Year 6 students' idiosyncratic notions of unitising, reunitising and regrouping decimal number places

Having flexible notions of the unit (e.g., 26 ones can be thought of as 2.6 tens, 1 ten 16 ones, 260 tenths, etc.) should be a major focus of elementary mathematics education. However, often these powerful notions are relegated to computations where the major emphasis is on "getting the right answer" thus procedural knowledge rather than conceptual knowledge becomes the primary focus. This paper reports on 22 high-performing students' reunitising processes ascertained from individual interviews on tasks requiring unitising, reunitising and regrouping; errors were categorised to depict particular thinking strategies. The results show that, even for high-performing students, regrouping is a cognitively complex task. This paper analyses this complexity and draws inferences for teaching.

Choreographing research : supervising the dancing thesis

The dancing doctorate is an interrogative endeavour which can but nurture the art form and forge a beneficial dynamism between those who seek and those who assess the emerging knowledges of dance’. (Vincs, 2009) From 2006-2008 three dance academics from Perth, Brisbane and Melbourne undertook a research project entitled Dancing between Diversity and Consistency: Refining Assessment in Postgraduate Degrees in Dance, funded by the ALTC Priority Projects Program. Although assessment rather than supervision was the primary focus of this research, interviews with 40 examiner/supervisors, 7 research deans and 32 candidates across Australia and across the creative arts, primarily in dance, provide an insight into what might be considered best practice in preparing students for higher research degrees, and the challenges that embodied and experiential knowledges present for supervision. The study also gained the industry perspectives of dance professionals in a series of national forums in 5 cities, based around the value of higher degrees in dance. The qualitative data gathered from these two primary sources was coded and analysed using the NVivo system. Further perspectives were drawn from international consultant and dance researcher Susan Melrose, as well as recent publications in the field. Dance is a young addition to academia and consequently there tends to be a close liaison between the academy and the industry, with a relational fluidity that is both beneficial and problematic. This partially explains why dance research higher degrees are predominantly practice-led (or multi-modal, referring to those theses where practice comprises the substantial examinable component). As a physical, embodied art form, dance engages with the contested territory of legitimising alternative forms of knowledge that do not sit comfortably with accepted norms of research. In supporting research students engaged with dance practice, supervisors traverse the tricky terrain of balancing university academic requirements with studies that are emergent, not only in the practice and attendant theory but in their methodologies and open-ended outcomes; and in an art form in which originality and new knowledge also arises from collaborative creative processes. Formal supervisor accreditation through training is now mandatory in most Australian universities, but it tends to be generic and not address supervisory specificity. This paper offers the kind of alternative proposed by Edwards (2002) that improving postgraduate supervision will be effective if supervisors are empowered to generate their own standards and share best practice; in this case, in ways appropriate to the needs of their discipline and alternative modes of thesis presentation. In order to frame the qualities and processes conducive to this goal, this paper will draw on both the experiences of interviewees and on philosophical premises which underpin the research findings of our study. These include the ongoing challenge of dissolving the binary oppositions of theory and practice, especially in creative arts practice where theory resides in and emerges from the doing as much as in articulating reflection about the doing through what Melrose (2003) terms ‘mixed mode disciplinary practices’. In guiding practitioners through research higher degrees, how do supervisors deal with not only different forms of knowledge but indeed differing modes of knowledge? How can they navigate tensions that occur between the ‘incompatible competencies’ (Candlin, 2000) of the ‘spectating’ academic experts with their ‘irrepressible drive ... to inscribe, interpret, and hence to practise temporal closure’, and practitioner experts who create emergent works of ‘residual unfinishedness’ (Melrose 2006) which are not only embodied but ephemeral, as in the case of live performance?

Strategies for comparing decimal numbers with the same whole-number part

The strategies employed by 130 Grade 5 Brisbane students in comparing decimal numbers which have the same whole-number part were compared with those identified in similar studies conducted in the USA, France and Israel. Three new strategies were identified. Similar to USA results, the most common comparison errors stemmed from the incorrect whole-number strategy in which length is confused with size. The findings of this present study tend to support Resnick et al.’s (1989) hypothesis that the introduction of decimal-fraction recording before common-fraction recording seems to promote better comparison of decimal numbers.

Construction of multiplicative abstract schema for decimal-number numeration

This paper reports on an intervention study planned to help Year 6 students construct the multiplicative structure underlying decimal-number numeration. Three types of intervention were designed from a numeration model developed from a large study of 173 Year 6 students’ decimal-number knowledge. The study found that students could acquire multiplicative structure as an abstract schema if instruction took account of prior knowledge as informed by the model.

Place value, multiplicativity, reunitizing and effective classroom teaching of decimal numeration

Student understanding of decimal number is poor (e.g., Baturo, 1998; Behr, Harel, Post & Lesh, 1992). This paper reports on a study which set out to determine the cognitive complexities inherent in decimal-number numeration and what teaching experiences need to be provided in order to facilitate an understanding of decimal-number numeration. The study gave rise to a theoretical model which incorporated three levels of knowledge. Interview tasks were developed from the model to probe 45 students’ understanding of these levels, and intervention episodes undertaken to help students construct the baseline knowledge of position and order (Level 1 knowledge) and an understanding of multiplicative structure (Level 3 knowledge). This paper describes the two interventions and reports on the results which suggest that helping students construct appropriate mental models is an efficient and effective teaching strategy.

Signalling layout for fixed-block railway lines with real-coded genetic algorithms.

Signalling layout design is one of the keys to railway operations with fixed-block signalling system and it also carries direct effect on overall train efficiency and safety. Based on an analysis to system objectives, this paper presents an optimization model with two objectives in order to devise an efficient signalling layout scheme. Taking into account the present railway line design practices in China, the paper describes steps of the computer-based signalling layout optimisation with real-coded genetic algorithms. A computer-aided system, based on train movement simulator, has also been employed to assist the optimisation process. A case study on a practical railway line has been conducted to make comparisons between the proposed GA-based approach and the current practices. The results illustrate the improved performance of the proposed approach in reducing signal block joints and shortening minimum train service headway.

Design of experiments for bivariate binary responses modelled by Copula functions

Optimal design for generalized linear models has primarily focused on univariate data. Often experiments are performed that have multiple dependent responses described by regression type models, and it is of interest and of value to design the experiment for all these responses. This requires a multivariate distribution underlying a pre-chosen model for the data. Here, we consider the design of experiments for bivariate binary data which are dependent. We explore Copula functions which provide a rich and flexible class of structures to derive joint distributions for bivariate binary data. We present methods for deriving optimal experimental designs for dependent bivariate binary data using Copulas, and demonstrate that, by including the dependence between responses in the design process, more efficient parameter estimates are obtained than by the usual practice of simply designing for a single variable only. Further, we investigate the robustness of designs with respect to initial parameter estimates and Copula function, and also show the performance of compound criteria within this bivariate binary setting.

Agrobacterium-mediated transformation of indica rice cultivars using binary and superbinary vectors

The Agrobacterium-mediated transformation system was extended to two indica cultivars: a widely cultivated breeding line IR-64 and an elite basmati cultivar Karnal Local. Root tips and shoot tips of seedlings, and scutellar-calli derived from mature seeds showed high-efficiency Agrobacterium tumefaciens infection and stable transformation. In addition to the superbinary vector pTOK233 in Agrobacterium strain LBA4404, almost equally high levels of transformation were achieved with a relatively much smaller (13.1 kb) binary vector (pCAMBIA1301) in a supervirulent host strain AGL1. In both cases, as well as in both cultivars, while 60–90% of the infected explants produced calli resistant to the selectable agent hygromycin, 59–75% of such calli tested positive for GUS. A high level (400 μM) of acetosyringone in the preinduction medium for Agrobacterium and a higher level (500 μM) in the cocultivation medium was necessary for an enhancement in transformation frequency of the binary vector to levels comparable to a superbinary. Hygromycin-resistant calli could be produced from all the explants used. Transformants could be regenerated for both cultivars using the superbinary and binary vector, but only for calli of scutellar origin. In addition to the molecular confirmation of hpt and gus gene transfer and transcription, absence of gene sequences outside the transferred DNA (T-DNA) region confirmed absence of any long T-DNA transfer.