Mathematical anxiety is linked to reduced cognitive reflection: A potential road from discomfort in the mathematics classroom to susceptibility to biases

Background
When asked to solve mathematical problems, some people experience anxiety and threat, which can lead to impaired mathematical performance (Curr Dir Psychol Sci 11:181–185, 2002). The present studies investigated the link between mathematical anxiety and performance on the cognitive reflection test (CRT; J Econ Perspect 19:25–42, 2005). The CRT is a measure of a person’s ability to resist intuitive response tendencies, and it correlates strongly with important real-life outcomes, such as time preferences, risk-taking, and rational thinking.

Methods
In Experiments 1 and 2 the relationships between maths anxiety, mathematical knowledge/mathematical achievement, test anxiety and cognitive reflection were analysed using mediation analyses. Experiment 3 included a manipulation of working memory load. The effects of anxiety and working memory load were analysed using ANOVAs.

Results
Our experiments with university students (Experiments 1 and 3) and secondary school students (Experiment 2) demonstrated that mathematical anxiety was a significant predictor of cognitive reflection, even after controlling for the effects of general mathematical knowledge (in Experiment 1), school mathematical achievement (in Experiment 2) and test anxiety (in Experiments 1–3). Furthermore, Experiment 3 showed that mathematical anxiety and burdening working memory resources with a secondary task had similar effects on cognitive reflection.

Conclusions
Given earlier findings that showed a close link between cognitive reflection, unbiased decisions and rationality, our results suggest that mathematical anxiety might be negatively related to individuals’ ability to make advantageous choices and good decisions.

Mathematics and reading difficulty subtypes: Minor phonological influences on mathematics for 5-7 year olds

Linguistic influences in mathematics have previously been explored throughsubtyping methodology and by taking advantage of the componential nature ofmathematics and variations in language requirements that exist across tasks. Thepresent longitudinal investigation aimed to examine the language requirements of mathematical tasks in young children aged 5-7 years. Initially, 256 children were screened for mathematics and reading difficulties using standardised measures. Those scoring at or below the 35th percentile on either dimension were classified as having difficulty. From this screening, 115 children were allocated to each of the MD (n=26), MDRD (n=32), reading difficulty (RD, n=22) and typically achieving (TA, n=35) subtypes. These children were tested at four time points, separated by six monthly intervals, on a battery of seven mathematical tasks. Growth curve analysis indicated that, in contrast to previous research on older children, young children with MD and MDRD had very similar patterns of development on all mathematical tasks. Overall, the subtype comparisons suggested that language played only a minor mediating role in most tasks, and this was secondary in importance to non-verbal skills. Correlational evidence suggested that children from the different subtypescould have been using different mixes of verbal and non-verbal strategies to solve the mathematical problems.

A-level Mathematics Options – Views of Secondary-level Teachers

The A-level Mathematics qualification is based on a compulsory set of pure maths modules and a selection of applied maths modules with the pure maths representing two thirds of the assessment. The applied maths section includes mechanics, statistics and (sometimes) decision maths. A combination of mechanics and statistics tends to be the most popular choice by far. The current study aims to understand how maths teachers in secondary education make decisions regarding the curriculum options and offers useful insight to those currently designing the new A-level specifications.

Semi-structured interviews were conducted with A-level maths teachers representing 27 grammar schools across Northern Ireland. Teachers were generally in agreement regarding the importance of pure maths and the balance between pure and applied within the A-level maths curriculum. A wide variety of opinions existed concerning the applied options. While many believe that the basic mechanics-statistics (M1-S1) combination is most accessible, it was also noted that the M1-M2 combination fits neatly alongside A-level physics. Lack of resources, timetabling constraints and competition with other subjects in the curriculum hinder uptake of A-level Further Maths.

Teachers are very conscious of the need to obtain high grades to benefit both their pupils and the school’s reputation. The move to a linear assessment system in England while Northern Ireland retains the modular system is likely to cause some schools to review their choice of exam board although there is disagreement as to whether a modular or linear system is more advantageous for pupils. The upcoming change in the specification offers an opportunity to refresh the assessment also and reduce the number of leading questions. However, teachers note that there are serious issues with GCSE maths and these have implications for A-level.

Mathematics Learning Support in Ireland in 2015

The provision of mathematics learning support in higher-level institutions on the island of Ireland has developed rapidly in recent times with the number of institutions providing some form of support doubling in the past seven years. The Irish Mathematics Learning Support Network aims to inform all mathematics support practitioners in Ireland on relevant issues. Consequently it was decided that a detailed picture of current provision was necessary. A comprehensive online survey was conducted to amass the necessary data. The ultimate aim of the survey is to benefit all mathematics support practitioners in Ireland, in particular those in third-level institutions who require further support to enhance the mathematical learning experience of their students. The survey reveals that the majority of Irish higher-level institutions provide mathematics learning support to some extent, with 65% doing so through a support centre. Learners of service mathematics are the primary users: first-year science, engineering and business undergraduates, with non-traditional students being a sizeable element. Despite the growing recognition for the need to offer mathematics learning support almost half of the centres are subject to annual review. Further, less than half the support offerings have a dedicated full-time manager, while 60% operate with a staff of five or fewer. The elevation of mathematics support as a viable and worthwhile career in order to attract and retain high quality staff is seen by many respondents as the crucial next phase of development.

O projeto PmatE e a aprendizagem da matemática no ensino superior

O contexto educacional exige renovação de paradigmas. Impõem-se profundas alterações ao nível do papel e da função do professor e dos alunos, devendo-se privilegiar metodologias de aprendizagem ativas, cooperativas e participativas, rompendo-se com o ensino magistral e a mera transmissão de ‘conhecimentos’. As ferramentas informáticas poderão constituir-se uma mais-valia no contexto educativo, promovendo uma aprendizagem significativa e autorregulada pelo aluno, sempre sob a adequada orientação do professor. Neste contexto, foi criado, na Universidade de Aveiro, o Projeto Matemática Ensino (PmatE), com o principal objetivo de combater, de uma forma inovadora, as causas do insucesso escolar a matemática. No entanto, tal plataforma ainda não foi alvo de uma avaliação sistemática, nomeadamente ao nível do ensino superior, que nos permita concluir da consecução dos seus propósitos. Assim, a questão de investigação subjacente ao estudo em curso é - Qual o impacte da utilização diferenciada, como complemento à abordagem didáctica, da plataforma de ensino assistido (PEA) desenvolvida pelo PmatE na aprendizagem de temas matemáticos ao nível do Ensino Superior, principalmente ao nível da autonomia, da construção e aplicação de conhecimentos e do desenvolvimento de apetências pela Matemática. Para se tentar dar resposta à mesma, implementou-se um estudo misto, quantitativo e qualitativo, com alunos da unidade curricular Análise Matemática I do Curso Engenharia Alimentar de um Instituto Politécnico português, a quem se propôs uma exploração prévia da plataforma extra-aula para que, nesse espaço, se pudesse conceptualizar os conceitos envolvidos e realizar tarefas variadas quanto à sua natureza. Usaram-se como principais técnicas de recolha de dados a análise documental, a inquirição e a observação direta, suportadas pelos diversos instrumentos: Questionário Inicial e Final; testes de avaliação, nas versões pré-teste, pós-teste1 e pós-teste2; produções de uma bateria de tarefas de natureza diversificada; registo computorizado do percurso dos alunos relativamente ao trabalho por eles desenvolvido na plataforma do PmatE; notas de campo; dossier dos alunos e entrevistas. Os resultados obtidos, a partir de uma análise estatística dos dados quantitativos e de conteúdo dos dados qualitativos, indicam, por um lado, que os alunos mais autónomos, mais persistentes e que obtêm os melhores resultados são os alunos que usaram a plataforma com frequência e, por outro, que a utilização da plataforma contribuiu para aumentar o gosto pela Matemática. Este estudo permitiu, também, obter informação importante sobre aspetos que poderão melhorar a plataforma, em particular, relativos à natureza das tarefas e à resolução dos exercícios propostos.

Exploração matemática de módulos interativos de ciências: um estudo de caso no "Jardim da Ciência" em articulação com a sala de aula com alunos do 1º ciclo do ensino básico

Orientações curriculares portuguesas para o 1.º Ciclo do Ensino Básico [CEB] preconizam o desenvolvimento de capacidades transversais como a resolução de problemas [RP] e a comunicação (em) matemática [CM], o estabelecimento de conexões Matemática–Ciências Físicas e Naturais [CFN] e a articulação de contextos de educação formal [EF] e de educação não formal [ENF]. Em Portugal, professores manifestam querer utilizar recursos didáticos com estes atributos. Contudo, tais recursos escasseiam, assim como investigação que se situa na confluência destas dimensões. Por conseguinte, na presente investigação, foram desenvolvidos recursos didáticos centrados na promoção de conexões Matemática-Ciências Físicas e Naturais e na articulação de contextos de EF e de ENF. Assim, a presente investigação tem por finalidade desenvolver (conceber, produzir, implementar e avaliar) recursos didáticos de exploração matemática de módulos interativos de ciências, articulando contextos de EF e ENF que, nomeadamente, apelem e possam desenvolver capacidades básicas ligadas à RP e à CM de alunos do 1.º CEB. Decorrente desta finalidade, definiram-se as seguintes questões de investigação: 1. Quais as repercussões dos recursos didáticos desenvolvidos na capacidade de RP de alunos do 4.º ano do 1.º CEB?; 2. Quais as repercussões dos recursos didáticos desenvolvidos na capacidade de CM de alunos do 4.º ano do 1.º CEB?. Além disso, procurou-se auscultar a opinião de alunos e professora sobre a exploração dos recursos didáticos desenvolvidos, principalmente, ao nível de conexões Matemática–CFN e articulação de contextos de EF e ENF de Ciências. Para tanto, foi realizado um estudo de caso com uma professora e seus alunos do 4.º ano do 1.º CEB, em sala de aula e num espaço de ENF de Ciências. A recolha de dados envolveu diversas técnicas e vários instrumentos. A técnica de análise documental incidiu nas produções dos alunos registadas em Guiões do Aluno e em Tarefas-Teste. No âmbito da técnica de inquirição foram administrados questionários a todos os alunos da turma – o Questionário Inicial e o Questionário Final – e entrevistas semiestruturadas à professora – a Entrevista Inicial à Professora e a Entrevista Final à Professora – e aos três alunos caso – Entrevista ao aluno caso. No que respeita à técnica de observação foi implementado o instrumento Notas de campo, onde foram efetuados registos de natureza descritiva e reflexiva. Os dados recolhidos foram objeto de análise de conteúdo e de análise estatística. Resultados da investigação apontam para que a exploração dos recursos didáticos desenvolvidos possa ter promovido o desenvolvimento de capacidades matemáticas de RP e, sobretudo, de CM dos alunos. Parecem ainda indicar que, genericamente, os alunos e a professora possam ter considerado que os recursos didáticos promoveram conexões Matemática– CFN e a articulação entre espaços de EF e ENF de Ciências.

Mathematics as Grammar

Tésis de Doctorado en Filosofía, Universidad de Indiana, Bloomington, 2000

Curriculum and beyond: Mathematics support for first year life science students

The move into higher education is a real challenge for students from all educational backgrounds, with the adaptation to a new curriculum and style of learning and teaching posing a daunting task. A series of exercises were planned to boost the impact of the mathematics support for level four students and was focussed around a core module for all students. The intention was to develop greater confidence in tackling mathematical problems in all levels of ability and to provide more structured transition period in the first semester of level 4. Over a two-year period the teaching team for Biochemistry and Molecular Biology provided a series of structured formative tutorials and “interactive” online problems. Video solutions to all formative problems were made available, in order that students were able to engage with the problems at any time and were not disadvantaged if they could not attend. The formative problems were specifically set to dovetail into a practical report in which the mathematical skills developed were specifically assessed. Students overwhelmingly agreed that the structured formative activities had broadened their understanding of the subject and that more such activities would help. Furthermore, it is interesting to note that the package of changes undertaken resulted in a significant increase in the overall module mark over the two years of development.

Percursos de vida profissional: da formação inicial à entrada na carreira

Dissertação apresentada à Escola Superior de Educação de Lisboa para a obtenção do Grau de Mestre em Ciências da Educação - especialidade Supervisão em Educação

Challenges in the creation and development of a mathematics MOOC

The fast development of distance learning tools such as Open Educational Resources (OER) and Massive Open Online Courses (MOOC or MOOCs) are indicators of a shift in the way in which digital teaching and learning are understood. MOOC are a new style of online classes that allow any person with web access, anywhere, usually free of charge, to participate through video lectures, computer graded tests and discussion forums. They have been capturing the attention of many higher education institutions around the world. This paper will give us an overview of the “Introduction to Differential Calculus” a MOOC Project, created by an engaged volunteer team of Mathematics lecturers from four schools of the Polytechnic Institute of Oporto (IPP). The MOOC theories and their popularity are presented and complemented by a discussion of some MOOC definitions and their inherent advantages and disadvantages. It will also explore what MOOC mean for Mathematics education. The Project development is revealed by focusing on used MOOC structure, as well as the quite a lot of types of course materials produced. It ends with a presentation of a short discussion about problems and challenges met throughout the development of the project. It is also our goal to contribute for a change in the way teaching and learning Mathematics is seen and practiced nowadays, trying to make education more accessible to as many people as possible and increase our institution (IPP) recognition.

Innovations in technology: a friendly math project and the learning analytics “challenge”

Currently the world around us "reboots" every minute and “staying at the forefront” seems to be a very arduous task. The continuous and “speeded” progress of society requires, from all the actors, a dynamic and efficient attitude both in terms progress monitoring and moving adaptation. With regard to education, no matter how updated we are in relation to the contents, the didactic strategies and technological resources, we are inevitably compelled to adapt to new paradigms and rethink the traditional teaching methods. It is in this context that the contribution of e-learning platforms arises. Here teachers and students have at their disposal new ways to enhance the teaching and learning process, and these platforms are seen, at the present time, as significant virtual teaching and learning supporting environments. This paper presents a Project and attempts to illustrate the potential that new technologies present as a “backing” tool in different stages of teaching and learning at different levels and areas of knowledge, particularly in Mathematics. We intend to promote a constructive discussion moment, exposing our actual perception - that the use of the Learning Management System Moodle, by Higher Education teachers, as supplementary teaching-learning environment for virtual classroom sessions can contribute for greater efficiency and effectiveness of teaching practice and to improve student achievement. Regarding the Learning analytics experience we will present a few results obtained with some assessment Learning Analytics tools, where we profoundly felt that the assessment of students’ performance in online learning environments is a challenging and demanding task.

The development and implementation of math projects in a HEI: expectations, objectives, experiences and analysis

An overwhelming problem in Math Curriculums in Higher Education Institutions (HEI), we are daily facing in the last decade, is the substantial differences in Math background of our students. When you try to transmit, engage and teach subjects/contents that your “audience” is unable to respond to and/or even understand what we are trying to convey, it is somehow frustrating. In this sense, the Math projects and other didactic strategies, developed through Learning Management System Moodle, which include an array of activities that combine higher order thinking skills with math subjects and technology, for students of HE, appear as remedial but important, proactive and innovative measures in order to face and try to overcome these considerable problems. In this paper we will present some of these strategies, developed in some organic units of the Polytechnic Institute of Porto (IPP). But, how “fruitful” are the endless number of hours teachers spent in developing and implementing these platforms? Do students react to them as we would expect? Do they embrace this opportunity to overcome their difficulties? How do they use/interact individually with LMS platforms? Can this environment that provides the teacher with many interesting tools to improve the teaching – learning process, encourages students to reinforce their abilities and knowledge? In what way do they use each available material – videos, interactive tasks, texts, among others? What is the best way to assess student’s performance in these online learning environments? Learning Analytics tools provides us a huge amount of data, but how can we extract “good” and helpful information from them? These and many other questions still remain unanswered but we look forward to get some help in, at least, “get some drafts” for them because we feel that this “learning analysis”, that tackles the path from the objectives to the actual results, is perhaps the only way we have to move forward in the “best” learning and teaching direction.

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.