Politics, Power, PISA: a genealogy of mathematics education policy at second level in Ireland at the beginning of the 21st century

This thesis traces a genealogy of the discourse of mathematics education reform in Ireland at the beginning of the twenty first century at a time when the hegemonic political discourse is that of neoliberalism. It draws on the work of Michel Foucault to identify the network of power relations involved in the development of a single case of curriculum reform – in this case Project Maths. It identifies the construction of an apparatus within the fields of politics, economics and education, the elements of which include institutions like the OECD and the Government, the bureaucracy, expert groups and special interest groups, the media, the school, the State, state assessment and international assessment. Five major themes in educational reform emerge from the analysis: the arrival of neoliberal governance in Ireland; the triumph of human capital theory as the hegemonic educational philosophy here; the dominant role of OECD/PISA and its values in the mathematics education discourse in Ireland; the fetishisation of western scientific knowledge and knowledge as commodity; and the formation of a new kind of subjectivity, namely the subjectivity of the young person as a form of human-capital-to-be. In particular, it provides a critical analysis of the influence of OECD/PISA on the development of mathematics education policy here – especially on Project Maths curriculum, assessment and pedagogy. It unpacks the arguments in favour of curriculum change and lays bare their ideological foundations. This discourse contextualises educational change as occurring within a rapidly changing economic environment where the concept of the State’s economic aspirations and developments in science, technology and communications are reshaping both the focus of business and the demands being put on education. Within this discourse, education is to be repurposed and its consequences measured against the paradigm of the Knowledge Economy – usually characterised as the inevitable or necessary future of a carefully defined present.

An investigation of post-primary students' images of mathematics

This research study investigates the image of mathematics held by 5th-year post-primary students in Ireland. For this study, “image of mathematics” is conceptualized as a mental representation or view of mathematics, presumably constructed as a result of past experiences, mediated through school, parents, peers or society. It is also understood to include attitudes, beliefs, emotions, self-concept and motivation in relation to mathematics. This study explores the image of mathematics held by a sample of 356 5th-year students studying ordinary level mathematics. Students were aged between 15 and 18 years. In addition, this study examines the factors influencing students‟ images of mathematics and the possible reasons for students choosing not to study higher level mathematics for the Leaving Certificate. The design for this study is chiefly explorative. A questionnaire survey was created containing both quantitative and qualitative methods to investigate the research interest. The quantitative aspect incorporated eight pre-established scales to examine students‟ attitudes, beliefs, emotions, self-concept and motivation regarding mathematics. The qualitative element explored students‟ past experiences of mathematics, their causal attributions for success or failure in mathematics and their influences in mathematics. The quantitative and qualitative data was analysed for all students and also for students grouped by gender, prior achievement, type of post-primary school attending, co-educational status of the post-primary school and the attendance of a Project Maths pilot school. Students‟ images of mathematics were seen to be strongly indicated by their attitudes (enjoyment and value), beliefs, motivation, self-concept and anxiety, with each of these elements strongly correlated with each other, particularly self-concept and anxiety. Students‟ current images of mathematics were found to be influenced by their past experiences of mathematics, by their mathematics teachers, parents and peers, and by their prior mathematical achievement. Gender differences occur for students in their images of mathematics, with males having more positive images of mathematics than females and this is most noticeable with regards to anxiety about mathematics. Mathematics anxiety was identified as a possible reason for the low number of students continuing with higher level mathematics for the Leaving Certificate. Some students also expressed low mathematical self-concept with regards to higher level mathematics specifically. Students with low prior achievement in mathematics tended to believe that mathematics requires a natural ability which they do not possess. Rote-learning was found to be common among many students in the sample. The most positive image of mathematics held by students was the “problem-solving image”, with resulting implications for the new Project Maths syllabus in post-primary education. Findings from this research study provide important insights into the image of mathematics held by the sample of Irish post-primary students and make an innovative contribution to mathematics education research. In particular, findings contribute to the current national interest in Ireland in post-primary mathematics education, highlighting issues regarding the low uptake of higher level mathematics for the Leaving Certificate and also making a preliminary comparison between students who took part in the piloting of Project Maths and students who were more recently introduced to the new syllabus. This research study also holds implications for mathematics teachers, parents and the mathematics education community in Ireland, with some suggestions made on improving students‟ images of mathematics.

Investigation of numerical atomic orbitals for first-principles calculations of the electronic and transport properties of silicon nanowire structures

This thesis is focused on the application of numerical atomic basis sets in studies of the structural, electronic and transport properties of silicon nanowire structures from first-principles within the framework of Density Functional Theory. First we critically examine the applied methodology and then offer predictions regarding the transport properties and realisation of silicon nanowire devices. The performance of numerical atomic orbitals is benchmarked against calculations performed with plane waves basis sets. After establishing the convergence of total energy and electronic structure calculations with increasing basis size we have shown that their quality greatly improves with the optimisation of the contraction for a fixed basis size. The double zeta polarised basis offers a reasonable approximation to study structural and electronic properties and transferability exists between various nanowire structures. This is most important to reduce the computational cost. The impact of basis sets on transport properties in silicon nanowires with oxygen and dopant impurities have also been studied. It is found that whilst transmission features quantitatively converge with increasing contraction there is a weaker dependence on basis set for the mean free path; the double zeta polarised basis offers a good compromise whereas the single zeta basis set yields qualitatively reasonable results. Studying the transport properties of nanowire-based transistor setups with p+-n-p+ and p+-i-p+ doping profiles it is shown that charge self-consistency affects the I-V characteristics more significantly than the basis set choice. It is predicted that such ultrascaled (3 nm length) transistors would show degraded performance due to relatively high source-drain tunnelling currents. Finally, it is shown the hole mobility of Si nanowires nominally doped with boron decreases monotonically with decreasing width at fixed doping density and increasing dopant concentration. Significant mobility variations are identified which can explain experimental observations.

First principles modelling of nucleation and growth during atomic layer deposition onto III-V substrates

Atomic layer deposition (ALD) is now used in semiconductor fabrication lines to deposit nanometre-thin oxide films, and has thus enabled the introduction of high-permittivity dielectrics into the CMOS gate stack. With interest increasing in transistors based on high mobility substrates, such as GaAs, we are investigating the surface treatments that may improve the interface characteristics. We focus on incubation periods of ALD processes on III-V substrates. We have applied first principles Density Functional Theory (DFT) to investigate detailed chemistry of these early stages of growth, specifically substrate and ALD precursor interaction. We have modelled the ‘clean-up’ effect by which organometallic precursors: trimethylaluminium (TMA) or hafnium and titanium amides clean arsenic oxides off the GaAs surface before ALD growth of dielectric commences and similar effect on Si3N4 substrate. Our simulations show that ‘clean-up’ of an oxide film strongly depends on precursor ligand, its affinity to the oxide and the redox character of the oxide. The predominant pathway for a metalloid oxide such as arsenic oxide is reduction, producing volatile molecules or gettering oxygen from less reducible oxides. An alternative pathway is non-redox ligand exchange, which allows non-reducible oxides (e.g. SiO2) to be cleaned-up. First principles study shows also that alkylamides are more susceptible to decomposition rather than migration on the oxide surface. This improved understanding of the chemical principles underlying ‘clean-up’ allows us to rationalize and predict which precursors will perform the reaction. The comparison is made between selection of metal chlorides, methyls and alkylamides precursors.

First principles simulation of amorphous silicon bulk, interfaces, and nanowires for photovoltaics

Amorphous silicon has become the material of choice for many technologies, with major applications in large area electronics: displays, image sensing and thin film photovoltaic cells. This technology development has occurred because amorphous silicon is a thin film semiconductor that can be deposited on large, low cost substrates using low temperature. In this thesis, classical molecular dynamics and first principles DFT calculations have been performed to generate structural models of amorphous and hydrogenated amorphous silicon and interfaces of amorphous and crystalline silicon, with the ultimate aim of understanding the photovoltaic properties of core-shell crystalline amorphous Si nanowire structures. We have shown, unexpectedly, from the simulations, that our understanding of hydrogenated bulk a-Si needs to be revisited, with our robust finding that when fully saturated with hydrogen, bulk a-Si exhibits a constant optical energy gap, irrespective of the hydrogen concentration in the sample. Unsaturated a-Si:H, with a lower than optimum hydrogen content, shows a smaller optical gap, that increases with hydrogen content until saturation is reached. The mobility gaps obtained from an analysis of the electronic states show similar behavior. We also obtained that the optical and mobility gaps show a volcano curve as the H content is varied from 7% (undersaturation) to 18% (mild oversaturation). In the case of mild over saturation, the mid-gap states arise exclusively from an increase in the density of strained Si-Si bonds. Analysis of our structures shows the extra H atoms in this case form a bridge between neighboring silicon atoms which increases the corresponding Si-Si distance and promotes bond length disorder in the sample. That has the potential to enhance the Staebler-Wronski effect. Planar interface models of amorphous-crystalline silicon have been generated in Si (100), (110) and (111) surfaces. The interface models are characterized by structure, RDF, electronic density of states and optical absorption spectrum. We find that the least stable (100) surface will result in the formation of the thickest amorphous silicon layer, while the most stable (110) surface forms the smallest amorphous region. We calculated for the first time band offsets of a-Si:H/c-Si heterojunctions from first principles and examined the influence of different surface orientations and amorphous layer thickness on the offsets and implications for device performance. The band offsets depend on the amorphous layer thickness and increase with thickness. By controlling the amorphous layer thickness we can potentially optimise the solar cell parameters. Finally, we have successfully generated different amorphous layer thickness of the a-Si/c-Si and a-Si:H/c-Si 5 nm nanowires from heat and quench. We perform structural analysis of the a-Si-/c-Si nanowires. The RDF, Si-Si bond length distributions, and the coordination number distributions of amorphous regions of the nanowires reproduce similar behaviour compared to bulk amorphous silicon. In the final part of this thesis we examine different surface terminating chemical groups, -H, - OH and –NH2 in (001) GeNW. Our work shows that the diameter of Ge nanowires and the nature of surface terminating groups both play a significant role in both the magnitude and the nature of the nanowire band gaps, allowing tuning of the band gap by up to 1.1 eV. We also show for the first time how the nanowire diameter and surface termination shifts the absorption edge in the Ge nanowires to longer wavelengths. Thus, the combination of nanowire diameter and surface chemistry can be effectively utilised to tune the band gaps and thus light absorption properties of small diameter Ge nanowires.

First-principles investigation of the alloy scattering potential in dilute Si1-xCx

A first-principles method is applied to find the intra and intervalley n-type carrier scattering rates for substitutional carbon in silicon. The method builds on a previously developed first-principles approach with the introduction of an interpolation technique to determine the intravalley scattering rates. Intravalley scattering is found to be the dominant alloy scattering process in Si1-xCx, followed by g-type intervalley scattering. Mobility calculations show that alloy scattering due to substitutional C alone cannot account for the experimentally observed degradation of the mobility. We show that the incorporation of additional charged impurity scattering due to electrically active interstitial C complexes models this residual resistivity well.

First principles calculation of electron-phonon and alloy scattering in strained SiGe

First-principles electronic structure methods are used to predict the mobility of n-type carrier scattering in strained SiGe. We consider the effects of strain on the electron-phonon deformation potentials and the alloy scattering parameters. We calculate the electron-phonon matrix elements and fit them up to second order in strain. We find, as expected, that the main effect of strain on mobility comes from the breaking of the degeneracy of the six Δ and L valleys, and the choice of transport direction. The non-linear effects on the electron-phonon coupling of the Δ valley due to shear strain are found to reduce the mobility of Si-like SiGe by 50% per % strain. We find increases in mobility between 2 and 11 times that of unstrained SiGe for certain fixed Ge compositions, which should enhance the thermoelectric figure of merit in the same order, and could be important for piezoresistive applications.

First-principles calculation of carrier-phonon scattering in n-type Si1−xGex alloys

First-principles electronic structure methods are used to find the rates of inelastic intravalley and intervalley n-type carrier scattering in Si1-xGex alloys. Scattering parameters for all relevant Delta and L intra- and intervalley scattering are calculated. The short-wavelength acoustic and the optical phonon modes in the alloy are computed using the random mass approximation, with interatomic forces calculated in the virtual crystal approximation using density functional perturbation theory. Optical phonon and intervalley scattering matrix elements are calculated from these modes of the disordered alloy. It is found that alloy disorder has only a small effect on the overall inelastic intervalley scattering rate at room temperature. Intravalley acoustic scattering rates are calculated within the deformation potential approximation. The acoustic deformation potentials are found directly and the range of validity of the deformation potential approximation verified in long-wavelength frozen phonon calculations. Details of the calculation of elastic alloy scattering rates presented in an earlier paper are also given. Elastic alloy disorder scattering is found to dominate over inelastic scattering, except for almost pure silicon (x approximate to 0) or almost pure germanium (x approximate to 1), where acoustic phonon scattering is predominant. The n-type carrier mobility, calculated from the total (elastic plus inelastic) scattering rate, using the Boltzmann transport equation in the relaxation time approximation, is in excellent agreement with experiments on bulk, unstrained alloys..

First-principles calculation of P-type alloy scattering in Si1-xGex

The p-type carrier scattering rate due to alloy disorder in Si1-xGex alloys is obtained from first principles. The required alloy scattering matrix elements are calculated from the energy splitting of the valence bands, which arise when one average host atom is replaced by a Ge or Si atom in supercells containing up to 128 atoms. Alloy scattering within the valence bands is found to be characterized by a single scattering parameter. The hole mobility is calculated from the scattering rate using the Boltzmann transport equation in the relaxation time approximation. The results are in good agreement with experiments on bulk, unstrained alloys..

First-principles calculation of alloy scattering in GexSi1-x

First-principles electronic structure methods are used to find the rates of intravalley and intervalley n-type carrier scattering due to alloy disorder in Si1-xGex alloys. The required alloy scattering matrix elements are calculated from the energy splitting of nearly degenerate Bloch states which arises when one average host atom is replaced by a Ge or Si atom in supercells containing up to 128 atoms. Scattering parameters for all relevant Delta and L intravalley and intervalley alloy scattering are calculated. Atomic relaxation is found to have a substantial effect on the scattering parameters. f-type intervalley scattering between Delta valleys is found to be comparable to other scattering channels. The n-type carrier mobility, calculated from the scattering rate using the Boltzmann transport equation in the relaxation time approximation, is in excellent agreement with experiments on bulk, unstrained alloys.

A first principles analysis of the effect of hydrogen concentration in hydrogenated amorphous silicon on the formation of strained Si-Si bonds and the optical and mobility gaps

In this paper, we use a model of hydrogenated amorphous silicon generated from molecular dynamics with density functional theory calculations to examine how the atomic geometry and the optical and mobility gaps are influenced by mild hydrogen oversaturation. The optical and mobility gaps show a volcano curve as the hydrogen content varies from undersaturation to mild oversaturation, with largest gaps obtained at the saturation hydrogen concentration. At the same time, mid-gap states associated with dangling bonds and strained Si-Si bonds disappear at saturation but reappear at mild oversaturation, which is consistent with the evolution of optical gap. The distribution of Si-Si bond distances provides the key to the change in electronic properties. In the undersaturation regime, the new electronic states in the gap arise from the presence of dangling bonds and strained Si-Si bonds, which are longer than the equilibrium Si-Si distance. Increasing hydrogen concentration up to saturation reduces the strained bonds and removes dangling bonds. In the case of mild oversaturation, the mid-gap states arise exclusively from an increase in the density of strained Si-Si bonds. Analysis of our structure shows that the extra hydrogen atoms form a bridge between neighbouring silicon atoms, thus increasing the Si-Si distance and increasing disorder in the sample.

Interaction of additive and quantization noises in digital PLLs

Phase-locked loops (PLLs) are a crucial component in modern communications systems. Comprising of a phase-detector, linear filter, and controllable oscillator, they are widely used in radio receivers to retrieve the information content from remote signals. As such, they are capable of signal demodulation, phase and carrier recovery, frequency synthesis, and clock synchronization. Continuous-time PLLs are a mature area of study, and have been covered in the literature since the early classical work by Viterbi [1] in the 1950s. With the rise of computing in recent decades, discrete-time digital PLLs (DPLLs) are a more recent discipline; most of the literature published dates from the 1990s onwards. Gardner [2] is a pioneer in this area. It is our aim in this work to address the difficulties encountered by Gardner [3] in his investigation of the DPLL output phase-jitter where additive noise to the input signal is combined with frequency quantization in the local oscillator. The model we use in our novel analysis of the system is also applicable to another of the cases looked at by Gardner, that is the DPLL with a delay element integrated in the loop. This gives us the opportunity to look at this system in more detail, our analysis providing some unique insights into the variance `dip' seen by Gardner in [3]. We initially provide background on the probability theory and stochastic processes. These branches of mathematics are the basis for the study of noisy analogue and digital PLLs. We give an overview of the classical analogue PLL theory as well as the background on both the digital PLL and circle map, referencing the model proposed by Teplinsky et al. [4, 5]. For our novel work, the case of the combined frequency quantization and noisy input from [3] is investigated first numerically, and then analytically as a Markov chain via its Chapman-Kolmogorov equation. The resulting delay equation for the steady-state jitter distribution is treated using two separate asymptotic analyses to obtain approximate solutions. It is shown how the variance obtained in each case matches well to the numerical results. Other properties of the output jitter, such as the mean, are also investigated. In this way, we arrive at a more complete understanding of the interaction between quantization and input noise in the first order DPLL than is possible using simulation alone. We also do an asymptotic analysis of a particular case of the noisy first-order DPLL with delay, previously investigated by Gardner [3]. We show a unique feature of the simulation results, namely the variance `dip' seen for certain levels of input noise, is explained by this analysis. Finally, we look at the second-order DPLL with additive noise, using numerical simulations to see the effects of low levels of noise on the limit cycles. We show how these effects are similar to those seen in the noise-free loop with non-zero initial conditions.

A study of the Hobson and Rogers volatility model

We firstly examine the model of Hobson and Rogers for the volatility of a financial asset such as a stock or share. The main feature of this model is the specification of volatility in terms of past price returns. The volatility process and the underlying price process share the same source of randomness and so the model is said to be complete. Complete models are advantageous as they allow a unique, preference independent price for options on the underlying price process. One of the main objectives of the model is to reproduce the `smiles' and `skews' seen in the market implied volatilities and this model produces the desired effect. In the first main piece of work we numerically calibrate the model of Hobson and Rogers for comparison with existing literature. We also develop parameter estimation methods based on the calibration of a GARCH model. We examine alternative specifications of the volatility and show an improvement of model fit to market data based on these specifications. We also show how to process market data in order to take account of inter-day movements in the volatility surface. In the second piece of work, we extend the Hobson and Rogers model in a way that better reflects market structure. We extend the model to take into account both first and second order effects. We derive and numerically solve the pde which describes the price of options under this extended model. We show that this extension allows for a better fit to the market data. Finally, we analyse the parameters of this extended model in order to understand intuitively the role of these parameters in the volatility surface.