RcppArmadillo : accelerating R with high-performance C++ linear algebra

The R statistical environment and language has demonstrated particular strengths for interactive development of statistical algorithms, as well as data modelling and visualisation. Its current implementation has an interpreter at its core which may result in a performance penalty in comparison to directly executing user algorithms in the native machine code of the host CPU. In contrast, the C++ language has no built-in visualisation capabilities, handling of linear algebra or even basic statistical algorithms; however, user programs are converted to high-performance machine code, ahead of execution. A new method avoids possible speed penalties in R by using the Rcpp extension package in conjunction with the Armadillo C++ matrix library. In addition to the inherent performance advantages of compiled code, Armadillo provides an easy-to-use template-based meta-programming framework, allowing the automatic pooling of several linear algebra operations into one, which in turn can lead to further speedups. With the aid of Rcpp and Armadillo, conversion of linear algebra centered algorithms from R to C++ becomes straightforward. The algorithms retains the overall structure as well as readability, all while maintaining a bidirectional link with the host R environment. Empirical timing comparisons of R and C++ implementations of a Kalman filtering algorithm indicate a speedup of several orders of magnitude.

ICT driven pedagogies and its impact on learning outcomes in high school mathematics

ICT (Information and Communication Technology) creates numerous opportunities for teachers to re-think their pedagogies. In subjects like mathematics which draws upon abstract concepts, ICT creates such an opportunity. Instead of a mimetic pedagogical approach, suitably designed activities with ICT can enable learners to engage more proactively with their learning. In this quasi-experimental designed study, ICT was used in teaching mathematics to a group of first year high school students (N=25) in Australia. The control group was taught predominantly through traditional pedagogies (N=22). Most of the variables that had previously impacted on the design of such studies were suitably controlled in this yearlong investigation. Quantitative and qualitative results showed that students who were taught by ICT driven pedagogies benefitted from the experience. Pre and post-test means showed that there was a difference between the treatment and control groups. Of greater significance was that the students (in the treatment group) believed that the technology enabled them to engage more with their learning.

How difficult are exams? A framework for assessing the complexity of introductory programming exams

Student performance on examinations is influenced by the level of difficulty of the questions. It seems reasonable to propose therefore that assessment of the difficulty of exam questions could be used to gauge the level of skills and knowledge expected at the end of a course. This paper reports the results of a study investigating the difficulty of exam questions using a subjective assessment of difficulty and a purpose-built exam question complexity classification scheme. The scheme, devised for exams in introductory programming courses, assesses the complexity of each question using six measures: external domain references, explicitness, linguistic complexity, conceptual complexity, length of code involved in the question and/or answer, and intellectual complexity (Bloom level). We apply the scheme to 20 introductory programming exam papers from five countries, and find substantial variation across the exams for all measures. Most exams include a mix of questions of low, medium, and high difficulty, although seven of the 20 have no questions of high difficulty. All of the complexity measures correlate with assessment of difficulty, indicating that the difficulty of an exam question relates to each of these more specific measures. We discuss the implications of these findings for the development of measures to assess learning standards in programming courses.

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.

Modeling as a bridge between real world problems and school mathematics

This article argues for an interdisciplinary approach to mathematical problem solving at the elementary school, one that draws upon the engineering domain. A modeling approach, using engineering model eliciting activities, might provide a rich source of meaningful situations that capitalize on and extend students’ existing mathematical learning. The study reports on the developments of 48 twelve-year old students who worked on the Bridge Design activity. Results revealed that young students, even before formal instruction, have the capacity to deal with complex interdisciplinary problems. A number of students created quite appropriate models by developing the necessary mathematical constructs to solve the problem. Students’ difficulties in mathematizing the problem, and in revising and documenting their models are presented and analysed, followed by a discussion on the appropriateness of a modeling approach as a means for introducing complex problems to elementary school students.

A comparison of Chinese and Australian early childhood mathematics

In the 21st century mathematics proficiency is synonymous with a numerate citizenry. In the past few decades young children’s ability to reason mathematically and develop mathematical proficiencies has been recognised. This paper explores the history of early childhood mathematics (ECME) that may explicate differences in Chinese and Australian contexts. Results of this review established that China and Australia are diametrically positioned in ECME. Influencing each countries philosophies and practices are their cultural beliefs. ECME in China and Australia must be culturally sustainable to achieve excellent outcomes for young children. Ongoing critique and review is necessary to ensure that ECME is meeting the needs of all teachers and children in their particular context. China and Australia with their rich contrasting philosophies can assist each other in their journeys to create exemplary ECME for the 21st century.

Mathematics learning in prep

Most teachers recognise the importance of mathematics teaching and learning in early years but there is not consensus on how and when this learning should occur. Young-Loveridge (cited in de Vries, Thomas, and Warren, 2010) suggests that quality early mathematical experiences are a key determinant to later achievement.

Efficient articulated trajectory reconstruction using dynamic programming and filters

This paper considers the problem of reconstructing the motion of a 3D articulated tree from 2D point correspondences subject to some temporal prior. Hitherto, smooth motion has been encouraged using a trajectory basis, yielding a hard combinatorial problem with time complexity growing exponentially in the number of frames. Branch and bound strategies have previously attempted to curb this complexity whilst maintaining global optimality. However, they provide no guarantee of being more efficient than exhaustive search. Inspired by recent work which reconstructs general trajectories using compact high-pass filters, we develop a dynamic programming approach which scales linearly in the number of frames, leveraging the intrinsically local nature of filter interactions. Extension to affine projection enables reconstruction without estimating cameras.

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.

Engaging students in authoring and playing computer games

Computer games have become a commonplace but engaging activity among students. They enjoy playing computer games as they can perform larger-than-life activities virtually such as jumping from great heights, flying planes, and racing cars; actions that are otherwise not possible in real life. Computer games also offer user interactivity which gives them a certain appeal. Considering this appeal, educators should consider integrating computer games into student learning and to encourage students to author computer games of their own. It is thought that students can be engaged in learning by authoring and using computer games and can also gain essential skills such as collaboration, teamwork, problem solving and deductive reasoning. The research in this study revolves around building student engagement through the task of authoring computer games. The study aims to demonstrate how the creation and sharing of student-authored educational games might facilitate student engagement and how ICT (information and communication technology) plays a supportive role in student learning. Results from this study may lead to the broader integration of computer games into student learning and contribute to similar studies. In this qualitative case study, based in a state school in a low socio-economic area west of Brisbane, Australia, students were selected in both junior and senior secondary classes who have authored computer games as a part of their ICT learning. Senior secondary students (Year 12 ICT) were given the task of programming the games, which were to be based on Mathematics learning topics while the junior secondary students (Year 8 ICT) were given the task of creating multimedia elements for the games. A Mathematics teacher volunteered to assist in the project and provided guidance on the inclusion of suitable Mathematics curricular content into these computer games. The student-authored computer games were then used to support another group of Year 8 Mathematics students to learn the topics of Area, Volume and Time. Data was collected through interviews, classroom observations and artefacts. The teacher researcher, acting in the role of ICT teacher, coordinated with the students and the Mathematics teacher to conduct this study. Instrumental case study was applied as research methodology and Third Generation Activity Theory served as theoretical framework for this study. Data was analysed adopting qualitative coding procedures. Findings of this study indicate that having students author and play computer games promoted student engagement and that ICT played a supportive role in learning and allowed students to gain certain essential skills. Although this study will suggest integrating computer games to support classroom learning, it cannot be presumed that computer games are an immediate solution for promoting student engagement.

Visual code annotations for cyberphysical programming

User interfaces for source code editing are a crucial component in any software development environment, and in many editors visual annotations (overlaid on the textual source code) are used to provide important contextual information to the programmer. This paper focuses on the real-time programming activity of ‘cyberphysical’ programming, and considers the type of visual annotations which may be helpful in this programming context.

Data and measurement in Year 4 of the national curriculum : Mathematics

The activities introduced here were used in association with a research project in four Year 4 classrooms and are suggested as a motivating way to address several criteria for Measurement and Data in the Australian Curriculum: Mathematics. The activities involve measuring the arm span of one student in a class many times and then of all students once.