Foolproof, and Other Mathematical Meditations by Brian Hayes

This research identifies important features of mathematics learning from the accounts of early school leavers. These students were enrolled in a Youth Reconnected Program at a Technical and Further Education [TAFE] college. Specifically, it examines the relationship between their experiences of mathematics learning in two different social contexts, the program and secondary school classrooms, the discourses and associated discursive practices of these two contexts, and the forms of identities of participation that are constructed in them. Drawing on a social theory of learning and critical discourse theory, the research identifies the processes by which membership in communities of learning is achieved and identities discursively constructed. It draws on the accounts of forty-three early school leavers and six staff members located in a Youth Reconnected Program, a program designed to re-engage early school leavers in education and or training, about their prior experiences of secondary school and their current experiences of TAFE. From these accounts, the thesis seeks to explicate their experiences of learning mathematics in both contexts and to consider the implications of different student experiences of learning for mathematics education. Critical discourse analysis provides the means for applying this framework to examine the discourses, discursive practices, identities, and forms of participation constructed in the accounts. It views the accounts as constituted by their social context. It shows how the language of the accounts works to construct particular versions of reality and how relations of power construct particular identities. It allows a richer qualitative exploration of the reported practices of mathematics classrooms with particular regard to linking the theoretical and practical concerns of this study. It makes transparent the relations between the social (including relations of power and domination), discourse, and discursive practice, thus enabling a more in-depth reading of the participants‘ accounts. Discourses about two different experiences of mathematics learning were identified—one about mathematics learning in the Youth Reconnected Program (TAFE) and the other about mathematics learning in secondary schools. The discourses and their associated discursive practices—differing application of instructional style, communication and interaction, source of authority, pace of instruction, grouping of students and assessment— were found to contribute significantly to the construction of particular student identities and forms of participation. Student experiences emphasised the importance of four characteristics for membership in a community of learners—mutual engagement, joint enterprise, shared repertoire and access. These characteristics and practices were shown to contribute substantially to the construction of identities of participation in mathematics learning for most of the students who participated in this study. There was strong evidence to indicate that small class sizes impacted significantly on student engagement and participation in learning. A crucial factor here rests with the role of the teachers and tutors and how, given the opportunities for reduced class size, they implement, sustain and maintain interactions with the students for effective teaching and learning to occur. If in this context, consistent interactions are provided that also includes clear and explicit explanations, discussion, and negotiation, students are more likely to maintain their interest and actively participate in mathematics learning. This study‘s significance is twofold. It lies in the critical importance of the issue it addresses—student participation in mathematics learning—and the articulation of a social theory of learning with critical discourse theory to enable a different and more effective understanding of the significance of discourses, identities and communities of participation in mathematics learning. Although the study cannot claim generalisability across other populations of learners or across all mathematics and TAFE classrooms because of the small select sample of early school leavers drawn upon, by highlighting these accounts of students‘ learning experiences in some mathematics classrooms and the TAFE program, and their associated experiences of them, it draws attention to a different perspective on mathematics learning. In this alternative framing, learning as a social process is contingent on access to the discourse of mathematics learning and the discursive practices that support learning, participation, and membership in a community of mathematics learners. An attention to the social processes of mathematics learning infers a significant role for mathematics teachers in enabling student access to certain discourses and communities of learning. Such a role is often neglected in current contexts of accountability that specify a focus on achieving syllabus outcomes.

This study investigated the impact of entry level of mathematics subject knowledge on student teachers` mathematics pedagogical content knowledge development and performance in mathematics teaching practice. The sample consisted of 24 mathematics student teachers, 12 of whom passed A-Level mathematics and 12 of whom only passed O-level mathematics. They were all studying in a 4-year bachelor of education (Honours/ Primary) programme; they were either majoring or minoring in mathematics. Results showed that student teachers` entry-level mathematics subject knowledge is not related to their mathematics pedagogical content knowledge development or their mathematics teaching performance. These findings may lead society to consider whether student teachers who have passed O-level mathematics are already eligible to be trained as professional primary mathematics teachers. As a consequence, this study raises the issues of how to develop student teachers` mathematics pedagogical content knowledge and whether we need to restructure our bachelor of education (Primary) programmes` curriculum in teacher professionalism.

The purpose of this study was conducted to determine the effect of Problem Based Learning aided interactive learning media on the ability of understanding the concepts, communication skills and mathematical reasoning abilities in the class X IPS MAN 1 City of Bengkulu. This research is quasi experiment research. The data collection is done by using a test sheet reasoning ability, the ability of understanding the concept and communicationskills. The results showed that the influences of the problem based learning model of interactive learning media aided the understanding of the concept of mathematical ability of students by 34.8%. There is the influence of problem based learning model of media-aided interactive learning to students' mathematical communication skills by 45,2 %. There is the influence of problem based learning model of media-aided interactive learning to students' mathematical reasoning abilities by 45%. There is a problem based learning model of the effect on the ability of students' mathematical reasoning by 42,5%. There is a model of the effect of problem based learning to the students' mathematical communication skills by 25.8%. There is a problem based learning model of the effect on the ability of students' understanding of mathematical concepts by 22.5%. There is the influence of the application of conventional learning aided interactive learning media on mathematical communication skills by 15,78 %. Effect of the application of conventional learning aided interactive learning media on mathematical reasoning skills was 24,89 %. Effect of the application of conventional learning aided interactive learning media on mathematical understanding the concepts was 16,74%. Keywords: problem based learning, interactive learning media, mathematical communication, understanding the concept, mathematical reasoning.

Advanced mathematics is a public basic course in science and engineering of colleges and universities. It is an important course to train students' thinking training and master mathematical theory and methods. Introducing mathematical experiments in higher mathematics teaching, giving play to the role of basic experiments and exploration, application and comprehensive modeling experiments, and reforming university mathematics courses can cultivate innovative talents. The idea of mathematical modeling is to combine practical problems with mathematical theories, to abstract them into concrete, and to solve practical problems in life by mathematical methods. Integrating the idea of mathematical modeling into today's higher mathematics teaching is a new teaching method for higher mathematics teaching and a very meaningful quality education method. Combining examples, the thesis analyzes the process of solving advanced mathematics problems by using higher mathematics modeling in the teaching process, and strengthens students' theoretical cognition ability through mathematical modeling, guides students' self-learning ability training, and enhances students' application ability and practical ability. Inspire students' enthusiasm for learning.

Este artigo é um ensaio que tem por objetivo problematizar e discutir o lugar da matemática na formação do futuro professor, em cursos de Licenciatura em Matemática. O ensaio é organizado em torno de duas questões básicas: de que matemática estamos falando, quando dizemos que o professor precisa saber bem matemática para ensiná-la? Que práticas formativas podem contribuir para que o futuro professor possa se apropriar dessa matemática fundamental para seu trabalho profissional? Para respondê-las, foram analisadas e problematizadas as diferentes práticas sociais do educador matemático, tentando perceber, nelas, indícios do tipo de saber matemático mobilizado e requerido. A seguir, foram trazidas algumas tentativas históricas de tratar o problema da formação matemática dos professores, tendo por base Felix Klein, Richard Courant e Bento de Jesus Caraça, e algumas pesquisas brasileiras e internacionais que têm como foco de estudo a formação matemática do professor. A partir desses estudos, o artigo destaca a existência, na formação inicial do professor de matemática, de uma quase tricotomia entre formação matemática, formação didático-pedagógica e prática profissional. Para romper com essa tradição tricotômica são sugeridas algumas mudanças em relação à prática e à pesquisa sobre formação de professores tais como: a formação do professor de matemática deve orientar-se pelas diferentes práticas sociais do educador matemático; adotar, na formação inicial, práticas e projetos nos quais os licenciandos possam integrar, fazendo contrastes, problematizações e investigações sobre as relações entre sua formação matemática na licenciatura, sua formação didático-pedagógica relacionada ao conteúdo, e a complexidade das práticas escolares.

Abstract: This study aims to find out how the signs of the Qur’an against arithmetic operations in mathematics education, determine the urgency of arithmetic operations in the Qur’an against mathematics education, and see the relevance of arithmetic operations in the Qur’an to modern mathematics education. The method used in this study is qualitative in library research design, because all that wants to be investigated is sourced from literature and in this study using maudhu’s interpretation. The results of the study of arithmetic operations in the Qur’anic cues to Mathematics Education are: first, the Qur’an implies arithmetic operations in mathematics education implicitly wherein the sum by stating the results and in subtraction, multiplication and division only shows the part of the arithmetic operations, the existence of arithmetic operations in the Qur’an as a means of human thinking in order to think mathematically. Second, the urgency of arithmetic operations in the Qur’an for mathematics education is stated that arithmetic operations in the Qur’an are very important for mathematics education as a means of thinking, cues for the need for innovation in learning, justification of mathematical principles, erasing thought of dichotomy in science, and as a cue for learning mathematics. Third, the relevance of arithmetic operations in the Qur’an to modern mathematics education is in line with the spirit of Islamic integration in science and learning carried out by Islamic education in this modern era. Qur’anic arithmetic operations can be material in integrated mathematics learning. The existence of arithmetic operations in the Qur’an for mathematics education is also very related in learning the 2013 curriculum which emphasizes the scientific approach.

Mathematics is an important skill that is taught to all children in the UK in a structured manner from a very early age. The purpose of this thesis was to examine how working memory (Baddeley & Hitch, 1974a; Baddeley & Hitch, 1994) and UK curricular mathematics are related, if specific components of working memory were more impactful upon performance in mathematics than others, and if we can predict mathematics outcomes using working memory measures. With reference to the influence of working memory on overall curricular mathematics performance, a cohort of 70 children from two primary schools in the North West of England was tested annually from their Reception year (mean age 5yrs 1m) at school to Year Two (mean age 6yrs 11m ). The study used a number of working memory tasks, a UK curricular mathematics test, and two Performance Measures. This allowed data to be analysed both in a cross-sectional manner and longitudinally (Chapter 5).The thesis also differentiates UK curricular mathematics into four separable “strands”, Number, Calculation, Measures, Shape and Space, and Problem Solving. These strands are described consistently throughout the UK mathematics curricular literature (DfEE, 1999; DfEE & QCA, 1999a; DfES, 2003a) and the cohort data was used to statistically analyse the relationships between working memory and each strand in turn using a correlational design in Chapters 6 to 9.Results indicated that working memory is a robust predictor of overall mathematics performance (Chapter 5), and of the Calculation Strand (Chapter 7). This finding was demonstrated in both the cross-sectional analyses and also in the longitudinal regression analyses. Of the working memory measures a distinct pattern of association was revealed. In particular the data imply that there is a strong role for the central executive at each age range, but in Year One verbal short-term memory emerges as an important predictor variable.Working memory also showed significant predictive influence over the remaining three curricular mathematics strands that were measured, particularly at the youngest age grouping, but working memory was not found to be a robust longitudinal predictor of Number, Problem Solving or Measures, Shape and Space.The overarching conclusion is that working memory, and in particular the central executive, may support the development of early curricular mathematical skills independent of the influence of age and Performance Measures. The practical and theoretical implications are considered.

This study aims to describe the representation of students' mathematical concepts in solving algebraic problems based on mathematical ability. This study is exploratory descriptive qualitative with four junior high school students as subjects. Data were collected using a task-based interview and analyzed using qualitative data analysis techniques through data reduction, data presentation, and drawing conclusions. The mathematical ability is categorized into high and low. The results show that differences in mathematical ability has no real effect on the representation of mathematical concepts of students in algebraic problem solving. Proved by all subjects (low and high mathematical abilities) are able to solve algebraic problems using verbal, symbolic, imagistic, and notational formal representations. However, there is a difference on the model of mathematical representation used. Students with high ability prefer to use conceptual representations using conventional standard notations, while those with low mathematical ability tend to use pictorial representations using non-standard illustrations.

Currently science and technology are changing so fast and college mathematics becomes more and more important. But the downturn of freshmen's scholastic performance has been intensified and this phenomenon leads to serious problems in managing college curriculums. During the recent years at a middle level engineering college, many freshmen had a lot of difficulties in their mathematics courses. In consequence, many of them had hard time to survive at their major curriculums. In this point of view, we analyse the situation of mathematical scholastic ability among engineering majored freshmen through the research on the actual state of mathematical background, mathematical scholastic ability test, college mathematics scores, and scholastic aptitude test scores. We study the relation between the mathematics score of scholastic aptitude test for the college entrance and mathematical scholastic ability of freshmen of a middle level engineering college. From this study we conclude that the essential reasons for the above situations are the curriculum of middle school mathematics and the system of scholastic aptitude test and the entrance examination of university. In order for improving mathematical accomplishment. we give suggestions such as a learning ability improvement program in mathematics.

There is an increasing use of and rapid advancements in technologies in education. In particular, there is a take-up of advanced calculators (graphing calculators and calculators with computer algebra system [CAS]) in senior secondary high-stakes examinations. Each year, tens of thousands of students in Australia, Singapore, and other regions, use advanced calculators in mathematics examinations that directly affect their entry into tertiary education. Thus, it was pertinent to investigate the ways in which students use these technologies when learning mathematics, and the factors that influenced them. This study was built upon a bricolage of theories from various fields such as mathematics education, educational technology, learning theories, learning styles, and gender studies. From a review of the literature in the different fields, it was hypothesised that factors such as students’ gender, beliefs about and attitudes toward mathematics and technology, approaches to studying mathematics, and learning preferences, influence their ways of using technology. A mixed methods research design was used, consisting of two parts: a quantitative large scale online survey of 964 Singaporean and 176 Victorian senior secondary mathematics students, and a qualitative small scale study of nine students in a Singaporean school. There were several main results found in this study: (i) How students interact with advanced calculators was influenced by their beliefs about and attitudes toward mathematics and their approaches to studying mathematics. Students’ ways of knowing and learning mathematics best explained their ways of interacting with the calculator. (ii) Students' learning preferences were situated, and the use of advanced calculators was associated with visual and kinesthetic modes and a preference to work cooperatively. (iii) There were regional differences (in favour of Victorian students) in how students use the advanced calculators and their attitudes toward calculators. (iv) There were gender differences (in favour of males) in students' attitudes toward mathematics and advanced calculators, and how they used the calculators. Although there were more student factors with gender differences in the Singaporean than the Victorian data, the effect sizes were larger in the Victorian sample. The findings have implications for mathematics teaching practice, in that deep understanding, intrinsic interest, beliefs about the creative, inter-connectedness and contextual aspects of mathematics are associated with using calculators as a partner and collaborator for learning. Additionally, students tend to employ visual and kinaesthetic modes when learning how to use the calculators. Students were also found to have a preference for working on the calculator cooperatively with friends. Therefore, teachers can employ methods such as encouraging students to try out the calculator keys and to work cooperatively in groups when learning how to use the calculators. Another significant contribution of this study was in the use of Facebook to recruit participants. More research can be done on the participation and response rates associated with this recruitment method. Further research can also be conducted to translate the implications of the study into practical classroom strategies and to investigate longitudinally the dynamic interplay between these factors and students’ interactions with the advanced calculators.

This paper focuses on the sub-field of study the philosophy of mathematics education from one perspective. The field is characterised in both narrow and broad terms, and from both bottom-up (questions and practices) and top-down perspectives (in terms of philosophy and its branches). From the bottom-up one can characterize the area in terms of questions, and I have asked: What are the aims and purposes of teaching and learning mathematics? What is mathematics? How does mathematics relate to society? What is learning mathematics? What is mathematics teaching? What is the status of mathematics education as knowledge field? I have characterized the sub-field using a ‘top down’ perspective using the branches of philosophy. Looking briefly into the contributions of ontology and metaphysics, aesthetics, epistemology and learning theory, social philosophy, ethics, and the research methodology of mathematics education reveals both how rich and deep the contributions of philosophy are to the theoretical foundations of our field of study. But even these two approaches leave many questions unanswered. For example: what are the responsibilities of mathematics and what is the responsibility of our own subfield, the philosophy of mathematics education? I conclude that the role of the philosophy of mathematics education is to analyse, question, challenge, and critique the claims of mathematics education practice, policy and research.

Mathematics has been viewed as the most difficult subject in the secondary school particularly by children in developing countries (Skemp 2008). And yet in spite of it being perceived as a difficult subject, mathematics is a fundamental part of human thought and logic and integral at attempts to understanding the world and ourselves. Evidence shows that students in many countries fail mathematics (Lynons 2010). Literature has a plethora of reasons for high failure rate of mathematics. For example, Polya (2011) states that shortage of well-trained teachers, inadequacy of teaching facilities, poor quality textbooks, large classes, poorly motivated teachers, poorly coordinated supervisory activities, incessant transfers of teachers and heads, automatic promotion of pupils, the negative role of public examinations, and inequality of educational opportunities, all hamper the smooth acquisition of mathematics knowledge. As a result of this situation, there has been a sudden deluge of supplementary tuition or extra-tuition. The practice of offering extra tuition has both social and economic implications on those involved. Parents pay the regular school fees and are then asked or volunteer to part with more money in the form of tuition for extra-lessons. This study sought to investigate teachers’ perceptions on the role played by supplementary-tuition or extra-lessons on the performance of pupils at ordinary level in secondary schools in Hwange District in Zimbabwe. The population comprised of 450 secondary school teachers in the district. Random sampling was used to arrive at a sample of 100 respondents who comprised of 58 females and 42 males. The study adopted the descriptive survey design and the questionnaire was the major instrument for collecting data. The main findings indicated that teachers felt that supplementary tuition in mathematics improved the results of those pupils who received it. The study also revealed that teachers felt that time allocated for mathematics was inadequate. The recommendations are that time allocated to mathematics should be increased right from the primary school to the secondary school level. Another recommendation was that properly qualified teachers to teach mathematics should be deployed in all the schools to avoid supplementary-tuition. Keywords: Supplementary tuition, Ordinary level, Student, Mathematics, Teachers, Tuition, Extra-tuition

According to published statistics regarding pharmaceutical industries, there has been a significant increase in the demand for cosmetics and skincare products. As a result, there has also been an increase in interest by researchers to conduct experimental and theoretical studies on these industries. The outermost layer of the skin, which is the stratum corneum, plays a key role in determining the skin permeability. This layer represents the first and main barrier against dehydration and screening from external elements and pathogens. Research efforts have been made to better understand skin permeability. Experimental methodologies play a key role in pharmaceutical and dermatology studies. However, ethical standards of using animal and human models in in-vitro permeability studies are very strict in their admission process and approval time. Furthermore, the lack of available material and financial resources for setting up, operating and repeating experiments has also limited the required progress. These problems can be addressed by developing better mathematical models on skin permeability. Progress, regarding the mathematical modelling of solute transport in skin, has been achieved in the last decades. The mathematical modelling strategy is used as a skin permeability predictor tool based on common physicochemical properties of the skin and penetrated solute molecules. Many factors play an important role in the formulation of a mathematical model and in determining how reliable the predicted results are. Therefore, the more skin physiological details are included in the model, the more computational efforts are required to interpret the solute permeation mechanisms in the dermal region and simulate a drug transport in the transdermal administration. The mathematical complexities associated with formulating a mathematical model of drug transport in skin significantly limit the utilities of such models in practice. Thus, a balance between the model’s complexity and how realistic it is in describing the solute transport processes in skin has to be achieved. Therefore, a mathematical model formulated on principles familiar to the pharmaceutical scientific community could be of value. In this thesis, we have developed a novel mathematical model using a compartmental approach. The proposed model enables for simulating solute transport across the stratum corneum (SC) in a variety of exposure scenarios in percutaneous drug delivery systems. The obtained results were compared with the diffusion model outcomes – which can be considered the current ‘golden standard’ for modelling skin transport – and showed a good agreement. The mathematical foundations of the model are relatively simple and better aligned with the physiology of the stratum corneum. In addition, this work shows that performing a numerical simulation using the compartmental model is less demanding than using the diffusing model, especially for complex exposure scenarios, and can be performed using free software, such as Python. The research findings of this thesis contribute towards a better understanding of transport phenomena across the skin. Also, the compartmental model proposed in this work can be used to describe transport phenomena in the membrane. Due to the similarity between the governing equations of diffusion and heat transfer, the proposed approach also offers a mathematical framework for studying heat transfer problems in biomaterials with minimum model development or computational efforts.

<p>As a powerful technique for holding relations in things, combinatorics has experienced rapidly development in the past century, particularly,enumeration of configurations, combinatorial design and graph theory. How-<br /> ever, the main objective for mathematics is to bring about a quantitative analysis for other sciences, which implies a natural question on combinatorics. Thus, how combinatorics can contributes to other mathematical sciences, not just in discrete mathematics, but metric mathematics and physics? After a long time speculation, I brought the CC conjecture for advancing mathematics by combinatorics, i.e., any mathematical science can be reconstructed from or made by combinatorialization in my postdoctoral report for Chinese Academy of Sciences in 2005, and then reported it at a few conferences of China. Clearly,<br /> CC conjecture is in fact a combinatorial notion and holds by a philosophical law, i.e., all things are inherently related, not isolated. The main purpose of this report is to survey the roles of CC conjecture in developing mathematical sciences with notions, such as those of its contribution to algebra, topology,Euclidean geometry and differential geometry, non-solvable differential equations or classical mathematical systems with contradictions to mathematics,quantum fields after it appeared 10 years ago. All of these show the importance of combinatorics to mathematical sciences in the past and future.</p>

Religion and mathematics are perceived as two things that are very reciprocally exclusive; this is due to the opinion asserting that mathematics is considered as valid and objective knowledge. On the other hand, religion is something interpretive and subjective. Historically speaking, religion and mathematics have a strong association and in certain aspects, both of them have points of contact that can permeate each other. The forms of intersection between mathematics and religion need to be analyzed so that we will be able to see more clearly the current religious phenomenon. This study investigates and discovers a potential encounter between religion and mathematics. To see the relationship between religion and mathematics, and even their fusion, the first thing to do is determine the definition of religion and mathematics. With a clear understanding of both, the encounter points will be easier to recognize. The research method used in this study is a literature review. This study collects systematically a variety of literature related to the theme under study. The encounter between religion and mathematics undergoes at five dimensions, they are ontological, epistemological, teleological, theoretical and application dimension. This study more focused on similarities to reveal the connection between religion and mathematics.