Literature describes different stances with respect to conceptual and procedural mathematical knowledge. The concept-driven versus skills-orientated perspectives have led to “math wars” between researchers, while some mathematics education specialists advocate that the five strands of mathematical proficiency should be seen as interconnected. Conceptual knowledge is the knowledge of concepts or principles, and procedural knowledge the knowledge of procedures. Both types of knowledge are critical components of mathematical proficiency. This study used a mixed methods design to analyse the relationship between conceptual and procedural knowledge. The qualitative content analysis investigated relations between procedural and conceptual knowledge within the solutions of 33 calculus items. The analysis included the number of procedural and conceptual steps needed to answer the item, item label and item classification into one of four knowledge classes based on the type and quality of knowledge. The items were included in a data collection instrument used for quantitative analysis. Rasch analysis was performed to measure item difficulty and person proficiency, and describe the underlying cognitive construct between items. The Rasch person–item map confirmed that items were not clustered together per class and that item difficulty was not linked to the number of procedural and/or conceptual steps needed to do the mathematics. Confirmatory factor analysis showed over-correlation between classes and that defined classes cannot be separated, confirming integration of procedural and conceptual cognitive processes. The relationship between procedural and conceptual knowledge within and between items is complex. Findings indicated that item solutions drew on both procedural and conceptual components that cannot be separated. Solutions could follow more than one approach and analyses could differ, since what is conceptual for one student could be procedural for another. Therefore, teaching strategies should navigate between concepts and procedures, methods and representations. / Thesis (PhD)--University of Pretoria, 2020. / Science, Mathematics and Technology Education / PhD / Unrestricted

abstract: This study investigates several students’ interpretations and meanings for negations of various mathematical statements with quantifiers, and how their meanings for quantified variables impact their interpretations and denials of these quantified statements. Eight students participated in three separate exploratory teaching interviews and were selected from Transition-to-Proof and advanced mathematics courses beyond Transition-to-Proof. In the first interview, students were asked to interpret mathematical statements from Calculus contexts and provide justifications and refutations for why these statements are true or false in particular situations. In the second interview, students were asked to negate the same set of mathematical statements. Both sets of interviews were analyzed to determine students’ meanings for the quantified variables in the statements, and then these meanings were used to determine how students’ quantifications influenced their interpretations, denials, and evaluations for the quantified statements. In the final interview, students were also be asked to interpret and negation statements from different mathematical contexts. All three interviews were used to determine what meanings comprised students’ interpretations and denials for the given statements. Additionally, students’ interpretations and negations across different statements in the interviews were analyzed and then compared within students and across students to determine if there were differences in student denials across different moments. / Dissertation/Thesis / Doctoral Dissertation Mathematics Education 2020

We study an energy functional able to describe low energy configurations of a two dimensional lattice with dislocations in a nonlinear elasticity regime. The main result can be described as follows: configurations of energy comparable to the lattice spacing consist of piecewise constant microrotations with small angle grain boundaries between them. Moreover, we also give bounds to the energy of particular configurations describing a small angle symmetric tilt grain boundary.

The purpose of this study was to examine the psychometric properties of the Precalculus Concept Assessment (PCA), a 25-item multiple-choice instrument designed to assess student reasoning abilities and understanding of foundational calculus concepts (Carlson et al., 2010). When this study was conducted, the extant research on the PCA and the PCA Taxonomy lacked in-depth investigations of the instruments' psychometric properties. Most notably was the lack of studies into the validity of the internal structure of PCA response data implied by the PCA Taxonomy. This study specifically investigated the psychometric properties of the three reasoning constructs found in the PCA taxonomy, namely, Process View of Function (R1), Covariational Reasoning (R2), and Computational Abilities (R3). Confirmatory Factor Analysis (CFA) was conducted using a total of 3,018 pretest administrations of the PCA. These data were collected in select College Algebra and Precalculus sections at a large private university in the mountain west and one public university in the Phoenix metropolitan area. Results showed that the three hypothesized reasoning factors were highly correlated. Rival statistical models were evaluated to explain the relationship between the three reasoning constructs. The bifactor model was the best fitting model and successfully partitioned the variance between a general reasoning ability factor and two specific reasoning ability factors. The general factor was the dominant factor accounting for 76% of the variance and accounted for 91% of the reliability. The omegaHS values were low, indicating that this model does not serve as a reliable measure of the two specific factors. PCA response data were retrofitted to diagnostic classification models (DCMs) to evaluate the extent to which individual mastery profiles could be generated to classify individuals as masters or non-masters of the three reasoning constructs. The retrofitting of PCA data to DCMs were unsuccessful. High attribute correlations and other model deficiencies limit the confidence in which these particular models could estimate student mastery. The results of this study have several key implications for future researchers and practitioners using the PCA. Researchers interested in using PCA scores in predictive models should use the General Reasoning Ability factor from the respecified bifactor model or the single-factor model in conjunction with structural equation modeling techniques. Practitioners using the PCA should avoid using PCA subscores for reasoning abilities and continue to follow the recommended practice of reporting a simple sum score (i.e., unit-weighted composite score).

abstract: Functions represented in the graphical register, as graphs in the Cartesian plane, are found throughout secondary and undergraduate mathematics courses. In the study of Calculus, specifically, graphs of functions are particularly prominent as a means of illustrating key concepts. Researchers have identified that some of the ways that students may interpret graphs are unconventional, which may impact their understanding of related mathematical content. While research has primarily focused on how students interpret points on graphs and students’ images related to graphs as a whole, details of how students interpret and reason with variables and expressions on graphs of functions have remained unclear. This dissertation reports a study characterizing undergraduate students’ interpretations of expressions in the graphical register with statements from Calculus, its association with their evaluations of these statements, its relation to the mathematical content of these statements, and its relation to their interpretations of points on graphs. To investigate students’ interpretations of expressions on graphs, I conducted 150-minute task-based clinical interviews with 13 undergraduate students who had completed Calculus I with a range of mathematical backgrounds. In the interviews, students were asked to evaluate propositional statements about functions related to key definitions and theorems of Calculus and were provided various graphs of functions to make their evaluations. The central findings from this study include the characteristics of four distinct interpretations of expressions on graphs that students used in this study. These interpretations of expressions on graphs I refer to as (1) nominal, (2) ordinal, (3) cardinal, and (4) magnitude. The findings from this study suggest that different contexts may evoke different graphical interpretations of expressions from the same student. Further, some interpretations were shown to be associated with students correctly evaluating some statements while others were associated with students incorrectly evaluating some statements. I report the characteristics of these interpretations of expressions in the graphical register and its relation to their evaluations of the statements, the mathematical content of the statements, and their interpretation of points. I also discuss the implications of these findings for teaching and directions for future research in this area. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

This paper investigated the sensitivity, specificity, accuracy, and precision of laser fluorescence and tactile probing for the detection of subgingival calculus. The gold standard for subgingival calculus detection has always been tactile probing. In this study 27 teeth were collected and 108 surfaces investigated, one tooth was excluded (group #13) where no calculus was observed on any surface, and three surfaces because of subgingival root caries to avoid confounding data, which left a total of 101 surfaces of 26 extracted teeth that meet the investigation criteria. The presence of subgingival calculus was observed on 75 tooth surfaces (74.25%). There was a correlation between tooth surface and the presence of calculus. Subgingival calculus was from most to least frequently observed on the Distal surface (92.0%), Lingual surface (76.9%), Mesial surface (70.8%) and Facial surface (57.7%). The amount of laser fluoresce increased according to the amount of subgingival calculus. There was a correlation between the amount of subgingival calculus and the amount of laser fluorescence. The tactile probing had a similar sensitivity compared to laser fluorescence for the detection of subgingival calculus. The laser fluorescence was more specific compared to tactile probing for the detection of subgingival calculus. The tactile probing had a similar accuracy compared to laser fluorescence for the detection of subgingival calculus. The laser fluorescence had more precision compared to tactile probing for the detection of subgingival calculus. These results show that by using both tactile probing and laser fluorescence the sensitivity, specificity, accuracy, and precision of detecting subgingival calculus can be increased. An increase in the sensitivity, specificity, accuracy, and precision of detecting subgingival calculus could help in the diagnosis and treatment of patients suffering from gingival recession and periodontal disease.

No description available.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013. / Recent studies of the dental remains from Malapa, a fossil-bearing karstic cave-site located in the Cradle of Humankind (Berger et al., 2010), have demonstrated the presence of dental calculus and associated plant material in the form of phytoliths, preserved on the teeth of Australopithecus sediba (MH1) (Henry et al., 2012). This discovery raised the probability that dental calculus and plant microremains may also be present on hominin material from other cave sites in southern Africa, where fossils are preserved under similar conditions. The aim of this study was to establish the presence of dental calculus and associated microfossils on the teeth of other southern African Plio-Pleistocene early hominins. The dental collection of the Plio-Pleistocene age hominin site of Sterkfontein was examined. Where fossils were observed with adherent material, several analyses were performed to determine whether this material was calculus or not. Where possible, comparisons with the texture of the sediment matrix surrounding the fossil were conducted. Small quantities of this material were removed and observed microscopically to determine if it included food particulates and microfossils. In these cases, we also looked for microfossils in the surrounding matrix as a control. Phytoliths were recovered from all tooth sample material tested. The establishment of phytoliths in the dental calculus is direct evidence that these two structures existed simultaneously, as the formation of calculus can only take place in the presence of saliva. A large number of phytolith morphotypes further indicated that A.africanus had an adaptable and diverse diet, and that monocotyledons and dicotyledons appear to have made up a considerable part of their diet. The results from this study will benefit future analyses, by not only providing new protocols for establishing the presence of dental calculus, but also for promoting better preservation of dental calculus in the future. Further, future studies may be able to obtain direct evidence of consumed food that can directly be associated with individual hominins’ feeding behaviours. This could result in significant clues to the diet and ecology of not only individual hominins, but populations, species and comparisons of diet and behaviour between species and genera.

We study energy-driven nonlocal pattern forming systems with opposing interactions. Selections are drawn from the area of Quantum Physics, and nonlocalities are present via Coulombian type interactions. More precisely, we study Thomas-Fermi-Dirac-Von Weizsacker (TFDW) type models, which are mass-constrained variational problems. The TFDW model is a physical model describing ground state electron configurations of many-body systems. First, we consider minimization problems of the TFDW type, both for general external potentials and for perturbations of the Newtonian potential satisfying mild conditions. We describe the structure of minimizing sequences, and obtain a more precise characterization of patterns in minimizing sequences for the TFDW functionals regularized by long-range perturbations. Second, we consider the TFDW model and the Liquid Drop Model with external potential, a model proposed by Gamow in the context of nuclear structure. It has been observed that the TFDW model and the Liquid Drop Model exhibit many of the same properties, especially in regard to the existence and nonexistence of minimizers. We show that, under a "sharp interface'' scaling of the coefficients, the TFDW energy with constrained mass Gamma-converges to the Liquid Drop model, for a general class of external potentials. Finally, we present some consequences for global minimizers of each model. / Thesis / Doctor of Philosophy (PhD)

Modern data analysis requires the study of tensors, or multi-way arrays. We consider the case where the dimension d is large and the order p is fixed. For dimension reduction and for interpretability, one considers tensor decompositions, where a tensor T can be decomposed into a sum of rank one tensors. In this thesis, I will describe some recent work that illustrate why and how to use decompositions for orthogonally decomposable tensors. Our developments are motivated by statistical applications where the data dimension is large. The estimation procedures will therefore aim to be computationally tractable while providing error rates that depend optimally on the dimension. A tensor is said to be orthogonally decomposable if it can be decomposed into rank one tensors whose component vectors are orthogonal. A number of data analysis tasks can be recast as the problem of estimating the component vectors from a noisy observation of an orthogonally decomposable tensor. In our first set of results, we study this decompositionproblem and derive perturbation bounds. For any two orthogonally decomposable tensors which are ε-perturbations of one another, we derive sharp upper bounds on the distances between their component vectors. While this is motivated by the extensive literature on bounds for perturbation of singular value decomposition, our work shows fundamental differences and requires new techniques. We show that tensor perturbation bounds have no dependence on eigengap, a quantity which is inevitable for matrices. Moreover, our perturbation bounds depend on the tensor spectral norm of the noise, and we provide examples to show that this leads to optimal error rates in several high dimensional statistical learning problems. Our results imply that matricizing a tensor is sub-optimal in terms of dimension dependence. The tensor perturbation bounds derived so far are universal, in that they depend only on the spectral norm of the perturbation. In subsequent chapters, we show that one can extract further information from how a noise is generated, and thus improve over tensor perturbation bounds both statistically and computationally. We demonstrate this approach for two different problems: first, in estimating a rank one spiked tensor perturbed by independent heavy-tailed noise entries; and secondly, in performing inference from moment tensors in independent component analysis. We find that an estimator benefits immensely— both in terms of statistical accuracy and computational feasibility — from additional information about the structure of the noise. In one chapter, we consider independent noise elements, and in the next, the noise arises as a difference of sample and population fourth moments. In both cases, our estimation procedures are determined so as to avoid accumulating the errors from different sources. In a departure from the tensor perturbation bounds, we also find that the spectral norm of the error tensor does not lead to the sharpest estimation error rates in these cases. The error rates of estimating the component vectors are affected only by the noise projected in certain directions, and due to the orthogonality of the signal tensor, the projected errors do not accumulate, and can be controlled more easily.