Research Report submitted to the Wits School of Education, Faculty of Science, University of the Witwatersrand, Johannesburg In partial fulfilment of the requirements For the degree of Master of Science (Mathematics Education) Johannesburg, 2017 / South Africa’s performance in mathematics education is ranked amongst the world’s worst. This performance is not only alarming at an international level, but also nationally. Annual National Assessments (ANA) conducted by the Department of Education have showed that the level of mathematics across the foundation and intermediate phase is poor with a pronounced dip in performance at a Grade 4 level (Department of Basic Education, 2014). Multiplication and division are common challenging areas that contribute to this poor performance. This is concerning as mathematics is globally recognised as a key competence for providing access to higher education and developing a country’s society and economy. My study, aimed at exploring multiplicative reasoning with Grade 4 learners through structured problem solving, is focused on the learning of multiplication and division within the context of an intervention concentrated on developing learners’ ability to model multiplicative situations. Shifts in the use of models were investigated following a smallscale intervention in which different modelling approaches (particularly ratio modelling) were introduced and developed. A control group was used to determine the usefulness of the intervention. Questions which I sought to answer were: (a) what kinds of multiplicative reasoning (models) are Grade 4 learners using prior to intervention, (b) what changes, if any, are seen in overall performance, across the intervention and control group, in the post-test, and, (c) what kinds of differences in model use were associated with the shifts in performance? The main dataset comprised of 61 pre- and post-test scripts across three Grade 4 classes in a former Model C school in a Johannesburg district. A sample of 15 interviews were also conducted across the classes. Document analysis and transcription notes were used to analyse data with a Realistic Mathematics Education (RME) framework informing my analysis. Findings from my study reveal that prior to intervention, Grade 4 learners presented limited multiplicative models which were predominantly confined to traditional algorithms. After the small-scale intervention, learners used a broader range of models with an emerging take up of ratio models. The success rate associated with the models presented by learners also improved. Limited and/or no changes in model use and their respective success rates were seen in the control group suggesting that the intervention program was useful. These findings suggest that, as a future recommendation, it would be worthwhile to investigate the outcomes of running a similar intervention in less privileged settings. / MT 2018

At last, this dissertation considers another marketing implication of this number-location association, namely the compatibility effect. In experiment 7, we find that people perceive a discount as more attractive when the two prices are actually posited in compatible locations (original price-right side; discounted price-left side) than in incompatible locations. Similarly, experiment 8 demonstrates that people are more likely to patronage a supermarket when the supermarket's slogan about low price is shown on the left side of a display than on the right side, and this effect is mediated by the subjective fluency feeling people felt at the time they process the advertisement. / Given a display, people usually think that large numbers should be located on the top or on the right hand side of the display, whereas small numbers should be posited at the bottom or on the left (Wood and Fischer 2008). / Given this number-location association, this dissertation secondly intends to apply it to the field of marketing, and to use three experiments to explore how and why location of product image can influence people's price judgment. The results of experiment 4 show that consumers think that the market price of a product is higher if the product's image is shown on the right side of a display than on the left side; experiment 5 and experiment 6 further indicate that the location of product image can only influence consumers' price judgment, but cannot influence quality judgment. / Key Words: Number-location association, Simulation, Perceptual Symbol Systems (PSS), Price perception. / This dissertation firstly aims to provide new evidence for this number-location association. Experiment 1 demonstrates that people incorrectly remember that large numbers appear to the right of the locations they actually were shown while small numbers appear to the left ofthe locations than they actually were presented; experiment 2 and experiment 3 show that people estimate there are more pieces in a pile of object when the pile of object is presented on the right side of a display than on the left side. / Cai, Fengyan. / Adviser: King Man Hui. / Source: Dissertation Abstracts International, Volume: 73-03, Section: A, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 139-144). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Probabilities, an essential dimension of risk communication, can be presented in various formats including frequencies (e.g., 1 in 10), percentages (e.g., 10%), or verbal phrases (e.g., unlikely); the literature is mixed concerning which format best supports comprehension. Additionally, it is not well understood how people who vary in their level of numeracy understand those probabilities. The goal of the present three-phase within-participant study was to understand how the factors of format and numeracy influence comprehension and mental representations of probabilities for younger and older adults. Overall, the results of this research clearly indicated that comprehension and mental representation of health risk probabilities are influenced by format, age, and numeracy. To best support comprehension and comparison of health risk probabilities for younger adults and healthy older adults with varying numeracy, percent format should be used.

Thesis (Ph. D.)--Ohio State University, 2008. / Title from first page of PDF file. Includes bibliographical references (p. 93-99).

Thesis (Ed.S.)--Mississippi State University. Department of Curriculum and Instruction. / Title from title screen. Includes bibliographical references.

Thesis (M.Phil.) - University of Queensland, 2005. / Includes bibliography.

Thesis (MEd (Education))--Cape Peninsula University of Technology, 2016. / The poor performance of learners in mathematics has long been a matter of concern in South Africa. One certain fact from the Annual National Assessment (ANA) results is that the problem starts in the Foundation Phase (FP) with number concepts. The aim of this study was to explore how five Foundation Phase teachers located in challenging socio-economic school contexts in the Western Cape used mathematical resources to promote teaching for understanding of the important number concept area in CAPS. These resources included humans, materials, culture and time. The research was located within the interpretive qualitative research paradigm and used a case study approach. The participants in the study included five FP teachers teaching Grades 1 to 3 at two schools in the Western Cape. Data was collected through lesson plan analysis, lesson observations and semi-structured interviews. The data collected was then analysed through the lens of Vygotsky’s socio-cultural theory. Socio-cultural theory maintains that knowledge is best acquired if it is mediated by language, more knowledgeable others and physical tools. Vygotsky believed that knowledge is first acquired interpersonally, then intrapersonally, as learners first learn from others, then internalise or individualise knowledge while going through the four stages of the Zone of Proximal Development (ZPD). The findings of this study revealed that teaching for understanding was often compromised by teaching to enable learners to pass assessments. Teachers understood the importance of using resources to teach number concepts in the Foundation Phase, but inclined to rote teaching with work drills in preparation for assessments such as the Annual National Assessment (ANA) and the systemic assessment. Resources were often used when learners struggled to understand concepts and as calculation tools. This study supports the view from the literature that the way in which resources are used affects the teaching and learning of number concepts. It recommends that teachers should read and follow the CAPS mathematics document, as it clearly states what resources to use and how. This study further recommends that more research on the use of resources to teach mathematics in other content areas should be done.

Thesis (DEd)--Stellenbosch University, 1995. / ENGLISH ABSTRACT: The rationale for this study essentially is the perceived and reported misconceptions in algebra that exist within pupils in the Junior Secondary phase. These misconceptions are the direct result of the incomplete mastery of algebra. The purpose of this research is to attempt to contribute towards pupils' complete mastery of algebra and the ensuing elimination of certain pupil misconceptions, by trying to increase primary school pupils' level of awareness of the general number properties. Primary school pupils who learn arithmetic in a problem based environment intuitively use the general number properties to execute arithmetical calculations, while pupils in the traditional teaching approach are taught standard algorithms which they must apply, often without comprehension. In the latter, the existence of the number properties is concealed. When these pupils officially encounter the general number properties for the first time in standard 4 or 5, they are probably not capable of integrating these new concepts into their existing knowledge structure. This may lead to incomplete mastery and ensuing misconceptions, which can become worse when pupils must apply exactly the same number properties in algebra to execute algebraic manipulations. The fact that primary school pupils in the problem based approach intuitively apply the number properties, earlier led to the hypothesis that these pupils possess a higher level of awareness of the number properties. However, research has indicated that these pupils possess only a moderately higher level of awareness. This has led to this study during which a specific attempt is made to take pupils' intuitive knowledge of the number properties, which they spontaneously apply, as a point of departure, and to develop it to such an extent that they become explicitly aware of the existence and nature of these fundamental concepts in mathematics. The technique how to link up with pupils' intuitive knowledge and to increase their level of awareness of the number properties inside a problem based environment, was developed during the first two years of this study. At the end of this period, a suitable teaching strategy was formulated. This strategy is essentially based on the following parameters: Group discussion, the use of calculators, and creating a cognitive disequilibrium. During the third year of this study this strategy was implemented at a number of schools. During this phase research concentrated on the distributive property only. Judging by the results, it appears as if a considerable increase in level of awareness has taken place within these pupils. During this study a hierarchical model of levels of awareness was also formulated. This model was used as a guide in the attempt to increase pupils' level of awareness of the number properties. / AFRIKAANSE OPSOMMING: Die rasionaal vir hierdie studie is hoofsaaklik gelee in die waargenome en gerapporteerde wanbegrippe in algebra wat by leerlinge in die Junior Sekondere fase bestaan. Hierdie wanbegrippe is die direkte gevolg van die onvolledige beheersing van algebra. Die doel van hierdie navorsing is om 'n hydrae te probeer lewer tot leerlinge se volledige beheersing van algebra, en die gepaardgaande uitskakeling van bepaalde leerlingwanbegrippe, deurdat gepoog word om laerskoolleerlinge se vlak van bewustheid van die algemene bewerkingseienskappe te verhoog. Laerskoolleerlinge wat probleemgebaseerde rekenkunde-onderrig ontvang, benut intu'itief die algemene bewerkingseienskappe ten einde rekenkundige berekeninge uit te voer, terwyl aan laerskoolleerlinge in die tradisionele onderrigbenadering standaardalgoritmes onderrig word wat hulle dikwels sonder begrip moet toepas. In laasgenoemde geval word die bestaan van die bewerkingseienskappe erg versluier. Wanneer hierdie leerlinge in standerd 4 of 5 vir die eerste keer amptelik met die algemene bewerkingseienskappe in aanraking kom, is hulle waarskynlik nie in staat om hierdie nuwe begrippe met hul bestaande kennisstruktuur te integreer nie. Dit kan tot onvolledige beheersing en gevolglike wanbegrippe lei, wat kan vererger wanneer leerlinge presies dieselfde eienskappe in algebra moet benut ten einde algebra!ese manipulasies uit te voer. Die feit dat laerskoolleerlinge in die probleemgebaseerde benadering intu'itief die bewerkingseienskappe toepas, het tevore reeds tot die hipotese dat hierdie leerlinge oor 'n hoer vlak van bewustheid van die bewerkingseienskappe beskik, aanleiding gegee. Navorsing het egter aangetoon dat daar slegs 'n matige hoer vlak van bewustheid by hulle bestaan. Dit het tot hierdie studie aanleiding gegee waartydens spesifiek gepoog word om leerlinge se intuitiewe kennis van die bewerkingseienskappe wat hulle spontaan aanwend ten einde rekenkundige bewerkings uit te voer, as vertrekpunt te neem, en te ontwikkel sodat hulle eksplisiet bewus sal raak van die bestaan en wese van hierdie grondliggende waarhede in wiskunde. Die tegniek hoe om by laerskoolleerlinge se intui'tiewe kennis aan te sluit en hul vlak van bewustheid van die bewerkingseienskappe binne 'n probleemgebaseerde omgewing te verhoog, is gedurende die eerste twee jaar van hierdie studie ontwikkel. Aan die einde van hierdie periode is 'n toepaslike onderrigstrategie geformuleer. Hierdie strategie steun sterk op die volgende parameters: Groepsbespreking , die benutting van sakrekenaars, en die skep van 'n kognitiewe disekwilibrium. Gedurende die derde jaar van hierdie studie is hierdie onderrigstrategie in verskeie skole toegepas. Daar is gedurende hierdie fase slegs op die distributiewe eienskap gekonsentreer. Volgens die resultate wil dit voorkom asof daar 'n aansienlike verhoging in die betrokke leerlinge se vlak van bewustheid van die distributiewe eienskap plaasgevind het. Gedurende hierdie studie is ook 'n hierargiese model van vlakke van bewustheid geformuleer aan die hand waarvan die poging aangewend is om leerlinge se vlak van bewustheid van bewerkingseienskappe te verhoog.

This study considered use of practical work as one of the strategies that may be used to teach and learn fraction concepts in primary school Mathematics. Although an educator and learners were participants in the study, the focus was mainly on the learners. The class educator’s perception of practical work was investigated and the results confirmed the assumption that most educators use minimal or no practical work when teaching learners fractions. The researcher carried out an experiment with learners to find out whether they saw any value in doing practical work. Data collection instruments used were an observation schedule which was collated by the researcher in teaching four lessons, written responses of learners to a series of activities they did as class work and their responses to interview questions. Data collected from learners confirmed that practical work did have value in the teaching of fraction concepts, especially addition of fractions. Other than confirming the value of practical work, much other valuable data emerged from the findings. The data have important implications for the teaching and learning of fractions, especially addition of fractions, teacher training in practical work and also further research. These are intended to improve teaching of fractions, particularly addition of fractions. / M. Ed. University of KwaZulu-Natal, Durban 2013.

We rely so much on vision that it is hard to imagine sensing the world differently. But most organisms primarily use other sensory information, even something as detached from our senses as electricity. Some fish, called weakly electric fish, generate electric pulses to sense their environment. Objects in their environment distort the electric pulses, and the fish use special receptors in their skin to process these distortions and identify the nearby objects. They detect the location, size, shape, and electric properties of nearby objects, enabling them to find preferred food. These fish use their discharges not only for sensing and foraging as described, but also for communication. Investigating this sensory system can provide insights into neural computations for sensory processing more broadly, and can expand our understanding of the complex stimuli present in our environment that we do not perceive.In the first half of this work, we investigated how the weakly electric fish Gnathonemus petersii processes the electric sensory information to interact with its environment. We also used the tools developed in this work to study social behavior in groups of freely swimming fish. Chapter 1 provides an in-depth introduction to this model organism and its prominent active electrolocation behavior. This introductory chapter is focused on the parts of the behavior that are relevant to the computational models developed in this work. We investigated the active electrolocation behavior using a comprehensive end-to-end model that contains multiple components, which will be detailed in the following chapters. Chapter 2 describes the physics model that simulates the fish and its environment to collect data. The physics model builds on previous work and extends it to a more general framework that can be used to simulate the fish in different environments. We developed and adequately documented an open code base that can be used to simulate various fish species and their interactions with nearby objects or electrical boundaries. Chapter 3, specifically Section 3.3, presents a data-based model of the electroreceptors that process the sensory input. We used machine learning techniques to develop a model that can predict the response of the receptors to distortions due to different objects. The model is based on local field potential data collected from the afferent layers of the electrosensory lobe, the first brain area that processes the sensory input. This data was collected by Abigail Zadina in Nathaniel Sawtell’s laboratory at Columbia University. Chapter 3, specifically Section 3.4, describes the neural network models that identify computations that help solve the behavior. We used data generated from the physics model as sensory input, we used our electroreceptor model to parse this data serving as first-stage input to down- stream brain areas, and we used neural network models to characterize the nearby objects’ spatial and electric properties based on the sensory input. Based on results from our neural network models, we set two hypotheses for how weakly electric fish sense their environment and motivate experiments on less studied brain areas to test these hypotheses. First, we suggest that decoding all spatial and electric properties of a nearby object distorting the electric discharge is very challenging due to interactions between these properties, but first decoding the spatial properties and then using the spatial properties as internal feedback to decode the electric properties helps solve the task by disentangling the interactions. Second, we suggest that the specialized Schnauzenorgan organ of the weakly electric G. petersii, previously described as an electric fovea due to the very high density of electroreceptors and believed to serve a primary role in close-range characterization, may also play a role in long-range detection of objects surrounding the fish. Chapter 4 explores social interactions in groups of freely swimming fish and starts to investigate how they use their electric discharges to navigate, interact and communicate. Here, we used our physics-based framework to accurately identify the fish that emitted each electric discharge in a group of fish. This work is currently in progress and we performed various preliminary analyses to investigate the social behavior and social rank of these fish, which we present here. Data for this project was collected by Federico Pedraja in Nathaniel Sawtell’s laboratory at Columbia University. The second half of this work addresses a variety of different research questions with loose connections in between them and in relation to the first half. The common factor present in all these projects can be generally described as investigating how computations may be used in neural circuits to produce successful behavior. We used a variety of computational models and tools to investigate these questions, and we present the results of these investigations in the following chapters. Chapter 5 provides a biologically plausible architecture alternative for the classical binary classification task. Typically, feed-forward models have been used to solve this task. However, neocortical circuits likely involved in decision making are recurrent and sparse. We used a recurrent neural network model with sparsity constraints to solve the binary classification task. We demonstrated that the sparse recurrent networks solve the task well, make use of dynamic computation similar to evidence accumulation, and distribute the information throughout the network despite the sparsity constraints. Chapter 6 explores syntactic differences of world languages and offers a potential neural computation mechanism that could account for those differences. We focused on differences in the basic word order of simple sentences because these have been extensively studied in the linguistic literature. These simple sentences only have three parts, subject, verb, and object, and the order of these parts varies across languages non-uniformly. We aimed to provide a possible language generation mechanism that could account for these differences. Chapter 7 investigates the computational journey from numerical cognition to arithmetic ability. This research direction was motivated by and based on experimental work that addressed whether bees (and later stingrays and cichlids) can learn simple arithmetic operations. This project was designed for introducing a Columbia SEAS undergraduate student, Katharyn Fatehi, to computational neuroscience research. I mentored Kat through the Women in Science at Columbia program, and provided detailed guidance, code base, tutorials and instructions for her to learn about computational neuroscience research and to contribute to this project. Chapter 8 represents my contribution to a large collaboration effort aimed at improving spike sorting techniques. This project quantified the impact on spike sorting quality of the geometry mis- match between typical recording probes (1D, or 2D at best) and the 3D structure of the brain. We leveraged the experimental setup, multi-electrode recording arrays with planar geometry recording the activity of 2D retinal tissue, to address this question. The work presented in this thesis is a collection of projects that investigate neural computations in different contexts. The first half of the work is focused on the weakly electric fish G. petersii and its active electrolocation behavior. The second half of the work explores a variety of different research questions related to computational mechanisms that could be implemented in neural circuits. The work presented here is a step towards understanding how computations in neural circuits can produce successful behavior in different contexts.