M.Ed. (Mathematics in Education) / This study aimed at investigating grade 8 learners‘ misconceptions and resulting errors in the learning of fractions with a view to expose the nature and origin of those errors and to make suggestions for classroom teaching. This study employed the theory of constructivism and a qualitative method to investigate the research questions. Purposive sampling was used in this study to provide a data that helped to answer the research questions of the study. Learners who were selected purposefully were able to provide rich source of data about the research problem and question. Data collection instruments which were used in the research were in the form of interviews, learners‘ classwork, homework and a test. These instruments were used to collect data so that it will assist in answering the research questions. Data analysis revealed the following errors:  Applying knowledge of like and unlike denominators to division of fractions.  Changing the division sign to multiplication without flipping the second fraction.  Finding the reciprocal of the first fraction and cross multiplied.  Cross cancelling without finding the reciprocal of the second fraction  Finding reciprocal of the second fraction and changing the division sign to subtraction This research revealed that errors emanates from misconceptions. The main reason for misconceptions was the lack of understanding of fractions‘ basic concepts, and learners‘ prior knowledge.

A major purpose of this study was to investigate the relationship between fraction-identification rule abstraction and training exemplars. Fractions can be identified with at least two different rules. One of these, the denominator rule, is general in that it yields correct responses across a wide variety of fraction identification problems. The second, the one-element rule, is appropriate only when the number of elements in the set and the denominator-specified number of subsets are equivalent. Because these rules are not equally serviceable, a question of major importance is what factors determine which of the two fraction identification rules a child will learn during training. The main hypothesis within this study specified that the nature of the fraction identification rule abstracted by a learner would be influenced by the nature of the examples used in training. It was further hypothesized that mastery of the denominator rule would positively affect performance on one-element problems, and that denominator-rule problem errors consistent with the one-element rule would occur significantly more frequently than would be expected by chance. The study addressed two additional questions. These related to recent work in the area of information processing and concerned both changes in learner behavior across training/posttesting sessions and consistency between verbal reports of thinking processes and the fraction rules hypothesized to control fraction identification. A pretest was used to determine eligibility to participate in the study. Eighty-two children incapable of set/subset fraction identification participated in the study. Two additional children were involved in an exploratory phase. The children ranged in age from six to ten and in grade level from one to four. The participants were mainly from middle and lower-class Anglo and Mexican-American homes. Children were randomly assigned to one of two training groups. In both training groups learners were provided, through symbolic (verbal) modeling, the general denominator rule for fraction identification. Children in one training group were also provided examples of fraction identification requiring the denominator rule. In the second training group children were provided simple examples in keeping with the denominator rule stated as part of instruction and yet also in keeping with the unstated one-element rule. A modified path analysis procedure was used to assess the effects of training group assignment on fraction identification performance. Results of this analysis suggest a training group main effect. That is, children's performance on fraction identification posttest problems was in keeping with rules associated with the training examples they had been provided. The results suggest that the strongest effects of training were related to performance on denominator rule problems in that the odds of passing the denominator rule posttest were 11.52 times greater for children taught with denominator-rule exemplars than for children taught with the one-element exemplars. The findings also suggest that performance on either of the fraction identification tasks influenced performance on the other. A further finding was that training with the one-element exemplars was associated with performance congruent with inappropriate use of the one-element rule. Recall that the one-element rule was never stated and that the exemplars, while compatible with the one-element rule were equally compatible with the stated denominator rule. Protocols from the children in the exploratory portion of the study suggest that the child taught with the ambiguous exemplars did abstract the one-element rule while the child taught with the denominator rule exemplars abstracted the denominator rule. The protocols also suggest that the child taught with the denominator rule made changes in his thinking as training and posttesting progress progressed.

No description available.

No description available.

published_or_final_version / Educational Psychology / Master / Master of Social Sciences

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The general understanding of mathematics as a subject and its implications is, in reality alarmingly low. Evidence of this is evident in learners’ performance and their reaction towards the subject. Fractions as a domain of Mathematics are no exception. The majority of the learners do not learn Fractions comfortably. The causes of this may be varied. However, it is believed that one way of ensuring meaningful teaching and learning is to make use of appropriate connections. The significance and the important role of the teacher in making mathematical connections in learning for understanding are well documented in the literature. This study focuses on the nature of mathematical connections selected Grade 7 teachers make when teaching Fractions, as well as their perceptions of the importance of making such connections. This qualitative case study was conducted in three schools in the Oshana region. The purpose was to investigate how mathematics teachers make connections in fractions. Underpinned by an interpretive paradigm, the study made use of observations and interviews to generate data. The framework borrowed from Businkas’ (2008) study was used in analysing and coding the nature of connections used in the lessons observed. An individual conversation on the nature and perceptions of the connections made in the observed lessons was undertaken with each teacher followed by a focus group discussion that aimed at analysing deeper perceptions on connections. The main findings of the study revealed that teachers made use of all the different types of connections as per Businkas’s framework. The frequency of occurrence showed that Instruction-Oriented Connection and Multiple Representation connections topped the list of connections used. Teachers pointed out that connections to prior knowledge and making multiple representations were most significant, as they related to learners’ existing knowledge and pointed to different ways of solving a problem. The teachers were, however, not familiar with the other connections identified as this was their first experience of interrogating connections. They, however, agreed on the importance of making those connections. The teachers agreed that meaningful connections indeed helped with their conceptual understanding of Mathematics. They believed that connections can increase learners’ interest in school and help reduce negative views of fractions, in particular, and mathematics in general. However, they felt that the limited number of resources, poor teaching approaches and the inability of creating fraction sense may hinder them from making appropriate connections.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-07-18T11:45:25Z No. of bitstreams: 1 Monique Lopes dos Santos Dias.pdf: 1428885 bytes, checksum: 6c52690071d28ac219840c2927010f6a (MD5) / Made available in DSpace on 2018-07-18T11:45:25Z (GMT). No. of bitstreams: 1 Monique Lopes dos Santos Dias.pdf: 1428885 bytes, checksum: 6c52690071d28ac219840c2927010f6a (MD5) Previous issue date: 2018-03-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This study aims to present the mapping of dissertations and theses produced by universities in the state of São Paulo between 2000 and 2016, related to the issue of fractional numbers. For the constitution of our corpus of research, we have selected 39 academic papers produced by the following universities: Pontifícia Universidade Católica de São Paulo (PUC-SP); Paulista State University "Júlio de Mesquita Filho" (UNESP); Federal University of São Carlos (UFSCar); Federal University of ABC (UFABC); State University of Campinas (Unicamp); Universidade Bandeirante de São Paulo (Anhanguera) and Cruzeiro do Sul University (UNICSUL). In the search for answers to the following research question "What are the main contributions of researches carried out by universities in the state of São Paulo, between the years 2000 and 2016, regarding the fractional numbers? What possible gaps and challenges are still presented on the subject?", we split the papers that compose our corpus into three distinct categories of analysis: approach of fractional numbers in official documents or in didactic materials; teacher’s training, initial training and continuing education; and teaching and learning of fractional numbers and their operations. As a result, we point out advances in the approach of fractional numbers through their different conceptions (part-whole, measure, quotient, operator and ratio), in all aspects observed in our mapping. Concerning the gaps and challenges in this field of study, we stress the exaggerated emphasis on the part-whole conception, albeit with all the criticisms imbued with it, besides the insufficiency of studies related to High School, Higher, Youth, Adult and Inclusive Education. We suggest that new research fill these gaps in order to broaden the contributions in the field of Mathematics Education / Este estudo tem o objetivo de apresentar o mapeamento de dissertações e teses produzidas por universidades do estado de São Paulo, entre os anos de 2000 e 2016, em relação ao tema números fracionários. Para a constituição de nosso corpus de pesquisa foram selecionados 39 trabalhos acadêmicos, produzidos pelas respectivas universidades: Pontifícia Universidade Católica de São Paulo (PUC-SP); Universidade Estadual Paulista “Júlio de Mesquita Filho” (UNESP); Universidade Federal de São Carlos (UFSCar); Universidade Federal do ABC (UFABC); Universidade Estadual de Campinas (Unicamp); Universidade Bandeirante de São Paulo (Anhanguera) e Universidade Cruzeiro do Sul (UNICSUL). Na busca de respostas para a seguinte questão de pesquisa “Quais as principais contribuições das pesquisas realizadas por universidades do estado de São Paulo, entre os anos de 2000 e 2016, no que diz respeito aos números fracionários? Que possíveis lacunas e desafios ainda são apresentados quanto ao tema?”, dividimos os trabalhos que compõem nosso corpus em três categorias distintas de análise: abordagem dos números fracionários em documentos oficiais ou em materiais didáticos; saberes docentes, formação inicial e formação continuada; e ensino e aprendizagem dos números fracionários e de suas operações. Como resultados, apontamos avanços quanto à abordagem dos números fracionários por meio de suas diferentes concepções (parte-todo, medida, quociente, operador e razão), em todos os aspectos observados em nosso mapeamento. Concernentes às lacunas e desafios nesse campo de estudo, destacamos a ênfase exagerada à concepção parte-todo, ainda que com todas as críticas imbuídas a ela, além da insuficiência de estudos correlatos ao Ensino Médio, ao Ensino Superior, à Educação de Jovens e Adultos e à Educação Inclusiva. Sugerimos que novas pesquisas sanem essas lacunas, a fim de ampliar as contribuições no âmbito da Educação Matemática

Lakoff and Nuñez (2000) argue that the origins of mathematical thinking arise from the progressive development of the human sensorium and experience. Cognitive science research in in education plays a big role in developing new pedagogies, especially those that leverage new “Cyberlearning” technologies. The current study employs two principle frameworks for creating pedagogy for learning mathematical fractions: (1) grounded and embodied cognition (Varela, Thompson & Rosch, 1991; Glenberg, 1997; 2003; Barsalou, 1999; 2008), (2) situated cognition (Lesh, 1981; Lave 1988, Greeno, 1998; Roth, 2002). Grounded and embodied cognition was operationalized through the gesture. Although gesture is traditionally discussed as a spontaneous co-articulation of speech (Kendon, 1972; McNeill & Levy, 1980; 1992; Goldin-Meadow, 1986) it is taking on a new role with the advent of 21st century technologies that utilize gestural interface. Using gestures as simulated action (Hostetter and Alibali, 2008), we developed two sets of gestural mechanics based on an exploratory study on the gestures elementary students used to explain mathematical fractions (Swart, 2014): (1) iconic gestures (I) – i.e., enactive of the processes to create objects, (2) deictic gestures (D) – i.e., index pointing to ground or identify objects or locations. Situated cognition was operationalized through narrative (Black and Bower, 1980; Graesser, Hauft-Smith, Cohen, and Pyles 1980; Graesser, Singer, Trabasso, 1994). Researchers crafted two types of narratives in order to create a situated learning environment (Hennessy, 1993): (1) strong narrative (S) – with a setting, characters and plot (based on the popular PBS Kids television show, Cyberchase, (2) weak narrative (W) – without an explicit setting, characters or plot. Combining these two factors together, the research team designed and developed Mobile Mathematics Movement (M3). Using the two independent variables, gesture (I vs. D) and narrative (S vs. W), M3 was crafted into 4 different versions: SI, SD, WI, WD. The first two iterations, M3:i1 and M3:i2, were tested in randomized factorial experiments in afterschool programs with high-needs populations. After completing these studies employing a design-based research (DBR) methodology, the tutor-game developed into its latest iteration, M3:i3. The curriculum of M3 had students employing a splitting objects (i.e., parts-to-whole) schema (Steffe, 2004) and was divided into two parts: (Part 1) object fracturing (x5 per level): estimating, denominating, numerating, re-estimating; (Part 2) object equivalency (comparing 5 fractions): comparing, ordering, verifying magnitudes, verifying positions on vertical number line. In the final dissertation study, 131 students (x̄age = 8.78 yrs, 52.6% Female; 39.7% Hispanic; 32.8% African-American; 19.9% South-East Asian; 3.8% Caucasian; 3.8% South Asian (Indian); 97.7 % received free/reduced lunch) from the Harlem and Lower East Side neighborhoods of New York City were consented and assented and completed the study. Students were randomly assigned to 1 of the 4 conditions, completed a direct pre-assessment of the curriculum as well as a transfer pre-assessment, played all seven levels of the tutor-game, completed an exit survey (free response and 5-point likert – motivation, self-efficacy, engagement, learning), completed a direct post-assessment of the curriculum as well as a transfer post-assessment (parallel forms) and a 7 minute semi-structured clinical interview. Factorial ANOVAs indicated a significant interaction between gesture and narrative (though all groups showed significant learning pre to post) on the direct assessment. Both the SI and WD groups significantly outperformed the other two groups, though were not different from each other. Though there was not a significant interaction between gesture and narrative on for the transfer assessment, pair-wise comparisons and planned contrasts showed that the SI group outperformed all the other groups. Follow up hierarchical linear regressions (HLR) showed that game play significantly mediated students’ learning. Specifically, students’ performances estimating and denominating were predictive of direct learning of the curriculum while estimating, denomination and numeration were all predictive of transfer. Further HLRs also found that students’ learning was moderated by their existing proficiencies for fractions. This finding helped clarify the nature of the narrative-gesture interaction, such that low-proficiency students improved more in the WD condition and high-proficiency students improved more in the SI condition. An exploratory factor analysis of the 5-point likert exit survey showed loaded on four factors as anticipated, with significant loadings for engagement and learning, but revealed no significant differences between the conditions. The significant interaction revealed that both a weak narrative (non-contextualized) environments using deictic (identity) gestures as well as strong narrative (contextualized) environments using iconic (enactive) gestures are differentially beneficial for learning. Contrary to our interaction hypothesis, learning for novices benefitted from a more abstract environment, supporting the work of (Kaminski, Sloutsky, Heckler, 2008) and learning for those with higher proficiencies at fractions was better in the more concrete environment (e.g., Moreno, Ozogul, & Reisslein (2011). The likert data supports research suggesting that students find digital platforms engaging and empowering, regardless of learning or not (for review see Wouters, van Nimwegen, van Oostendorp, & van der Spek, 2013). Together, these results have important implications for the design of learning environments and a digital pedagogy and follow-up work is necessary for expounding on the interactions between gestures and narratives as well as the possible mediation by task complexity.

The researcher's personal conviction that major problems in the teaching of mathematics are inherited from elementary levels inspired the investigation of the contribution of practical work in the teaching of fraction division in grade seven. The all encompassing approach of the study dictated the involvement of teachers and learners as participants. Teachers' perceptions of practical work and their classroom practices were investigated to confirm or refute existing assumptions and literature claims. Questionnaires in which teachers expressed their views on practical work and fraction teaching were administered to teachers. Lessons on the division of fractions were observed to determine teachers' practices in relation to the researcher's assumptions and claims by literature. Data yielded by these research instruments confirmed or refuted assumptions and literature claims. Learners underwent an experiment and their views were sought to establish the value of practical work in the teaching of fractions and fraction division. Instruments used for the experiment were the pre-test, post-test and worksheets. Data from these instruments gave an indication of the value of practical work in enhancing learners' understanding of fraction division. Learners' responses to interview questions further elucidated and confirmed the valuable role played by practical work in learners ' understanding of fraction division. Learners' responses also provided deeper insight into facets of learners ' cognitive development as they engaged with different aspects of practical work in the division of fractions . Besides confirmation and refutation of some established assumptions and literature claims, previously unknown realities about aspects of practical work and fraction division also emerged from findings. This wealth of the data carried crucial implications for teacher training, the teaching of fractions and fraction division, and further research. A look at these implications hopes to contribute to the enhancement and improvement of the teaching of fractions and fraction division. Teacher training institutions, designers of INSET programmes, policy makers and teachers should all benefit from findings of this study. / Thesis (M.Ed.)-University of Kwazulu-Natal, 2005.