Development of fourth-grade students’ understanding of experimental and theoretical probability

Students explored variation and expectation in a probability activity at the end of the first year of a 3-year longitudinal study across grades 4-6. The activity involved experiments in tossing coins both manually and with simulation using the graphing software, TinkerPlots. Initial responses indicated that the students were aware of uncertainty, although an understanding of chance concepts appeared limited. Predicting outcomes of 10 tosses reflected an intuitive notion of equiprobability, with little awareness of variation. Understanding the relationship between experimental and theoretical probability did not emerge until multiple outcomes and representations were generated with the software.

Bhutanese eighth grade students' and teachers' perceptions of their classroom learning environment in relation to the new mathematics curriculum

This thesis examined how Bhutanese eighth grade students and teachers perceived their classroom learning environment in relation to a new standards-based mathematics curriculum. Data were gathered from administering surveys to a sample of 608 students and 98 teachers, followed by semi-structured interviews with selected participants. The findings of the study indicated that participants generally perceived their learning environments favorably. However, there were differences in terms of gender, school level, and school location. The study provides teachers, educational leaders, and policy-makers in Bhutan new insights into students' and teachers' perceptions of their mathematics classroom environments.

Evolutionary games with two timescales

We consider a two timescale model of learning by economic agents wherein active or 'ontogenetic' learning by individuals takes place on a fast scale and passive or 'phylogenetic' learning by society as a whole on a slow scale, each affecting the evolution of the other. The former is modelled by the Monte Carlo dynamics of physics, while the latter is modelled by the replicator dynamics of evolutionary biology. Various qualitative aspects of the dynamics are studied in some simple cases, both analytically and numerically, and its role as a useful modelling device is emphasized.

Playing Games on Sets and Models

The most prominent objective of the thesis is the development of the generalized descriptive set theory, as we call it. There, we study the space of all functions from a fixed uncountable cardinal to itself, or to a finite set of size two. These correspond to generalized notions of the universal Baire space (functions from natural numbers to themselves with the product topology) and the Cantor space (functions from natural numbers to the {0,1}-set) respectively. We generalize the notion of Borel sets in three different ways and study the corresponding Borel structures with the aims of generalizing classical theorems of descriptive set theory or providing counter examples. In particular we are interested in equivalence relations on these spaces and their Borel reducibility to each other. The last chapter shows, using game-theoretic techniques, that the order of Borel equivalence relations under Borel reduciblity has very high complexity. The techniques in the above described set theoretical side of the thesis include forcing, general topological notions such as meager sets and combinatorial games of infinite length. By coding uncountable models to functions, we are able to apply the understanding of the generalized descriptive set theory to the model theory of uncountable models. The links between the theorems of model theory (including Shelah's classification theory) and the theorems in pure set theory are provided using game theoretic techniques from Ehrenfeucht-Fraïssé games in model theory to cub-games in set theory. The bottom line of the research declairs that the descriptive (set theoretic) complexity of an isomorphism relation of a first-order definable model class goes in synch with the stability theoretical complexity of the corresponding first-order theory. The first chapter of the thesis has slightly different focus and is purely concerned with a certain modification of the well known Ehrenfeucht-Fraïssé games. There we (me and my supervisor Tapani Hyttinen) answer some natural questions about that game mainly concerning determinacy and its relation to the standard EF-game

Basic components in the scienctific didactical training of the secondary school mathematics teachers

Secondary mathematics teacher training in Spain is currently the subject of a heated revision debate. The speed of social, cultural, scientific and economic changes have left a hundred years old teacher training model well behind. However, academical inertia and professional interests are impeding a real new training of the mathematics teacher as an autonomous mathematical educator. Teachers of Didactic of Mathematics and the Spanish Associations of mathematics teachers have recently been discussing the issue. Their conclusions are included here.

[Book review] Chance encounters: a first course in data analysis and inference

Book review of: Chance Encounters: A First Course in Data Analysis and Inference by Christopher J. Wild and George A.F. Seber 2000, John Wiley & Sons Inc. Hard-bound, xviii + 612 pp ISBN 0-471-32936-3

Adaptation of a questionnaire to assess the epistemological beliefs of mathematics in secondary school teachers

The objective of the study is to determine the psychometric properties of the Epistemological Beliefs Questionnaire on Mathematics. 171 Secondary School Mathematics Teachers of the Central Region of Cuba participated. The results show acceptable internal consistency. The factorial structure of the scale revealed three major factors, consistent with the Model of the Three Constructs: beliefs about knowledge, about learning and teaching. Irregular levels in the development of the epistemological belief system about mathematics of these teachers were shown, with a tendency among naivety and sophistication poles. In conclusion, the questionnaire is useful for evaluating teacher’s beliefs about mathematics.

On the applicability of mathematics : philosophical and historical perspectives

The present thesis is a contribution to the debate on the applicability of mathematics; it examines the interplay between mathematics and the world, using historical case studies. The first part of the thesis consists of four small case studies. In chapter 1, I criticize "ante rem structuralism", proposed by Stewart Shapiro, by showing that his so-called "finite cardinal structures" are in conflict with mathematical practice. In chapter 2, I discuss Leonhard Euler's solution to the Königsberg bridges problem. I propose interpreting Euler's solution both as an explanation within mathematics and as a scientific explanation. I put the insights from the historical case to work against recent philosophical accounts of the Königsberg case. In chapter 3, I analyze the predator-prey model, proposed by Lotka and Volterra. I extract some interesting philosophical lessons from Volterra's original account of the model, such as: Volterra's remarks on mathematical methodology; the relation between mathematics and idealization in the construction of the model; some relevant details in the derivation of the Third Law, and; notions of intervention that are motivated by one of Volterra's main mathematical tools, phase spaces. In chapter 4, I discuss scientific and mathematical attempts to explain the structure of the bee's honeycomb. In the first part, I discuss a candidate explanation, based on the mathematical Honeycomb Conjecture, presented in Lyon and Colyvan (2008). I argue that this explanation is not scientifically adequate. In the second part, I discuss other mathematical, physical and biological studies that could contribute to an explanation of the bee's honeycomb. The upshot is that most of the relevant mathematics is not yet sufficiently understood, and there is also an ongoing debate as to the biological details of the construction of the bee's honeycomb. The second part of the thesis is a bigger case study from physics: the genesis of GR. Chapter 5 is a short introduction to the history, physics and mathematics that is relevant to the genesis of general relativity (GR). Chapter 6 discusses the historical question as to what Marcel Grossmann contributed to the genesis of GR. I will examine the so-called "Entwurf" paper, an important joint publication by Einstein and Grossmann, containing the first tensorial formulation of GR. By comparing Grossmann's part with the mathematical theories he used, we can gain a better understanding of what is involved in the first steps of assimilating a mathematical theory to a physical question. In chapter 7, I introduce, and discuss, a recent account of the applicability of mathematics to the world, the Inferential Conception (IC), proposed by Bueno and Colyvan (2011). I give a short exposition of the IC, offer some critical remarks on the account, discuss potential philosophical objections, and I propose some extensions of the IC. In chapter 8, I put the Inferential Conception (IC) to work in the historical case study: the genesis of GR. I analyze three historical episodes, using the conceptual apparatus provided by the IC. In episode one, I investigate how the starting point of the application process, the "assumed structure", is chosen. Then I analyze two small application cycles that led to revisions of the initial assumed structure. In episode two, I examine how the application of "new" mathematics - the application of the Absolute Differential Calculus (ADC) to gravitational theory - meshes with the IC. In episode three, I take a closer look at two of Einstein's failed attempts to find a suitable differential operator for the field equations, and apply the conceptual tools provided by the IC so as to better understand why he erroneously rejected both the Ricci tensor and the November tensor in the Zurich Notebook.

Adolescent girls and mathematics: classroom performance, attitudes toward mathematics and perceptions of parental attitudes /

The purpose of this study was to determine the extent to which gender differences exist in student attitudes toward mathematics and in their performance in mathematics at the Grade Seven and Eight level. The study also questioned how parents influence the attitudes of this grade level of male and female students toward mathematics. Historically, the literature has demonstrated gender differences in the attitudes of students toward mathematics, and in parental support for classroom performance in mathematics. This study was an attempt to examine these differences at one senior public school in the Peel Board of Education. One hundred three Grade Seven and Eight students at a middle school in the Peel Board of Education volunteered to take part in a survey that examined their attitudes toward mathematics, their perceptions of their parents' attitudes toward mathematics and support for good performance in the mathematics classroom, parental expectations for education and future career choices. Gender differences related to performance levels in the mathematics classroom were examined using Pearson contingency analyses. Items from the survey that showed significant differences involved confidence in mathematics and confidence in writing mathematics tests, as well as a belief in the ability to work on mathematics problems. Male students in both the high and low performance groups demonstrated higher levels of confidence than the females in those groups. Female students, however, indicated interest in careers that would require training and knowledge of higher mathematics. Some of the reasons given to explain the gender differences in confidence levels included socialization pressures on females, peer acceptance, and attribution of success. Perceived parental support showed no significant differences across gender groups or performance levels. Possible explanations dealt with the family structure of the participants in the study. Studies that, in the past, have demonstrated gender differences in confidence levels were supported by this study, and discussed in detail. Studies that reported on differences in parental support for student performance, based on the gender of the parent, were not confirmed by this study, and reasons for this were also discussed. The implications for the classroom include: 1) build on the female students' strengths that will allow them to enjoy their experiences in mathematics; 2) stop using the boys as a comparison group; and 3) make students more aware of the need to continue studying mathematics to ensure a wider choice of future careers.

Structuralism and the Applicability of Mathematics

In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.

The Use of Vocabulary in an Eighth Grade Mathematics Classroom: Improving Usage of Mathematics Vocabulary in Oral and Written Communication

In this action research study of my mathematics classroom of eighth grade students, I investigated the use of mathematics vocabulary by focusing on improving the usage of this vocabulary in both oral and written communication. I discovered oral communication tended to show more improvements compared to written communication done by the same group of students. As a result of this research, I plan to continue to focus my teaching on the use of mathematics vocabulary in an effort to help my students gain a greater understanding of the daily use of that vocabulary.

Analysis of sport psychology consultancy at three Olympic Games: Facts and figures

Objectives Despite many reports on best practises regarding onsite psychological services, little research has attempted to systematically explore the frequency, issues, nature and client groups of onsite sport psychology consultancy at the Olympic Games. The present paper will fill this gap through a systematic analysis of the sport psychology consultancy of the Swiss team for the Olympic Games of 2006 in Turin, 2008 in Beijing and 2010 in Vancouver. Design Descriptive research design. Methods The day reports of the official sport psychologist were analysed. Intervention issues were labelled using categories derived from previous research and divided into the following four intervention-issue dimensions: “general performance”, “specific Olympic performance”, “organisational” and “personal” issues. Data were analysed using descriptive statistics, chi square statistics and odds ratios. Results Across the Olympic Games, between 11% and 25% of the Swiss delegation used the sport psychology services. On average, the sport psychologist provided between 2.1 and 4.6 interventions per day. Around 50% of the interventions were informal interventions. Around 30% of the clients were coaches. The most commonly addressed issues were performance related. An association was observed between previous collaboration, intervention likelihood and intervention theme. Conclusions Sport psychologists working at the Olympic Games are fully engaged with daily interventions and should have developed ideally long-term relationships with clients to truly help athletes with general performance issues. Critical incidents, working with coaches, brief contact interventions and team conflicts are specific features of the onsite consultancy. Practitioners should be trained to deal with these sorts of challenges.

A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0

We partially solve a long-standing problem in the proof theory of explicit mathematics or the proof theory in general. Namely, we give a lower bound of Feferman’s system T0 of explicit mathematics (but only when formulated on classical logic) with a concrete interpretat ion of the subsystem Σ12-AC+ (BI) of second order arithmetic inside T0. Whereas a lower bound proof in the sense of proof-theoretic reducibility or of ordinalanalysis was already given in 80s, the lower bound in the sense of interpretability we give here is new. We apply the new interpretation method developed by the author and Zumbrunnen (2015), which can be seen as the third kind of model construction method for classical theories, after Cohen’s forcing and Krivine’s classical realizability. It gives us an interpretation between classical theories, by composing interpretations between intuitionistic theories.

Design and implementation of a mathematical videogame

This paper presents videogames as a very useful tool in high studies with respect to mathematical matters. It describes the implementation of a videogame developed by its authors which makes it possible for students to reinforce mathematical concepts in a motivating environment. With this work we intend to contribute to the process of engaging a bigger number of university teaching professionals and researchers in the use of serious games and the study of their theoretical frameworks, design, development and application of scientific education. With this idea the authors of the present paper have created and developed the videogame “The Math Castle” which consists in a series of tests through which various aspects of Mathematics are dealt with, especially in the areas of Discrete Mathematics, which due to its nature can be particularly well adapted to this kind of activity, Analysis or Geometry. In this paper there lies a complete description of the game developed and the results obtained with it.