Dissertations / Theses on the topic 'Fractions, Decimals'

Thesis (M.S.)--Central Connecticut State University, 1999.<br>Thesis advisor: Dr. Philip Halloran. " ... in partial fulfillment of the requirements for the degree of Master of Science [in Mathematics]." Includes bibliographical references (leaves 74-75).

The purpose of this thesis is to seek evidence of commonalities in the mental representations of fractions and decimals between zero and one. The focus is on the mental representations of non-familiar fractions and decimals in adults. In addition, individual differences in the extent of common fraction and decimal mental representations are explored and their links to mathematical understanding of numbers between zero and one. For whole numbers, number comparison tasks have found evidence of an ordered, magnitude mental representation known as the mental number line through which the magnitude of a whole number is automatically processed. This evidence consists of phenomena such as the distance effect and SNARC effect. Here, indications of a similar magnitude representation common to both fractions and decimals are sought through a task in which a fraction is compared with a decimal. Substantial evidence of a distance effect is presented but not a SNARC effect, indicating that fractions and decimals can have mental representations containing or accessing a common magnitude but that this magnitude is not automatically processed. In addition, two emergent phenomena are reported. The first is an effect of location which is contrasted with the size effect in whole numbers and a previously reported anchor-point effect. The second is a larger-stimulus effect which is an indication of differences in the mental representations of fractions and decimals. These effects are explored in two additional, simple magnitude and location tasks. Furthermore, success but not speed within the comparison task is linked to strength of the distance effect for individuals. Therefore the number comparison task is repeated in series with a test designed to uncover common misconceptions of fractions and decimals. Patterns with the individual differences in responses to the test and comparison task are explored. By making links between the features and commonalities of individuals’ mental representations of fractions and decimals and quality of their understanding, this research hopes to be of value to mathematical educators.

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.<br>M.Ed.<br>Department of Teaching and Learning Principles<br>Education<br>K-8 Math and Science MEd

Thesis (Ph. D.)--Illinois State University, 2006.<br>Title from title page screen, viewed on June 8, 2007. Dissertation Committee: Norma C. Presmeg (chair), Cynthia W. Langrall, Beverly S. Rich, Janet Warfield. Includes bibliographical references (leaves 177-187) and abstract. Also available in print.

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.<br>M.Ed.<br>Department of Teaching and Learning Principles<br>Education<br>K-8 Math and Science MEd

Over the past decade, technology has become a prominent feature in our lives. Technology has not only been integrated into our lives, but into the classroom as well. Teachers have been provided with a tremendous amount of technology related tools to educate their students. However, many of these technologically enhanced tools have little to no research supporting their claims to enhance learning. This study focuses on one aspect of technology, the iBook, to complete homework relating to fractions, decimals, and percents in a sixth grade classroom. An iBook is a digital textbook that allows the user to interact with the book through various features. Some of these features include galleries, videos, review quizzes, and links to websites. These interactive features have the potential to enhance comprehension through interactivity and increased motivation. Prior to this study, two pilot iterations were conducted. During each pilot study, students in two sixth grade classrooms used the iBook to supplement learning of fractions, decimals, and percents. A comparison group was not included during either iteration, as the goal was to fine-tune the study prior to implementation. The current study was the third iteration, which included a comparison and treatment group. During this study, three research questions were considered: 1) When learning fractions, decimals, and percents, in what ways, if any, do students achieve differently on a unit test when using an interactive iBook for homework as compared to students who have access to the same homework questions in an online static PDF format? 2) What are students' perceptions of completing homework regarding fractions, decimals, and percents with an interactive iBook compared to students who complete homework in an online static PDF format? 3) In what ways does students' achievement on homework differ when completing homework related to fractions, decimals, and percents from an interactive iBook and a static PDF online assignment? Thirty students from a small charter school in southeast Florida participated in the third iteration of this study. Fifteen students were in the comparison group and fifteen were in the treatment group. Students in both groups received comparable classroom instruction, which was determined through audio recordings and similar lesson plans. Treatment group students were provided with a copy of the iBook for homework. Comparison group students were provided with a set of questions identical to the iBook questions in a static digital PDF format. The comparison group students also had access to the textbook, but not the iBook nor the additional resources available within the iBook. The study took place over three weeks. At the commencement of the study, all students were given a pretest to determine their prior knowledge of fractions, decimals, and percents. Students were also asked to respond to questions regarding typical homework duration, level of difficulty, overall experience, and additional resources used for support. During the study, both classes received comparable instruction, which included mini lessons, manipulative based activities, mini quizzes, and group activities. Nightly homework was assigned to each group. At the conclusion of the study, both groups were given a posttest, which was identical to the pretest. Students were asked identical questions about their homework perceptions as prior to the study, but were asked to respond in regards to the study alone. All participating students completed a questionnaire to describe their perceptions of completing homework regarding fractions, decimals, and percents with an iBook as opposed to static digital PDF homework. Lastly, six students from the comparison group participated in a focus group and six students from the treatment group participated in a separate focus group. Data were collected from the pretest and posttest, pre and post homework responses, collected homework, mini quizzes, audio recordings, teacher journal, questionnaires, and the focus group. No difference in achievement was found between the two groups. However, both groups improved significantly from the pretest to posttest. Based on the questionnaires and focus groups, both groups of students felt they learned fractions, decimals, and percents effectively. However, the questionnaire data showed the treatment group found the iBook more convenient than the comparison group did the textbook. Data from this study provide a baseline for future studies regarding iBooks in middle school mathematics. Although the data show no difference in achievement between the two groups, further studies should be conducted in regards to the iBook. Questionnaire and focus group data suggest, with modifications, students may be more inclined to use the resources within the iBook, which may enhance achievement with fractions, decimals, and percents.

Understanding the relationship between fractions and decimals is an important step in developing an overall understanding of rational numbers. Research has demonstrated the feasibility of technology in the form of virtual manipulatives for facilitating students’ meaningful understanding of rational number concepts. This exploratory dissertation study was conducted for the two closely related purposes: first, to investigate a sample of fifth-grade students’ reasoning regarding the relationship between fractions and decimals for fractions with terminating decimal representations while using virtual manipulative incorporating parallel number lines; second, to investigate the affordances of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship. The study employed qualitative methods in which the researcher collected and analyzed data from fifth-grade students’ verbal explanations, hand gestures, and mouse cursor motions. During the course of the study, four fifth-grade students participated in an initial clinical interview, five task-based clinical interviews while using the number line-based virtual manipulatives, and a final clinical interview. The researcher coded the data into categories that indicated the students’ synthetic models, their strategies for converting between fractions and decimals, and evidence of students’ accessing the affordances of the virtual manipulatives (e.g., students’ hand gestures, mouse cursor motions, and verbal explanations). The study yielded results regarding the students’ conceptions of the decimal-fraction relationship. The students’ synthetic models primarily showed their recognition of the relationship between the unit fraction 1/8 and its decimal 0.125. Additionally, the students used a diversity of strategies for converting between fractions and decimals. Moreover, results indicate that the pattern of strategies students used for conversions of decimals to fractions was different from the pattern of strategies students used for conversions of fractions to decimals. The study also yielded results for the affordances of the virtual manipulatives for supporting the students’ reasoning regarding the decimal-fraction relationship. The analysis of students’ hand gestures, mouse cursor motions, and verbal explanations revealed the affordances of alignment and partition of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship. Additionally, the results indicate that the students drew on the affordances of alignment and partition more frequently during decimal to fraction conversions than during fraction to decimal conversions.

The aim of this study is to investigate how the connection between fractions, decimals and percent are taught in grade 4-6 with more focuson the fractions. The empirical data was obtained by qualitative methods comprising interviews with four mathematic elementary school teachers, in addition to two observations with two classrooms in grade 6. The data presented is from one school. The theoretical framework is based on Liping Ma profound understanding of fundamental mathematics and theories of subject didactic concepts of Kilborn, Löwing, Karlsson &amp; Kilborn and MacIntosh. The results of the interviews and observations show that the connection between fractions, decimals and percent is being taught without illuminating how the mentioned are connected. The aspect of fractions, which has been taught to show the relation between fractions and decimals, was division as metaphor. While there was no aspect of fractions has been taught to show the relation between it and percent except that a percent is a hundredth. Such as 40% is equal with 40/100. In addition, fractions has been taught by using visual aids, but never taught by using number line. In conclusion the connection between fractions, decimals and percent has not been related clearly with basic concept fractions.

Made available in DSpace on 2014-06-11T19:24:21Z (GMT). No. of bitstreams: 0 Previous issue date: 2003Bitstream added on 2014-06-13T18:52:14Z : No. of bitstreams: 1 valera_ar_me_mar.pdf: 594283 bytes, checksum: 7fa747413b18f73739f058ca4ea1146e (MD5)<br>Os números racionais apresentam-se como conteúdo que os alunos do Ensino Fundamental e Médio têm dificuldades para aprender. Parte dessas dificuldades decorre da diferença instituída entre o uso cotidiano dos números racionais pelo aluno e a maneira como são ensinados na escola e, também pelo desconhecimento, por parte da escola, da multiplicidade dos significados dos racionais. Enquanto o uso social centra-se na forma decimal o uso escolar recai mais sobre a forma fracionária dos números racionais. É uma separação indesejável que as práticas escolares trataram de acentuar ao longo do tempo. A partir de pesquisa bibliográfica e de estudo documental procurou-se caracterizar, nesse trabalho, a dicotomização existente entre o uso e o ensino da Matemática, que acabam sendo responsáveis por prejuízos na aprendizagem dos alunos. Isto pode ser verificado nos erros que os alunos cometeram nas provas oficiais (SARESP, SAEB...). Procurou-se analisar como essa separação vem sendo reforçada nos documentos oficiais, por meio das propostas pedagógicas e curriculares. Verificaram-se como diferentes documentos e publicações oficiais abordam os números racionais e tratam da articulação entre a perspectivas do uso escolar e a do uso cotidiano dos números racionais. Essa análise possibilitou compreender diferentes tipos de argumentações e justificativas para o ensino das frações, presentes nos currículos oficiais, bem como explicitar os conteúdos e metodologias adequadas às concepções apresentadas em tais documentos. Tudo isso possibilitou conhecer parte dos problemas que ocorrem com o ensino de frações e suas causas e por isso sugerir propostas que sinalizam para a sua superação. Embora o estabelecimento de relações entre o uso social e uso escolar ainda não ocorra de maneira efetiva, reconhece-se que aquelas orientações...

Orientador: Vinício de Macedo Santos<br>Banca: Célia Maria Carolino Pires<br>Banca: José Carlos Miguel<br>Abstract: The rational numbers are shown as a subject that the students of the Elementary and High School have difficulties to learn. Some of these difficulties are due to the difference established between the daily use of the rational numbers by the student and the way it is taught at the school and, also for the ignorance, on the part of the school, of the multiplicity of their meanings. While the social use is centered in the decimal form, the school use lies more on the fractional form of the rational numbers. It is an undesirable separation that the school practices have accentuated through time. This study tried to characterize the existent dichotomization between it the use and the teaching of the Mathematics, starting from bibliographical research and of documental study that end up being responsible for damages in the students' learning.. This can be verified in the mistakes committed in the official tests (SARESP, SAEB...). It was sought to analyze how that separation has been reinforced in the official documents, by the pedagogic proposals and curricula. It was verified how the different documents and official publications deal with the rational numbers and the articulation among perspectives of the school use and the daily use of the rational numbers. That analysis made possible to understand different types of arguments and justifications for the teaching of the fractions, present in the official curricula, as well as explain the contents and the most appropriate methodologies of the conceptions presented in such documents. All this made possible to know part of the problems that happen with the teaching of fractions and their causes, and so, make suggestions on how these problems can be solved. Although the establishment of relationships between the social use and school use still doesn't happen in an effective way, it is recognized... (Complete abstract, click electronic address below)<br>Resumo: Os números racionais apresentam-se como conteúdo que os alunos do Ensino Fundamental e Médio têm dificuldades para aprender. Parte dessas dificuldades decorre da diferença instituída entre o uso cotidiano dos números racionais pelo aluno e a maneira como são ensinados na escola e, também pelo desconhecimento, por parte da escola, da multiplicidade dos significados dos racionais. Enquanto o uso social centra-se na forma decimal o uso escolar recai mais sobre a forma fracionária dos números racionais. É uma separação indesejável que as práticas escolares trataram de acentuar ao longo do tempo. A partir de pesquisa bibliográfica e de estudo documental procurou-se caracterizar, nesse trabalho, a dicotomização existente entre o uso e o ensino da Matemática, que acabam sendo responsáveis por prejuízos na aprendizagem dos alunos. Isto pode ser verificado nos erros que os alunos cometeram nas provas oficiais (SARESP, SAEB...). Procurou-se analisar como essa separação vem sendo reforçada nos documentos oficiais, por meio das propostas pedagógicas e curriculares. Verificaram-se como diferentes documentos e publicações oficiais abordam os números racionais e tratam da articulação entre a perspectivas do uso escolar e a do uso cotidiano dos números racionais. Essa análise possibilitou compreender diferentes tipos de argumentações e justificativas para o ensino das frações, presentes nos currículos oficiais, bem como explicitar os conteúdos e metodologias adequadas às concepções apresentadas em tais documentos. Tudo isso possibilitou conhecer parte dos problemas que ocorrem com o ensino de frações e suas causas e por isso sugerir propostas que sinalizam para a sua superação. Embora o estabelecimento de relações entre o uso social e uso escolar ainda não ocorra de maneira efetiva, reconhece-se que aquelas orientações... (Resumo completo, clicar acesso eletrônico abaixo)<br>Mestre

A total of 84 students from a lower economic area, aged 8 to 14, were interviewed about their understanding of decimal fractions. Results showed that most students could give a context in which they saw decimal fractions outside of school. The vast majority could draw a diagram of how a cake or field could be divided equally among 10 or 100 people. However, few students under 14 could give either decimal fraction symbols or common fraction symbols to represent these divisions. Less than half of the students at ages 10, 11 and 12 could visualize what might come between 0 and 1. About half of the students aged 11 and 12 could indicate what 0.1 or 0.01 meant. It was inferred that difficulty in relating these symbols to referents might be an important source of difficulty in understanding decimal fractions. Therefore, these interviews were followed by an intervention study that examined if working with contextualized decimal fractions aided understanding of these numbers when they were presented without context. Half of a group of 16 similar students, aged 11 and 12, were asked to solve problems in which numbers that incorporated decimal fractions were contextualized, and the other half were asked to solve similar problems given in purely numerical form. Students worked in pairs, on problems which incorporated common misconceptions. The group who worked on contextualized problems gained significantly more understanding than did the group that worked on purely numerical problems, as measured by the difference between pretest and posttest scores. Transcripts of the students' discussions were analysed for the effect of prior learning, aspects of peer collaboration that appeared to be beneficial to learning, and the effect of cognitive conflict. The students who gained most from collaboration were not too distant in initial expertise, showed a degree of social equity, and worked on contextualized problems. Much of students' learning appeared to result from needing to reconsider their views following a conflict between their expectations and the results of operating on a calculator or in writing, or hearing an alternative view.<br>Subscription resource available via Digital Dissertations only.

Made available in DSpace on 2014-06-11T19:24:54Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-04-01Bitstream added on 2014-06-13T20:13:19Z : No. of bitstreams: 1 lima_cw_me_rcla.pdf: 3093037 bytes, checksum: 82ceff562d5a32cc23b45ec23e51ab60 (MD5)<br>See-Sp<br>Essa pesquisa investiga as contribuições que a exploração dos números racionais como medidas de segmentos, em um programa de geometria dinâmica, podem trazer ao entendimento de frações, decimais e da reta numérica entre outras representações dos racionais. A pesquisa se fundamenta em evidências históricas e resultados de pesquisas que mostram a importância do significado de medida para o entendimento dos números. Através das tecnologias informáticas viu-se uma alternativa para a exploração da medida de segmentos. Essa pesquisa é baseada no processo de medição de segmentos, em teorias sobre visualização, experimentação e representações múltiplas. Também se inspira em preceitos construcionistas. Essa investigação qualitativa se baseou na metodologia de experimentos de ensino, em que foram formados dois grupos com alunos de 6ª série / 7º ano do ensino fundamental de uma escola pública estadual do interior de São Paulo. Esses grupos participaram de encontros em que foram desenvolvidas atividades que envolviam: divisão de segmentos; frações como medidas de segmentos; operações de adição e subtração de frações utilizando os segmentos; processo de medição para criação dos números decimais; relações entre decimais e frações; adição e subtração dos números decimais; adição e subtração de frações e decimais. As atividades realizadas se basearam nos recursos de visualização e experimentação proporcionadas pelo software de geometria dinâmica Régua e Compasso. O trabalho evidenciou a importância da aprendizagem das representações múltiplas dos números racionais e como as tecnologias informáticas (computadores, software de geometria e calculadoras) podem atuar nessa aprendizagem. A pesquisa também evidência que a utilização de recursos tecnológicos pode modificar a matemática da sala de aula, proporcionando aos estudantes...<br>This research investigates the contributions that the exploration of rational numbers as measure of segments, using geometry dynamic software, can introduce into the understanding of fractions, decimal numbers and the number line, amongst other rational number representations. The research is motivated by both historical evidence and evidence from the research literature showing the importance of the measure meaning to the understanding of rational numbers. Digital technologies offer an alternative method for the exploration of segments measure, as yet underexplored in the field of mathematics education. This research is based on an approach to numbers as measurements of segments, which draws from theories emphasizing the role of visualization, experimentation and multiple representations in mathematics learning. It is also inspired by a constructionist perspective. The qualitative investigation made use of the teaching experiment methodology, in that two groups were formed with students of 6th grade / 7th year within an elementary school of a public school in the state of São Paulo. These groups took part in research sessions where they developed activities that involve: division of segments; fractions as measure of segments; operations of addition and subtraction of fraction using segments; measurement for decimal numbers creation; relations between decimal numbers and fractions; addition and subtraction of decimal numbers; addition and subtraction of fractions and decimal numbers. The activities exploited the resources visualization and experimentation proportioned by the dynamic geometry software “Compass and Rule”. Analyses of the data collected pointed to the importance of the understanding of multiple representations for rational numbers and to the role that digital technologies (computers, geometry software and calculators) can play in this learning. This research, also, ... (Complete abstract click electronic access below)

Thesis (MEd)--University of Stellenbosch, 2005.<br>ENGLISH ABSTRACT: Research shows that misconceptions about calculations develop in many classrooms without being noticed and these are not corrected by repeated routine exercises. The misconceptions formed are at times the result of inappropriate models used to solve problems. An even bigger concern is that these particular models sometimes provide the correct answers by accident. This may result in the learner's belief in the models being reinforced, as described by Swan (n.d.). The aim of this study is to identify the misconceptions related to the use of decimal fractions by Grade 8 and 9 learners and then, through the use of an intervention program, to address the learners' misconceptions and attempt to correct them. Two schools were involved in this study. The group of learners from school A served as a control group to determine the success of the intervention in learners from school B. The results of school A, the frequency and nature of errors were compared with the test results of school B as well as described by interviews with learners from school B. After the diagnostic tests and interview, the learners' answers were compared with those already described in literature. The learners from school B participated voluntarily in the intervention program. Learners from both schools wrote a post-test and the results were compared with those of a pre-test. The conclusion of this study is that there are misconceptions concerning calculations with decimal fractions at Grade 8 and 9 level. These misconceptions are formed during the intermediate phase and are not suitably corrected. The intervention program, for various reasons, had limited success. These reasons are discussed and recommendations are made for future intervention programs.<br>AFRIKAANSE OPSOMMING: Navorsing toon dat wanbegrippe ten opsigte van berekeninge in baie klaskamers onopgemerk verbygaan en dat dit nie reggestel word deur herhaalde roetine oefeninge nie. Wanbegrippe wat kinders vorm is onder andere die gevolg van onvanpaste modelle wat gebruik word vir die oplos van probleme. 'n Groter gevaar is dat hierdie onvanpaste modelle toevallig die regte antwoord lewer. Dit kan dan veroorsaak dat die leerder se vertroue op die modelle net versterk word, soos Swan (s.j.) dit beskryf. Die doel van hierdie studie is om wanbegrippe ten opsigte van bewerkings met desimale breuke by Graad 8 en 9 leerders te identifiseer en dan deur middel van 'n intervensieprogram die leerders se wanbegrippe aan te spreek en te probeer regstel. Twee skole is by hierdie studie betrek. Die groep leerders van skool A sou dien as 'n kontrolegroep om die intervensie-sukses van die leerders van skool B te bepaal. Die skool A resultate en frekwensie van foute asook die aard daarvan is vergelyk met die toetse van skool B en beskryf op grond van onderhoude met die leerders van skool B. Ná die diagnostiese toets en onderhoud is die leerders se antwoorde vergelyk met dié wat reeds in die literatuur beskryf is. Die leerders van skool B is op vrywillige basis by 'n intervensieprogram betrek. Beide skole se leerders het daarna 'n natoets geskryf en die resultate is vergelyk met dié van die voortoets. Die gevolgtrekking wat uit hierdie studie gemaak word, is dat daar wanbegrippe ten opsigte van bewerkings met desimale breuke op graad 8 en 9 vlak aanwesig is. Hierdie wanbegrippe is in die intermediêre fase gevorm en nie reggestel nie. Die intervensieprogram het om verskeie redes slegs beperkte sukses gehad. Hierdie redes word bespreek en aanbevelings word gemaak vir toekomstige intervensieprogramme.

There are many researchers that emphasize the importance of how teachers’ knowledge will affect students’ learning. However, not much research is focused on an international comparison between preservice mathematics teachers’ procedural knowledge and conceptual knowledge. There were 91 preservice mathematics teachers involved in this study. A test on the operations on fractions, decimals, percentages, and integers knowledge showed a) the significant differences between the United States and Chinese preservice teachers’ (PTS) procedural knowledge, b) the significant differences between the United States and Chinese PTS’ conceptual knowledge, and c) the relationships between the United States and Chinese PTS’ procedural knowledge and conceptual knowledge. By comparing the results, the researcher determined the strengths and weaknesses of preservice mathematics teachers in the two countries. The researcher will provide PTS some information based on the results of the knowledge test.

Submitted by Raniere Barreto (raniere.barros@ufvjm.edu.br) on 2018-04-12T16:51:05Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) raphael_neves_matos.pdf: 4286914 bytes, checksum: 4faddab9001b8b035017adfd9a2d6d75 (MD5)<br>Approved for entry into archive by Rodrigo Martins Cruz (rodrigo.cruz@ufvjm.edu.br) on 2018-04-20T14:12:28Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) raphael_neves_matos.pdf: 4286914 bytes, checksum: 4faddab9001b8b035017adfd9a2d6d75 (MD5)<br>Made available in DSpace on 2018-04-20T14:12:28Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) raphael_neves_matos.pdf: 4286914 bytes, checksum: 4faddab9001b8b035017adfd9a2d6d75 (MD5) Previous issue date: 2017<br>Este trabalho teve como objetivo principal apresentar uma contribui??o para o ensino aprendizagem dos n?meros racionais, destacando principalmente a rela??o entre d?zimas peri?dicas e progress?es geom?tricas. A metodologia utilizada permitiu a an?lise da abordagem e sequ?ncia did?tica dos t?picos D?zima peri?dica e Progress?o Geom?trica Infinita, contemplada nos livros did?ticos aprovados pelo Programa Nacional do Livro Did?tico. Nesta abordagem as fra??es e os n?meros decimais, especialmente os decimais infinitos e peri?dicos, e por consequ?ncia o c?lculo de sua fra??o geratriz, foram objetos de estudo centrais e instigadores dessa pesquisa. Realizou-se um estudo mais detalhado sobre a representa??o decimal dos n?meros racionais e analisando a compreens?o destes n?meros em n?vel fundamental e m?dio. Foi ainda proposto uma abordagem das maneiras mais usuais do c?lculo da fra??o geratriz, bem como, explorado a rela??o entre os decimais infinitos e peri?dicos e as progress?es geom?tricas. Durante o desenvolvimento deste trabalho, foi poss?vel perceber que h? mais de uma abordagem did?tica dos t?picos de ensino inerentes ao tema central analisado. O reconhecimento de que a parte decimal das d?zimas peri?dicas pode ser expressa como uma soma infinita de parcelas que, a partir de certo ponto, descreve uma progress?o geom?trica infinita de raz?o compreendida entre zero e um, ? um ponto chave na proposta de interven??o apresentada para a sala de aula. Diante desse quadro, foi verificado a ordem atualmente seguida pelos professores do 1? Ano do Ensino M?dio, o que permitiu constatar que os conte?dos D?zimas Peri?dicas e Progress?es Geom?tricas Infinitas s?o tratados sem liga??o significativa e, diante disso, foi proposta uma altera??o na ordem de abordagem desses conte?dos no Ensino M?dio. Ao final foram propostas algumas sugest?es de atividades resolvidas e outras para serem desenvolvidas em sala de aula.<br>Disserta??o (Mestrado Profissional) ? Programa de P?s-Gradua??o Matem?tica, Universidade Federal dos Vales do Jequitinhonha e Mucuri, 2017.<br>The aim of this work was to present a contribution to the teaching of rational numbers, emphasizing mainly the relation between periodic tithe and geometric progression. The methodology used allowed the analysis of the approach and didactic sequence of the topics Periodic Dizima and Infinite Geometric Progression, contemplated in textbooks approved by the National Textbook Program. In this approach fractions and decimal numbers, especially the infinite and periodic decimals, and consequently the calculation of their generative fraction, were central objects and instigators of this research. A more detailed study on the decimal representation of rational numbers was carried out and the understanding of these numbers at the fundamental and medium levels was analyzed. It was also proposed an approach of the most usual ways of calculating the generative fraction, as well as exploring the relationship between infinite and periodic decimals and geometric progressions. During the development of this work, it was possible to perceive that there is more of a didactic approach of the teaching topics inherent to the central theme analyzed. The recognition that the decimal part of the periodic tithe can be expressed as an infinite sum of plots which, from a certain point, describes an infinite geometric progression of ratio between zero and one, is a key point in the proposal of intervention presented for the classroom. In view of this situation, we verified the order currently being followed by teachers of the 1? Year of High School, which allowed to verify that the Periodic Dictionaries and Infinite Geometric Progressions are treated without significant connection and, accordingly, a change was proposed in order to approach these contents in High School. At the end, some suggestions for solved activities and others to be developed in the classroom were proposed.

Orientador: Marcus Vinicius Maltempi<br>Banca: Marcelo de Carvalho Borba<br>Banca: Siobhan Victoria Healy<br>Resumo: Essa pesquisa investiga as contribuições que a exploração dos números racionais como medidas de segmentos, em um programa de geometria dinâmica, podem trazer ao entendimento de frações, decimais e da reta numérica entre outras representações dos racionais. A pesquisa se fundamenta em evidências históricas e resultados de pesquisas que mostram a importância do significado de medida para o entendimento dos números. Através das tecnologias informáticas viu-se uma alternativa para a exploração da medida de segmentos. Essa pesquisa é baseada no processo de medição de segmentos, em teorias sobre visualização, experimentação e representações múltiplas. Também se inspira em preceitos construcionistas. Essa investigação qualitativa se baseou na metodologia de experimentos de ensino, em que foram formados dois grupos com alunos de 6ª série / 7º ano do ensino fundamental de uma escola pública estadual do interior de São Paulo. Esses grupos participaram de encontros em que foram desenvolvidas atividades que envolviam: divisão de segmentos; frações como medidas de segmentos; operações de adição e subtração de frações utilizando os segmentos; processo de medição para criação dos números decimais; relações entre decimais e frações; adição e subtração dos números decimais; adição e subtração de frações e decimais. As atividades realizadas se basearam nos recursos de visualização e experimentação proporcionadas pelo software de geometria dinâmica "Régua e Compasso". O trabalho evidenciou a importância da aprendizagem das representações múltiplas dos números racionais e como as tecnologias informáticas (computadores, software de geometria e calculadoras) podem atuar nessa aprendizagem. A pesquisa também evidência que a utilização de recursos tecnológicos pode modificar a matemática da sala de aula, proporcionando aos estudantes ... (Resumo completo, clicar acesso eletrônico abaixo)<br>Abstract: This research investigates the contributions that the exploration of rational numbers as measure of segments, using geometry dynamic software, can introduce into the understanding of fractions, decimal numbers and the number line, amongst other rational number representations. The research is motivated by both historical evidence and evidence from the research literature showing the importance of the measure meaning to the understanding of rational numbers. Digital technologies offer an alternative method for the exploration of segments measure, as yet underexplored in the field of mathematics education. This research is based on an approach to numbers as measurements of segments, which draws from theories emphasizing the role of visualization, experimentation and multiple representations in mathematics learning. It is also inspired by a constructionist perspective. The qualitative investigation made use of the teaching experiment methodology, in that two groups were formed with students of 6th grade / 7th year within an elementary school of a public school in the state of São Paulo. These groups took part in research sessions where they developed activities that involve: division of segments; fractions as measure of segments; operations of addition and subtraction of fraction using segments; measurement for decimal numbers creation; relations between decimal numbers and fractions; addition and subtraction of decimal numbers; addition and subtraction of fractions and decimal numbers. The activities exploited the resources visualization and experimentation proportioned by the dynamic geometry software "Compass and Rule". Analyses of the data collected pointed to the importance of the understanding of multiple representations for rational numbers and to the role that digital technologies (computers, geometry software and calculators) can play in this learning. This research, also, ... (Complete abstract click electronic access below)<br>Mestre

I denna studie har tidigare internationell forskning inom området tal i decimalform undersökts i en svensk kontext. I den matematikdidaktiska forskningen har ett teoretiskt ramverk för elevers olika uppfattningar av tal i decimalform tagits fram. Tidigare studier har gjort flera försök till att kategorisera elevers olika förståelse av tal i decimalform (Moloney &amp; Stacey, 1997; Resnik et al., 1989; Sackur-Grisvard &amp; Léonard, 1985; Stacey &amp; Steinle, 1998). Sackur-Grisvard och Léonards (1985) kategorisering utgår ifrån elevernas förkunskaper inom andra matematiska områden. Deras teoretiska ramverk består av elevers användning utav tre olika regler; heltalsregeln, bråkregeln och nollregeln. Sackur-Grisvard och Léonards (1985) teoretiska ramverk har inte använts i någon högre utsträckning i det svenska forskningsfältet. Ramverket har i denna studie använts för att ta reda på om det kan användas som ett verktyg för att kategorisera elevers olika förståelse av tal i decimalform i årskurserna 4-6. I studien har metoden triangulering använts med både ett skriftligt test och semistrukturerade intervjuer. Alla elever har genomfört ett skriftligt test där de fått lösa uppgifter genom att jämföra och storleksordna olika tal i decimalform. Elevernas resultat användes sedan där några få elever ifrån årskurserna 4 och 5 valdes ut till semistrukturerade intervjuer genom ett målstyrt urval.  Resultatet visade att det teoretiska ramverket hade vissa begränsningar och att flera elever inte kunde kategoriseras till enbart en kategori utan flertalet använde sig av flera regler på det skriftliga testet. Elevernas resultat visade även en progression inom ämnesområdet där elever i årskurs 6 presterade bäst efterföljt av årskurs 5 och elever i årskurs 4 presterade sämst.<br>In this study earlier international research has been used from a Swedish perspective to investigate the field of decimal numbers. A theoretical framework for students’ various perceptions of decimal number has developed from the mathematical didactic research field. Earlier studies have done different attempts to categories students’ various perceptions of decimal numbers (Moloney &amp; Stacey, 1997; Resnik et al., 1989; Sackur-Grisvard &amp; Léonard, 1985; Stacey &amp; Steinle, 1998). Sackur-Grisvard and Léonard (1985) categorization focus on students’ earlier knowledge in the mathematical field. Their theoretical framework involves the use of three different rules; the whole number rule, the fraction rule and the zero rule. Sackur-Grisvard and Léonard’s (1985) theoretical framework has not been used much in the Swedish research field. In this study the framework has been used to investigate if it can be used as a tool to categorise students in grade 4 and grade 5 various perception of decimal numbers. In this study the method triangulation has been used which involves a written test and semi-structure interviews. In the written test all students got tasks where they would compare and order different decimal numbers.  The students result from the test were used to choose a few students from grade 4 and grade 5 to do the semi-structured interviews through a target-driven selection. The result showed that the theoretical framework did have some limits and several students´ did not belong to only one category, several students did use more than one of the three rules in the written test. The students result showed a progression where students from grade 6 performed best on the test followed by students from grade 5, students in grade 4 performed worst.

Using both quantitative and qualitative data collection and analyses techniques, this study examined representations used by sixteen (n = 16) teachers while teaching the concepts of converting among fractions, decimals, and percents. The classroom videos used for this study were recorded as part of the Middle School Mathematics Project (MSMP). The study also compared teacher-selected and textbook representations and examined how teachers‘ use of idiosyncratic representations influenced representational choices on the number test by the teachers‘ five hundred eighty-one (N = 581) students. In addition to using geometric figures and manipulatives, a majority of the teachers used natural language such as the words nanny, north, neighbor, dog, cowboy, and house to characterize fractions and mathematical procedures or algorithms. Coding of teacher-selected representations showed that verbal representations deviated from textbook representations the most. Some teachers used the words or phrases bigger, smaller, doubling, tripling, breaking-down, and building-up in the context of equivalent fractions. There was widespread use of idiosyncratic representations by teachers, such as equations with missing or double equal signs, numbers and operators written as superscripts, and numbers written above and below the equal sign. Although use of idiosyncratic representations by teachers influenced representational choices by students on the number test, no evidence of a relationship between representational forms and degree of correctness of solutions was found. The study did reveal though that teachers‘ use of idiosyncratic representations can lead to student misconceptions such as thinking that multiplying by a whole number not equal to 1 gives an equivalent fraction. Statistical tests were done to determine if frequency of representation usage by teachers was related to the textbook, highest degree obtained by teacher, certification, number of years spent teaching mathematics, number of years teaching mathematics at grade level, number of hours completed on professional development related to their textbook, and total number of days spent on the Interagency Education Research Initiative (IERI) professional development. The results showed representation usage was related to all the above variables, except the highest degree obtained and the total number of days spent on the IERI professional development.

碩士<br>國立屏東教育大學<br>數理教育研究所<br>100<br>The purpose of this study is to compare and analyze the problems types of fractions and decimals in most popular textbooks used in three countries: ‘Kang-Hsuan mathematics’ in Taiwan, ‘Laskutaito in English’ in Finland. and ‘My Pals are Here! Maths’ in Singapore. Content analysis was used as method and mathematics problem was as unit to analyze the similarities and differences among them. The analytic categories were followed the classification of Stein, Remilliard and Smith (2000) that classified problems base on cognitive demand when problem solving, the design of the contexts by Lesh and Lemon(1992), and the types of representations by Zhu and Fan(2006). The finding of this study indicated the numbers of question in three edition is different, Kang-Hsuan has 920 questions, Laskutaito has 2512 , and My Pals are Here has 1280. Kang-Hsuan and My Pals are Here presented a complete problem-solving process for each example task, Laskutaito only offers a brief question narrative for each example and exercise. Three editions of textbooks all emphasize tasks in low cognitive demand, especially using processes without connections as the most. The number of exercise/example in Laskutaito is about 27, it is the highest in three country. Kang-Hsuan and My Pals are Here! Math&apos;&apos;s only provide about 2. Considering the dimentions of task contexts, there is about 50% tasks designed into context in Kang-Hsuan, but Laskutaito and My Pals are Here! Math&apos;&apos;s both use tasks without context more then into context. Considering the approach to present questions, Laskutaito and My Pals are Here! Math&apos;&apos;s hold representational types of question mostly in the mathematical attribute, and Kang-Hsuan use mathematical but also verbal attribute.

碩士<br>國立臺北教育大學<br>教育經營與管理學系<br>99<br>An Action Study of Implementing an After-School Alternative Program at an Elementary School in New Taipei City -Math Fractions and Decimals Diagnostic Instruction- Abstract This research applies diagnostic instruction to an After-School Alternative Program to identify the misconceptions of some fourth-grade students regarding mathematical fractions and decimals. It reviews the changes in the students’ learning throughout the implementation of the action study program, and concludes with some reflections on professional growth. The subjects were fourth-grade students from Sing-Sing Elementary School, New Taipei City, where the researcher is employed. The results of the pretest on fractions and decimals, diagnostic instruction, and an interview, indicated that the students’ common misconceptions/problems about fractions and decimals included: (1) wrongly thinking of fractions as having the same unit partition in the same context, (2) difficulty identifying the unit when identifying fractions in the part-whole model, (3) incorrect understanding about unit partition when comparing the size of fractions, (4) mistakenly thinking of fractions in terms of the one with a greater denominator having greater value, (5) a weak understanding of fractional equality, (6) neglecting the given units when solving fractional questions, (7) misusing integers to process fractions by treating the numerator and the denominator as two independent values, (8) incorrectly calculating the fractions resulting from different partitions and combinations, (9) innumeracy when comparing the size of fractions, (10) feeling confused about the meaning of tenth decimals and conversion between decimals and fractions, (11) mistaking the content of a tenth or a hundredth for that of the number one, (12) completely ignoring the decimal point, (13) neglecting the units corresponding to decimals, (14) being affected by the place names and values of integers and wrongly applying them to decimals, and (15) misusing the operational concept for integers to calculate decimals by justifying numeral figures in disregard of any decimal points. After the implementation of the diagnostic instruction, the students showed a significant improvement in their understanding of “the meaning of simple fractions,” “the meaning of halving, namely equally dividing,” “the part-whole model when identifying fractional units,” “the meaning of decimals, conversion between decimals and fractions, and the relation between 0.1 and 1,” and “calculation with decimals.” However, some still found it difficult to answer relatively straight-forward questions, such as ones involving “equal division,” “fractional units,” “equivalent fractions” and “applications of decimals.” The researcher found from the process of diagnostic instruction that one of its priorities should be the development of students’ confidence and interests. It is thus believed that teaching students by providing diverse and ample opportunities for practical applications is important. It is also believed that teachers should equip themselves with adequate knowledge in mathematics, so as to properly determine and facilitate in-class discussion. Ideally, it is argued that teachers can use reflective journals to understand their students’ real feelings and learning outcomes. Finally, some recommendations are proposed, consistent with the results, for school administrators, teachers, and future researchers.

碩士<br>國立臺南大學<br>應用數學系碩士在職專班<br>106<br>This study aimed to investigate the effects of team-ranking-tournament and cooperative problem-posing instruction on six graders’ learning achievement and learning motivation. In addition, students’ learning feelings and teachers’ implementation difficulties were also explored. The study was expected to be reference for teachers to adopt teaching strategies. Both qualitative and quantitative methods were utilized to collect and analyze the data in this study. The participants were 41 six graders from two classes of an elementary school in Nansi district of Tainan city. One class was the experimental group receiving team-ranking-tournament and cooperative problem-posing instruction for eight periods in two weeks while the other was the controlled group receiving traditional teaching method. The teaching materials were the unit “Four Arithmetic Operations of Fractions and Decimals” from the twelfth volume of Kang Hsuan Edition. After the experimental teaching, students’ scores of “Learning Achievement Quizzes for Four Arithmetic Operations of Fractions and Decimals” and “Scale of Mathematical Learning Motivation” were collected and analyzed by SPSS 21.0. Semi-structured interviews were further conducted to understand students’ learning feelings. The results were summarized as follows: 1. In terms of the performances on the quizzes of four arithmetic operations of fractions and decimals, sixth graders showed the highest correctness to the questions of solving four arithmetic operations of decimals while they showed the lowest correctness to the questions of solving four arithmetic operations of decimals. 2. Students’ performances towards learning fractions and decimals in the experimental class were not significantly enhanced after receiving team-ranking-tournament and cooperative problem-posing instruction. However, the mean score of the experimental class was higher than the one of the controlled class. 3. Sixth graders’ learning motivation of math was not significantly influenced by team-ranking-tournament and cooperative problem-posing instruction. However, the average of post-test was higher than the one of pre-test. 4. According to the data from the interviews, students’ learning feelings were as follows: (1) Math classes became diverse and interesting. (2) Posing math questions could promote thinking. (3) Students felt their grade progress. (4) Students would like to learn math through team-ranking-tournament and cooperative problem-posing instruction. Based on the experimental teaching, teachers’ implementation difficulties were as follows: (1) Some of the students could not get along with their peers, causing grouping concerns. (B) Teaching time was limited, leading to high similarity of students’ posing questions. (C) Disadvantaged students lacked experience of presentation, resulting in their feeling of powerless.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1997.<br>Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 135-139).

Thesis (M. Ed. (Curriculum Studies)) -- University of Limpopo, 2020<br>The purpose of this study was to explore the Pedagogical Content Knowledge (PCK) of Intermediate Phase teachers in the teaching of decimal fractions to Grade 6 learners. The study followed a qualitative research approach whereby a case study design was adopted. Three Grade 6 teachers were selected using a purposive sampling strategy to form part of the study. Shulman‟s (1986) Theory of Teacher Knowledge was used to guide the entire study. Data were collected through lesson observations, semi-structured interviews and document analysis. Data were analysed and interpreted using the Argyris, Putman and Smith‟s Ladder of Inference. The study established that Grade 6 teachers lacked PCK in the teaching of decimal fractions. Teachers lacked confidence in the teaching of decimals. The analysis of data also revealed that teachers‟ knowledge of decimal fractions was poor, and that teachers experienced challenges in teaching decimal fractions. Generally, decimal fractions were found to be difficult for teachers to teach. This led to the conclusion that teachers lack Pedagogical Content Knowledge in the teaching of decimal fractions. These findings, though not generalizable to a wider population, provide useful information for further research and insights of what Grade 6 mathematics teachers may be experiencing in their classrooms. The findings may help teachers improve their teaching. They also have implications for teacher-education institutions as they may restructure their teaching programmes, both for pre-service and in-service teachers.

碩士<br>國立臺南大學<br>應用數學研究所碩士班<br>101<br>This study is about an innovative teaching method. The purpose of the study is to explore the learning process when students learned the concept of Decimal Division by Equivalent Fractions concept and Metacognition strategy. The researcher believes that the best way to learn mathematics concepts is to connect the new concept to the old relevant ones and to make the mathematics concepts from a whole connected net. Only Metacognition can make it happen. In this study, we used Equivalent Fractions concept to turn the Decimal Division into the Integer Division. And used Metacognition strategy to help students learn and understand the Decimal Division more easily. In preview-work sheet, we helped students remind relevant concepts, and then ask them to solve the new concept problems by themselves. In classroom, the teacher and students discussed the ideas on the preview-work sheet. Through the discussion, the teacher could help students clarify and correct their wrong concepts. After class, the teacher ask students to write another worksheet in order to combine and to connect the concepts. The important findings of this study is as follows: First: When students learning new concept, they can search the latest relevant concept to solve the new concept problem actively. Second: By using Equivalent Fractions concept and Metacognition strategy, students learn the Decimal Division more understandble. Third: Learning metacognitvely, student can through the solve process (using Equivalent Fractions) to deduce the adult way, such as “the decimal point of quotient has to align the new decimal point of dividend” and “the decimal point of remainder has to align the old decimal point of dividend”. Fourth: In post test, the experimental group are better than the control group. It can prove the innovative teaching method is an effective teaching method.

碩士<br>國立臺南大學<br>應用數學系數學科教學碩士班<br>102<br>This study aimed to investigate the effectiveness of teaching after the fourth grade students for acceptance to mention in this study, "the concept of teaching a unit volume and fractional units of fractional divide research," learning retention effect and problem solving performance history, architecture through teaching experiment teaching, as well as pre-test, post-test and retention tests, semi-structured interviews to collect data and analyze statistical learning. The results are summarized as follows: 1, The experimental group students in decimal and fractional division of conceptual understanding and problem solving have been enhanced in the application of experimental teaching problem solving strategies are cited using the "unit volume concept 'conduct problem-solving. 2, The experimental group and the control group to test two independent sample t-test post-test scores, the results reached significant difference, indicating that the real Experimental group than the control group learning. 3, Students can learn from the experimental group in the post-test and delayed test scores statistics, the use of "unit volume concept 'study. For the fourth grade students in the integration of the concept of fractional division and fractions learning, can really enhance learning and concept retention scenario.

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!<br>MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

碩士<br>國立中正大學<br>課程研究所<br>100<br>The aim of the research is mainly on to assist disadvantaged children by mathematics remedial instruction in the alternative after- school program. Two cycles of the action research were conducted in January 10 2010 to January 27of 2010 and April 12 to May 6 of 2010. The objectives of this research include the following four points: 1.To analysis the encountered teaching scene problems during implementing mathematics remedial instruction course in “after school alternative program”. 2. to search and plan appropriate activities related to "decimal division" and " fractions and decimals of multiplication and division” for the class. 3. to carry out the action plan of reflecting the progress of "decimal division" and " fractions and decimals of multiplication and division” for the class. 4. to evaluate and to reflect from the action plan of "decimal division" and " fractions and decimals of multiplication and division” for the class. To solve the research above, firstly, the principal investigator analyze the misconceptions of "decimal division" and " fractions and decimals of multiplication and division” from the disadvantaged sixth-grader. Secondly , the researcher develops and implements the remedial instructional materials. The findings include the following sections:1.The remedial instruction model could improve the teachers' abilities to face in the alternative after-school program.2. Via collecting the math misconceptions, the author develop and adjust the remedial teaching materials. 3.The remedial instruction should be considered as continuous cycle process in the action research. 4. The remedial instruction is a continuous process including diagnostics, teaching, and diagnostics instead of focusing on scores. The principal investigator 's reflections cover the special needs of disadvantaged children, but mention about the profession of afterschool teachers. The remedial instruction should not be only focused on disciplines. The construction of math concept couldn't be established and effective in a short time, and the most important sections of the remedial instruction are to provide and encourage students continually. The performance and outcome of disadvantaged students are diversified and the afterschool program teachers should pay attentions on their other positive learning outcomes.

In 1978, only 24% of 8th grade students in the United States correctly answered whether 12/13+7/8 was closest to 1, 2, 19, or 21 (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980). In 2014, only 27% of 8th grade students selected the correct answer to the same problem, despite the ensuing forty years of effort to improve students’ conceptual understanding (Lortie-Forgues, Tian, & Siegler, 2015). This is troubling, given that 5th grade students’ fraction knowledge predicts mathematics achievement in secondary school (Siegler et al, 2012) and that achievement in math is linked to greater life outcomes (Murnane, Willett, & Levy, 1995). General rational number knowledge (fractions, decimals, percentages) has proven problematic for both children and adults in the U.S. (Siegler & Lortie-Forgues, 2017). Though there is debate about which type of rational number instruction should occur first, it seems it would be beneficial to use an integrated approach to numerical development consisting of all rational numbers (Siegler, Thompson, & Schneider, 2011). Despite numerous studies on specific types of rational numbers, there is limited information about how students translate one rational number notation to another (Tian & Siegler, 2018). The present study seeks to investigate middle school students’ understanding of the relations among fraction, decimal, and percent notations and the influence of a daily, brief numerical magnitude translation intervention on fraction arithmetic estimation. Specifically, it explores the benefits of Simultaneous presentation of fraction, decimal, and percent equivalencies on number lines versus Sequential presentation of fractions, decimals, and percentages on number lines. It further explores whether rational number review using either Simultaneous or Sequential representation of numerical magnitude is more beneficial for improving fraction arithmetic estimation than Rote practice with fraction arithmetic. Finally, it seeks to make a scholarly contribution to the field in an attempt to understand students’ conceptions of the relations among fractions, decimals, and percentages as predictors of estimation ability. Chapter 1 outlines the background that motivates this dissertation and the theories of numerical development that provide the framework for this dissertation. In particular, many middle school students exhibit difficulties connecting magnitude and space with rational numbers, resulting in implausible errors (e.g., 12/13+7/8=1, 19, or 21, 87% of 10>10, 6+0.32=0.38). An integrated approach to numerical development suggests students’ difficulty in rational number understanding stems from how students incorporate rational numbers into their numerical development (Siegler, Thompson, & Schneider, 2011). In this view, students must make accommodations in their whole number schemes when encountering fractions, such that they appropriately incorporate fractions into their mental number line. Thus, Chapter 1 highlights number line interventions that have proven helpful for improving understanding of fractions, decimals, and percentages. In Chapter 2, I hypothesize that current instructional practices leave middle school students with limited understanding of the relations among rational numbers and promote impulsive calculation, the act of taking action with digits without considering the magnitudes before or after calculation. Students who impulsively calculate are more likely to render implausible answers on problems such as estimating 12/13+7/8 as they do not think about the magnitudes (12/13 is about equal to one and 7/8 is about equal to one) before deciding on a calculation strategy, and they do not stop to judge the reasonableness of an answer relative to an estimate after performing the calculation. I hypothesize that impulsive calculation likely stems from separate, sequential instructional approaches that do not provide students with the appropriate desirable difficulties (Bjork & Bjork, 2011) to solidify their understanding of individual notations and their relations. Additionally, in Chapter 2, I hypothesize that many middle school students are unable to view equivalent rational numbers as being equivalent. This hypothesis is based on the documented tendency of many students to focus on the operational rather than relational view of equivalence (McNeil et al., 2006). In other words, students typically focus on the equal sign as signal to perform an operation and provide an answer (e.g., 3+4=7) rather than the equal sign as a relational indicator (e.g., 3+4=2+5). Moreover, this hypothesis is based on the documented whole number bias exhibited by over a quarter of students in 8th grade, such that students perceived equivalent fractions with larger parts as larger than those with smaller parts (Braithwaite & Siegler, 2018b). If middle school students are unable to perceive equivalent values within the same notation as equivalent in size, it seems probable that they might also struggle perceiving equivalent rational numbers as equivalent across notations. This is especially true in light of evidence that many teachers often do not use equal signs to describe equivalent values expressed as fractions, decimals, and percentages (Muzheve & Capraro, 2012). Chapter 2 underscores the importance of highlighting the connections among notations by discussing the pivotal role of notation connections in prior research (Moss & Case, 1999) and the benefit of interleaved practice in math (Rohrer & Taylor, 2007). Finally, I propose a plan for improving students’ understanding of rational numbers through linking notations with number line instruction, as an integrated theory of numerical development (Siegler et al, 2011) suggests that all rational numbers are incorporated into one’s mental number line. Chapter 3 details two experiments that yielded empirical evidence consistent with the hypotheses that students do not perceive equivalent rational numbers as equivalent in size and that this lack of integrated number sense influences estimation ability. The findings identify a discrepancy in performance in magnitude comparison across different rational number notations, in which students were more accurate when presented with problems where percentages were larger than fractions and decimals than when they were presented with problems where percentages were smaller than fractions and decimals. Superficially, this finding of a percentages-are-larger bias suggests students have a bias towards perceiving percentages as larger than fractions and decimals; however, it appears this interpretation is not true on all tasks. If students always perceive percentages as larger than fractions and decimals, then their placement of percentages on the number line should be larger than the equivalent fractions or decimals. However, this was not the case. The experiments revealed that students’ number line estimation was most accurate for percentages rather than the equivalent fraction and decimal values, demonstrating that students who are influenced by the percentages-are-larger bias are most likely not integrating understanding of fractions, decimals, and percentages on a single mental number line. Furthermore, empirical evidence provided support for the theory of impulsive calculation defined earlier, such that many students perform worse when presented with distracting information (“lures”) meant to elicit the use of flawed calculation strategies than in situations without such lures. Importantly, integrated number sense, as measured by the composite score of all cross-notation magnitude comparison trials, was shown to be an important predictor of estimation ability in the presence of distracting information on number lines and fraction arithmetic estimation tasks, often above and beyond number line estimation ability and general math ability. The experiments reported in Chapter 3 also evaluated whether Simultaneous, integrated instruction of all notations improved integration of rational number notations more than Sequential instruction of the three notations or a control condition with Rote practice in fraction arithmetic. The experiments also evaluated whether the instructional condition influenced fraction arithmetic estimation ability. The findings supported the hypothesis that a Simultaneous approach to reviewing rational numbers provides greater benefit for improving integrated number sense, as measured by more improvement in the composite score of magnitude comparison across notations. However, there was no difference among conditions in fraction arithmetic estimation ability at posttest. The experiments point to potential areas for improvement in future work, which are described subsequently. Chapter 4 attempts to explore further students’ understanding of the relations among notations. For this analysis, a number of data sources were examined, including student performance on assessments, interview data, analysis of student work, and classroom observations. Three themes emerged: (1) students are employing a flawed translation strategy, where students concatenate digits from the numerator and denominator to translate the fraction to a decimal such that a/b=0.ab (e.g., 3/5=0.35). (2) percentages can serve as a useful tool for students to judge magnitude, and (3) students equate math with calculation rather than estimation (e.g., in response to being asked to estimate addition of fractions answers, a student responded, “I can’t do math, right?”). Moreover, case studies investigated the differential effect of condition (Simultaneous, Sequential, or Control) on students’ strategy use. The findings suggest that the Simultaneous approach facilitated a more developed schema for magnitude, which is crucial given that a student’s degree of mathematical understanding is determined by the strength and accuracy of connections among related concepts (Hiebert & Carpenter, 1992). Chapter 5 concludes the dissertation by discussing the contributions of this work, avenues for future research, and educational implications. Ultimately, this dissertation advances the field of numerical cognition in three important ways: (1) by documenting a newly discovered bias of middle school students perceiving percentages as larger than fractions and decimals in magnitude comparisons across notations and positing that a lack of integrating notations on the same mental number line is a likely mechanism for this bias; (2) by demonstrating that students exhibit impulsive calculation, as measured by the difference in performance between situations where students are presented with distracting information (“lures”) meant to elicit the use of flawed calculation strategies and situations that do not involve lures; and (3) by finding that integrated number sense, as measured by the composite score for magnitude comparison across notations, is a unique predictor of estimation ability, often above and beyond general mathematical ability and number line estimation. In particular, students with higher integrated number sense are more than twice as likely to correctly answer the aforementioned 12/13+7/8 estimation problem than their peers with the same number line estimation ability and general math ability. This finding suggests that integrated number sense is an important inhibitor for impulsive calculation, above estimation ability for individual fractions and a general standardized test of math achievement. Finally, this dissertation advances the field of mathematics education by suggesting instruction that connects equivalent values with varied notations might provide superior benefits over a sequential approach to teaching rational numbers. At a minimum, this dissertation suggests that more careful attention must be paid to relating rational number notations. Future work might examine the origins of impulsive calculation and the observed percentages-are-larger bias. Future research might also examine whether integrated number sense is predictive of estimation ability beyond general number sense within notations. From these investigations, it might be possible to design a more impactful intervention to improve rational number outcomes.

碩士<br>國立屏東大學<br>科普傳播學系數理教育碩士班<br>105<br>The purpose of this study is to investigate the implementation process and the effect of fraction and decimal remedial instruction with embodied materials on 5th grade. Action research is adopted as research method. The remedial course with embodied materials is according to the specifications of mathematics curriculum. There are 5 students who attended this study and the researcher is the teacher. The research implemented the remedial teaching activities twice a week which including seven classes on fraction and six classes on decimal. 　　Before the remedial instruction, the study understands students’ performance of fraction and decimal through the pre-test. During the remedial instruction, learning sheets, questionnaires, teacher reflection journals and post-test were collected for analyzing the effects of remedial instruction. 　　The results indicate that, before attended the research, students’ performance level of fraction and decimal learning are at 3rd grade. However, through implementing the embodied materials in remedial instruction, the students could improve their learning performance on the specifications of mathematics curriculum more than 60% in fraction and 70% in decimal. Therefore, this research shows that using embodied materials in remedial instruction could improve the learning performance on fraction and decimal.

碩士<br>世新大學<br>資訊傳播學研究所(含碩專班)<br>104<br>Technology has been in progress, use of information technology into the teaching has become a trend. The ability to use technology and information to students should develop basic skills is expressed in Grade 1-9 curriculum guidelines. This study aims to explore the influence on third grade students’ mathematics learning achievement and the learning attitude by using Interactive Response System, and also investigates the satisfaction of schoolchildren toward the Interactive Response System learning. A quasi-expermental method was adopted in this study. Participants in this study included third grade elementary schoolchildren in 2 class from an elementary school in New Taipei City. The experimental group, using Interactive Response System learning two units of mathematics (including Fraction Addition and Subtraction, Decimal). The other class was designated as the control group which traditional lecture-based teaching was administered. The tools used in this study were: mathematics learning achievement test, mathematics learning attitude scale, and satisfaction questionnaires, statistical software for data processing and analysis. Finally, satisfaction questionnaires and interviews were conducted to survey the satisfaction levels and learning experiences of the students who participated in Interactive Response System learning.According to the results, the schoolchildren can enhance mathematics learning achievement, improve the learning attitude by using Interactive Response system learning, and schoolchildren exhibited positive attitude in using interactive response system learning.

碩士<br>國立臺中教育大學<br>教育測驗統計研究所<br>95<br>During math teaching process, it must research the relation between each concept of unit. The most convenient is taking test.The purpose of this research is making a set of topics of decimal to fraction conversion with reliability and validity, analyze the results of the test, and form the structural picture through IRS and then explore the 4th grade students of elementary school cognitive structure in decimal to fraction conversion concepts. I hope to gain the message from the item relative structural picture of the 4th grade students of elementary school in decimal to fraction conversion concepts. Then according to the results and suggestions, it could be the references of teaching and curriculum designing in the future. According to the information from IRS, it shows: (1) “The test of decimal to fraction conversion for the 4th grade students of elementary school” is not only a test with reliability and validity, but also a practical test. (2) For the professional point of view, it should be equal between the testand IRS by giving the same structure test questions; however, the result from students’ evaluation shows the different cognition. (3) The concept develop sequence of students start from “ decimal fraction to mixed fraction” to “mixed decimal to mixed fraction” , and finally develop to “mixed decimal to improper fraction.” (4) The number of fractional part effect students’ cognition when students answer the questions. (5) The concept of tenths decimal or hundredths decimal to fraction conversion is not the real reason to students’ cognition to resolve questions. (6) Students who are given test not familiar with some concepts of hundredths decimal to fraction conversion, such as “when hundreds decimal place is zero, it will not effect a result no matter the fraction will be reduced or not”. Those concepts need the upper level cognitive development for students.

碩士<br>國立臺中教育大學<br>數學教育學系在職進修教學碩士學位班<br>99<br>Abstract This study explores mathematics games for a concept of fraction and decimal of the fourth-grade students in the elementary school. The subjects of study are eleven underrepresented children from a school in Changhua County. After pre-test, teaching activities and post-test, the study uses the dependency sample of SPSS 12.0 paired t-test analysis. The results of this study are concluded as follows. 1. The students’ average score is 51.27 in the pre-test before the experiment. And the score is 75.27 after the experiment. 2. The students promote their high studying achievements after their studying with mathematics games in after-school teaching activities. 3. The students improve their studying attitudes in mathematics after mathematics games in teaching activities. According to the results of this study to be discussed,some-suggestions were proposed in this study as a reference for teaching and future research. Keywords ：“game teaching”,“underrepresented children”, “a concept of fraction”,“a concept of decimal”

碩士<br>國立屏東教育大學<br>數理教育研究所<br>95<br>The purpose of the study is to research translation performance among fraction symbolic, decimal symbolic, and rectangular grids of six-graders in primary school. The research method of this work is to analyze the result of written tests. Also, the interviews between researcher and six-graders are included as research data. The objects of written tests included two hundred and forty-six six-graders in nine classes from four primary schools in Kaohsiung. To understand students’ problem-solving ideas, the researcher interviewed two students in high level and two in medium level from one of the nine classes. The analysis of data mainly focused on describing statistics and assaying contents of interviews. According to the results of data analysis, the following conclusions were made: 1.The six-graders’ performance of answering questions of “translations between fractional symbolic and rectangular grids” was not good enough; especially questions of “translate the fractional symbolic into rectangular grids.” And the four correct problem-solving strategies used by the four case students are as following. First, the application of the meaning of fractional symbolic. Second, the application of conception of equivalent fraction. Third, the application of division, and combination of grids. Last, the application of conception of sub regions of the whole region in rectangular grids. 2.The six-graders’ performance of answering questions of “translations between decimal symbolic and rectangular grids” was terrible; especially questions of “translate rectangular grids into decimal symbolic.” And the five correct problem-solving strategies used by the four case students are as following. First, the application of the meaning of decimal symbolic. Second, the connection of decimal symbolic. Third, the division and combination of grids. Forth, the application of the equalizing division of rectangular grids. Fifth, the application of the relationship of sizes among different divisions and sub divisions. 3.The six-graders’ performance of answering questions of “transformations between fraction symbolic and decimal symbolic” was bad. The worst part was “transform fraction symbolic into decimal symbolic.” And the five correct problem-solving strategies used by the four case students are as following. First, the application of concept of equivalent division. Second, the application of concept of equivalent fraction. Third, the application of the meaning of decimal symbolic through the concept of equivalent division. 4.Students performed badly at answering questions of “translation between rectangular grids and symbolic representation.” And “translate between rectangular grids and decimal symbolic.” part was even worse. 5.Four wrong ideas that case students have are as following: unable to recognize the amount of units correctly, answering the questions merely according to the blacked grids, the lack of right concept of equivalent fraction, and directly interchange fraction and decimal outwardly according to the incomprehension of the meaning of the symbols.

碩士<br>國立臺中教育大學<br>數學教育學系在職進修教學碩士學位班<br>100<br>The purpose of this study is to analyze concept structure of the conversion concepts of fractions and decimals for fourth-graders by S-P chart(student-problem chart) and CAISM(Concept Advanced Interpretive Structural Modeling). There were 266 fourth graders tested by self-designed test according to mathematical competence indicators of grade 1-9 curriculum, and these students were classified into 6 groups by S-P chart. We then compared the CAISM graphs’ characteristics and differences among these groups. We also compared the individualized concept hierarchy structure of the students who got the same scores with different response patterns. Thus the results are as follow: 1.Applying S-P chart is helpful for teachers to understand the situation of studying of students and classify the studying style of students. The CAISM graphs among these group are different. 2.For different learning types more difficult concept to mastery is fractions converted to decimals and more difficult kinds of questions is the two pure decimal converted to a proper fraction, and decimal’s percentile is 0 or omitted. 3.The CAISM graphs among the students who got the same total scores but different response patterns are different, and teachers can implement remedial instruction properly according to the information of links among concepts.

碩士<br>國立臺中教育大學<br>數學教育學系在職進修教學碩士學位班<br>101<br>The purpose of this research was to investigate the effects and the learning attitude of the game integrated into the mathematic teaching of fourth graders on two sections, fraction and decimal. The student problem chart and the ordering theory were used to analyze the learning types and hierarchical structures of items. Based on the quasi-experimental design of fourteen hours of teaching, the subjects were fourth graders students from two different classes in an elementary school in the Taichung city. One class was assigned as the experimental group and the other was the control group. The design of fourteen hours of teaching, and traditional teaching strategies were applied in the control group while the game were incorporated in the experimental group.The results of this study are as follows. 1. As the result from analysis of covariance, there was no significant difference found between the higher achievers from the experimental group and those from the contral group, under-achiever s’ learning effect of experimental group was significantly better than that of the contral group in the fraction and decimal learning activities. 2. According to the S-P chart regarding students’ learning types, neither C-typed students nor C’-typed ones were found in the fraction learning activities.During the decimal learning activities,all of the students in the experimental group belonged either to the A type or to the A’ type,it appeared that the learning condition with the game integrated into the mathematic teaching was better than that of traditional teaching strategies. 3. The S-P chart and the ordering theory were applicable when it came to analyzing the students’ learning results in the fraction and decimal units. Therefore the current study results can provide ideas for teaching material writers and teachers when they design curricula and remedial instruction programs. 4. The game integrated into the mathematic teaching not only enhanced the effects in the academic performance but also effectively improved the learning interests.

碩士<br>國立屏東教育大學<br>數理教育研究所<br>101<br>The present study aimed at exploring the performance of the fifth grade students of elementary school in the subject of decimal fraction in Pingtung County. The study, moreover, aimed at exploring the differences of the performance in decimal fraction for students of different ethnic groups, different regions, different social status, and different sexes. The subjects of the study were chosen by the hierarchy cluster sampling method. The 390 subjects were from general schools, schools in suburban(or outlying ) areas, and schools in remote areas. The instruments of the study were self-written tests on the subject of decimal fraction. Based on the knowledge of mathematics, the tests were classified into three sections: stating concepts, using procedures, and making connections. The main data collecting method was written tests. The data analysis of the study was through descriptive statistics and one-way multivariate analysis of variance. The results of the study showed that firstly the fifth-grade students of elementary school performed significantly best in using procedures, relatively good in making connections, and worst in stating concepts. Secondly, different ethnic students were significantly different in stating concepts. The Han students performed significantly better than the new immigrant students and indigenous students There were no significant differences between the new immigrant students and indigenous students. Thirdly, students of different areas showed significantly different performance in stating concepts, using procedures, and making connections of decimal fraction. In the aspect of stating concepts, the students of general schools performed significantly better than the students of suburban areas and remote areas, whereas, there were no significant differences between the students of suburban areas and remote areas. In the aspects of using procedures and making connections, the students of general schools performed significantly better than the students of suburban areas while there were no significant differences in performance between the students of general schools and those of schools of remote areas , and between the students of schools of suburban areas and those of schools of remote areas. Fourthly, students of different social status showed significantly different performance in stating concepts, using procedures and making connections of decimal fraction. In the three aspects of decimal fraction, the students without lunch subsidies performed significantly better than the students with lunch subsidies. Fifthly, there were no significant differences between the students of different sexes in the three aspects of decimal fraction.

碩士<br>亞洲大學<br>資訊工程學系碩士班<br>94<br>Abstract We get nothing but scores from traditional paper test. It can not show the details such as weaknesses and mistaken types. According to the circumstance, teachers can only offer the remedy instructions based on the most mistaken types. In fact, variety weaknesses cause to different mistaken types. The main idea of the study is to research the ability index of decimal on Grade 5 and demonstrate the applicable of students mistaken types based on Bayesian Networks which is a probability analysis method. Students attend the test through the Learning Educational Program online, which is developed based on the index of Grade 1-9 Curriculum. The system can show the subject comprehension and begin to the Learning Educational Program on the basis of the mistaken type distribution. Due to the analysis result, there are two conclusions as below. 1. Computerized diagnosing test part Using the dynamic classified decision-point to identify the mistaken types and ability & sub-skills classified index, the scores are up to 95.69% and 92.33% respectively. It is suitable to use the ability index of decimal on Grade 5 by Bayesian Networks. 2. Computerized remedy instructions part One of the program advantages is to give individual remedy instructions based on mistaken types, and it is un-available by traditional paper test. According to the t-test result on the scores which come from adopting and un-adopting remedy instructions groups, it is significantly effective to use the program. Keywords: Mistaken Types、Computerized Diagnosing Test、Adaptive Remedy Instructions

This thesis presents comprehensive study material relating the subject of rational numbers for students trained to be teachers at primary school. I am mainly concentrating on qualities of racional numbers and their methodology in the theoretic part. I am checking the level of school knowledge in the subject of racional numbers in the operative part. I was concentrated on pupils at 2nd and 4th classes of Primary School in Choustník and on students of the pedagogical faculty at South Bohemia University in České Budějovice. Task collection for students trained to be teachers at primary school and also for teachers in profession is part of my thesis.

碩士<br>亞洲大學<br>資訊工程學系碩士班<br>94<br>The main purpose of the research is to explore the educational assessment on the basis of Evidence-Centered Design（ECD） to build a convenient and effective diagnosis system. We use Bayesian networks for modeling assessment data and identifying bugs and sub-skills in The “Fraction And Decimal” of Mathematics in Grade 6. This research integrates the opinion of the experts, scholars, and primary school teachers. Also, The multimedia computer is devised for diagnostic testing and computerizes adaptive remedial instruction with the system. Students can receive not only individual diagnostic tests but adequate and in-time computerized adaptive remedial instruction. Evaluation Diagnosis and remedy can be achieved simultaneously. The results： 1.The Bayesian networks evaluation mode and evidence-centered assessment design apply effectively to the diagnosis of students’ mistakes and sub-skills. 2.To set up a complete and effective Bayesian network. one has to do some research to build a Bayesian networks at first and then modifies it by analyzing practical materials. 3.Both testing and remedial teaching can be achieved simultaneously. 4.The progress of students is significant after taking the Computerize adaptive remedial instruction.

碩士<br>臺北市立大學<br>數學系數學教育碩士在職專班<br>106<br>To evaluate the variation in mathematics sense for sixth grade students, the present study aims to investigate both of the five core connotations of mathematics sense and the optimization multiple teaching integrating teaching units of “Fractional division” and “Decimal division”. In this study, an examination for those 2 teaching units is designed based on “Give an Example”, “Drawing”, “Ask Why”, and “Rethinking”, which are 4 of five core connotations of mathematics sense. An experimental group of 27 students and a control group of other 82 students come from other 3 classes in the same school are tested before and after the teaching to evaluate their variation of mathematics sense and effect of learning retention, and also to understand the feedback from the students in experimental group through how they feel about the teaching integrating mathematics sense. The results are listed as below. 1.The students in experimental group showed obvious improvement in conceptual understanding of ‘Give an example’,‘Drawing’, and ‘Ask why’ for those 2 teaching units, procedural calculation and ability to solve the problem compared to the students in control group. 2.The students in experimental group showed obvious improvement in rethinking context of knowledge compared to the students in control group. 3. With respect to the effect of learning retention, the students in experimental group showed a little better than the students in control group for the parts of conceptual understanding and problem solving. For procedural calculation; furthermore, the control group is better than the experimental group. 4. Higher percentage of students in experimental group showed positive feedback about the teaching integrating mathematics sense. In conclusion, the students in experimental group showed improvement in mathematics sense compared to other students, and also get the interest and confidence back for mathematics learning.

This diploma thesis deals with the number line as a tool to improve pupil's understanding of various mathematical concepts, structures, properties and algorithms. The aim of the thesis is to determine how are lower secondary pupils able to use the number line and to connect findings with the content of their textbooks. The thesis consists of a theoretical and a practical part. The theoretical part introduces the topic of number line, deals with the problems used for testing in the Czech Republic and presents several researches on this topic. The theoretical part, together with the analysis of current state of the number line usage in teaching, serves as a basis for an experimental investigation that is covered in the practical part. After a pilot testing with 19 pupils, the main testing was conducted in 7 classes of two Prague primary schools. The total of 156 pupils participated in the main testing and 8 of the pupils were later interviewed about their tests. The results are presented in the practical part. It turned out that there is a correspondence between textbook content and pupils' errors while working with number line and that pupils struggle with representing of fractions on the number line. In the conclusion some recommendation for teachers were put forward concerning the usage of the...