Tutoring with higher mathematics and the use of technology

We discuss three approaches to the use of technology as a teaching and learning tool that we are currently implementing for a target group of about one hundred second level engineering mathematics students. Central to these approaches is the underlying theme of motivating relatively poorly motivated students to learn, with the aim of improving learning outcomes. The approaches to be discussed have been used to replace, in part, more traditional mathematics tutorial sessions and lecture presentations. In brief, the first approach involves the application of constructivist thinking in the tertiary education arena, using technology as a computational and visual tool to create motivational knowledge conflicts or crises. The central idea is to model a realistic process of how scientific theory is actually developed, as proposed by Kuhn (1962), in contrast to more standard lecture and tutorial presentations. The second approach involves replacing procedural or algorithmic pencil-and-paper skills-consolidation exercises by software based tasks. Finally, the third approach aims at creating opportunities for higher order thinking via "on-line" exploratory or discovery mode tasks. The latter incorporates the incubation period method, as originally discussed by Rubinstein (1975) and others.

Archaea in the Mycorrhizosphere of Boreal Forest Trees

Archaea were long thought to be a group of ancient bacteria, which mainly lived in extreme environments. Due to the development of DNA sequencing methods and molecular phylogenetic analyses, it was shown that the living organisms are in fact divided into three domains; the Archaea, Bacteria and the Eucarya. Since the beginning of the previous decade, it was shown that archaea generally inhabit moderate environments and that these non-extremophilic archaea are more ubiquitous than the extremophiles. Group 1 of non-extreme archaea affiliate with the phylum Crenarchaeota. The most commonly found soil archaea belong to the subgroup 1.1b. However, the Crenarchaeota found in the Fennoscandian boreal forest soil belong to the subgroup 1.1c. The organic top layer of the boreal forest soil, the humus, is dominated by ectomycorrhizal fungal hyphae. These colonise virtually all tree fine root tips in the humus layer and have been shown to harbour distinct bacterial populations different from those in the humus. The archaea have also been shown to colonise both boreal forest humus and the rhizospheres of plants. In this work, studies on the archaeal communities in the ectomycorrhizospheres of boreal forest trees were conducted in microcosms. Archaea belonging to the group 1.1c Crenarchaeota and Euryarchaeota of the genera Halobacterium and Methanolobus were detected. The archaea generally colonised fungal habitats, such as ectomycorrhizas and external mycelia, rather than the non-mycorrhizal fine roots of trees. The species of ectomycorrhizal fungus had a great impact on the archaeal community composition. A stable euryarchaeotal community was detected especially in the mycorrhizas, of most of the tested Scots pine colonising ectomycorrhizal fungi. The Crenarchaeota appeared more sporadically in these habitats, but had a greater diversity than the Euryarchaeota. P. involutus mycorrhizas had a higher diversity of 1.1c Crenarchaeota than the other ectomycorrhizal fungi. The detection level of archaea in the roots of boreal trees was generally low although archaea have been shown to associate with roots of different plants. However, alder showed a high diversity of 1.1c Crenarchaeota, exceeding that of any of the tested mycorrhizas. The archaeal 16S rRNA genes detected from the non-mycorrhizal roots were different from those of the P. involutus mycorrhizas. In the phylogenetic analyses, the archaeal 16S rRNA gene sequences obtained from non-mycorrhizal fine roots fell in a separate cluster within the group 1.1c Crenarchaeota than those from the mycorrhizas. When the roots of the differrent tree species were colonised by P. involutus, the diversity and frequency of the archaeal populations of the different tree species were more similar to each other. Both Cren- and Euryarchaeota were enriched in cultures to which C-1 substrates were added. The 1.1c Crenarchaeota grew anaerobically in mineral medium with CH4 and CO2 as the only available C sources, and in yeast extract media with CO2 and CH4 or H2. The crenarchaeotal diversity was higher in aerobic cultures on mineral medium with CH4 or CH3OH than in the anaerobic cultures. Ecological functions of the mycorrhizal 1.1c Crenarchaeota in both anaerobic and aerobic cycling of C-1 compounds were indicated. The phylogenetic analyses did not divide the detected Crenarchaeota into anaerobic and aerobic groups. This may suggest that the mycorrhizospheric crenarchaeotal communities consist of closely related groups of anaerobic and aerobic 1.1c Crenarchaeota, or the 1.1c Crenarchaeota may be facultatively anaerobic. Halobacteria were enriched in non-saline anaerobic yeast extract medium cultures in which CH4 was either added or produced, but were not detected in the aerobic cultures. They may potentially be involved in anaerobic CH4 cycling in ectomycorrhizas. The CH4 production of the mycorrhizal samples was over 10 times higher than for humus devoid of mycorrhizal hyphae, indicating a high CH4 production potential of the mycorrhizal metanogenic community. Autofluorescent methanogenic archaea were detected by microscopy and 16S rRNA gene sequences of the genus Methanolobus were obtained. The archaeal community depended on both tree species and the type of ectomycorrhizal fungus colonising the roots and the Cren- and Euryarchaeota may have different ecological functions in the different parts of the boreal forest tree rhizosphere and mycorrhizosphere. By employing the results of this study, it may be possible to isolate both 1.1c Crenarchaeota as well as non-halophilic halobacteria and aerotolerant methanogens from mycorrhizospheres. These archaea may be used as indicators for change in the boreal forest soil ecosystem due to different factors, such as exploitations of forests and the rise in global temperature. More information about the microbial populations with apparently low cell numbers but significant ecological impacts, such as the boreal forest soil methanogens, may be of crucial importance to counteract human impacts on such globally important ecosystems as the boreal forests.

A playful side to twelfth-century mathematics

As editors of the book Lilavati's Daughters: The Women Scientists of India, reviewed by Asha Gopinathan (Nature 460, 1082; 2009), we would like to elaborate on the background to its title. Lilavati was a mathematical treatise of the twelfth century, composed by the mathematician and astronomer Bhaskaracharya (1114–85) — also known as Bhaskara II — who was a teacher of repute and author of several other texts. The name Lilavati, which literally means 'playful', is a surprising title for an early scientific book. Some of the mathematical problems posed in the book are in verse form, and are addressed to a girl, the eponymous Lilavati. However, there is little real evidence concerning Lilavati's historicity. Tradition holds that she was Bhaskaracharya's daughter and that he wrote the treatise to console her after an accident that left her unable to marry. But this could be a later interpolation, as the idea was first mentioned in a Persian commentary. An alternative view has it that Lilavati was married at an inauspicious time and was widowed shortly afterwards. Other sources have implied that Lilavati was Bhaskaracharya's wife, or even one of his students — raising the possibility that women in parts of the Indian subcontinent could have participated in higher education as early as eight centuries ago. However, given that Bhaskara was a poet and pedagogue, it is also possible that he chose to address his mathematical problems to a doe-eyed girl simply as a whimsical and charming literary device.

A model eliciting framework for integrating mathematics and robotics learning

Robotics is taught in many Australian ICT classrooms, in both primary and secondary schools. Robotics activities, including those developed using the LEGO Mindstorms NXT technology, are mathematics-rich and provide a fertile round for learners to develop and extend their mathematical thinking. However, this context for learning mathematics is often under-exploited. In this paper a variant of the model construction sequence (Lesh, Cramer, Doerr, Post, & Zawojewski, 2003) is proposed, with the purpose of explicitly integrating robotics and mathematics teaching and learning. Lesh et al.’s model construction sequence and the model eliciting activities it embeds were initially researched in primary mathematics classrooms and more recently in university engineering courses. The model construction sequence involves learners working collaboratively upon product-focussed tasks, through which they develop and expose their conceptual understanding. The integrating model proposed in this paper has been used to design and analyse a sequence of activities in an Australian Year 4 classroom. In that sequence more traditional classroom learning was complemented by the programming of LEGO-based robots to ‘act out’ the addition and subtraction of simple fractions (tenths) on a number-line. The framework was found to be useful for planning the sequence of learning and, more importantly, provided the participating teacher with the ability to critically reflect upon robotics technology as a tool to scaffold the learning of mathematics.

On the Structure of Contractible Edges in k-connected Partial k-trees

Contraction of an edge e merges its end points into a new single vertex, and each neighbor of one of the end points of e is a neighbor of the new vertex. An edge in a k-connected graph is contractible if its contraction does not result in a graph with lesser connectivity; otherwise the edge is called non-contractible. In this paper, we present results on the structure of contractible edges in k-trees and k-connected partial k-trees. Firstly, we show that an edge e in a k-tree is contractible if and only if e belongs to exactly one (k + 1) clique. We use this characterization to show that the graph formed by contractible edges is a 2-connected graph. We also show that there are at least |V(G)| + k - 2 contractible edges in a k-tree. Secondly, we show that if an edge e in a partial k-tree is contractible then e is contractible in any k-tree which contains the partial k-tree as an edge subgraph. We also construct a class of contraction critical 2k-connected partial 2k-trees.

Building teachers' pedagogy practices in reasoning, to improve students' dispositions towards mathematics

Drawing on participatory action research, this study identifies the pedagogies necessary to advance reasoning, which is one of the proficiencies from the Australian Curriculum Mathematics, and explores how reasoning leads to greater productive disposition. With the current emphasis on STEM in schools, this research is timely. This thesis makes an original and substantive contribution to the understanding of why and how teachers can most effectively advance student proficiency in reasoning through targeted instructional strategies and style of instruction. The study explores the ways in which teacher practices, when focused on reasoning, enhance the disposition of students towards greater mathematical proficiency.

Mapping of fuelwood trees using geoinformatics

Rural population of India constitutes about 70% of the total population and traditional fuels account for 75% of the rural energy needs. Depletion of woodlands coupled with the persistent dependency on fuel wood has posed a serious problem for household energy provision in many parts. This study highlights that the traditional fuels still meet 85-95% of fuel needs in rural areas of Kolar district: people prefer fuel wood for cooking and agriculture residues for water heating and other purposes. However, rapid changes in land cover and land use in recent times have affected these traditional fuels availability necessitating inventorying, mapping and monitoring of bioresources for sustainable management of bioresources. Remote sensing data (Multispectal and Panchromatic), Geographic Information System (GIS), field surveys and non-destructive sampling were used to assess spatially the availability and demand of energy. Field surveys indicate that rural household depends on species such as Prosopis juliflora, Acacia nilotica, Acacia auriculiformis to meet fuel wood requirement for domestic activities. Hence, to take stock of fuel wood availability, mapping was done at species level (with 88% accuracy) considering villages as sampling units using fused multispectral and panchromatic data. (C) 2009 Elsevier Ltd. All rights reserved.

Branch-branch connections in trees analogous to hyphal fusions in fungal colonies

the leopard tree Caesalpinia ferrea (Leguminosae) a native of eastern Brazil-some of the leader branches connect to and fuse with neighbouring branches of the same tree. The bridge initials project out as pegs or protuberances and apparently extend in a coordinated manner, connecting branches up to 4 ft apart. The fusion of two branches of the same tree implies intra-plant communication involving signaling factor(s). The bridges resemble fusions between hyphae in a fungal colony. Whereas hyphal fusions are common and the process is apparently completed in <1 h, branch fusions in C. ferrea tree are limited and a slow process, apparently requiring several months to years to complete. Branch fusions in C. ferrea are in accord with Claus Mattheck's analysis that tree branches actually seek contact rather than avoid contacts.

Mathematics, nature and cryptography: Insights from philosophy of information

One influential image that is popular among scientists is the view that mathematics is the language of nature. The present article discusses another possible way to approach the relation between mathematics and nature, which is by using the idea of information and the conceptual vocabulary of cryptography. This approach allows us to understand the possibility that secrets of nature need not be written in mathematics and yet mathematics is necessary as a cryptographic key to unlock these secrets. Various advantages of such a view are described in this article.

Prosopis, an Alien among the Sacred Trees of South India

The problematic of invasive species in an alien environment has aroused the attention of scientists all over the world for quite some time. One of the exotic tree species that has provoked special attention in the tropical drylands is Prosopis juliflora. Originating in South America, prosopis (hereafter referred to as prosopis) has been introduced in the hot and semi-arid zones of the world particularly to provide fuelwood, to stabilize sand dunes and to combat desertification. The tree has become an essential source for fuelwood and a provider of several other products and services in areas where it has become established. However, despite the numerous benefits the tree provides to rural people, in several regions prosopis has become a noxious weed with a negative impact on the environment and to the economy of farmers and landowners. In India, prosopis was introduced in Andhra Pradesh in 1877. The tree was then proclaimed as the precious child of the plant world by scientists and local people alike. The purpose of this study was to investigate the overall impact of prosopis on local rural livelihoods in the drylands of South India. Of particular interest was the examination of the different usages of the tree, especially as fuelwood, and people s perceptions of it. Furthermore, the study examined the negative impacts of the uncontrolled invasion of prosopis on croplands, and its occupation of the banks of irrigation canals and other water sources. As another central theme, this study analysed the Hindu classification system for nature and for trees in particular. In India, several tree species are regarded as sacred. This study examined the position of the exotic prosopis among sacred trees, such as the bodhi, banyan and neem trees. The principle method for collecting the field data was by using individual and thematic group interviews. These interviews were semi-structured with open ended questions. Moreover, unstructured interviews as well as general observations provided complementary information. The data were gathered during two fieldwork periods in the states of Andhra Pradesh and Tamil Nadu, in South India. The results confirmed that prosopis both provides benefits and causes hazards to different stakeholders. Farmers and agriculturalists suffer economic losses in areas where prosopis has invaded crop fields and competes with other plants for water and nutrients. On the other hand, for a significant number of poor rural people, prosopis has become an important source of livelihood benefits. This tree, which grows on government wastelands, is commonly a free resource for all and has thus become a major local source of fuelwood. It also provides several other goods and services and cash income that contributes to improve livelihoods in rural communities. Prosopis ranked lowest in the tree classificatioin system of the Hindus of South India. Although it is appreciated for many benefits it provides for poor people, it has remained an outsider compared with the indigenous tree species. On the other hand, the most sacred trees, such as the bodhi or the banyan, are completely excluded from extraction and it is seen as a sacrilege to even cut branches from any of these trees. An unexpected finding was that, in a few cases, prosopis had also been elevated to the status of a sacred tree. Goods and services from prosopis are not utilized in the most beneficial way. Silvicultural management practices are suggested that would provide additional income and employment opportunities. Interventions are recommended to control further invasion of the tree that might cause serious negative effects in the future. For Hindus, the sacred always ranks highest, even above economic gain. The conservation of sacred groves and sacred trees is a tradition that has its roots in ancient history. These socio-religious practices need to be respected and continued. Successful management of tree and forest resources depends on the willingness of the local people to manage their natural resources, and this willingness exists and has always existed in South India. Keywords: South India, drylands, livelihood, fuelwood, invasive, resource, silviculture.

A Geometric Algorithm for Learning Oblique Decision Trees

In this paper we present a novel algorithm for learning oblique decision trees. Most of the current decision tree algorithms rely on impurity measures to assess goodness of hyperplanes at each node. These impurity measures do not properly capture the geometric structures in the data. Motivated by this, our algorithm uses a strategy, based on some recent variants of SVM, to assess the hyperplanes in such a way that the geometric structure in the data is taken into account. We show through empirical studies that our method is effective.

Reducing the babel in plant volatile communication: using the forest to see the trees

While plants of a single species emit a diversity of volatile organic compounds (VOCs) to attract or repel interacting organisms, these specific messages may be lost in the midst of the hundreds of VOCs produced by sympatric plants of different species, many of which may have no signal content. Receivers must be able to reduce the babel or noise in these VOCs in order to correctly identify the message. For chemical ecologists faced with vast amounts of data on volatile signatures of plants in different ecological contexts, it is imperative to employ accurate methods of classifying messages, so that suitable bioassays may then be designed to understand message content. We demonstrate the utility of `Random Forests' (RF), a machine-learning algorithm, for the task of classifying volatile signatures and choosing the minimum set of volatiles for accurate discrimination, using datam from sympatric Ficus species as a case study. We demonstrate the advantages of RF over conventional classification methods such as principal component analysis (PCA), as well as data-mining algorithms such as support vector machines (SVM), diagonal linear discriminant analysis (DLDA) and k-nearest neighbour (KNN) analysis. We show why a tree-building method such as RF, which is increasingly being used by the bioinformatics, food technology and medical community, is particularly advantageous for the study of plant communication using volatiles, dealing, as it must, with abundant noise.

Splittings of free groups, normal forms and partitions of ends

Splittings of a free group correspond to embedded spheres in the 3-manifold M = # (k) S (2) x S (1). These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in pi (2)(M) can be represented by an embedded sphere.