongruent with the Wholist-Verbaliser style, would be expected to split attention between the text and the diagrams. The information contained in the images would also be a source of redundancy and not easily ignored in the integrated structured-pictorial format. Despite a number of authors having emphasised the need to include individual differences as a fundamental component of problem solving within domainspecific subjects such as mathematics, few studies have attempted to investigate a relationship between mathematical or science instructional method, cognitive style, and problem solving. Cognitive style theory proposes that the structure and presentation of learning material is likely to affect each of the four cognitive styles differently. No study could be found which has used Riding's (1997) model of cognitive style as a framework for examining the interaction between the structural presentation of worked examples and an individual's cognitive style. 269 Year 12 Mathematics B students from five urban and rural secondary schools in Queensland, Australia participated in the main study. A factorial (three treatments by four cognitive styles) between-subjects multivariate analysis of variance indicated a statistically significant interaction. As the difficulty of the posttest components increased, the empirical evidence supporting the research hypotheses became more pronounced. The rigour of the study's theoretical framework was further tested by the construction of a measure of instructional efficiency, based on an index of cognitive load, and the construction of a measure of problem-solving efficiency, based on problem-solving time. The consistent empirical evidence within this study that learning from worked examples is affected by an interaction of cognitive style and the structure and presentation of the worked examples emphasises the need to consider individual differences among senior secondary mathematics students to enhance educational opportunities. Implications for teaching and learning are discussed and recommendations for further research are outlined.

Working together for Toby : early childhood student teachers engaging in collaborative problem-based learning around child abuse and neglect

Early childhood teacher education programs have a responsibility, amongst many, to prepare teachers for decision-making on real world issues, such as child abuse and neglect. Their repertoire of skills can be enhanced by engaging with others, either face-to-face or online, in authentic problem-based learning. This paper draws on a study of early childhood student teachers who engaged in an authentic learning experience, which was to consider and to suggest how they would act upon a real-life case of child abuse encountered in an early childhood classroom in Queensland. This was the case of Toby (a pseudonym), who was suspected of being physically abused at home. Students drew upon relevant legislation, policy and resource materials to tackle Toby’s case. The paper provides evidence of students grappling with the complexity of a child abuse case and establishing, through collaboration with others, a proactive course of action. The paper has a dual focus. First, it discusses the pedagogical context in which early childhood student teachers deal with issues of child abuse and neglect in the course of their teacher education program. Second, it examines evidence of students engaging in collaborative problem-solving around issues of child abuse and neglect and teachers’ responsibilities, both legal and professional, to the children and families they work with. Early childhood policy-makers, practitioners and teacher educators are challenged to consider how early childhood teachers are best equipped to deal with child protection and early intervention.

Numerical solution to the Saffman-Taylor finger problem with kinetic undercooling regularisation

The Saffman-Taylor finger problem is to predict the shape and,in particular, width of a finger of fluid travelling in a Hele-Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem suffciently diffcult that asymptotic or numerical methods must be used. In this paper we adapt numerical methods used to solve the Saffman-Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele-Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as with the surface tension problem.

Students' perceptions of the relevance of mathematics in engineering

In this paper, we report on the findings of an exploratory study into the experience of students as they learn first year engineering mathematics. Here we define engineering as the application of mathematics and sciences to the building and design of projects for the use of society (Kirschenman and Brenner 2010)d. Qualitative and quantitative data on students' views of the relevance of their mathematics study to their engineering studies and future careers in engineering was collected. The students described using a range of mathematics techniques (mathematics skills developed, mathematics concepts applied to engineering and skills developed relevant for engineering) for various usages (as a subject of study, a tool for other subjects or a tool for real world problems). We found a number of themes relating to the design of mathematics engineering curriculum emerged from the data. These included the relevance of mathematics within different engineering majors, the relevance of mathematics to future studies, the relevance of learning mathematical rigour, and the effectiveness of problem solving tasks in conveying the relevance of mathematics more effectively than other forms of assessment. We make recommendations for the design of engineering mathematics curriculum based on our findings.

Infinite-horizon policy-gradient estimation

Gradient-based approaches to direct policy search in reinforcement learning have received much recent attention as a means to solve problems of partial observability and to avoid some of the problems associated with policy degradation in value-function methods. In this paper we introduce GPOMDP, a simulation-based algorithm for generating a biased estimate of the gradient of the average reward in Partially Observable Markov Decision Processes (POMDPs) controlled by parameterized stochastic policies. A similar algorithm was proposed by Kimura, Yamamura, and Kobayashi (1995). The algorithm's chief advantages are that it requires storage of only twice the number of policy parameters, uses one free parameter β ∈ [0,1) (which has a natural interpretation in terms of bias-variance trade-off), and requires no knowledge of the underlying state. We prove convergence of GPOMDP, and show how the correct choice of the parameter β is related to the mixing time of the controlled POMDP. We briefly describe extensions of GPOMDP to controlled Markov chains, continuous state, observation and control spaces, multiple-agents, higher-order derivatives, and a version for training stochastic policies with internal states. In a companion paper (Baxter, Bartlett, & Weaver, 2001) we show how the gradient estimates generated by GPOMDP can be used in both a traditional stochastic gradient algorithm and a conjugate-gradient procedure to find local optima of the average reward. ©2001 AI Access Foundation and Morgan Kaufmann Publishers. All rights reserved.

Learning the kernel matrix with semidefinite programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive semidefinite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space - classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -using the labeled part of the data one can learn an embedding also for the unlabeled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method for learning the 2-norm soft margin parameter in support vector machines, solving an important open problem.

Complex modelling in the primary/middle school years

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by continuing education as usual (Katehi, Pearson, & Feder, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualisation. These technologies have led to significant changes in the forms of mathematical and scientific thinking that are required beyond the classroom. Modelling, in its various forms, can develop and broaden children’s mathematical and scientific thinking beyond the standard curriculum. This paper first considers future competencies in the mathematical sciences within an increasingly complex world. Next, consideration is given to interdisciplinary problem solving and models and modelling. Examples of complex, interdisciplinary modelling activities across grades are presented, with data modelling in 1st grade, model-eliciting in 4th grade, and engineering-based modelling in 7th-9th grades.

Reflection in Integrated Problem Based Learning : 20 years of continuing application to the teaching of architecture

In 1984 the School of Architecture and Built Environment within the University of Newcastle, Australia introduced an integrated program based on real design projects and using Integrated Problem Based Learning (IPBL) as the teaching method. Since 1984 there have been multiple changes arising from the expectations of the architectural fraternity, enrolling students, lecturers, available facilities, accreditation authorities and many others. These challenges have been successfully accommodated whilst maintaining the original purposes and principles of IPBL. The Architecture program has a combined two-degree structure consisting of a first degree, Bachelor of Science (Architecture), followed by a second degree, Bachelor of Architecture. The program is designed to simulate the problem-solving situations that face a working architect in every day practice. This paper will present the degree structure where each student is enrolled in a single course per semester incorporating design integration and study areas in design studies, professional studies, historical studies, technical studies, environmental studies and communication skills. Each year the design problems increase in complexity and duration set around an annual theme. With 20 years of successful delivery of any program there are highlights and challenges along the way and this paper will discuss some of the successes and barriers experienced within the School of Architecture and Built Environment in delivering IPBL. In addition, the reflective process investigates the currency of IPBL as an appropriate vehicle for delivering the curriculum in 2004 and any additional administrative or staff considerations required to enhance the continuing application of IPBL.

Problem-based learning : does accounting education need it?

Problem-based learning (PBL) has been used successfully in disciplines such as medicine, nursing, law and engineering. However a review of the literature shows that there has been little use of this approach to learning in accounting. This paper extends the research in accounting education by reporting the findings of a case study of the development and implementation of PBL at the Queensland University of Technology (QUT) in a new Accountancy Capstone unit that began in 2006. The fundamentals of the PBL approach were adhered to. However, one of the essential elements of the approach adopted was to highlight the importance of questioning as a means of gathering the necessary information upon which decisions are made. This approach can be contrasted with the typical ‘give all the facts’ case studies that are commonly used. Another feature was that students worked together in the same group for an entire semester (similar to how teams in the workplace operate) so there was an intended focus on teamwork in solving unstructured, real-world accounting problems presented to students. Based on quantitative and qualitative data collected from student questionnaires over seven semesters, it was found that students perceived PBL to be effective, especially in terms of developing the skills of questioning, teamwork, and problem solving. The effectiveness of questioning is very important as this is a skill that is rarely the focus of development in accounting education. The successful implementation of PBL in accounting through ‘learning by doing’ could be the catalyst for change to bring about better learning outcomes for accounting graduates.