Scaffolding students’ thinking in mathematical investigations

This article describes some of the issues that teachers might encounter when scaffolding students’ thinking during mathematical investigations. It describes four episodes where a teacher’s scaffolding failed to support students’ mathematical thinking and explores the reasons why the scaffolding was ineffective. Understanding what is ineffective and why is one way to improve pedagogical practice. As a background to these episodes, we first provide an overview of the mathematical investigation. Our paper concludes with some recommendations for judicious scaffolding during investigations.

Numerical investigations of linear least squares methods for derivative estimation

The results of a numerical investigation into the errors for least squares estimates of function gradients are presented. The underlying algorithm is obtained by constructing a least squares problem using a truncated Taylor expansion. An error bound associated with this method contains in its numerator terms related to the Taylor series remainder, while its denominator contains the smallest singular value of the least squares matrix. Perhaps for this reason the error bounds are often found to be pessimistic by several orders of magnitude. The circumstance under which these poor estimates arise is elucidated and an empirical correction of the theoretical error bounds is conjectured and investigated numerically. This is followed by an indication of how the conjecture is supported by a rigorous argument.

Real time detectionof airborne fungal spores and investigations into their dynamics in indoor air

Concern regarding the health effects of indoor air quality has grown in recent years, due to the increased prevalence of many diseases, as well as the fact that many people now spend most of their time indoors. While numerous studies have reported on the dynamics of aerosols indoors, the dynamics of bioaerosols in indoor environments are still poorly understood and very few studies have focused on fungal spore dynamics in indoor environments. Consequently, this work investigated the dynamics of fungal spores in indoor air, including fungal spore release and deposition, as well as investigating the mechanisms involved in the fungal spore fragmentation process. In relation to the investigation of fungal spore dynamics, it was found that the deposition rates of the bioaerosols (fungal propagules) were in the same range as the deposition rates of nonbiological particles and that they were a function of their aerodynamic diameters. It was also found that fungal particle deposition rates increased with increasing ventilation rates. These results (which are reported for the first time) are important for developing an understanding of the dynamics of fungal spores in the air. In relation to the process of fungal spore fragmentation, important information was generated concerning the airborne dynamics of the spores, as well as the part/s of the fungi which undergo fragmentation. The results obtained from these investigations into the dynamics of fungal propagules in indoor air significantly advance knowledge about the fate of fungal propagules in indoor air, as well as their deposition in the respiratory tract. The need to develop an advanced, real-time method for monitoring bioaerosols has become increasingly important in recent years, particularly as a result of the increased threat from biological weapons and bioterrorism. However, to date, the Ultraviolet Aerodynamic Particle Sizer (UVAPS, Model 3312, TSI, St Paul, MN) is the only commercially available instrument capable of monitoring and measuring viable airborne micro-organisms in real-time. Therefore (for the first time), this work also investigated the ability of the UVAPS to measure and characterise fungal spores in indoor air. The UVAPS was found to be sufficiently sensitive for detecting and measuring fungal propagules. Based on fungal spore size distributions, together with fluorescent percentages and intensities, it was also found to be capable of discriminating between two fungal spore species, under controlled laboratory conditions. In the field, however, it would not be possible to use the UVAPS to differentiate between different fungal spore species because the different micro-organisms present in the air may not only vary in age, but may have also been subjected to different environmental conditions. In addition, while the real-time UVAPS was found to be a good tool for the investigation of fungal particles under controlled conditions, it was not found to be selective for bioaerosols only (as per design specifications). In conclusion, the UVAPS is not recommended for use in the direct measurement of airborne viable bioaerosols in the field, including fungal particles, and further investigations into the nature of the micro-organisms, the UVAPS itself and/or its use in conjunction with other conventional biosamplers, are necessary in order to obtain more realistic results. Overall, the results obtained from this work on airborne fungal particle dynamics will contribute towards improving the detection capabilities of the UVAPS, so that it is capable of selectively monitoring and measuring bioaerosols, for which it was originally designed. This work will assist in finding and/or improving other technologies capable of the real-time monitoring of bioaerosols. The knowledge obtained from this work will also be of benefit in various other bioaerosol applications, such as understanding the transport of bioaerosols indoors.

Biomechanical aspects of bone microstructure in vertebrates : potential approach to paleontological investigations

The biomechanical or biophysical principles can be applied to study biological structures in their modern or fossil form. Bone is an important tissue in paleontological studies as it is a commonly preserved element in most fossil vertebrates, and can often allow its microstructures such as lacuna and canaliculi to be studied in detail. In this context, the principles of Fluid Mechanics and Scaling Laws have been previously applied to enhance the understanding of bone microarchitecture and their implications for the evolution of hydraulic structures to transport fluid. It has been shown that the microstructure of bone has evolved to maintain efficient transport between the nutrient supply and cells, the living components of the tissue. Application of the principle of minimal expenditure of energy to this analysis shows that the path distance comprising five or six lamellar regions represents an effective limit for fluid and solute transport between the nutrient supply and cells; beyond this threshold, hydraulic resistance in the network increases and additional energy expenditure is necessary for further transportation. This suggests an optimization of the size of bone’s building blocks (such as osteon or trabecular thickness) to meet the metabolic demand concomitant to minimal expenditure of energy. This biomechanical aspect of bone microstructure is corroborated from the ratio of osteon to Haversian canal diameters and scaling constants of several mammals considered in this study. This aspect of vertebrate bone microstructure and physiology may provide a basis of understanding of the form and function relationship in both extinct and extant taxa.

Reconstruction of falsified computer logs for digital forensics investigations

Digital forensics investigations aim to find evidence that helps confirm or disprove a hypothesis about an alleged computer-based crime. However, the ease with which computer-literate criminals can falsify computer event logs makes the prosecutor's job highly challenging. Given a log which is suspected to have been falsified or tampered with, a prosecutor is obliged to provide a convincing explanation for how the log may have been created. Here we focus on showing how a suspect computer event log can be transformed into a hypothesised actual sequence of events, consistent with independent, trusted sources of event orderings. We present two algorithms which allow the effort involved in falsifying logs to be quantified, as a function of the number of `moves' required to transform the suspect log into the hypothesised one, thus allowing a prosecutor to assess the likelihood of a particular falsification scenario. The first algorithm always produces an optimal solution but, for reasons of efficiency, is suitable for short event logs only. To deal with the massive amount of data typically found in computer event logs, we also present a second heuristic algorithm which is considerably more efficient but may not always generate an optimal outcome.

Morphological investigations of supported titania photocatalysts

Due in no small part to an increasing need to augment existing water purification strategies, the synthesis of titania photocatalysts has been under considerable examination. However, in order to make the use of titania photocatalysts commercially viable there needs to be an increase in the efficiency of the catalysts while decreasing the potential toxicity. Due to its high porosity and novel optical properties, inverse opal titania derived from colloidal crystal templating offers one of the most efficient solutions. While a number of synthesis methods for inverse opal titania have been presented in the literature, the co�]deposition method offers the most effective method of generating the relative large areas of inverse opal material. The factors which affect the codeposition method and the mechanism by which titania inverse opals form in general remain relatively unstudied. This manuscript presents an examination of the morphology of inverse opals generated by the co�]deposition method while proposing a mechanism by which the inverse structures form.

Mechanistic investigations into interactions between IGF-I and IGFBPs and their impact on facilitating cell migration on vitronectin

Numerous studies have reported links between insulin-like growth factors (IGFs) and the extra-cellular matrix protein vitronectin (VN). We ourselves have reported that IGF-I binds to VN via IGF-binding proteins (IGFBPs) to stimulate HaCaT and MCF-7 cell migration. Here, we detail the functional evaluation of IGFBP-1, -2, -3, -4 and -6 in the presence and absence of IGF-I and VN. The data presented here, combined with our prior data on IGFBP-5, suggest that IGFBP-3, -4 and -5 are the most effective at stimulating cell migration in combination with IGF-I and VN. In addition, we demonstrate that different regions within IGFBP-3 and -4 are critical for complex formation. Furthermore, we examine whether multi-protein complexes of IGF-I and IGFBPs associated with fibronectin and collagen IV are also able to enhance functional biological responses.

Spatiotemporal investigations of DNA damage repair using microbeams

Cellular response to radiation damage is made by a complex network of pathways and feedback loops whose spatiotemporal organisation is still unclear despite its decisive role in determining the fate of the damaged cell. Revealing the dynamic sequence of the repair proteins is therefore critical in understanding how the DNA repair mechanisms work. There are also still open questions regarding the possible movement of damaged chromatin domains and its role as trigger for lesion recognition and signalling in the DNA repair context. The single-cell approach and the high spatial resolution offered by microbeams provide the perfect tool to study and quantify the dynamic processes associated with the induction and repair of DNA damage. We have followed the development of radiation-induced foci for three DNA damage markers (i.e. γ-H2AX, 53BP1 and hSSB1) using normal fibroblasts (AG01522), human breast adenocarcinoma cells (MCF7) and human fibrosarcoma cells (HT1080) stably transfected with yellow fluorescent protein fusion proteins following irradiation with the QUB X-ray microbeam (carbon X-rays <2 µm spot). The size and intensity of the foci has been analysed as a function of dose and time post-irradiation to investigate the dynamics of the above-mentioned DNA repair processes and monitor the remodelling of chromatin structure that the cell undergoes to deal with DNA damage.

Noising around : investigations in mobile learning

In this paper we present an account of children's interactions with a mobile technology prototype within a school context. The Noise Detectives trial was conducted in a school setting with the aim of better understanding the role of mobile technology as a mediator within science learning activities. Over eighty children, aged between ten and twelve, completed an outdoor data gathering activity using a mobile learning prototype that included paper and digital components. They measured and recorded noise levels at a range of locations throughout the schools. We analyzed the activity to determine how the components of the prototype were integrated into the learning activity, and to identify differences in behavior that resulted from using these components. We present design implications that resulted from observed differences in prototype use and appropriation.