Fractions are very hard to learn. As the joke goes, “Three out of two people have trouble with fractions”. Yet the invention of a notation for fractions is very ancient, dating back to Babylonians and Egyptians. Moreover, it is thought that ratio representation is innate. And obviously, fractions are part of our everyday life. We read them in recipes, we need them to estimate distances on maps or rebates in shops. In addition, fractions play a key role in science and mathematics, in probabilities, proportions and algebraic reasoning. Then why is it so hard for pupils to understand and use them? What is so special about fractions? As in other areas of numerical cognition, a fast-developing field in cognitive science, we tackled this paradox through a multi-pronged approach, investigating both adults and children.<p>Based on some recent research questions and intense debates in the literature, a first behavioural study examined the mental representations of the magnitude of fractions in educated adults. Behavioural observations from adults can indeed provide a first clue to explain the paradox raised by fractions. Contrary perhaps to most educated adults’ intuition, finding the value of a given fraction is not an easy operation. Fractions are complex symbols, and there is an on-going debate in the literature about how their magnitude (i.e. value) is processed. In a first study, we asked adult volunteers to decide as quickly as possible whether two fractions represent the same magnitude or not. Equivalent fractions (e.g. 1/4 and 2/8) were identified as representing the same number only about half of the time. In another experiment, adults were also asked to decide which of two fractions was larger. This paradigm offered different results, suggesting that participants relied on both the global magnitude of the fraction and the magnitude of the components. Our results showed that fraction processing depends on experimental conditions. Adults appear to use the global magnitude only in restricted circumstances, mostly with easy and familiar fractions. <p>In another study, we investigated the development of the mental representations of the magnitude of fractions. Previous studies in adults showed that fraction processing can be either based on the magnitude of the numerators and denominators or based on the global magnitude of fractions and the magnitude of their components. The type of processing depends on experimental conditions. In this experiment, 5th, 6th, 7th-graders, and adults were tested with two paradigms. First, they performed a same/different task. Second, they carried out a numerical comparison task in which they had to decide which of two fractions was larger. Results showed that 5th-graders do not rely on the representations of the global magnitude of fractions in the Numerical Comparison task, but those representations develop from grade 6 until grade 7. In the Same/Different task, participants only relied on componential strategies. From grade 6 on, pupils apply the same heuristics as adults in fraction magnitude comparison tasks. Moreover, we have shown that correlations between global distance effect and children’s general fraction achievement were significant.<p>Fractions are well known to represent a stumbling block for primary school children. In a third study, we tried to identify the difficulties encountered by primary school pupils. We observed that most 4th and 5th-graders had only a very limited notion of the meaning of fractions, basically referring to pieces of cakes or pizzas. The fraction as a notation for numbers appeared particularly hard to grasp. <p>Building upon these results, we designed an intervention programme. The intervention “From Pies to Numbers” aimed at improving children’s understanding of fractions as numbers. The intervention was based on various games in which children had to estimate, compare, and combine fractions represented either symbolically or as figures. 20 game sessions distributed over 3 months led to 15-20% improvement in tests assessing children's capacity to estimate and compare fractions; conversely, children in the control group who received traditional lessons improved more in procedural skills such as simplification of fractions and arithmetic operations with fractions. Thus, a short classroom intervention inducing children to play with fractions improved their conceptual understanding. <p>The results are discussed in light of recent research on the mental representation of the magnitude of fractions and educational theories. The importance of multidisciplinary approaches in psychology and education was also discussed. <p>In sum, by combining behavioural experiments in adults and children, and intervention studies, we hoped to have improved the understanding how the brain processes mathematical symbols, while helping teachers get a better grasp of pupils’ difficulties and develop classroom activities that suit the needs of learners.<p> / Doctorat en Sciences Psychologiques et de l'éducation / info:eu-repo/semantics/nonPublished

Thesis (M. Ed. (Curriculum Studies)) -- University of Limpopo, 2020 / 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.

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Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Syftet med denna studie är att ta reda på hur lärare i grundskolans tidigare år introducerar och arbetar med området bråkräkning. Utifrån detta syfte så formulerades tre stycken frågeställningar: Hur beskriver lärare att de introducerar området för sina elever? Hur beskriver lärare i grundskolans tidigare år att de arbetar med området? Samt är lärare medvetna om någon svårighet med området bråk? För att kunna besvara dessa tre frågeställningar genomfördes kvalitativa intervjuer med sju stycken lärare som arbetar runt om i Sverige. Studiens resultat visar att bråkräkning är någonting som upplevs som svårt av många elever samt att grunden till förståelse för området ligger vid en tydlig introduktion av både området i sig, men även av väsentliga begrepp. De intervjuade lärarna har även beskrivit hur de introducerar och arbetar med området bråkräkning och detta diskuteras sedan i enighet med tidigare forskning.

Using 3 fb-1 of pp collision data collected at √s = 7 and 8 TeV by the LHCb experiment in the 2011 and 2012 data taking periods, the decays B⁰s → ϕπ+π- and B⁰d → ϕπ+π- have been studied in the π+π- invariant mass range below 1600 MeV/c². The B⁰s,d → ϕπ+π- branching fractions are determined to be: B(B⁰s → ϕπ+π-;mππ < 1600) = [3:72 ± 0:18 ± 0:38 ± 0:38] x 10-6 B(B⁰d → ϕπ+π-,mππ < 1600) = [1:75 ± 0:25 ± 0:42 ± 0:14] x 10-7 where the first uncertainty is statistical, the second is systematic, and the third comes from the normalisation mode B⁰s → ϕϕ. From the π+π- mass distribution and fits to angular distributions, the resonant decay mode B⁰s → ϕf₀(980) is observed and the branching fraction is measured to be: B(B⁰s → ϕf₀(980); f₀(980) → π+π-) = [1:23 ± 0:15 ± 0:12 ± 0:12] x 10-6 The fit also requires contributions from B⁰s → ϕf2(1270). A search for a P-wave contribution from B⁰s → ϕρ⁰(770) finds evidence at ~ 4σ but confirmation will require more data. An upper limit of the B⁰s → ϕρ⁰(770) decay branching fraction at 90% C.L. is measured to be: B(B⁰s → ϕρ⁰(770)) < 4 x 10-7.

>Magister Scientiae - MSc / Antimicrobials are chemical compounds that destroy or inhibit the growth of microorganisms. The majority of these antimicrobials are actually natural products or natural product derived with key examples being the pioneer antibiotics penicillin and cephalosporin. Antimicrobials are an extremely important class of therapeutic agents; however, the development of drug resistance and slow pace of new antibiotic discovery is one of the major health issues facing the world today. There is therefore a crucial need to discover and develop new antibacterial agents. In this study, the potential of marine algae as a source of new antibiotics was explored. Crude organic extracts and chromatographic fractions obtained from small-scale extraction of 17 different marine algae were used to prepare a pre-fractionated library that would be tested against several disease causing microorganisms. The activity of the pre-fractionated library and purified compounds was determined against a panel of drug resistant microorganisms namely Acinetobacter baumannii ATCCBAA®-1605™, Enterococcus faecalis ATCC® 51299™, Escherichia coli ATCC® 25922™, Staphylococcus aureus subsp. aureus ATCC® 33591™ and Candida albicans ATCC® 24433™. Finally, cytotoxicity tests of 50 selected library extracts and isolated compounds were done against two cell lines namely MCF-7 (breast cancer) and HEK-293 (kidney embryonic).

Secondary students, especially those with learning disabilities, often lack an understanding of computations involving fractions. Much of the secondary math core, especially algebra, requires an understanding of fractions to be able to successfully complete core classes. Instruction on fraction concepts in not part of the secondary core standards. These students are expected to already have this knowledge. There is a need for students with learning disabilities who struggle with fraction computations to receive instruction on fraction concepts in addition to their core instruction. This study used direct instruction and fluency practice as an intervention to teach basic fraction skills to two secondary students with learning disabilities. A multiple probe multiple baseline design was used. Results suggest that fluency training has a positive impact on secondary students' acquisition and retention of basic fraction skills. The implications of this study suggest that this intervention is a viable option to help students acquire fraction skills in a minimal amount of time.

In the classical case we make use of Pells equation to compute units in the ring OF. Consider the parallel to the classical case and the quadratic field extension that creates the ring OK. We use the generalized Pell's equation to find the units in this ring since they are solutions. Through the use of continued fractions we may further characterize this ring and compute its units.

This study examined the different types of mathematical errors exhibited by primary level pupils in Brunei when working with fractions. In addition, the study examined pupils' attitudes towards the learning of fractions and investigated if there were gender differences among Bruneian pupils' performances with fractions and with their attitudes towards fractions. The study was longitudinal in nature and its two phases involved a single cohort of Primary 5 pupils followed through a full year period in four government-funded primary schools in Brunei Darussalam. Pupils' mathematical errors were assessed by means of researcher-developed paper-and-pencil tests, while pupils' attitudes towards the learning of fractions were measured by means of an adapted version of attitude questionnaire that has been used previously with Bruneian pupils. Guided by six research questions, a number of statistical analyses were carried out to ensure the validity and reliability of the instruments used. These included piloting and revising the instruments, the use of Cronbach's alpha with the items in the attitude questionnaire, and the calculation of the Pearson Product Correlation Coefficient between scales of the questionnaire. The data was analysed by calculating the percentages and means of occurrences of each type of error. Paired and independent sample t-tests were carried out in order to investigate gender differences in pupils' errors and the impact of further instruction on fraction at the P6 level, while the GLM test was administered in order to investigate if there were significant change in pupils' attitudes towards fractions from the pre- to the posttests. Qualitative information obtained through pupils' interviews, field notes and lesson observations was used to support the quantitative data. / The study revealed that though pupils' achievement in the post-test improved, their performances on fraction work remained generally unsatisfactory. Many pupils in the study continued to have difficulty with the basic operations on fractions and resorted to the use of keyword strategies in dealing with word problems. Despite the pupils' unsatisfactory performance in the diagnostic tests, they generally held very positive attitudes towards the learning of fractions. No significant gender differences were observed either in pupils' performance in working with fractions tasks nor with their attitudes towards the learning of fractions. The findings of this study also highlight a number of issues for mathematics teachers to consider when dealing with fractions, and the findings also have implications for the quality of the instructional activities provided by the teachers, for the impact of language transfer in the medium of instruction - that is, from Bahasa Melayu to English at the pupils' Primary 4 level- and for the quality of the teacher training program in Brunei.