Thesis (Ph. D.)--Illinois State University, 1996. / Title from title page screen, viewed May 26, 2006. Dissertation Committee: Jane O. Swafford (chair), John A. Dossey, Roger P. Day, Michael Marsalli, Jeffrey J. Walczyk. Includes bibliographical references (leaves 131-138) and abstract. Also available in print.

Thesis (M.Ed. (Education)) - Rhodes University, 2009. / A thesis submitted in partial fulfilment of the requirements for the degree of Master of Education.

This study traces the professional learning of student teachers who have lived and studied outside the UK, and successfully applied to follow a Post Graduate Certificate in Education (PGCE) course in London to become teachers of mathematics in English schools. It draws upon Bourdieu’s theory of habitus and field to discuss how these student teachers adapt their capitals, as described in migration studies by Erel (2010) and Nowicka (2015) and how, during initial teacher training (ITT), they develop professional capitals for the teaching of mathematics (Nolan, 2012). Recent migration flows have led to a growth of diversity, as measured by countries of origin, in London and other cities around the world, resulting in what Vertovec (2006) has called superdiversity. Through a series of semi-structured interviews with 16 PGCE student teachers hailing from 13 different countries, this study explores the implications of superdiversity for the practices of training teachers. The focus of the research is on the complications of ‘bring[ing] off’ (MacLure, 2003:55) the embodied performance of becoming a teacher, and on how student teachers develop ‘enough’ (Blommaert and Varis, 2011:5) professional capital to pass the course. This leads to a reassessment of the category ‘highly skilled migrant’, which is used to define those who have academic qualifications for teaching from outside the UK. The study uses instead the term ‘highly qualified migrant’, to argue that a mathematical degree needs to be complemented by knowledge of the national mathematics curriculum, national pedagogies and local communicative resources. It shows how London can become an ‘escalator region’ (Fielding, 1992:1), as the student teachers achieve a working life that matches their academic qualifications, and also their own aspirations and those of their families, in the UK and elsewhere. In so doing, they become part of a teaching workforce that reflects the growing superdiversity of the region’s school pupils.

When teachers plan to teach mathematics, they draw on many examples to either demonstrate a concept or provide opportunities for learners to practise skills and procedures. The examples used by primary trainee teachers, it is suggested, are often chosen without suitable consideration of learners' strengths, weaknesses or misconceptions. Whilst there has been research on the choice of examples by teachers in secondary mathematics, detailed empirical research of primary mathematics or for trainee teachers is relatively scarce. In this study, two cohorts of final year trainee primary teachers were invited to submit lesson plans for analysis and a sample group was interviewed to try to identify the theoretical frameworks trainees use for planning mathematics and their approaches to choosing examples for learning. The data collected was then analysed using a multiple case study approach against a conceptual framework based on the Knowledge Quartet research of Rowland et al. (2009) and the development of the notion of example spaces by Watson and Mason (2005). The analysis sought to identify commonalities in the way the group of trainees approached planning mathematics and draw insights on their rationales for choosing mathematical examples. Each trainee's planning was scrutinized against the theoretical background in the literature and conclusions were drawn regarding the methods of planning adopted, the examples chosen and the possible links between these actions and the trainees' levels of mathematical subject knowledge. Evidence from the study appears to show that trainees do not make use of theoretical frameworks when planning mathematics lessons, examples are chosen from existing sources such as textbooks and websites, and any modifications are made with differentiation as a key factor rather than mathematics pedagogy, with trainees' subject knowledge playing a minimal role in the planning process.

The purpose of my research is to examine and explore the ways that undergraduate students understand the concept of recursion. In order to do this, I have designed computer-based software, which provides students with a virtual and interactive environment where they can explore the concept of recursion, and demonstrate and develop their knowledge of recursion through active engagement. I have designed this computer-based software environment with the aim of investigating how students think about recursion. My approach is to design digital tools to facilitate students' understanding of recursion and to expose that thinking. My research investigates students' understanding of the hidden layers and inherent complexity of recursion, including how they apply it within relevant contexts. The software design embedded the idea of functional abstraction around two basic principles of: 'functioning' and 'functionality'. The functionality principle focuses on what recursion achieve, and the functioning dimension concerns how recursion is operationalised. I wanted to answer the following crucial question: How does the recursive thinking of university students evolve through using carefully designed digital tools? In the process of exploring this main question, other questions emerged: 1. Do students understand the difference between recursion and iteration? 2. How is tail and embedded recursion understood by the students? 3. To what extent does prior knowledge of the concept of iteration influence students' understanding of tail and embedded recursion? 4. Why is it important to have a clear understanding of the control passing mechanisms in order to understand recursion? 5. What is the role of functional abstraction in both, the design of computer-based tools and the students' understanding of recursion? 6. How are students' mental models of recursion shaped by their engagement with computer-based tools? From a functional abstraction point of view almost all previous research into the concept of recursion has focused on the functionality dimension. Typically, it has focused on procedures for the calculation of the factorial of a natural number, and students were tested to see if they are able to work out the values of the a function recursively (Wiedenbeck, 1988; Anazi and Uesato, 1982) or if they are able to recognize a recursive structure (Sooriamurthi, 2001; Kurland and Pea, 1985). Also, I invented the Animative Visualisation in the Domain of Abstraction (AVDA) which combines the functioning and functionality principles regarding the concept of recursion. In the AVDA environment, students are given the opportunity to explore the hidden layers and the complicated behaviour of the control passing mechanisms of the concept of recursion. In addition, most of the textbooks in mathematics and computer sciences usually fail to explain how to use recursion to solve a problem. Although it is also true that text books do not typically explain how to use iteration to solve problems, students are able to draw on to facilitate solving iterative problems (Pirolli et al, 1988). My approach is inspired by how recursion can be found in everyday life and in real world phenomena, such as fractal-shaped objects like trees and spirals. This research strictly adheres to a Design Based Research methodology (DBR), which is founded on the principle of the cycle of designing, testing (observing the students' experiments with the design), analysing, and modifying (Barab and Squire, 2004; Cobb and diSessa, 2003). My study was implemented throughout three iterations. The results showed that in the AVDA (Animative Visualisation in the Domain of Abstraction) environment students' thinking about the concept of recursion changed significantly. In the AVDA environment they were able to see and experience the complicated control passing mechanism of the tail and embedded recursion, referred to a delegatory control passing. This complicated control passing mechanism is a kind of generalization of flow in the iterative procedures, which is discussed later in the thesis. My results show that, to model a spiral, students prefer to use iterative techniques, rather than tail recursion. The AVDA environment helped students to appreciate the delegatory control passing for tail recursive procedures. However, they still demonstrated difficulties in understanding embedded recursive procedures in modelling binary and ternary trees, particularly regarding the transition of flow between recursive calls. Based on the results of my research, I have devised a model of the evolution of students' mental model of recursion which I have called – the quasi-pyramid model. This model was derived from applying functional abstraction including both functionality and functioning principles. Pedagogic implications are discussed. For example, the teaching of recursion might adopt 'animative' visualization, which is of vitally important for students' understanding of latent layers of recursion.

This thesis is concerned with how a group of student teachers make sense of trigonometry. There are three main ideas in this study. This first idea is about the theoretical framework which focusses on the growth of mathematical thinking based on human perception, operation and reason. This framework evolves from the work of Piaget, Bruner, Skemp, Dienes, Van Hiele and others. Although the study focusses on trigonometry, the theory constructed is applicable to a wide range of mathematics topics. The second idea is about three distinct contexts of trigonometry namely triangle trigonometry, circle trigonometry and analytic trigonometry. Triangle trigonometry is based on right angled triangles with positive sides and angles bigger than 0 [degrees] and less than 90 [degrees]. Circle trigonometry involves dynamic angles of any size and sign with trigonometric ratios involving signed numbers and the properties of trigonometric functions represented as graphs. Analytic trigonometry involves trigonometric functions expressed as power series and the use of complex numbers to relate exponential and trigonometric functions. The third idea is about supportive and problematic conceptions in making sense of mathematics. This idea evolves from the idea of met‐before as proposed in Tall (2004). In this case, the concept of ‘met‐before’ is given a working definition as ‘a trace that it leaves in the mind that affects our current thinking’. Supportive conception supports generalization in a new contexts whereas problematic conception impedes generalization. Furthermore, a supportive conception might contain problematic aspects in it and a problematic conception might contain supportive aspects in it. In general, supportive conceptions will give the learner a sense of confidence whereas problematic conceptions will give the learner of sense of anxiety. Supportive conceptions may occur in different ways. Some learners might know how to perform an algorithm without a grasp of how it can be related to different mathematical concepts and the underlying reasons for using such an algorithm.

This research is devoted to focussing on the influence of different learning style on the performance of undergraduate students in various parts of calculus. In carrying out the study, calculus materials were classified into four main categories (Z4,Z5,Z6,Cals) and, for the Iranian students, the results of their mathematical performance in the university entrance examination is labelled (En) to identify their grounding in high school mathematics at the beginning of the calculus course in higher education. Also, in the present study, students' performance (weakness) in the manipulation of mathematical notation and logical discussion is called (Z1) category and (Cal) indicates students' total achievement in calculus examination which is, in fact, the students' performance on the combination of the categories (Z4,Z5,Z6). These calculus categories are described in Chapter 5. However in short term, multi-conceptual and procedural tasks are classified as (Z4). The (Z5) category is defined as the translation processes between mathematical abstraction (analytic/symbolic) and (pictorial/visual) forms of calculus materials. Moreover, multi-skilled, transferable and procedural skills are labelled as (Z6) category. It should be noted that these categories are interrelated in a scheme to exhibit activities in calculus. 572 students participated in the experimental part of this study and were selected from two Iranian universities (Sabzvar University and Mashhad University) and Glasgow University in Scotland, U.K. During the period of the study, the samples of students were subjected to some psychological tests in order to assign their Field-dependent/Field-independent and Convergent/Divergent learning styles. It was found throughout the study that the most effective combination of learning styles which emerged from the interacting picture of all the psychological factors used in the research, were field-independent/convergent (F1+Con) in Iran, and field-independent/divergent (FI+Div) in Scotland in performing on the calculus. On the other hand, the combination of field-dependent and convergent styles (FD+Con) could lessen achievement in calculus by mathematics/physics students, and field-dependent and divergent styles (FD+Div) would lessen attainment in calculus by engineering students. In addition, when the mean scores in calculus categories were calculated for various groups of students with different learning styles, the convergent thinkers (Con) were found to be best in (Z6), while divergent thinkers (Div) exhibited higher performance in (Z5). These findings demonstrate that the Con/Div way of thinking is the most effective in influencing performance in different areas of calculus, the FI/FD factor takes the second position. All these findings have been combined to form a model which emerges at the end of this thesis. Moreover, in Chapters 3 and 4, a comparison is made between calculus in secondary (high school) and higher education in Iran and Scotland, focussing on content, teaching order, learning objectives and teaching methods.

We consider classification problems in a variant of the Probably Approximately Correct (PAC)-learning framework, in which an unsupervised learner creates a discriminant function over each class and observations are labeled by the learner returning the highest value associated with that observation. Consideration is given to whether this approach gains significant advantage over traditional discriminant techniques. It is shown that PAC-learning distributions over class labels under Ll distance or KL-divergence implies PAC classification in this framework. We give bounds on the regret associated with the resulting classifier, taking into account the possibility of variable misclassification penalties. We demonstrate the advantage of estimating the a posteriori probability distributions over class labels in the setting of Optical Character Recognition. We show that unsupervised learners can be used to learn a class of probabilistic concepts (stochastic rules denoting the probability that an observation has a positive label in a 2-class setting). This demonstrates a situation where unsupervised learners can be used even when it is hard to learn distributions over class labels - in this case the discriminant functions do not estimate the class probability densities. We use a standard state-merging technique to PAC-learn a class of probabilistic automata and show that by learning the distribution over outputs under the weaker L1 distance rather than KL-divergence we are able to learn without knowledge of the expected length of an output. It is also shown that for a restricted class of these automata learning under L1 distance is equivalent to learning under KL-divergence.

Home-education, also known as home-schooling, is an educational choice made by families to facilitate learning at home rather than in school. Research by Rothermel (2002) and Rudner (1999) shows that, on average, home-educated children far outperform school-educated children on standard mathematics tests. But at present, no study has yet investigated the key reasons behind this phenomenon – indeed, no research has taken an in-depth look into the ways in which parents facilitate the learning of mathematics at home and the resultant effects on their children’s mathematical development. Therefore, in this study, we will consider the nature of mathematics education through the eyes of the home-educating parent and their children. Through questionnaires, this research examines the relationship between the educational and mathematical beliefs of home-educating parents. Parental views are compared with the children’s perceptions of the home learning environment, their mathematical beliefs and their mathematical understanding. Furthermore, the children’s mathematical understanding is addressed through consideration of their responses to a series of mathematical questions set within the context of Key Stages 1-3 of the National Curriculum. To obtain the research sample, home-educating families from across the United Kingdom were contacted via the Internet, and information was collected through both email and postal response. From the parental data, three categories of home-educator were highlighted: (1) Structured, (2) Semi-Formal and (3) Informal (as described by Lowe and Thomas, 2002). The children’s questionnaire responses were then analysed, using illustrative case studies to demonstrate how different home-educating approaches of their parents could result in different perceptions of mathematics and mathematical learning in the children. For example, children learning via a ‘structured’ approach were less likely to be able to measure their own level of mathematical ability than children from the other families; they also mentioned limited resources and less independence when learning mathematics. When examining the children’s assessed work, selective case studies, together with detailed analysis, revealed a strong link between the home-educating approach and the problem-solving strategies of the children. Children from structured families were often competent when solving more routine, ‘calculation-type’ problems, but less able to adapt their knowledge to problems that required a ‘deeper’ understanding of the concept. Children from families where the parent themselves had a mathematical background (e.g. mathematician or mathematics teacher) typically used formal mathematical reasoning in their work. On the other hand, children learning from ‘informal’ families (where emphasis was placed on ‘child-directed’ learning) seldom used ‘standard procedural’ type approaches to solve problems, but instead displayed a range of creative strategies. The findings suggested that a home-educating parent’s conception of mathematics not only influenced the way in which they attempt to teach mathematics but also their children’s mathematical beliefs and learning style. Furthermore, there was evidence to suggest that certain home-educating approaches encouraged a ‘type’ of mathematical understanding that could be applied in a range of situations, whereas other approaches, particularly where both the learning materials and interaction with others was restricted, resulted in a more limited level of mathematical understanding.

This thesis explores children’s conceptions of the equals sign from the vantage point of notating task design. The existing literature reports that young children tend to view the equals sign as meaning “write the result here”. Previous studies have demonstrated that teaching an “is the same as” meaning leads to more flexible thinking about mathematical notation. However, these studies are limited because they do not acknowledge or teach children that the equals sign also means “can be exchanged for”. The thesis explores the “sameness” and “exchanging” meanings for the equals sign by addressing four research questions. The first two questions establish the distinction, in terms of task design, between the two meanings. Does the “can be exchanged for” meaning for the equals sign promote attention to statement form? Are the “can be exchanged for” and “is the same as” meanings for the equals sign pedagogically distinct? The final two research questions seek to establish how children might coordinate the two meanings, and connect them with their existing implicit knowledge of arithmetic principles. Can children coordinate “can be exchanged for” and “is the same as” meanings for the equals sign? Can children connect their implicit arithmetical knowledge with explicit transformations of notation? The instrument used is a specially designed notational computer-microworld called Sum Puzzles. Qualitative data are generated from trials with pairs of Year 5 (9 and 10 years), and in one case Year 8 (12 and 13 years), pupils working collaboratively with the microworld toward specified task goals. It is discovered that the “sameness” meaning is useful for distinguishing equality statements by truthfulness, whereas the “exchanging” meaning is useful for distinguishing statements by form. Moreover, a duality of both meanings can help children connect their own mental calculation strategies with transformations of properly formed notation.