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This study characterizes the mathematical knowledge upon which secondary school algebra teachers draw when pondering problem situations that could arise in the teaching and learning of solving algebraic equations, as well as examines the potential connections between teachers' knowledge and their academic backgrounds and teaching experiences. Seventy-two middle school and high school algebra teachers in Texas participated in the study by completing an academic background questionnaire and a written-response assessment instrument. Eight participants were then invited for followup semi-structured interviews. The results revealed three topic areas in equation solving in which teachers' mathematical subject matter understanding should be strengthened: (a) the balancing method, (b) the concept of equivalent equations, and (c) the properties of linear equations in their general forms. The participants provided a wide range of instances of student misconceptions and difficulties in learning how to solve linear and quadratic equations, as well as a variety of strategies for helping students to improve their understanding. Teachers' subject matter knowledge played a central or prerequisite role in their reasoning and decision-making in specific contexts. When the problem contexts became broader or more general, teachers drew from across the three basic domains of mathematical knowledge for teaching (knowledge of the mathematical subject matter, knowledge of learners' conceptions, and knowledge of didactic representations) and showed individual preferences. Overall, teachers tended to rely more heavily upon their knowledge of students' specific or general learning characteristics. Statistical analyses suggest that teachers who majored in mathematics and who had the most experience in teaching first-year or more advanced algebra courses performed significantly higher on the assessment than their counterparts, and there is a linear relationship between teachers' performance and the number of advanced mathematics course they have taken. Neither course-taking in mathematics education nor number of years of algebra teaching made a significant difference in their performance. Results are either unclear or inconsistent about the role of teachers' (a) use of algebra textbooks, (b) prior experience with a method or a manipulative, and (c) participation in professional development activities. Teachers also rated (a) collaborating with and learning from colleagues and (b) dealing with student conceptions and questions as highly influential on their professional knowledge growth.

The pass rate of first-year university mathematics students at the University of KwaZulu-Natal (Pietermaritzburg Campus) has been low for many years. One cause may be weak algebra skills. At the time of this study, revision of high school algebra was not part of the major first year mathematics course. This study set out to investigate if it would be worthwhile to spend tutorial time on basic algebra when there is already an overcrowded calculus syllabus, or if students refresh their algebra skills sufficiently as they study first year mathematics. Since it was expected that remediation of algebra skills would be found to be worthwhile, two other questions were also investigated: Which remediation strategy is best? Which errors are the hardest to remediate? Five tutorial groups for Math 130 were randomly assigned one of four remediation strategies, or no remediation. Three variations of using cognitive conflict to change students’ misconceptions were used, as well as the strategy of practice. Pre- and post-tests in the form of multiple choice questionnaires with spaces for free responses were analysed. Comparisons between the remediated and non-remediated groups were made based on pre- and post-test results and Math 130 results. The most persistent errors were determined using an 8-category error classification developed for this purpose. The best improvement from pre- to post-test was 12.1% for the group remediated with cognitive conflict over 5 weeks with explanations from the tutor. Drill and practice gave the next-best improvement of 8.1%, followed by self-guided cognitive conflict over 5 weeks (7.8% improvement). A once-off intervention using cognitive conflict gave a 5.9% improvement. The group with no remediation improved by 2.3%. The results showed that the use of tutorintensive interventions more than doubled the improvement between pre-and post-tests but even after remediation, the highest group average was 80%, an unsatisfactory level for basic skills. The three most persistent errors were those involving technical or careless errors, errors from over-generalising and errors from applying a distorted algorithm, definition or theorem. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2009.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-07-18T11:45:16Z No. of bitstreams: 1 Maria Elizabeth de Oliveira Borges.pdf: 2082871 bytes, checksum: 4fd49893bd758a45d3d74eb2e80706aa (MD5) / Made available in DSpace on 2018-07-18T11:45:16Z (GMT). No. of bitstreams: 1 Maria Elizabeth de Oliveira Borges.pdf: 2082871 bytes, checksum: 4fd49893bd758a45d3d74eb2e80706aa (MD5) Previous issue date: 2018-04-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The present study, with a qualitative bibliographical approach, aims to present the state of the art of academic research whose subjects involve the teaching and learning processes of Algebra in the final years of Elementary and Secondary Education, published in the period between 2008 and 2017. The corpus of analysis consists of 42 researches obtained, mainly, in digital media, from the following bases: Bank of Thesis and dissertations of the Coordination of Higher Level Personnel Improvement - CAPES and repositories of theses and dissertations of the Programs post graduate. In order to compose the analysis data, each work was recorded, based, in particular, its bibliographic data and abstracts. Among the main results obtained, we highlight that of the 42 surveys collected, three were PhD’s theses and 39 were Master's dissertations. The mapping of these surveys revealed the quantity of productions per higher education institution, by regions and by post-graduate programs. The analysis of the productions was done through the categories: Algebraic Language and Thinking; Relation between Algebra / Arithmetic / Geometry; Multiple Representations; Materials / didactic resources; Conceptions / Knowledge; Teaching Methodology and Pedagogical Practice. We could observe that research focuses more on students in the final years of Elementary School than on high school and continues to point out the difficulties of students in constructing algebraic knowledge, particularly those involving variability. The few works that deal with teachers point to their dissatisfaction with the lack of connection between the initial formation and the practice in the classroom, besides divergences regarding their knowledge for the teaching of Algebra. Of the analyzed works few discuss the official documents or textbooks regarding what they propose for the teaching of algebra. In addition, in the works that deal with specific contents it is noticed that some are more worked, as is the case of equations of first and second degrees, while others are not treated. Thus, for a better understanding of the state of Algebra teaching and learning, it is necessary to develop research that addresses the teaching and/or learning of other algebraic subjects in Basic Education, as well as in-depth analyzes of the proposed curriculum for teaching arranged in documents officers and/or textbooks, as well as to investigate the knowledge necessary for teachers to be able to teach this subject, among other possible topics / O presente estudo, de abordagem qualitativa de cunho bibliográfico, tem como objetivo apresentar o estado da arte de pesquisas acadêmicas cuja temática envolve os processos de ensino e aprendizagem de Álgebra nos anos finais do Ensino Fundamental e do Ensino Médio, publicadas no período entre 2008 e 2017. O corpus de análise constitui-se de 42 pesquisas obtidas, sobretudo, em meio digital, a partir das seguintes bases: Banco de teses e dissertações da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – CAPES e repositórios de teses e dissertações dos Programas de Pós-Graduação. Para compor os dados de análise, cada trabalho foi fichado, tendo por base, em especial, seus dados bibliográficos e resumos. Dentre os principais resultados obtidos, destacam-se que, das 42 pesquisas coligidas, três eram teses de doutorado e 39 dissertações de mestrado. O mapeamento dessas pesquisas revelou a quantidade de produções por instituição de ensino superior, por regiões e por programas de pós-graduação. A análise das produções foi realizada por meio das categorias: Linguagem e Pensamento algébrico; Relação entre Álgebra / Aritmética / Geometria; Múltiplas Representações; Materiais / Recursos didáticos; Concepções / Conhecimento; Metodologia de Ensino e Prática Pedagógica. Pudemos observar que as pesquisas focam mais em alunos dos anos finais do Ensino Fundamental, do que do Ensino Médio e continuam apontando para as dificuldades dos alunos em construir conhecimentos algébricos, em particular, os que envolvem variabilidade. Os poucos trabalhos que tratam de professores apontam para sua insatisfação com a falta de conexão entre a formação inicial e a prática em sala de aula, além de divergências a respeito de seus conhecimentos para o ensino de Álgebra. Dos trabalhos analisados poucos discutem os documentos oficiais ou livros didáticos a respeito do que propõem para o ensino de álgebra. Além disso, nos trabalhos que tratam de conteúdos específicos nota-se que alguns são mais trabalhados, como é o caso de equações de primeiro e segundo graus, enquanto outros nem são tratados. Assim, para uma melhor compreensão do estado do ensino e da aprendizagem de Álgebra se faz necessário desenvolver pesquisas que tratem do ensino e/ou aprendizagem de outros temas algébricos do Ensino Básico, bem como de análises aprofundadas do currículo proposto para o ensino dispostos em documentos oficiais e/ou livros didáticos, além de investigar os conhecimentos necessários para que os professores tenham condições para ensinar tal assunto, entre outros tantos temas possíveis

One purpose of this study was to determine whether any differences in immediate achievement or retention existed between students using manipulatives and students not using manipulatives. Also addressed in this study is whether or not the use of manipulatives is more beneficial for girls than boys and whether the use of manipulatives is more beneficial for low-ability students than for high-ability students. Students selected for this study were from a large suburban school district in Texas. The students were from eight intact classes, four of which were designated as the experimental group and the other four as the control group. The sample consisted of one hundred eighty-seven students. All students were tested with a test developed by the researcher. This same test was administered as a pretest, posttest, and retention test. The following supplemental data were also gathered on the students: mathematics scores from the California Test of Basic Skills and scores from the mathematics section of the Texas Educational Assessment of Minimum Skills test. Analysis of the data revealed no statistical difference in the mean scores of students instructed with or without manipulatives when the test was administered immediately after instruction. Nor was there any statistical difference in the mean scores when the test was administered two months after instruction. There was no statistical difference in the mean gain scores from the pretest to the posttest between boys and girls or between high- and low-achieving students. Nor was there any statistical difference between the mean gain scores from the pretest to the retention test between boys and girls or between high- and low-achieving students. It is recommended that further studies be conducted to investigate achievement and retention of students using manipulatives at the secondary level. It is also recommended that variables other than achievement be studied to determine the effects of manipulatives on secondary students.

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Science in Information Technology in Education

The study explored the relationship between student academic performance in an exit-level, developmental mathematics course and subsequent academic performance in a college-level mathematics course. Using an ex post facto research design, the study focused specifically on the influence of three sets of factors: (a) demographic characteristics, (b) "stopping-out," and (c) the developmental course. The criterion variables were college-level performance, defined in terms of the student's course grade, and college-level persistence, defined in terms of whether or not the student officially withdrew from the course. A convenience sample of 824 community college students who had completed both the exit-level developmental mathematics course and the entry-level college course during a three-year period from fall 1989 to summer 1992 was used for the data set; the students in the set were shown to be similar to several populations of developmental students. Discriminant function analysis indicated that the data supported the hypotheses. The discriminant function was calibrated on 364 cases randomly selected from the data set; the remainder of the cases were used to cross-validate the results. Cross-validated correct classification rates of 76.74% for academic success and 81.09% for persistence were obtained. The major conclusions of the study were: (1) Developmental course performance is a significant discriminator of college-level mathematics performance and persistence. (2) The length of time a student allows to pass between exiting the developmental course and entering the college-level course is a negatively related discriminator of both college-level performance and persistence. (3) Student age is a positively related discriminator of college-level mathematics performance. (4) The number of attempts at the developmental course is a negatively related discriminator of persistence. (5) African American completers of developmental mathematics appear to be more likely to withdraw from entry-level college mathematics than developmental completers in other ethnic groups. (6) Poor performance in exit-level developmental mathematics greatly increases the risk of failure or attrition for students in entry-level college mathematics. The implications of these results and those of several post hoc analyses were discussed in terms of their theoretical and applied contributions, the limitations of the study were detailed, and suggestions made for future research. / text

<p>This thesis investigates grade seven learners perceived experiences in learning algebra.Things that learners do and say during algebra lessons and about algebra were investigated. The study was done at one of the previously disadvantaged schools in Cape Town, South Africa.The data were collected through observations, a questionnaire and interviews. Observations were made from the day the topic was started in two grade seven classes. Two different teachers taught the two classes. Focus group interviews were conducted, two group of learners, ten learners from each of the two classes were interviewed. Learners devised a number of strategies for solving problems related to sums and differences. The principal learning difficulties experienced by learners in algebra related to the transition from arithmetic conventions to those of algebra, the meaning of literal symbols and the recoginition of structures. It became obvious then that developing algebraic thinking is not necessarily dependent upon algebraic notation and that the presence of algebraic notation says little about the level of problem solving.</p>

The purpose of this study was to determine whether students taught by certified teachers and those taught by uncertified teachers had significantly different achievement on a state Algebra I End of Course examination. The specific research questions were: (1) Does type of teacher certification impact Algebra I End of Course (EOC) Exam scores for high school freshman when controlling for students' past mathematics success as measured by 8th grade TAAS mathematics test scores and teachers' years of experience? (2) Does type of teacher certification impact Algebra I End of Course (EOC) Exam passage rates for high school freshman when controlling for students' past mathematics success as measured by 8th grade TAAS mathematics test scores, socio-economic status, ethnicity, gender, and teachers' years of experience? This research was conducted in a large north Texas suburban school district. The entire population (N=1,433) of freshman students enrolled in year-long Algebra I was included for this study. Three statistical tests were used in data analysis for the first question. Analysis of covariance using student as well as teacher as the unit of analysis and hierarchical multiple regression were used to analyze students' specific scores. Logistic regression was used for the second research question. This study found that students in classes with non-certified teachers scored eight points lower on the Algebra I EOC Exam than those in classes with certified teachers. However, when controlling for students' prior mathematics achievement and other variables, the difference was of no practical significance. There was no practical significance in a student's odds of passing the examination between students in certified teachers' classrooms and those in uncertified teachers' classrooms. The results of this study offer further understanding of the debate over type of certification.