Thesis (Ph.D. (Mathematics Education)) -- University of Limpopo, 2018 / The purpose of the study is to explore the effectiveness of Annual National Assessment (ANA) in monitoring the standard of mathematics education and to assess the mathematical proficiencies tested and exhibited by Grade 9 learners in South Africa. The research problem was premised on the dearth of data that justifies ANA as an evaluative assessment. As such, the study utilised five strands which were; procedural fluency, conceptual understanding, strategic competence, adaptive reasoning and productive dispositions as a theoretical framework to assess mathematics that was tested and exhibited by learners. To explore the research problem, the study used mixed methods in the context of exploratory sequential design. Document analysis was used first to capture mathematics content and cognitive levels examined by ANA. Second, learner responses were explored using four variables of achievement levels; no response, correctly answered, incorrectly answered and partially answered. First, the results from the analysis of ANA questions indicated that ANA mostly tested questions of low complexity. Second, the results from the learners’ responses revealed that the majority of learners were not proficient to ANA irrespective of low complexity testing. Third, the Porter’s alignment index for ANA and TIMSS was between moderate and perfect. Subsequently, content and cognitive levels were misaligned in the three consecutive years of ANA testing. It implies that learners were most likely to show a deficit of higher order problems solving skills which are a prerequisite of courses in advanced mathematics. Additionally, the results suggest that ANA had challenges of reliability and validity as an evaluative assessment due to inconsistency in the testing. As such, it is recommended that the complexity of ANA be addressed, the content areas where learners are not proficient be addressed and the alignment of ANA must be frequently calculated to monitor the standard of mathematics education in South Africa effectively. Keywords Alignment index, mean deviation, strands of mathematical proficiency, systemic assessment.

No presente trabalho, são estudadas propriedades de uma distribuição pertencente à classe de distribuições Kumaraswamy generalizadas, denominada Kumaraswamy normal, formulada a partir da distribuição Kumaraswamy e da distribuição normal. Algumas propriedades estudadas são: expansão da função densidade de probabilidade em série de potências, função geradora de momentos, momentos, função quantílica, entropia de Shannon e de Rényi e estatísticas de ordem. São construídos dois modelos de regressão lineares do tipo localização-escala para a distribuição Kumaraswamy normal, um para dados sem censura e o outro com a presença de observações censuradas. Os parâmetros dos modelos são estimados pelo método de máxima verossimilhança e algumas medidas de diagnóstico, como influência global, influência local e resíduos são desenvolvidos. Para cada modelo de regressão é realizada uma aplicação a um conjunto de dados reais. / In this work, properties of a distribution belonging to the class of generalized Kumaraswamy distributions, called Kumaraswamy normal, are studied. The Kumaraswamy normal distribution is formulated from the Kumaraswamy distribution and from the normal distribution. Some properties studied are: expansion of the probability density function in power series, moment generating function, moments, quantile function, Shannon and Rényi entropy, and order statistics. Two location-scale linear regression models are constructed for the Kumaraswamy-normal distribution, one for datas uncensored and the other with the presence of censoreds observations. The parameters of these models are estimated by the maximum likelihood method and some diagnostic measures such as global influence, local influence and residuals are developed. For each regression model an application is made to a real data set.

為達成「精熟學習」的教與學過程中強調的個別化教學、學生學習的回饋、校正學生學習錯誤等高效能的教學理想。本研究自行發展一套數位化診斷系統，在不改變學校政策、班級作息、教室管理實施情況下，對準備參加科大入學測驗高工三年級的學生，在考前六週進行數學科總複習的「精熟學習配合數位化診斷」實驗教學。經由學生的作答反應，本診斷系統完成以下功能： (1)能快速的篩選出優良試題，並提供試題的難度、鑑別度、答對率與評鑑試題的品質。 (2)能診斷出個別學生與試題的差異性，並完成試題卷的信度、效度評鑑與S-P分析表。 (3)能提供完整的學習成就遷移記錄，並可歸類出學生學習潛能與整體學習表現。 (4)能彙整出學生學習成效與具體教學建議。 教師藉診斷系統所提供的資訊，對學生進行個別化校正或充實教學，期以達成精熟學習的教學理想。並在實驗教學後分析學生學習回饋單，大多數學生對實驗教學法持正面態度，且自認學習態度更積極，並建議若能提早實施精熟學習配合數位化診斷教學，應會有更好成績表現。 本研究並探討「實驗教學」的實驗A組學生，分別對「不同科別」的實驗B組學生與對「傳統教學」的控制C組學生的數學學習態度、學習成就、學習成就絕對均差的差異性，經單因子共變數分析，排除前測影響後，具體結論如下： (1)實驗A、B組與控制C組學生在數學學習態度後測得分並無明顯差異。 (2)實驗A組與控制C組在數學學習成就後測成績有明顯差異，且實驗A組分數明顯高於控制C組。 (3)不同科別實驗組的A組電機類與B組機械類學生，在數學學習成就後測成績有明顯差異，且電機類分數明顯高於機械類，但機械類排除未完成實驗的學生後，則不同科別的實驗A組、實驗B組，在數學學習成就後測成績無明顯差異。 (4)實驗A組與控制C組學生在數學學習成就後測成績的絕對均差有明顯差異，且實驗A組絕對均差小於控制C組。 最後為使實驗教學能更順利進行，建議教學前先營造愉快與舒適教學環境與良好的師生互動情誼再實施，這將會是實驗教學成功的關鍵 / Mastery Learning Strategy is a highly effective teaching method, emphasizing the importance of individualized tutoring, student feedback and learning error correction. In this study, a self-devised computerized diagnostic system was developed for the third-year vocational high school students who were preparing for the technological university entrance examination. An experimental teaching method using Mastery Learning Strategy with computerized diagnostic system was given to the aforementioned students as a math review lesson six weeks before the entrance examination, following the original school policy, class schedule or class management. From test responses, the diagnostic system achieved the following: (1)Quick selecting of good quality test items. It also evaluates the test items and provides the difficulty, discrimination, and correction rates of them. (2)Individualized diagnosis of each student vs. test items. It also completes the reliability, validity and S-P charts of the test. (3)Offering a comprehensive record of the learning process. It also analyzes the learning process of the students and sorts out their learning potential. (4)Analyzing students’ learning effects and coming up with teaching suggestions. In this experiment, the data gained from the diagnosis were used to provide each student with individualized instruction or error correction with the hope of attaining the goals of Mastery Learning. The analysis of the students’ feedback after the experimental teaching method showed that most students thought positively of the experiment, admitting that they developed a more aggressive learning attitude, and suggested that this computerized diagnostic system, had it been implemented earlier, would have helped them achieve even better grades. The students in this study consisted of three groups. Group A and B were both experimental groups except that group A were students from electrical engineering department while group B were students from mechanical engineering department. Group C was the control group receiving only conventional teaching method. Group A was compared with group B and group C respectively in terms of differences in math-learning attitude, math-learning achievement, and in the absolute mean deviation of learning achievement. By means of one-way analysis of covariance, with pre-test effects eliminated, the conclusions were as follows: (1)None of the groups exhibited significant grade differences on the math-learning attitude post-test. (2)Group A scored significantly higher on the math-learning achievement post-test than did Group C, the control group. (3)Group A of electrical engineering students scored significantly higher on the math-learning achievement post-test than did Group B of mechanical engineering students. However, if the grades of the Group B students who didn’t complete the experiment were unaccounted, no grade difference was noticeable on the math-learning achievement post-test between Group A and Group B. (4)Group A, the experimental group, had significantly lower absolute mean deviation on the math-learning achievement post-test than did Group C, the control group. Finally, it was suggested that creating a comfortable environment for teaching and learning as well as an advance establishment of a good teacher-student rapport are the key to a smooth-going and successful experimental teaching method.