A multiplication curriculum : a short outline

Teachers not only know how to multiply, they also know how to teach children multiplication. But at any one time, most teachers are only concerned with a small section of the whole large process of teaching multiplication. It is easy to lose sight of the wood, because of the close attention being given to individual trees. What is the larger picture? How do children learn to multiply, asks the author? The author discusses the progressive stages of ideas and processes that are involved in learning to multiply. He also provides questions to assist teachers with identifying how far students have progressed in their understanding of multiplication.

Legionella pneumophila multiplication is enhanced by chronic AMPK signalling in mitochondrially diseased dictyostelium cells

Human patients with mitochondrial diseases are more susceptible to bacterial infections, particularly of the respiratory tract. To investigate the susceptibility of mitochondrially diseased cells to an intracellular bacterial respiratory pathogen, we exploited the advantages of Dictyostelium discoideum as an established model for mitochondrial disease and for Legionella pneumophila pathogenesis. Legionella infection of macrophages involves recruitment of mitochondria to the Legionella-containing phagosome. We confirm here that this also occurs in Dictyostelium and investigate the effect of mitochondrial dysfunction on host cell susceptibility to Legionella. In mitochondrially diseased Dictyostelium strains, the pathogen was taken up at normal rates, but it grew faster and reached counts that were twofold higher than in the wild-type host. We reported previously that other mitochondrial disease phenotypes for Dictyostelium are the result of the activity of an energy-sensing cellular alarm protein, AMP-activated protein kinase (AMPK). Here, we show that the increased ability of mitochondrially diseased cells to support Legionella proliferation is suppressed by antisense-inhibiting expression of the catalytic AMPKα subunit. Conversely, mitochondrial dysfunction is phenocopied, and intracellular Legionella growth is enhanced, by overexpressing an active form of AMPKα in otherwise normal cells. These results indicate that AMPK signalling in response to mitochondrial dysfunction enhances Legionella proliferation in host cells.

Dean's great discovery : multiplication, division and fractions

Multiplication, division and fractions are 'hotspots' for students in the middle years with many students experiencing difficulty with these concepts (Siemon, Virgona & Cornielle, 2001). Arrays effectively model multiplication and help children develop multiplicative thinking and learn multiplication facts (Young-Loveridge, 2005). In this article we show how an open-ended array problem enabled a Grade 5/6 student to think about the relationship between multiplication, division and fractions. In the article we describe the project and 'hot spot' mathematical tasks that we used and provide some background on multiplicative thinking before presenting the case and a commentary (Western Melbourne Roundtable, 1997) of one student's exploration. This case was documented whilst we were working on a collaborative project with a team of upper primary teachers and a group of pre-service teachers at a local primary school.

Enabling efficient publicly verifiable outsourcing computation for matrix multiplication

Outsourcing heavy computational tasks to remote cloud server, which accordingly significantly reduce the computational burden at the end hosts, represents an effective and practical approach towards extensive and scalable mobile applications and has drawn increasing attention in recent years. However, due to the limited processing power of the end hosts yet the keen privacy concerns on the outsourced data, it is vital to ensure both the efficiency and security of the outsourcing computation in the cloud computing. In this paper, we address the issue by developing a publicly verifiable outsourcing computation proposal. In particular, considering a large amount of applications of matrix multiplication in large datasets and image processing, we propose a publicly verifiable outsourcing computation scheme for matrix multiplication in the amortized model. Security analysis demonstrates that the proposed scheme is provable secure by blinding input and output in a simple way. By comparing the developed scheme with existing proposals, we show that our proposal is more efficient in terms of functionality, as well as the computation, communication and storage overhead.

Students` impressions of the value of games for the learning of mathematics

The use of mathematical games in primary classrooms is commonplace in Australia. This paper reports on key findings from a larger investigation exploring the impact of games on mathematical learning, student attitudes, and behaviours. 222 Grade 5 and 6 children were taught multiplication and division of decimal numbers using calculator games. This paper raises questions about the students’ attitudes towards games as a vehicle for learning mathematics. One aspect reported in this paper is an apparent difference between students’ attitudes to games usage when data were
collected quantitatively compared with qualitatively.

Extending abelian groups to rings

For any abelian group G and any function f : G → G we define a commutative binary operation or `multiplication' on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p-group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p-group of odd order p².

Students' conflicting attitudes towards games as a vehicle for learning mathematics : a methodological dilemma

Mathematics games are widely employed in school classrooms for such reasons as a reward for early finishers or to enhance students' attitude towards mathematics. During a four week period, a total of 222 Grade 5 and 6 (9 to 12 years old) children from Melbourne, Australia, were taught multiplication and division of decimal numbers using calculator games or rich mathematical activities. Likert scale surveys of the children's attitudes towards games as a vehicle for learning mathematics revealed unexpectedly high proportions of negative attitudes at the conclusion of the research. In contrast, student interview data revealed positive associations between games and mathematical learning. This article reports on the methodological dilemma of resultant conflicting attitudinal data related to game- playing. Concerns arising from the divergence in the results are raised in this article. Implications based on the experience of this study may inform educational researchers about future methodological choices involving attitudinal research.

Diversions

The article provides information on the misconceptions in solving perimeter, area, volume and mass and on their respective interventions. Solving the perimeter of a rectangle of a lesser area through multiplication using its formula but may double the sum of its length and width to get its answer. Working out for the area of an irregular figure or a loop is solved by counting the unit-squares contained in the loop.

As fire burns : functional work and praxis in ontology

The context: the historical and philosophical demise of the Marxist model of praxis as a unity of theory and practice organized by a Party in service of a Cause. The task: to remodel praxis by distinguishing it from functional work. The proving ground: the discourse of ontology. The thesis works through four types of ontology in its attempt to construct different ontological schemas for praxis and functional work. In the first three ontologies, Platonic, Aristotelian and relativist, ontological impasses occur in the accounts of the relation between one and the multiple, and of the existence of order. They prevent the successful construction of a schema for functional work. It is in the set-theory ontology of Alain Badiou that the means arise for the passage through these impasses and the definitive construction of distinct ontological schemas for functional work and praxis. This results in a new concept of praxis and a multiplication of its domains beyond politics to science, art and love

Inbreeding and testicular abnormalities in a bottlenecked population of koalas (Phascolarctos cinereus)

Habitat destruction and fragmentation, interactions with introduced species or the relocation of animals to form new populations for conservation purposes may result in a multiplication of population bottlenecks. Examples are the translocations of koalas to French Island and its derivative Kangaroo Island population, with both populations established as insurance policies against koala extinction. In terms of population size, these conservation programs were success stories. However, the genetic story could be different. We conducted a genetic investigation of French and Kangaroo Island koalas by using 15 microsatellite markers, 11 of which are described here for the first time. The results confirm very low genetic diversity. French Island koalas have 3.8 alleles per locus and Kangaroo Island koalas 2.4. The present study found a 19% incidence of testicular abnormality in kangaroo Island animals. Internal relatedness, an individual inbreeding coefficient, was not significantly different in koalas with testicular abnormalities from that in other males, suggesting the condition is not related to recent inbreeding. It could instead result from an unfortunate selection of founder individuals carrying alleles for testicular abnormalities, followed by a subsequent increase in these alleles’ frequencies through genetic drift and small population-related inefficiency of selection. Given the low diversity and possible high prevalence of deleterious alleles, the genetic viability of the population remains uncertain, despite its exponential growth so far. This stands as a warning to other introductions for conservation reasons.

Engineering molecular self-assembled fibrillar networks by ultrasound

The architecture of self-organized three-dimensionally interconnected nanocrystal fibrillar networks has been achieved by ultrasound from a solution consisting of separate spherulites. The ultrasound stimulated structural transformation is correlated to the striking ultrasonic effects on turning nongelled solutions or weak gels into strong gels instantly, with enhancement of the storage modulus up to 3 magnitudes and up to 4 times more gelling capability. The basic principle involved in the ultrasound-induced structural transformation is established on the basis of the nucleation-and-growth model of a fiber network formation, and the mechanism of seeding multiplication, aggregation suppressing, and fiber distribution and growth promotion is proposed. This novel technique enables us to produce self-supporting gel functional materials possessing significantly modified macroscopic properties, from materials previously thus far considered to be “useless”, without the use of chemical stimuli. Moreover, it provides a general strategy for the engineering of self-organized fiber network architectures, and we are consequently able to achieve the supramolecular functional materials with controllable macroscopic properties.

Testing the effectiveness of mathematical games as a pedagogical tool for children's learning

In an effort to engage children in mathematics learning, many primary teachers use mathematical games and activities. Games have been employed for drill and practice, warm-up activities and rewards. The effectiveness of games as a pedagogical tool requires further examination if games are to be employed for the teaching of mathematical concepts. This paper reports research that compared the effectiveness of non-digital games with non-game but engaging activities as pedagogical tools for promoting mathematical learning. In the classrooms that played games, the effects of adding teacher-led whole class discussion was explored. The research was conducted with 10–12-year-old children in eight classrooms in three Australian primary schools, using differing instructional approaches to teach multiplication and division of decimals. A quasi-experimental design with pre-test, post-test and delayed post-test was employed, and the effects of the interventions were measured by the children’s written test performance. Test results indicated lesser gains in learning in game playing situations versus non-game activities and that teacher-led discussions during and following the game playing did not improve children’s learning. The finding that these games did not help children demonstrate a mathematical understanding of concepts under test conditions suggests that educators should carefully consider the application and appropriateness of games before employing them as a vehicle for introducing mathematical concepts.

Dematerialised data and human desire : the Internet and copy culture

Since Licklider in the 1960s [27] influential proponents of networked computing have envisioned electronic information in terms of a relatively small (even singular) number of 'sources', distributed through technologies such as the Internet. Most recently, Levy writes, in Becoming Virtual, that "in cyberspace, since any point is directly accessible from any other point, there is an increasing tendency to replace copies of documents with hypertext links. Ultimately, there will only need to be a single physical exemplar of the text" [13 p.61]. Hypertext implies, in theory, the end of 'the copy', and the multiplication of access points to the original. But, in practice, the Internet abounds with copying, both large and small scale, both as conscious human practice, and also as autonomous computer function. Effective and cheap data storage that encourages computer users to keep anything of use they have downloaded, lest the links they have found, 'break'; while browsers don't 'browse' the Internet - they download copies of everything to client machines. Not surprisingly, there is significant regulation against 'copying' - regulation that constrains our understanding of 'copying' to maintain a legal fiction of the 'original' for the purposes of intellectual property protection. In this paper, I will firstly demonstrate, by a series of examples, how 'copying' is more than just copyright infringement of music and software, but is a defining, multi-faceted feature of Internet behaviour. I will then argue that the Internet produces an interaction between dematerialised, digital data and human subjectivity and desire that fundamentally challenges notions of originality and copy. Walter Benjamin noted about photography: "one can make any number of prints [from a negative]; to ask for the 'authentic' print makes no sense" [4 p.224]. In cyberspace, I conclude, it makes no sense to ask which one is the copy.

Second-year pre-service teachers’ responses to proportional reasoning test items

A recent international study of pre-service teachers identified that proportional reasoning was problematic for pre-service teachers. Proportional reasoning is an important topic in the middle years of schooling and therefore it is critical that teachers understand this topic and can rely on their Mathematical Content Knowledge (MCK) when teaching. The focus of this paper is second-year Australian primary pre-service teachers’ MCK of real number items related to ratio, rate, proportion and proportional reasoning. This paper reports on strengths and weakness of pre-service teachers’ MCK when responding to test items; including a method suitable for analysing responses to five items and ranked by three levels of difficulty. The results revealed insights into their correct methods of solutions and common incorrect responses, identifying difficulty, where multiplication and division were required. The method of coding test items by difficulty ranking may assist with developing an appropriate learning trajectory, which will assist pre-service teachers develop their MCK of this and other difficult topics.