Modern theory of arithmetic instruction supports the idea that the development of understandings of basic mathematical principles produces a desirable type of learning. This is a reaction against the traditional method of instruction which places emphasis upon mechanical drill procedures, devoid of meanings. This study is an attempt to deter-mine what relationship, if any, exists between computational ability and understanding of fundamental processes. The investigation has been limited to the area of decimal fractions. Two tests -were developed for the purpose of the investigation. The test in computation was constructed and validated using pupils of the junior high school level as testees. Student-teachers constituted the personnel for the construction and validation of the test in understandings. The investigation, of relationship was performed using 236 Normal School students as testees. The tests, which had been constructed for use in the study, were administered at the beginning of the school term. The data obtained from the investigation were analyzed and the following conclusions were formulated: 1. There is a positive correlation of considerable magnitude between the scores on the test in computation and the scores on the test in understandings. ( r = .640 ). This is an indication that there is a tendency for the scores to vary in the same direction. 2. When the factor of intelligence is held constant, there is a net correlation of marked magnitude -which is somewhat less than, the apparent coefficient. This indicates that the common factor of intelligence has an influence upon the relationship between the two variables. 3. The magnitude of the relationship between scores in understanding and intelligence test scores is an indication of common elements in both these tests. 4. The relationship between, the scores in computation and the intelligence test scores is not high. A high intelligence does not appear to be a prerequisite for high achievement in computation. 5. There is evidence that ability in computation is not essential for high achievement in understandings and vice versa, nor do high scores in one of these factors guarantee high scores in the other. 6. Although a study of the scatter diagram suggests that success in computation is more probable if it is accompanied by a high degree of understanding, it cannot be inferred from the data that one variable is the cause or the effect of the other. / Education, Faculty of / Graduate

M.Ed. (Subject Didactics) / Each child has his own personality and individuality. Children are learning in different ways, at different tempo's and achieve different heights of success with.their efforts. The degree to which the learner is able to master new concepts, is closely related to the reference framework and given pre-knowledge. However, the learning process is not always successful. Various reasons for this phenomena can be identified. This study focuses on the role which misconceptions play in this regard. In general, misconceptions can be defined as a distortion or misinterpretation of the learned concepts. synonyms used to describe this phenomena includes words. like "previous knowledge", "preconceptions" and "alternative frameworks" Misconceptions in Mathematics are numerous. In various studies conducted, misconceptions were identified in almost all areas of Mathematics. Likewise a great deal of misconceptions were found existing among students concerning the handling of fractions. It is an impossible task to research all misconceptions in Mathematics in one study. For this reason it was decided to do research on only one aspect, namely fractions where possible misconceptions can occur. With the empirical research which was conducted, certain misconceptions in the area of fractions were identified. These misconceptions include, amongst other, the following: 1. The sum of and difference between two fractions. There is very little or no notion of the smallest denominator. 2. Multiplying and division of fractions. The student is uncertain about the role which the numerator and the denominator play in the solution. As fractions play such an important role in Mathematical success, it is suggested that a plan of action will be set as soon as possible in order to prevent misconceptions influencing the student learning process.

The preparation of root nodule extract by Virtanen, et. al., has created a good deal of interest in the study of plant hemoglobin (legoglobin). In the preparation of any new substances, it is of interest to determine the physical constants of the new species.

This master thesis is focused on fractions as a problematic area of mathematics education. The goal of the thesis is to find out which mistakes students make and which wrong strategies they adopt in dealing with fractions. Another aim is to study how the historical ways of dealing with fractions could help the students nowadays. The thesis consists of a theoretical part introduces the matematical thinking of the Ancient Mesopotomia and Egypt periods, as well as present day teaching of fractions, using analysis of the textbooks and a brief glimpse into Czech as well as international researches. The practical part is based on an investigation in three different seventh grade classes in two primary schools and one high school class of the prima grade. In total 73 students participated in the research which cotains a diagnostic test, a series of individual interviews and an educational experiment with historical tasks. The result of this thesis is the finding that most of the mistaken strategies are not connected with only one particular school, because they appear in all of the schools and the Ancient Egyptian methods could be helpful in teaching fractions today as well. For this reason I recommend to spend more time in the educational proces with unit fractions (fractions of the form 1 n ), which are...

Thesis Supervisor: Dr. Marv Westrom The purpose of this study was to investigate the effectiveness of the software program Visual Fractions in teaching basic fraction concepts and the effect that student control over the construction of fraction diagrams had on their learning. The Visual Fractions program provides a diagram and two fractions in numeric form. The diagram consists of a figure divided into partitions with some of the partitions shaded. One fraction represents the shaded parts of the whole and the other represents the unshaded parts. Students can control the total number of partitions and whether each is shaded. Manipulating the diagram changes the value of the fractions. A Non-interactive (crippled) version of the software was designed to eliminate the user-control aspect of the program. Users of this program could click to generate a new fraction, but had no control over the choice of fraction. The computer randomly generated a new fraction and displayed the corresponding diagram each time. A third treatment, Fraction Flash Cards, was designed to simulate the Noninteractive version of the program, without the computer. The students received Flash Cards containing images of the computer-generated fraction diagrams. The study consisted of a pilot project during which data collection techniques were tested and revised and the main study. Sixty-four subjects were taken from four intact classes of grade four students. The students were randomly assigned to one of the three Treatment Groups or the Control Group. Three different sets of data were collected: a pretest and postest on fractions, structured interviews, and field notes taken by the researcher during the treatment process. In Treatment Group One, students used the Interactive Version of Visual Fractions. Here, students could create fractions at their command. There is evidence to suggest that this type of interactive control is a critical factor in learning (Merrill, 1987). In Treatment Group Two, students used the Noninteractive version of the software. Students could control the rate of observing fractions and fraction diagrams, but not the value of the fraction. Students in Treatment Group Three used the Flash Cards. Motivation appears to strongly affect one's ability to learn and children appear to be highly motivated to use computers. The purpose of this treatment was to control for any achievement gain that may have been due to the novelty of using computers. The four Groups were compared using analysis of variance with repeated measures. Significance at the 0.01 level was found for the tests and the interaction. A study of the interaction showed that there was no significant difference between the gains of the Visual Fractions Noninteractive Group, the Flash Card Group, or the Control Group. However the gain achieved by the Visual Fractions Interactive Group was significant. From this study, it is clear that the Visual Fractions Interactive program which provides students the opportunity to construct fraction diagrams with immediate feedback, is an effective method of teaching fractions. / Education, Faculty of / Graduate

Flagellate protozoa from the order Trypanosomatida have developed a range of strategies to survive in their mammalian hosts. A consequence is that the glycolytic pathway has assumed an important role, especially in bloodstream-form Trypanosoma brucei, where it is essential as the sole producer of ATP. The seventh enzyme in the pathway, 2,3-bisphosphoglycerate-independent phosphoglycerate mutase (iPGAM) is particularly attractive as a drug target because it shares no common properties with the corresponding enzyme in humans. This enzyme catalyses the conversion of 3PGA to 2PGA, with the requirement for metal ions to assist the catalytic function. In this study, two important biochemical and structural aspects of the enzyme were investigated: i) The in vitro and in vivo requirements for biologically relevant metal ions to support the activity of iPGAM, and ii) The ability of trypanosomatid iPGAM to exist in multiple conformations and oligomeric states in solution. The maximum activity of iPGAM in vitro requires Co2+, but this cannot be the case in vivo where ICP-OES analyses confirmed that Co2+ was essentially undetectable in T. brucei cytosolic fractions. The activity of iPGAM in vivo is therefore one of the lowest among the glycolytic enzymes. By contrast, Mg2+ and Zn2+ were found to be the most abundant metals in both cytosolic fractions and in purified bacterially expressed iPGAM. Our newly-developed multimode-plate reader discontinuous assay further revealed that of the biologically relevant metals, only Mg2+ can support iPGAM activity, but at less than 50% of the level of Co2+. By contrast, Zn2+ strongly inhibits iPGAM. This assay which was developed with minimal metal interference on the coupling enzymes, also showed that in solution, the ratio of the concentrations of 3PGA:2PGA (substrate:product) at equilibrium is not 1:1 as observed in the crystal structure, but is in fact 12:1, which may be due to the tighter binding of 2PGA to the enzyme. A series of biophysical analyses, notably by SEC-MALS showed that iPGAM from Leishmania mexicana, another trypanosomatid protozoan parasite exists in different forms and oligomeric states in solution, either as the closed-form monomer, openiii form monomer, or closed/open-form dimer which can be successfully separated by ion-exchange chromatography. The open-form LmiPGAM is particularly relevant for drug development, as the catalytic site in the closed-form structure is poorly inaccessible. Both virtual and high-throughput screening approaches were used to identify novel potential inhibitors. Out of a collection of 11 compounds tested at 1 mM, two showed substantial inhibition with 49% and 14% remaining activity. Taken together, the findings from this study demonstrated the potential of iPGAM to be a key modulator in controlling glycolytic flux in trypanosomes, and thus further validated it as an important drug target.

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 research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand. Johannesburg, 2016. / The purpose of this study, conducted at a mathematics clinic, was to investigate the misconceptions that learners display through errors they make when solving algebraic equations involving fractions. A teaching intervention to address those errors and misconceptions was done at a mathematics clinic. A mathematics clinic is a remedial facility where low-attaining students attend sessions, by choice or by referrals. In this study teaching intervention was used to address learners’ errors and misconceptions. The assumption of the study was that learners are knowledge constructors that use previously-learned knowledge as the basis of new knowledge. Since their previous knowledge contains errors and misconceptions, the construction of new knowledge results in errors. This research was mainly qualitative. Data were collected, using a sample of 17 grade 10 learners, though the work of only 13 of them was analysed. Two participants wrote the pre-test, but did not participate in the subsequent data collection, and the other two did not solve some of the equations in the pre- and post-tests. There were three stages of data collection; pre-test, teaching intervention and post-test. Pre- and post-tests were analysed for errors committed by learners, and the teaching intervention sessions were analysed for opportunities of learning provided. Transcripts were produced from the teaching intervention sessions. They were also analysed to check how students participated in constructing mathematical meanings, and also how effectively their attention was focused on the object of learning. The errors found in learners’ equation-solving were like-term errors, lowest common denominator errors, careless errors, sign errors and restriction errors. The comparison of the number of learners who committed these errors in the pre- and the post-test was insightful. Of 13 learners, 4 committed like-term errors in the pre-test and just 1 in the post-test; 4 committed LCD errors both in the pre- and post-tests; 9 committed careless errors (other errors) in the pre-test, and 6 learners in the post-test; 7 committed sign errors in the pre-test and 1 in the post-test; and 12 committed restriction errors in the pre-test, and 9 in the post-test. These findings suggest that teaching intervention is a necessary pedagogical technique, and needs to be employed when addressing learners’ errors and misconceptions in mathematics. Reduction in learners’ errors and misconceptions was evident after the teaching intervention suggesting that the mathematics clinic provided learning opportunities for participants. / LG2017

Thesis advisor: Sara Cordes / Human infants show relatively sophisticated abilities to track and use proportional information. However, by the age of 6, children tend to make predictable errors in their proportional reasoning and later encounter significant challenges in many aspects of formal fraction learning. Thus, one of the central questions motivating this research is to identify the factors leading to these difficulties, in light of evidence of early intuitions about these concepts. In the current dissertation, I address this question by investigating the tradeoff between attending to proportional magnitude information and discrete numerical information about the components (termed “numerical interference”) across both spatial (i.e., area models, number lines) and symbolic (fractions, decimals) representations of proportion information. These explorations focus on young children (5-7 year olds) who have yet to receive formal fraction instruction, older children (9-12 year olds) who are in the process of learning these concepts, and adults who have already learned formal fractions. In Project 1, I investigated how older children and adults map between symbolic and spatial representations, particularly focusing on their strategies in highlighting componential information versus magnitude information when solving these mapping tasks. In Projects 2 and 3, I explore the malleability of individual differences in this numerical interference in 4- to 7-year-old children. Across the three projects, I suggest that although numerical interference does impact proportional reasoning, this over-attention to number can be reduced through modifying early experiences with proportional information. These findings have implications for education and the way we conceptualize numerical interference more generally. / Thesis (PhD) — Boston College, 2017. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Psychology.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013. / Continued fractions have been extensively studied in number theoretic ways. These continued fractions are expressed as compositions of M¨obius maps in the Picard group PS L(2;C) that act, by Poincar´e’s extension, as isometries on H3. We investigate the Picard group with its generators and derive the fundamental domain using a direct method. From the fundamental domain, we produce an ideal octahedron, O0, that generates the Farey tessellation of H3. We explore the properties of Farey neighbours, Farey geodesics and Farey triangles that arise from the Farey tessellation and relate these to Ford spheres. We consider the Farey addition of two rationals in R as a subdivision of an interval and hence are able to generalise this notion to a subdivision of a Farey triangle with Gaussian Farey neighbour vertices. This Farey set allows us to revisit the Farey triangle subdivision given by Schmidt [44] and interpret it as a theorem about adjacent octahedra in the Farey tessellation of H3. We consider continued fraction algorithms with Gaussian integer coe cients. We introduce an analogue of Series [45] cutting sequence across H2 in H3. We derive a continued fraction expansion based on this cutting sequence generated by a geodesic in H3 that ends at the point in C that passes through O0.