The purpose of this research project was to investigate the effects of a short-term teaching experiment on the learning of decimal numbers by primary students. The literature describes this area of mathematics as highly problematic for students. The content first covered student understanding of decimal symbols, and how this impacted upon their ability to order decimal numbers and carry out additive operations. It was then extended to cover the density of number property, and the application of multiplicative operations to situations involving decimals. In doing so, three areas of cognitive conflict were encountered by students, the belief that longer decimal numbers are larger than shorter ones (irrespective of the actual digits), that multiplication always makes numbers bigger, and that division always makes numbers smaller. The use of a microgenetic approach yielded data was able to be presented that provides details of the environment surrounding the moments where new learning was constructed. The characteristics of this environment include the use of physical artifacts and situational contexts involving measurement that precipitate student discussion and reflection. The methodology allowed for the collection of evidence regarding the highly complex nature of the learning, with evidence of 'folding back' to earlier schema and the co-existence of competing schema. The discussion presents reasons as to why the pedagogical approach that was employed facilitated learning. One of the main findings was that the use of challenging problems situated in measurement contexts that involved direct student participation promoted the extension and/or re-organization of student schema with regard to decimal numbers. The study has important implications for teachers at the upper primary level wanting to support student learning about the decimal numbers system.

This study explores the pedagogical approaches used by fifth grade teachers to introduce division with decimals and the resultant understandings of students in their classrooms. The study is important because of the need for students to gain conceptually-based understandings in mathematics and the limited research on instruction and related learning of the very difficult and complex concept of division with decimals. In particular, there is limited research on strategies teachers use to develop students’ conceptual understanding of division with decimals. Therefore, the research questions are as follows. What strategies do teachers use to introduce division with decimals? When first learning to divide decimal numbers, how do fifth-grade students explain the strategies they use? The study is grounded in social constructivist learning theory and uses a collective case study methodology. Following the study design, three fifth-grade teachers from three schools were interviewed before and after an introductory lesson to division with decimals. They also were observed teaching the study lesson. Following the lesson, one to three students from each class (six in all) were interviewed on their understandings of division with decimals using their classwork from the lesson as a point of entry. The design includes three sources of data: transcriptions from semi-structured interviews of teachers and students, field notes from classroom observations, and artifacts from lessons. Results suggest that instruction of division with decimals varies such that the differences can be captured along a continuum of traditional to reform practices. The placement of the decimal point in the quotient is the focus of the discussion regardless of where the instruction lies on the continuum. Interestingly, as instruction moves towards the traditional end of the continuum, student engagement was a result of interaction with the teacher, whereas closer to the reform end of the spectrum students were engaged with the mathematics.

Research study on sixth grade problem-posing instruction: Case of addition, subtraction and number comparison on decimals Abstract The aim of this research project is to investigate the implementation of problem-posing instruction on decimals to one sixth-grade mathematics class. There are four research objectives: 1) design and implement problem-posing instruction on decimals; 2) discuss the status of children¡¦s performance in problem-solving; 3) analyze the type of problems posed by children; and, 4) display categories of misconceptions exhibited when children did problem posing. The stages for instructions were three: 1) children solved the problem given by the instructor; 2) children referred to given problem and posed a problem; and, 3) children solved their own problem. In this study, the type of problem posing chosen for instruction is ¡§similar problem¡¨, which is adapted from Tsubota, a Japan scholar. The researcher collected data by using: own constructed decimal problems question sheet, worksheet on problem solving, worksheet on problem posing, children¡¦s diaries and teachers¡¦ notes on instruction. There are four findings. First, the implementation of sixth grade problem-posing instruction on decimals is feasible. Second, 96.9% of students¡¦ problems are plausible and contain sufficient information for problem solvers. Most students could change the number and content of the question but few revised the structure of the question. There was also multiple development for those problems. Third, children¡¦s performance in posing/solving stage was better than that in problem-solving stage. Finally, the researcher reported that the teacher faced problems such as difficulty in control of time, establishing children¡¦s habit in reporting, and collecting misconceptions of children. Key word : problem solving; problem posing; addition, subtraction and number comparison on decimals

Social norms are patterns of behavior expected within a particular society in a given situation. Social norms can be shared belief of what is normal and acceptable shapes and enforces the actions of people in a society. In the educational classroom, they are characteristics that constitute the classroom participation structure. Sociomathematical norms are fine-grained aspects of general social norms specifically related to mathematical practices. These can include, but are not limited to, a student-centered classroom that includes the expectation that the students should present their solution methods by describing actions on mathematical objects rather than simply accounting for calculational manipulations. For this action research study, my goal was to determine if the role of the teacher would influence the social and sociomathematical norms in a mathematics classroom and in what ways are sociomathematical norms reflected in students' written work. I focused specifically on students' mathematics journal writing and taped conversations. I discovered that students tended to not justify their work. Also, I discovered that my idea of justification was not really justification. I learned from this and was able to change my idea of justification. By encouraging the students to socialize in mathematics class, I found that the quality of their dialogue improved. Students readily discussed mathematical concepts within small groups and whole class discussions.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2014-08-28T17:49:12Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Numeros Reais Um Corpo Ordenado Completo.pdf: 4328358 bytes, checksum: 5062827ca2822fd04229310850171740 (MD5) / Made available in DSpace on 2014-08-28T17:49:12Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Numeros Reais Um Corpo Ordenado Completo.pdf: 4328358 bytes, checksum: 5062827ca2822fd04229310850171740 (MD5) Previous issue date: 2013-03-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper aims to expand knowledge about the real numbers, providing a new perspective on their conceptual construction. Initially, covers up some historical facts that were of utmost importance in the process of conceptual evolution of the real numbers. Secondly, through the development of theories of abstract algebra, sets and mathematical analysis, is used a axiomatic method to expose the complete ordered field of real, stating and proving some of its properties. Finally, we discuss some relevant aspects of the correspondence between the real field and line, and also the correspondence between the real field and decimals. / Este trabalho tem como objetivo ampliar os conhecimentos sobre os números reais, proporcionando uma nova perspectiva sobre sua construção conceitual. Inicialmente, aborda-se alguns fatos históricos que foram de maior importância no processo da evolução conceitual dos números reais. Posteriormente, por meio do desenvolvimento das teorias de álgebra, de conjuntos e de análise matemática, utiliza-se de um método axiomático para expor uma construção do corpo ordenado e completo dos reais, enunciando e provando algumas de suas propriedades. Finalmente, abordam-se alguns aspectos relevantes da correspondência entre o corpo dos reais e a reta, e ainda da correspondência entre o corpo dos reais e os decimais.

This study investigated the effects of a direct instruction program on the ability of elementary school students identified as at risk for math failure to add and subtract numbers with decimals, and complete addition and subtraction word problems with decimals. Direct instruction has previously been shown to increase the math skills of special education and general education students. This study examined the extent to which these students could master these skills in six hours of instruction, with carefully designed sequences of examples and strategy instruction in word problems. The study took place in two elementary schools. The participants were fifth grade students who had received low math scores on a school wide test and placed in a math group accordingly. The students were given a pretest and placed into two different groups, iv based on a stratified random process. The students in the treatment group received six lessons in decimals and word problems. After the six lessons, the groups were given a posttest. Student progress was assessed by comparing the groups on posttest results, comparing the students’ pretest and posttest scores, and using the ANOVA to determine statistical significance. On the posttest, the students in the treatment group scored 35 percentage points higher than the students in the control group – this difference was statistically significant. The increase was largest in their ability to add and subtract decimals, however many of the students also made considerable progress in their ability to solve word problems.

Within a low-performing seventh grade mathematics classroom, communication techniques including discourse, collaborative groups, listening, reading, and writing were implemented during a six week period. This study shows how the use of these techniques led to the twenty four students' conceptual understanding of fraction and decimal concepts. This research study provides insight to the deep-seeded beliefs of low-performing students. It provides a record of how the teacher used communication techniques in the classroom and had a strong positive impact on the attitudes and performance of these struggling students.

Made available in DSpace on 2018-06-27T19:13:42Z (GMT). No. of bitstreams: 3 Sabrina Londero da Silva Rossato.pdf: 3438965 bytes, checksum: eb7c9a323abe55da5adcc7c931cb8283 (MD5) Sabrina Londero da Silva Rossato.pdf.txt: 171609 bytes, checksum: 150c6ac91c113bde2652f66413567bf5 (MD5) Sabrina Londero da Silva Rossato.pdf.jpg: 3639 bytes, checksum: 60022a5df896a5e3c59566e0ad171aa3 (MD5) Previous issue date: 2014-01-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This study had as its theme the operation of division in the set Q, focusing on decimal numbers. The analysis focused on the errors in division that students make when they resolve issues using the usual division algorithm with decimal results. This study was based in authors who write about error analysis in mathematics and also in David Ausubel Meaningful Learning Theory. The work developed throughout 2013 aimed to analyze the errors presented by the students of 6th grade of elementary school to solve exercises of division of decimal numbers and evaluate teaching strategies to construct meaning for the operation of division of decimals. The study followed a quantitative-qualitative approach and after analyzing the errors, we developed a workshop employing teaching techniques with the support of learning objects and manipulatives as Golden Material and Table of Value Place. The results showed that, from the analysis of the errors made by students in diagnostic testing, the application of a didactic teaching sequence to help reduce the errors made by the students, allowed for overcoming these errors, at least in part. / Este estudo teve como tema a operação divisão no conjunto, com foco nos números decimais. A análise concentrou-se nos erros de divisão que os alunos cometem ao resolver questões utilizando o algoritmo usual da divisão com resultados decimais. O estudo foi embasado nos autores que escrevem sobre análise de erros na Matemática e também na Teoria da Aprendizagem Significativa de David Ausubel. O trabalho, desenvolvido ao longo de 2013, teve como objetivo analisar os erros apresentados pelos alunos de 6º ano do Ensino Fundamental ao resolverem exercícios de divisão de números decimais e avaliar uma estratégia de ensino para construção de significados para a operação de divisão de decimais. A pesquisa seguiu uma abordagem quanti-qualitativa e, após a análise dos erros, foi desenvolvida uma oficina empregando técnicas de ensino com apoio de Objetos de Aprendizagem e Materiais Manipuláveis como Material Dourado e Quadro Valor de Lugar. Os resultados mostraram que, a partir da análise dos erros cometidos pelos alunos no teste diagnóstico, a aplicação de uma sequência didática de ensino para ajudar a reduzir os erros cometidos pelos alunos permitiu uma superação desses erros, pelo menos em parte.

Doctor of Philosophy / Curriculum and Instruction Programs / Jennifer M. Bay-Williams / This study examined end-of-the-year seventh grade students’ rational number knowledge using comparison tasks and rational number subconstruct tasks. Comparison tasks included: comparing two decimals, comparing two fractions and comparing a fraction and a decimal. The subconstructs of rational number addressed in this research include: part-whole, measure, quotient, operator, and ratio. Between eighty-six and one-hundred-one students were assessed using a written instrument divided into three sections. Nine students were interviewed following the written instrument to probe for further understanding. Students were classified by error patterns using decimal comparison tasks. Students were initially to be classified into four groups according to the error pattern: whole number rule (WNR), zero rule (ZR), fraction rule (FR) or apparent expert (AE). However, two new patterns emerged: ignore zero rule (IZR) and money rule (MR). Students’ knowledge of the subconstructs of rational numbers was analyzed for the students as a whole, but also analyzed by classification to look for patterns within small groups of students and by individual students to create a thick, rich description of what students know about rational numbers. Students classified as WNR struggled across almost all of the tasks. ZR students performed in many ways similar to WNR but in other ways performed better. FR and MR students had more success across all tasks compared to WNR and ZR. On average apparent experts performed significantly better than those students classified by errors. However, further analysis revealed hidden misconceptions and deficiencies for a number of apparent experts. Results point to the need to make teachers more aware of the misconceptions and deficiencies because in many ways errors reflect the school experiences of students.

Underachievement in Mathematics hangs over South African Mathematics learners like a dark cloud. TIMSS studies over the past decade have confirmed that South African learners‟ results (Grades 8 and 9 in 2011) remained at a low ebb, denying them the opportunity to compete and excel globally in the field of Mathematics. It is against this backdrop that the researcher investigated the meaningful understanding of the important yet challenging algebraic concept of Proportion. The theoretical as well as the empirical underpinnings of the fundamental idea of Proportion are highlighted. The meaningful learning of Algebra was explored and physical, effective and cognitive factors affecting meaningful learning of Algebra, views on Mathematics and learning theories were examined. The research narrowed down to the meaningful understanding of Proportion, misconceptions, and facilitation in developing Proportional reasoning. This study was embedded in an interpretive paradigm and the research design was qualitative in nature. The qualitative data was collected via task sheets and interviews. The sample informing the central phenomenon in the study consisted of a heterogeneous group of learners and comprised a kaleidoscope of nationalities, both genders, a variety of home languages, differing socio-economic statuses and varying cognitive abilities. The findings cannot be generalised. Triangulation of the literature review, the analysis of task sheets and interviews revealed that overall the participants have a meaningful understanding of the Proportion concept. However, a variety of misconceptions were observed in certain cases. Finally, recommendations are made to address the meaningful learning of Proportion and its associated misconceptions. It is hoped that teachers read and act on the recommendations as it is the powerful mind and purposeful teaching of the teacher that can make a difference in uplifting the standard of Mathematics in South African classrooms! / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014