Investigation into the skill levels achieved by mathematics students in the V.C.E. and the H.S.C. mathematics courses

This study examines whether recent changes to the mathematics courses offered in the final year of secondary school (Year 12) in the state of Victoria, Australia have affected the learning outcomes of students in terms of then: skill levels in algebra, calculus and problem solving; and in terms of their preparation for a tertiary mathematics unit. The impact of these changes on the transition from secondary to tertiary mathematics is also considered. A comparison is made between students who attempted a first year mathematics unit at the University of Melbourne (U. of M.) having completed the new V.C.E. (Victorian Certificate of Education) mathematics courses and mathematics courses from the previous H.S.C. (Higher School Certificate) system. The comparison involves the use of tests administered upon entrance to a tertiary mathematics unit at the U. of M., and questionnaires. In 1991, V.C.E, students and H.S.C. students attempted the same mathematics test at the U. of M. and their results were compared. In 1992, the tests were attempted by V.C.E. students only. To compare new V.C.E. students and H.S.C. students, questions on the 1991 test were matched with similar questions on the 1992 tests and a panel of experts determined what the H.S.C. students who attempted the 1991 test would have been expected to average on these matched questions on the 1992 tests had they attempted them. These expected average scores were then compared with the actual scores of the new V.C.E. students. The scores of the groups were scaled when necessary. Questionnaires were administered to 1991 U. of M, mathematics students who were part of the V.C.E. pilot group in 1990, secondary mathematics educators, tertiary mathematics educators, and 1991 V.C.E. (1992 U. of M.) students. The mathematical misconceptions exhibited by new V.C.E. students are discussed and their frequencies stated. The research indicates that the new V.C.E. mathematics courses have provided the V.C.E. mathematics students in this study with significantly lower skill levels and a significantly poorer preparation for a tertiary mathematics unit than those which were previously provided by the H.S.C. mathematics courses.

Revealing conceptions of rate of change

Rate of change is an important mathematical concept. Research referring to students’ difficulties with this concept spans more than twenty years. Research suggests that problems experienced by some calculus students are likely a result of pre-existing limited or incorrect conceptions of rate of change. This study investigated 23 Victorian Year 10 students’ understanding of rate as revealed by phenomenographic analysis of interviews. Eight conceptions of rate of change emerged. Four important aspects of the concept were identified and gaps in students’ thinking defined. In addition, the employment of phenomenography, to reveal conceptions of rate, is described in detail.

Syzygies of the polynomial invariants of the Riemann tensor

This thesis introduces a novel way of writing polynomial invariants as network graphs, and applies this diagrammatic notation scheme, in conjunction with graph theory, to derive algorithms for constructing relationships (syzygies) between different invariants. These algorithms give rise to a constructive solution of a longstanding classical problem in invariant theory.

Where is the rate in the rule?

A well-developed understanding of rate is foundational to conceptual understanding of introductory calculus. Many students achieve procedural competence with the application of rules for differentiation without developing an awareness of the connection between derivative and rate. In addition, rate-related reasoning is needed to make informed decisions in many everyday applications of rate. This paper reports on additional data collected during interviews for a project investigating the different ways rate may be experienced by pre-calculus students. Many researchers (for example Kaput, 1999) have suggested that the conceptual understanding of function may be enhanced through the presentation and exploration of multiple representations of a variety of functions. In this paper, one section of each interview is considered in detail to evaluate the participants’ understanding in a specific rate context. Participants were asked to discuss a dynamic geometry simulation of a blind on two different windows one rectangular and the other not. Detailed analysis of the video-record of each participant’s interview provides insights into their perceptions of rate in several different representations. In the sections below, the conceptual framework is described; details of the interviews and the computer-based simulation are provided; and the analysis of the data is discussed.

Impact of context and representation on year 10 students' expression of conceptions of rate

Rate is an important, but difficult mathematical concept. More than twenty years of research, especially with calculus students, report difficulties with this concept. This paper reports on an alternative analysis, from the perspective of multiple representations and context, of interviews probing twenty Victorian Year 10 students’ conceptions of rate. This analysis shows that multiple representations of functions provide different rate-related
information for different students. Understandings of rate in one representation or context are not necessarily transferred to another representation or context.

The play of power and politics in innovation and HRM

This paper analyses the human resource management (HRM) practices involved in the implementation of a process innovation approach to product development (concurrent engineering (CE)) in the Australian subsidiary of a multinational firm engaged in military defence electronics. According to the research literature, almost all aspects of managing product development under a CE approach are linked to people management. Yet in this particular case, other than project team structure, the prescriptive HRM dimensions of CE were conspicuously absent in the implementation process. This absence is explained by the play of power and politics involving stakeholders analysed over an 18 month period. The implications of this analysis for understanding the embedded, interdependent and political nature of HRM and process innovation are addressed.

Innovation and HRM : absences and politics

This article analyses the role of HRM practices in the implementation of an innovative cross-functional approach to new product development (concurrent engineering, CE) in Eurotech Industries. Contrary to CE methodology stipulations, and despite supportive conditions, HRM received scant attention in the implementation process. Organizational power and politics were clearly involved in this situation, and this article explores how their play created such HRM ‘absences’. The article builds on a four-dimensional view of power in order to provide a deeper understanding of the embedded, interdependent and political nature of HRM practice and innovation.

Differential modulation of spinal and corticospinal excitability during drop jumps

Previously it was shown that spinal excitability during hopping and drop jumping is high in the initial phase of ground contact when the muscle is stretched but decreases toward takeoff. To further understand motor control of stretch-shortening cycle, this study aimed to compare modulation of spinal and corticospinal excitability at distinct phases following ground contact in drop jump. Motor-evoked potentials (MEPs) induced by transcranial magnetic stimulation (TMS) and H-reflexes were elicited at the time of the short (SLR)-, medium (MLR)-, and long (LLR, LLR2)-latency responses of the soleus muscle (SOL) after jumps from 31 cm height. MEPs and H-reflexes were expressed relative to the background electromyographic (EMG) activity. H-reflexes were highly facilitated at SLR (172%) and then progressively decreased (MLR = 133%; LLR = 123%; LLR2 = 110%). TMS showed no effect at SLR, MLR, and LLR, whereas MEPs were significantly facilitated at the LLR2 (122%; P = 0.003). Background EMG was highest at LLR and lowest at LLR2. Strong H-reflex facilitation at the beginning of the stance phase indicated significant contribution of Ia-afferent input to the α-motoneurons during this phase that then progressively declined toward takeoff. Conversely, corticospinal excitability was exclusively increased at the phase of push off (LLR2, ∼120 ms). It is argued that corticomotoneurons increased their excitability at LLR2. At LLR (∼90 ms), Ia-afferent transmission as well as corticospinal excitability was low, whereas background EMG was high. Therefore it is speculated that other sources, presumably subcortical in origin, contributed to the EMG activity at LLR in drop jumps.

An investigation of middle secondary students' mathematical conceptions of rate

This study identified four educationally critical aspects of ‘rate’ through analysis of phenomenographic, video-recorded interviews with Year 10 students. Further analysis indicated understandings in one representation or context were not necessarily transferable.

What is rate? Does context or representation matter?

Rate is an important, but difficult, mathematical concept. Despite more than 20 years of research, especially with calculus students, difficulties are reported with this concept. This paper reports the results from analysis of data from 20 Australian Grade 10 students. Interviews targeted students’ conceptions of rate, focussing on the influence of representation and context on their expression of their understanding of rate. This analysis shows that different representations of functions provide varying levels of rate-related information for individual students. Understandings of rate in one representation or context are not necessarily transferred to another representation or context. Rate is an important, but commonly misunderstood, mathematical concept with many everyday applications (Swedosh, Dowsey, Caruso, Flynn, & Tynan, 2007). It is a complicated concept comprising many interwoven ideas such as the ratio of two numeric, measurable quantities but in a context where both quantities are changing. In mathematics classes, this is commonly expressed as change in the dependent variable resulting from a unit change in the independent variable, and variously described as constant or variable rate; average or instantaneous rate. In addition, rate may be seen as a purely abstract mathematical notion or embedded in the understanding of real-world applications. This paper explores the research question: Are students’ expressions of their conceptions of rate affected by either context or mathematical representation? This question was part of a larger study (Herbert, 2010) conducted with Grade 10 students from the Australian state of Victoria.

The capacity of criminal sanctions to shape the behaviour of offenders : specific deterrence doesn’t work, rehabilitation might and the implications for sentencing

There is a considerable gap between the law and knowledge regarding the efficacy of state-imposed sanctions to achieve several key sentencing objectives. Two sentencing objectives which often carry considerable weight in the sentencing calculus are rehabilitation and specific deterrence, despite the fact that neither has been proven to be attainable. This article examines the empirical data on whether specific deterrence and rehabilitation are attainable, and consequently whether they should be retained or abolished as sentencing objectives.

The paradox of parity in sentencing in Australia: the pursuit of equal justice that highlights the futility of consistency in sentencing

Parity in sentencing is the principle that offenders who are parties to a crime should, all things being equal, receive the same penalty. While it is a well-established principle, the reality is that its scope is greatly limited by the largely unfettered nature of the sentencing calculus. Things are rarely equal between offenders due to the large number of variables that current orthodoxy maintains are relevant to sentencing. This makes application of the parity principle unpredictable, resulting in the paradox that parity highlights the unfairness that it is meant to mitigate: inconsistency in sentencing. This article contends that parity will remain an aspiration, as opposed to a concrete principle, until the instinctive synthesis approach to sentencing yields to a more transparent and precise decision-making process. The article focuses on Australian jurisprudence, but the analysis applies to all jurisdictions where sentencing has a considerable discretionary component (including the UK and the USA--apart from the limited circumstances where mandatory sentences apply).