This study has examined how the attainment of theoretical frameworks may create the conditions for and support subsequent learning of related material. In this regard, it has investigated a particular conception of Vygotsky‟s proposal that learning only occurs in the zone of proximal development, which he defined as the gap between what can be performed independently and what can be achieved with assistance. Specifically, it used a multi-pronged, mixed method research approach to probe the relationship between the actual level of development, as reflected by an ability to do proportional reasoning, and potential development, which was measured as the ability to perform certain strategic procedural operations in the molecular biosciences which were underpinned by proportionality. This four phase study which was carried out on a class of 106 second year students registered for Basic Molecular Biosciences II in the School of Molecular and Cell Biology, at the University of the Witwatersrand, Johannesburg, South Africa, initially measured proportional reasoning ability by posing a generative question requiring proportional reasoning to the class during a lecture and established that only 49% of the students who participated were able to answer the question. It could be shown statistically that these students were more adept at answering a contextual question based on proportion than those who had answered the generative question incorrectly, which suggested that actual development created the conditions for future learning. A paper and pencil test developed from Fleener (1993) which claimed to measure the hierarchical development of proportional reasoning ability was administered to the class and was used to select two groups for comparative purposes. The first group (group one) was comprised of the 23 students who scored 50 % or less, and the control group (group two) consisted of the 15 students who scored 100 %. Using these two groups, it was shown that the control group performed better than group one on specific questions underpinned by proportion which had been included in pre-laboratory tests and in summative assessments. Moreover, the control group‟s general performance in the course, as assessed by their marks in the examination at the end of the first semester, was substantially better than that of group one (67 % as opposed to a 51% average mark). These results were supported by findings where conceptual development of proportion had been judged from student‟s informal written accounts of the concept. Drawing on biological evidence, it was concluded that the actual level produces the structures necessary for further development. The second phase of the study utilized two focus groups constituted from students who iv had been randomly selected from the two groups compared in phase one of the research. Facilitated guided informal discussions probed which of factors like play and leisure activities, early childhood enrichment, schooling, mathematical ability and practices, instruction in proportional reasoning, and parental involvement, might have augmented the development of proportional reasoning ability. In phase three, the factors which emerged from the discussions were interrogated in a specially designed questionnaire which was administered to a sub-set of students who were concurrently registered for Basic Molecular Biosciences II and Biochemistry and Cell Biology II. Statistical analysis of the questionnaire which occurred in phase four of the research led to the conclusion that enrichment in early childhood, and having learnt proportion at school were the two factors that contributed most to attainment of the actual level of development which would enable subsequent learning of more elaborate procedural knowledge constructs based on the concept of proportion. These results supported the view that mediation results in internalisation of the embedded knowledge which can be drawn on for further learning in that domain. Therefore, in the final analysis of the research, it was concluded that actual levels of development create conditions for potential development as conceived by Vygotsky‟s zone of proximal development.

No description available.

A study of pre-service primary school teachers in Singapore and the United States revealed superior performance by the Singaporeans on proportional reasoning problems. Analysis of solutions showed the Singapore future teachers were more likely to use unitary and benchmark approaches than were their American counterparts. Conclusions include suggestions for programs intended to improve the performance of prospective elementary school teachers on proportional reasoning problems.

Submitted by Irene Nascimento (irene.kessia@ufpe.br) on 2016-08-26T18:19:38Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertacao do Mestrado Edna Porto.pdf: 1288631 bytes, checksum: 3e6f2db35e44707de753eec0c1fbddd2 (MD5) / Made available in DSpace on 2016-08-26T18:19:38Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertacao do Mestrado Edna Porto.pdf: 1288631 bytes, checksum: 3e6f2db35e44707de753eec0c1fbddd2 (MD5) Previous issue date: 2015-02-26 / CNPQ / Compreender o raciocínio dos estudantes ao resolverem problemas matemáticos têm ocupado muitos pesquisadores de áreas como a Psicologia e a Educação Matemática, em especial no que tange identificar as dificuldades e erros conceituais atrelados a um conceito, como também a investigação de suportes que facilitem a articulação entre conhecimentos prévios à educação formal. O presente estudo consistiu em investigar o raciocínio proporcional de estudantes da educação de adultos, cursando a 4ª Fase (que corresponde ao 8° e ao 9° ano do ensino fundamental); bem como de forma específica, (i) as estratégias utilizadas para solucionar problemas envolvendo o conceito de proporção; (ii) se existem diferenças nos desempenhos e nas estratégias em função dos temas que perpassam vida social apresentados nos problemas, neste estudo em particular, as Eleições presidenciais e a Copa do Mundo e (iii) se existem diferenças no desempenho e nas estratégias em função do tipo de problema. Para tal, participaram 34 estudantes, de idades variando de 18 a 47 anos, de uma escola pública da cidade de Petrolina-PE. Todos os participantes resolveram 18 problemas, envolvendo seis tipos de situações (valor omisso; conversão entre razão, taxa e representações; os que envolvem unidade de medidas e números; comparação; transformação; e conversão entre sistemas de representação). Estes foram apresentados, individualmente, em duas sessões, durante as quais foi utilizado o método clínico Piagetiano para melhores esclarecimentos sobre as formas de resolução e ao final foi realizada uma entrevista. Os dados foram analisados em função de dois aspectos: números de acertos e as estratégias adotadas na resolução. Na avaliação do desempenho foram controladas as variáveis internas: tipos de problemas, tipos de problemas associados ao contexto (Copa do Mundo, Eleições Presidenciais e Prototípicos) como também a variável externa afinidade com o contexto. Os resultados obtidos foram analisados à luz da teoria dos Campos Conceituais de Gérard Vergnaud e mostraram que estudantes da 4ª fase, mesmo não tendo estudado formalmente o conceito de proporcionalidade conseguem resolver alguns problemas envolvendo relações proporcionais. Foi verificada a influência do contexto apenas quando comparado os problemas da Copa do Mundo e os Prototípicos, e foi observado desempenho semelhante quando comparado o contexto Copa do Mundo e Eleições, e também entre este último e o desempenho nos problemas Prototípico. No que tange às diferentes situações de proporcionalidade resolvidas, constatou-se que aquelas que envolvem o julgamento qualitativo são mais facilmente resolvidas do que as que envolvem outros sistemas de representação. As respostas dos estudantes demonstraram o uso de vários tipos de estratégias, que foram classificadas como: Tipo 1(imprecisa ou ausente); Tipo 2 (conhecimento de mundo); Tipo 3 (sentido numérico); Tipo 4 (operações aditivas); Tipo 5 (campo multiplicativo associado a operações aditivas) e Tipo 6 (campo multiplicativo). Concluiu-se com este estudo que nem sempre ao resolver e acertar problemas proporcionais o estudante apresenta o raciocínio proporcional e que este é mais facilmente desenvolvido em algumas situações que em outras, evidenciando que o domínio da proporcionalidade se dá de forma gradativa e requer o desenvolvimento de outros conceitos, representações e procedimentos. / Understanding the reasoning of students when resolving mathematical problems has occupied many researchers of the Psychology and Mathematics Education fields, especially in that which regards to identify the difficulties and conceptual errors tied to a concept, as well as an investigation of supporting materials that facilitate the link between prior knowledge and formal education. This study consisted in investigating the proportional reasoning of Adults in initial schooling who are taking the 4th stage ( which corresponds to 7th and 8th grade of elementary school); specifically, (i) the strategies utilized to resolve problems involving the concept of proportion; (ii) if there are differences in performances and in the strategies in view of topics that spans social life presented in the problems, particularly in this study, the presidential elections and the Fifa World Cup and (iii) if there are differences in the performance and strategies in view of the type of problem. For this study, 34 students between the ages of 18 and 47, from a public school in Petrolina- PE, participated. All participants resolved 18 problems involving six types of situations (missing value; conversion of ratio, rate and representations; those which involve units of measurements and numbers; comparisons; transformation; and conversions of system of representation). These were presented, individually in two sessions, in which the Piaget clinical method was used for the better understanding of the forms of solution and at the end an interview was conducted. The data was analyzed on the basis of two aspects: number of correct answers and the strategies adopted in the resolution. In the performance evaluation the following independent variables were controlled: types of problems, types of problems associated to context (Fifa World Cup, presidential elections and prototypes) as well as the dependent variable affinity to context. The acquired results were analyzed in view of the Conceptual Fields of Gérard Vergnaud and they showed that students in the 4th stage, even without having formally studied the concept of proportionality, can resolve some problems involving proportional relationships. Influence was verified only when the context of the World Cup and Prototypes were compared, and it was observed similar performance when compared the World Cup and Elections context, furthermore among the latter and the performance problems in Prototype. Regarding the different situations of proportionality resolved, it was confirmed that the problems that involve qualitative judgment were easier to resolve than those that involve other systems of representation. The students’ answers demonstrate the usage of various types of strategies which were classified as: Type 1 (inaccurate or absent); Type 2 (knowledge of the world); Type 3 (number sense); Type 4 (operations of addition); Type 5 (multiplication associated with operations of addition) and Type 6 (multiplication). It was concluded that by resolving and acquiring correct answers in proportional problems, the student presents proportional reasoning and that it is more easily developed in some situations than others proving that the domain of proportionality is given in a gradual manner and requires development of other concepts, representations and procedures.

Submitted by Irene Nascimento (irene.kessia@ufpe.br) on 2016-08-26T19:41:05Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertacao do Mestrado Edna Porto.pdf: 1288631 bytes, checksum: 3e6f2db35e44707de753eec0c1fbddd2 (MD5) / Made available in DSpace on 2016-08-26T19:41:05Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertacao do Mestrado Edna Porto.pdf: 1288631 bytes, checksum: 3e6f2db35e44707de753eec0c1fbddd2 (MD5) Previous issue date: 2015-02-26 / CNPQ / Compreender o raciocínio dos estudantes ao resolverem problemas matemáticos têm ocupado muitos pesquisadores de áreas como a Psicologia e a Educação Matemática, em especial no que tange identificar as dificuldades e erros conceituais atrelados a um conceito, como também a investigação de suportes que facilitem a articulação entre conhecimentos prévios à educação formal. O presente estudo consistiu em investigar o raciocínio proporcional de estudantes da educação de adultos, cursando a 4ª Fase (que corresponde ao 8° e ao 9° ano do ensino fundamental); bem como de forma específica, (i) as estratégias utilizadas para solucionar problemas envolvendo o conceito de proporção; (ii) se existem diferenças nos desempenhos e nas estratégias em função dos temas que perpassam vida social apresentados nos problemas, neste estudo em particular, as Eleições presidenciais e a Copa do Mundo e (iii) se existem diferenças no desempenho e nas estratégias em função do tipo de problema. Para tal, participaram 34 estudantes, de idades variando de 18 a 47 anos, de uma escola pública da cidade de Petrolina-PE. Todos os participantes resolveram 18 problemas, envolvendo seis tipos de situações (valor omisso; conversão entre razão, taxa e representações; os que envolvem unidade de medidas e números; comparação; transformação; e conversão entre sistemas de representação). Estes foram apresentados, individualmente, em duas sessões, durante as quais foi utilizado o método clínico Piagetiano para melhores esclarecimentos sobre as formas de resolução e ao final foi realizada uma entrevista. Os dados foram analisados em função de dois aspectos: números de acertos e as estratégias adotadas na resolução. Na avaliação do desempenho foram controladas as variáveis internas: tipos de problemas, tipos de problemas associados ao contexto (Copa do Mundo, Eleições Presidenciais e Prototípicos) como também a variável externa afinidade com o contexto. Os resultados obtidos foram analisados à luz da teoria dos Campos Conceituais de Gérard Vergnaud e mostraram que estudantes da 4ª fase, mesmo não tendo estudado formalmente o conceito de proporcionalidade conseguem resolver alguns problemas envolvendo relações proporcionais. Foi verificada a influência do contexto apenas quando comparado os problemas da Copa do Mundo e os Prototípicos, e foi observado desempenho semelhante quando comparado o contexto Copa do Mundo e Eleições, e também entre este último e o desempenho nos problemas Prototípico. No que tange às diferentes situações de proporcionalidade resolvidas, constatou-se que aquelas que envolvem o julgamento qualitativo são mais facilmente resolvidas do que as que envolvem outros sistemas de representação. As respostas dos estudantes demonstraram o uso de vários tipos de estratégias, que foram classificadas como: Tipo 1(imprecisa ou ausente); Tipo 2 (conhecimento de mundo); Tipo 3 (sentido numérico); Tipo 4 (operações aditivas); Tipo 5 (campo multiplicativo associado a operações aditivas) e Tipo 6 (campo multiplicativo). Concluiu-se com este estudo que nem sempre ao resolver e acertar problemas proporcionais o estudante apresenta o raciocínio proporcional e que este é mais facilmente desenvolvido em algumas situações que em outras, evidenciando que o domínio da proporcionalidade se dá de forma gradativa e requer o desenvolvimento de outros conceitos, representações e procedimentos. / Understanding the reasoning of students when resolving mathematical problems has occupied many researchers of the Psychology and Mathematics Education fields, especially in that which regards to identify the difficulties and conceptual errors tied to a concept, as well as an investigation of supporting materials that facilitate the link between prior knowledge and formal education. This study consisted in investigating the proportional reasoning of Adults in initial schooling who are taking the 4th stage ( which corresponds to 7th and 8th grade of elementary school); specifically, (i) the strategies utilized to resolve problems involving the concept of proportion; (ii) if there are differences in performances and in the strategies in view of topics that spans social life presented in the problems, particularly in this study, the presidential elections and the Fifa World Cup and (iii) if there are differences in the performance and strategies in view of the type of problem. For this study, 34 students between the ages of 18 and 47, from a public school in Petrolina- PE, participated. All participants resolved 18 problems involving six types of situations (missing value; conversion of ratio, rate and representations; those which involve units of measurements and numbers; comparisons; transformation; and conversions of system of representation). These were presented, individually in two sessions, in which the Piaget clinical method was used for the better understanding of the forms of solution and at the end an interview was conducted. The data was analyzed on the basis of two aspects: number of correct answers and the strategies adopted in the resolution. In the performance evaluation the following independent variables were controlled: types of problems, types of problems associated to context (Fifa World Cup, presidential elections and prototypes) as well as the dependent variable affinity to context. The acquired results were analyzed in view of the Conceptual Fields of Gérard Vergnaud and they showed that students in the 4th stage, even without having formally studied the concept of proportionality, can resolve some problems involving proportional relationships. Influence was verified only when the context of the World Cup and Prototypes were compared, and it was observed similar performance when compared the World Cup and Elections context, furthermore among the latter and the performance problems in Prototype. Regarding the different situations of proportionality resolved, it was confirmed that the problems that involve qualitative judgment were easier to resolve than those that involve other systems of representation. The students’ answers demonstrate the usage of various types of strategies which were classified as: Type 1 (inaccurate or absent); Type 2 (knowledge of the world); Type 3 (number sense); Type 4 (operations of addition); Type 5 (multiplication associated with operations of addition) and Type 6 (multiplication). It was concluded that by resolving and acquiring correct answers in proportional problems, the student presents proportional reasoning and that it is more easily developed in some situations than others proving that the domain of proportionality is given in a gradual manner and requires development of other concepts, representations and procedures.

Ratio is one of the key mathematics concepts included in the South African Mathematics curriculum. It is applied in other topics of the Grade 7 curriculum, including geometry, functions and relationships, algebra, similarity and congruency. The aim of this qualitative research study was to explore the difficulties that learners experience in learning ratio. The primary research question for the study was: What is Grade 7 learners’ knowledge of ratio? This research question was answered through the following secondary research questions: How do learners solve problems involving ratio? What is learners’ conceptual knowledge of ratio? And what learning difficulties do learners experience when learning about ratio? The study was informed by Kilpatrick, Swafford and Findell’s (2001) five strands of mathematical proficiency; however, the focus was on conceptual and procedural knowledge of ratio. The interpretivist paradigm and the single exploratory case study design were used to gain insight into the learning of ratio. Data was collected from Grade 7 learners (23 of the 35 learners originally sampled) through a self-developed test that followed the prescripts of the Grade 7 Mathematics curriculum in South Africa and through semi-structured interviews. The test scripts were analysed using the Atlas.tiTM windows coding system and the results were used to construct questions for the semi-structured interviews. The interviews were used to corroborate data emerging from the test. The results of the study indicated that Grade 7 learners can do simple and routine manipulations of ratio as well as non-proportional ratio problems but struggle to solve problems that require multiplicative thinking and proportional reasoning skills. Although there could be other factors contributing to learners’ struggle to tackle proportional ratio problems requiring multiplication and proportional reasoning, a lack of conceptual knowledge seemed to contribute significantly. / Dissertation (MEd)--University of Pretoria, 2020. / Science, Mathematics and Technology Education / MEd / Unrestricted

Proportional reasoning instruction is prevalent in elementary, secondary, and post-secondary schooling. The concept of proportional reasoning is used in a variety of contexts for solving real-world problems. One of these contexts is the solving of dosage calculation proportional problems in the healthcare field. On the job, nurses perform drug dosage calculations which carry fatal consequences. As a result, nursing students are required to meet minimum competencies in solving proportion problems. The goal of this research is to describe the lived experiences of nurses in connection to their use of proportional reasoning in order to impact instruction of the procedures used to solve these problems. The research begins by clarifying and defining the conceptual field of proportional reasoning. Utilizing Vergnaud*s theory of conceptual fields and synthesizing the differing organizational frameworks used in the literature on proportional reasoning, the concept is organized and explicated into three components: concepts, procedures, and situations. Through the lens of this organizational structure, data from 44 registered nurses who completed a dosage calculation proportion survey were analyzed and connected to the framework of the conceptual field of proportional reasoning. Four nurses were chosen as a focus of in-depth study based upon their procedural strategies and ability to vividly describe their experiences. These qualitative results are synthesized to describe the lived experiences of nurses related to their education and use of proportional reasoning. Procedural strategies that are supported by textbooks, instruction, and practice are developed and defined. Descriptive statistics show the distribution of procedures used by nurses on a five question dosage calculation survey. The most common procedures used are the nursing formula, cross products, and dimensional analysis. These procedures correspond to the predominate procedures found in nursing dosage calculation texts. Instructional implications focus on the transition between elementary and secondary multiplicative structures, the confusion between equality and proportionality, and the difficulty that like quantities present in dealing with proportions.

No description available.

abstract: Industry, academia, and government have spent tremendous amounts of money over several decades trying to improve the mathematical abilities of students. They have hoped that improvements in students' abilities will have an impact on adults' mathematical abilities in an increasingly technology-based workplace. This study was conducted to begin checking for these impacts. It examined how nine adults in their workplace solved problems that purportedly entailed proportional reasoning and supporting rational number concepts (cognates). The research focused on four questions: a) in what ways do workers encounter and utilize the cognates while on the job; b) do workers engage cognate problems they encounter at work differently from similar cognate problems found in a textbook; c) what mathematical difficulties involving the cognates do workers experience while on the job, and; d) what tools, techniques, and social supports do workers use to augment or supplant their own abilities when confronted with difficulties involving the cognates. Noteworthy findings included: a) individual workers encountered cognate problems at a rate of nearly four times per hour; b) all of the workers engaged the cognates primarily via discourse with others and not by written or electronic means; c) generally, workers had difficulty with units and solving problems involving intensive ratios; d) many workers regularly used a novel form of guess & check to produce a loose estimate as an answer; and e) workers relied on the social structure of the store to mitigate the impact and defuse the responsibility for any errors they made. Based on the totality of the evidence, three hypotheses were discussed: a) the binomial aspect of a conjecture that stated employees were hired either with sufficient mathematical skills or with deficient skills was rejected; b) heuristics, tables, and stand-ins were maximally effective only if workers individually developed them after a need was recognized; and c) distributed cognition was rejected as an explanatory framework by arguing that the studied workers and their environment formed a system that was itself a heuristic on a grand scale. / Dissertation/Thesis / Doctoral Dissertation Curriculum and Instruction 2015

A study of pre-service primary school teachers in Singapore and the United States revealed superior performance by the Singaporeans on proportional reasoning problems. Analysis of solutions showed the Singapore future teachers were more likely to use unitary and benchmark approaches than were their American counterparts. Conclusions include suggestions for programs intended to improve the performance of prospective elementary school teachers on proportional reasoning problems.