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.

Modeling as a means for making powerful ideas accessible to children at an early age

The SimCalc Vision and Contributions Advances in Mathematics Education 2013, pp 419-436 Modeling as a Means for Making Powerful Ideas Accessible to Children at an Early Age Richard Lesh, Lyn English, Serife Sevis, Chanda Riggs … show all 4 hide » Look Inside » Get Access Abstract In modern societies in the 21st century, significant changes have been occurring in the kinds of “mathematical thinking” that are needed outside of school. Even in the case of primary school children (grades K-2), children not only encounter situations where numbers refer to sets of discrete objects that can be counted. Numbers also are used to describe situations that involve continuous quantities (inches, feet, pounds, etc.), signed quantities, quantities that have both magnitude and direction, locations (coordinates, or ordinal quantities), transformations (actions), accumulating quantities, continually changing quantities, and other kinds of mathematical objects. Furthermore, if we ask, what kind of situations can children use numbers to describe? rather than restricting attention to situations where children should be able to calculate correctly, then this study shows that average ability children in grades K-2 are (and need to be) able to productively mathematize situations that involve far more than simple counts. Similarly, whereas nearly the entire K-16 mathematics curriculum is restricted to situations that can be mathematized using a single input-output rule going in one direction, even the lives of primary school children are filled with situations that involve several interacting actions—and which involve feedback loops, second-order effects, and issues such as maximization, minimization, or stabilizations (which, many years ago, needed to be postponed until students had been introduced to calculus). …This brief paper demonstrates that, if children’s stories are used to introduce simulations of “real life” problem solving situations, then average ability primary school children are quite capable of dealing productively with 60-minute problems that involve (a) many kinds of quantities in addition to “counts,” (b) integrated collections of concepts associated with a variety of textbook topic areas, (c) interactions among several different actors, and (d) issues such as maximization, minimization, and stabilization.

From solving puzzles to designing solutions : integrating design thinking into evidence based practice

Evidence based practice (EBP) focuses on solving ‘tame’ problems, where literature supports question construction toward determining a solution. What happens when there is no existing evidence, or when the need for agility precludes a full EBP implementation? How might we build a more agile and innovative practice that facilitates the design of solutions to complex and wicked problems, particularly in cases where there is no existing literature? As problem solving and innovation methods, EBP and design thinking overlap considerably. The literature indicates the potential benefits to be gained for evidence based practice from adopting a human-centred rather than literature-focused foundation. The design thinking process is social and collaborative by nature, which enables it to be more agile and produce more innovative results than evidence based practice. This paper recommends a hybrid approach to maximise the strengths and benefits of the two methods for designing solutions to wicked problems. Incorporating design thinking principles and tools into EBP has the potential to move its applicability beyond tame problems and continuous improvement, and toward wicked problem solving and innovation. The potential of this hybrid approach in practice is yet to be explored.

Exploring the conditions to support assessment for more equitable learning outcomes

NAPLAN RESULTS HAVE gained socio-political prominence and have been used as indicators of educational outcomes for all students, including Indigenous students. Despite the promise of open and in-depth access to NAPLAN data as a vehicle for intervention, we argue that the use of NAPLAN data as a basis for teachers and schools to reduce variance in learning outcomes is insufficient. NAPLAN tests are designed to show statistical variance at the level of the school and the individual, yet do not factor in the sociocultural and cognitive conditions Indigenous students’ experience when taking the tests. We contend that further understanding of these influences may help teachers understand how to develop their classroom practices to secure better numeracy and literacy outcomes for all students. Empirical research findings demonstrate how teachers can develop their classroom practices from an understanding of the extraneous cognitive load imposed by test taking. We have analysed Indigenous students’ experience of solving mathematical test problems to discover evidence of extraneous cognitive load. We have also explored conditions that are more supportive of learning derived from a classroom intervention which provides an alternative way to both assess and build learning for Indigenous students. We conclude that conditions to support assessment for more equitable learning outcomes require a reduction in cognitive load for Indigenous students while maintaining a high level of expectation and participation in problem solving.

Young children's statistical reasoning : a tale of two contexts

This thesis explored the knowledge and reasoning of young children in solving novel statistical problems, and the influence of problem context and design on their solutions. It found that young children's statistical competencies are underestimated, and that problem design and context facilitated children's application of a wide range of knowledge and reasoning skills, none of which had been taught. A qualitative design-based research method, informed by the Models and Modeling perspective (Lesh & Doerr, 2003) underpinned the study. Data modelling activities incorporating picture story books were used to contextualise the problems. Children applied real-world understanding to problem solving, including attribute identification, categorisation and classification skills. Intuitive and metarepresentational knowledge together with inductive and probabilistic reasoning was used to make sense of data, and beginning awareness of statistical variation and informal inference was visible.

Complex Modelling in the Primary and Middle School Years : An Interdisciplinary Approach

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. With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualisation. These technologies have led to signifi cant changes in the forms of mathematical and scientifi c thinking required beyond the classroom. Modelling, in its various forms, can develop and broaden students’ mathematical and scientific thinking beyond the standard curriculum. This chapter first considers future competencies in the mathematical sciences within an increasingly complex world. Consideration is then given to interdisciplinary problem solving and models and modelling, as one means of addressing these competencies. Illustrative case studies involving complex, interdisciplinary modelling activities in Years 1 and 7 are presented.

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.

Reduction of amplitude equations by the renormalization group approach

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence { Zi } i=1 . Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation. © 2008 The American Physical Society.

Learning through modelling in the primary years

Many nations are highlighting the need for a renaissance in the mathematical sciences as essential to the well-being of all citizens (e.g., Australian Academy of Science, 2006; 2010; The National Academies, 2009). Indeed, the first recommendation of The National Academies’ Rising Above the Storm (2007) was to vastly improve K–12 science and mathematics education. The subsequent report, Rising Above the Gathering Storm Two Years Later (2009), highlighted again the need to target mathematics and science from the earliest years of schooling: “It takes years or decades to build the capability to have a society that depends on science and technology . . . You need to generate the scientists and engineers, starting in elementary and middle school” (p. 9). Such pleas reflect the rapidly changing nature of problem solving and reasoning needed in today’s world, beyond the classroom. As The National Academies (2009) reported, “Today the problems are more complex than they were in the 1950s, and more global. They’ll require a new educated workforce, one that is more open, collaborative, and cross-disciplinary” (p. 19). The implications for the problem solving experiences we implement in schools are far-reaching. In this chapter, I consider problem solving and modelling in the primary school, beginning with the need to rethink the experiences we provide in the early years. I argue for a greater awareness of the learning potential of young children and the need to provide stimulating learning environments. I then focus on data modelling as a powerful means of advancing children’s statistical reasoning abilities, which they increasingly need as they navigate their data-drenched world.

Academic potential and cognitive functioning of long-term survivors after childhood liver transplantation

This cross-sectional study assessed intellect, cognition, academic function, behaviour, and emotional health of long-term survivors after childhood liver transplantation. Eligible children were >5 yr post-transplant, still attending school, and resident in Queensland. Hearing and neurocognitive testing were performed on 13 transplanted children and six siblings including two twin pairs where one was transplanted and the other not. Median age at testing was 13.08 (range 6.52-16.99) yr; time elapsed after transplant 10.89 (range 5.16-16.37) yr; and age at transplant 1.15 (range 0.38-10.00) yr. Mean full-scale IQ was 97 (81-117) for transplanted children and 105 (87-130) for siblings. No difficulties were identified in intellect, cognition, academic function, and memory and learning in transplanted children or their siblings, although both groups had reduced mathematical ability compared with normal. Transplanted patients had difficulties in executive functioning, particularly in self-regulation, planning and organization, problem-solving, and visual scanning. Thirty-one percent (4/13) of transplanted patients, and no siblings, scored in the clinical range for ADHD. Emotional difficulties were noted in transplanted patients but were not different from their siblings. Long-term liver transplant survivors exhibit difficulties in executive function and are more likely to have ADHD despite relatively intact intellect and cognition.