Thesis--University of Florida. / Description based on print version record. Typescript. Vita. Includes bibliographical references (leaves 104-105).

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg, 2014. / This study investigates to what extent and how teaching informed by variation theory could improve the Grade 11 learners’ interpretation of algebraic functions. The study adopted a learning study approach, where learner difficulties are elicited in a pre-test, and on the basis of the results of the pre-test, a lesson is planned, informed by variation theory, to make it possible for learners to discern what they found difficult. In this study, a pre-test on functions was given to three groups of Grade 11 learners (85 learners in all) in the researcher’s school that enabled the identification of aspects of functions learners found to be most problematic. The lesson was then taught successively to each group. A post-test at the end of each lesson, together with reflection on the lesson led to refinements for the next lesson to the next group. The study describes the changes made to the lessons, and the results of the pre and post tests for each of the three groups. The results showed that while each group improved in the post test, the third group outperformed the others, confirming that an intervention, informed by variation theory, did improved learning. All learners were afforded the opportunity to discern the object of learning and their interpretation of functions improved, with the third group improving the most. Keywords: object of learning, patterns of variation theory, interpretation of functions.

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Thesis (Ph. D.)--Ohio State University, 1984. / Includes bibliographical references (leaves 183-188). Available online via OhioLINK's ETD Center

Thesis (M.S.)--University of Wisconsin--Madison, 1982. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 63-64).

The results of a qualitative analysis of 170 precalculus students' interpretations of mathematical variables, constituted the foundation for a teaching intervention in a precalculus course at the University of Victoria. Some serious misconceptions of variables were identified. The possible effects of the intervention were investigated in a retrospective analysis of students' mathematics course grades. Students' interpretations of variables were extrapolated from their written explanations of answers to three algebra questions and from interview responses (N = 17). The subjects seldom interpreted variables as representing generalized sets of numbers or as co-variants. Their interpretations of variables were context-dependent, and generally inappropriate. In simplifications and equation solving most subjects appeared to use arbitrary rules to manipulate non-numeric symbols. When forced to consider numerical interpretations many described the variables as single numbers occurring in different instances. Some subjects appeared to substitute instances of variable use for the generalized number interpretation of variables, and patterns across instances for variable change. The interpretation of the variable as a single value in multiple instances can account for responses ranging from denial that variables change values to apparently correct descriptions of variable change. Some students interpreted letters as concrete objects or as units. The intervention, which was incorporated into the researcher's precalculus course lectures, consisted of making explicit the contextual interpretations of mathematical variables as single, generalized, or co-varying numbers, and of expressions as actions or as variable objects. Student response to the intervention content was very positive. The effect of the intervention was investigated quantitatively using log-linear models of the distributions of students' precalculus grades and, more important, their subsequent calculus grades. The models controlled for student changes over time, for instructor effects, and for differences in class composition based on students' year classifications. For students continuing to calculus there was a possible association between the intervention and better calculus grades (N = 166, p = 0.0008) but the confound of year standing prevented conclusions being drawn for their precalculus grades. For the subjects who did not continue on to calculus ( N = 524), there was no association between grade distributions and the experimental and control groups. / Graduate

This thesis considers the problem of estimating limited dependent variable models when the latent residuals are heteroskedastic normally distributed random variables. Commonly used estimators are generally inconsistent in such situations. Two estimation methods that allow consistent estimation of the parameters of interest are presented and shown to be consistent when the latent residuals are heteroskedastic of unknown form. Both estimators use recent advances in the econometric literature on nonparametric estimation and deal with the unknown form of heteroskedasticity by approximating the variance using a Fourier series type approximation. The first estimator is based on the method of maximum likelihood and involves maximising the likelihood function by choice of the parameters of the variance function approximation and the other parameters of interest. Consistency is shown to hold in the three most popular limited dependent variable models — the Probit, Tobit, and sample selection models — provided that the number of terms in the approximation increases with the sample size. The second estimator, which can be used to estimate the Tobit and sample selection models, is based on a two-step procedure, using Fourier series approximations in both steps. Consistency and asymptotic normality are proven under restrictions on the rate of increase of the number of parameters in the approximating functions. Finally, a small Monte Carlo experiment is conducted to examine the small sample properties of the estimators. The results show that the estimators perform quite well and in many cases reduce the bias, relative to the commonly used estimators, with little or no efficiency loss. / Arts, Faculty of / Vancouver School of Economics / Graduate

by Kwok-leung Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 60-63). / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Model --- p.7 / Chapter 2.1 --- Basic Beliefs on the vaguely classified variable --- p.7 / Chapter 2.2 --- Properties of the contingency table --- p.13 / Chapter 2.3 --- Mathematical model --- p.21 / Chapter Chapter 3 --- Simulation --- p.32 / Chapter 3.1 --- Likelihood function for the model and the simulation method --- p.32 / Chapter 3.2 --- Simulation result --- p.48 / Chapter Chapter 4 --- Discussion --- p.57 / Reference --- p.60

In general model selection so far considered in literature, the parameter estimation loss and the prediction loss from the model selected are considered to be the same. In this thesis, the methods of parameter estimation may vary with different estimation loss, and the model selection may be based on different prediction loss. Under some regularized conditions, a model selection criterion, called generalized risk criterion (GRC), is proposed with a closed form. For multivariate linear regression model, and Cox regression model for ranking data, our studies that this criterion is an extension of the model selection criterion AIC. We also demonstrate that GRC performs better than AIC in a practical semi-parametric regression problem involving investments on horse racing. / Keywords: Variable selection; Model selection criterion; AIC; GRC; Loss function; Risk function; Multinomial Choice Model; Cox model for ranking data. / Searching for the true model based on the limited data is usually an impossible task. More and more attention in research has been focused on how to find an optimal model based on some special objective, such as focused information criterion (FIC, Hjort and Claeskens, 2003 [15]), Subspace Information criterion (Sugiyama and Ogawa, 2001 [43]) in statistical learning, etc. These ideas also motivate us to find an optimal subset of variables based on some objective. Different objectives may result in different choices of subset of variables. / Variable selection, an important aspect of model selection, is applied widely in real practices to explore the latent relationship between the random phenomena and various factors. Many model selection criteria, such as Mallow's Cp (Mallows, 1964 [28]). PRESS (Allen, 1971 [3]). AIC (Akaike, 1973 [2]), are proposed for seeking the optimal subset of the variables. Most of them try to find a criterion based on the observed data such that the selected models perform well both for fitting and for prediction. / Zuo, Guo Xin. / "July 2007." / Adviser: Ming Gao Gu. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0402. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 71-75) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Leung, Kin Pang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 70-71). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Model --- p.4 / Chapter 2.1 --- The Basic Thurstonian Model --- p.4 / Chapter 2.2 --- The Thurstonian Model with Mean Structure in 3-object Ranking Data --- p.6 / Chapter 3 --- Implementation Using the Mx --- p.13 / Chapter 4 --- Simulation Study --- p.21 / Chapter 4.1 --- 2 covariate values --- p.23 / Chapter 4.2 --- 4 covariate values --- p.23 / Chapter 4.3 --- 10 covariate values --- p.23 / Chapter 4.4 --- 50 covariate values --- p.24 / Chapter 5 --- Discussion --- p.37 / Chapter A --- Sample Mx script-2 covariate values --- p.39 / Chapter B --- Sample Mx script-50 covariate values --- p.60 / Bibliography --- p.70