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Full-Text Articles in Mathematics
Adults’ Perceptions Of Risk In The Big Data Era, Theodosia Prodromou
Adults’ Perceptions Of Risk In The Big Data Era, Theodosia Prodromou
The Mathematics Enthusiast
The present digital era has seen rapid growth in the availability of big data; we were curious about whether such availability of data changes perceptions and assessments of risk. In this paper, we investigate adults’ (35-63 years old) perceptions of risk in the big-data era and how it figures in their everyday life. We developed decision-making scenarios for socio-economic, environmental and health topics that involve modelling with personal value systems alongside Gapminder word map data. Going beyond the idea of risk in statistical theory, we attempt to gain an understanding of the processes by which adults assess risks.
A Conversation With Herbert Tate: Mathematics Educator And Builder, Christian Genest
A Conversation With Herbert Tate: Mathematics Educator And Builder, Christian Genest
The Mathematics Enthusiast
Herbert Tate was a Professor of Mathematics at McGill University (Montréal, Canada) from 1921 to 1964. As the author of four textbooks, and in his capacity as Chairman of the Department of Mathematics from 1948 to 1960, he played a key role in structuring the institution’s research and study programs in mathematics during an important period of growth. McGill’s current position as a hub of mathematical research owes much to him. In this interview given shortly after his retirement, Herbert Tate describes his career and shares some of his views about mathematics and related topics. Beyond its archival value, this …
Mathematical Creativity: The Unexpected Links, Amine El-Sahili, Nour Al-Sharif, Sahar Khanafer
Mathematical Creativity: The Unexpected Links, Amine El-Sahili, Nour Al-Sharif, Sahar Khanafer
The Mathematics Enthusiast
Creativity in mathematics is identified in many forms or we can say is made up of many components. One of these components is The Unexpected Links where one tries to solve a mathematical problem in a nontraditional manner that requires the formation of hidden bridges between distinct mathematical domains or even between seemingly far ideas within the same domain. In this article, we design problems that express unexpected links in mathematics and suit students of intermediate and secondary levels. We prove their feasibility through teachers’ testimonies and through introducing them in classrooms and collecting students’ attitudes with respect to understanding …
Editorial: The Economics Of Risk, Bharath Sriraman
Editorial: The Economics Of Risk, Bharath Sriraman
The Mathematics Enthusiast
No abstract provided.
Understanding Risk Through Board Games, Joshua T. Hertel
Understanding Risk Through Board Games, Joshua T. Hertel
The Mathematics Enthusiast
In this article, I describe a potential avenue for investigating individual’s understanding of and reactions to risk using the medium of board games. I first discuss some challenges that researchers face in studying risk situations. Connecting to the existing probabilistic reasoning literature, I then present a rationale for using board games to model these situations. Following this, I draw upon intuition and dual-process theory to outline an integrated theoretical perspective for such investigations. The article concludes with two vignettes demonstrating how this perspective might be used to analyze thinking about risk in a board game setting.
Risk Education: A Worldview Analysis Of What Is Present And Could Be, Gale L. Russell
Risk Education: A Worldview Analysis Of What Is Present And Could Be, Gale L. Russell
The Mathematics Enthusiast
Risk, risk analysis, risk management and risk-based decision-making are ubiquitous ideas in the modern world. Consequently, risk education is emerging as a new field of research. However, just as the defining of risk and what it entails is a contested topic, so too is the field of risk education research open to many possible approaches. In this paper notions of risk, particularly as they play out in research on risk education, are analyzed (within an ethical space) using a theoretical framework based on the Traditional Western and an Indigenous worldview. Through this analysis, along with the identification of the kinds …
Good Models And Good Representations Are A Support For Learners’ Risk Assessment, Laura Martignon, Sebastian Kuntze
Good Models And Good Representations Are A Support For Learners’ Risk Assessment, Laura Martignon, Sebastian Kuntze
The Mathematics Enthusiast
When learners have to make sense of risky situations, they can use mathematical models and representations which facilitate successful risk assessment. Based on theoretical considerations on the benefits of specific models and specific representations in such contexts, we present empirical findings of a study which examined whether students use such models and representations in their risk assessment. We conclude that the availability of adequate models to learners may help them gain transparency when facing risks and thus foster their decision-making.
Risk And Decision Making: The “Logic” Of Probability, Manfred Borovcnik
Risk And Decision Making: The “Logic” Of Probability, Manfred Borovcnik
The Mathematics Enthusiast
Risk is a hot topic. There is an international trend to use examples of risk or the concept of risk in the early teaching of probability. It enriches the problems, it widens the contexts, and it motivates the students to learn probability. This paper illustrates the notion of risk as a multi-faceted concept. The diverse perceptions start with language where risk is used in very different ways. The overlap of risk and hazard is not restricted to the technical context of safety and reliability; Knight’s seminal work on risk and uncertainty has its definite impact on today’s perception of the …
A Dialectical Invariant For Research In Mathematics Education, Mauro García Pupo, Juan E. Nápoles Valdes
A Dialectical Invariant For Research In Mathematics Education, Mauro García Pupo, Juan E. Nápoles Valdes
The Mathematics Enthusiast
Many current problems in research in mathematics education emerge from pairs of contradictory dialectical categories. In effect, these pairs characterize the problems. When an epistemological study is made to determine the object of research in which a problem is immersed, it is possible to find essential pairs of dialectical categories that become more profound and thus provide enough elements for the determination of appropriate didactic actions to solve the problem under research.
Risks Worth Taking? Social Risks And The Mathematics Teacher, Ami Mamolo, Laura Elizabeth Pinto
Risks Worth Taking? Social Risks And The Mathematics Teacher, Ami Mamolo, Laura Elizabeth Pinto
The Mathematics Enthusiast
In this article, we explore notions of risk as perceived or experienced by individuals involved in mathematical education. We present this exploration in the form of vignettes, each illustrating a form of risk: a parent’s reaction to classroom “propaganda”; a teacher trying to do justice by her students; a teacher confronted by his administration; and a college professor who believes university policy to be unjust. Each vignette sheds light on areas in which teacher education may offer additional support in fostering the mathematical knowledge, pedagogical sensitivity, and social awareness required to foster, what are in our view, much needed risks …
Probability, Justice, And The Risk Of Wrongful Conviction, Jeffrey S. Rosenthal
Probability, Justice, And The Risk Of Wrongful Conviction, Jeffrey S. Rosenthal
The Mathematics Enthusiast
We consider the issue of standards of proof in legal decisions from the point of view of probability. We compare ``balance of probabilities'' and ``beyond a reasonable doubt'' to the statistical use of p-values. We point out various fallacies which sometimes arise in legal reasoning. And we provide several examples of legal cases which involved probabilities, including some in which incorrect decisions were made and defendants were wrongfully convicted.
Risk: Mathematical And Otherwise, John Adams
Risk: Mathematical And Otherwise, John Adams
The Mathematics Enthusiast
What role might mathematicians have to play in the management of risk? The idea of turning a risk, a possibility of loss or injury, into a “calculated” risk, a quantified probability of loss or injury, is one that has obvious appeal not just to statisticians and mathematicians – but to large numbers of others who would like to know the probability of failure before pursuing some intended course of action. Conclusion: even when risks can be calculated with great precision, they can only be used to inform judgment, but not substitute for it. And it matters who is making the …
Worth The Risk? Modeling Irrational Gambling Behavior, Matt Lane
Worth The Risk? Modeling Irrational Gambling Behavior, Matt Lane
The Mathematics Enthusiast
In math class, expected value is often used when deciding whether or not a game is worth playing. A common refrain is that games with negative expected value should be avoided. However, nearly all games of chance have a negative expected value, and a simple expected value analysis fails to explain why these games are so popular. In this article, we consider three psychological factors leading to irrational gambling behavior – the illusion of control, hypersensitivity to reward, and beginner’s luck – and explore how these factors affect an otherwise purely rational model of gambling behavior.
A Cognitive Framework For Normative Reasoning Under Uncertainty, And Reasoning About Risk, And Implications For Educational Practice, Sylvia Kuzmak
A Cognitive Framework For Normative Reasoning Under Uncertainty, And Reasoning About Risk, And Implications For Educational Practice, Sylvia Kuzmak
The Mathematics Enthusiast
Clarifying what is normative or appropriate reasoning under various circumstances provides a valuable reference for guiding what should be taught, and, in contrast, what should not be. This paper proposes a cognitive framework for viewing normative reasoning and behavior under uncertainty, including the applying of knowledge of probability and statistics in real world situations; and identifies implications for educational practice. Factors relevant to normative reasoning under uncertainty that are addressed within the framework include: risk of misapplying statistics knowledge, involvement of mathematical and non-mathematical reasoning, knowledge of real world domains and situation/application detail, and existence of expert consensus. The cognitive …
Risk As An Explanatory Factor For Researchers’ Inferential Interpretations, Rink Hoekstra
Risk As An Explanatory Factor For Researchers’ Inferential Interpretations, Rink Hoekstra
The Mathematics Enthusiast
Logical reasoning is crucial in science, but we know that this is not something that humans are innately good at. It becomes even harder to reason logically about data when there is uncertainty, because there is always a chance of being wrong. Dealing with uncertainty is inevitable, for example, in situations in which the evaluation of sample outcomes with respect to some population is required. Inferential statistics is a structured way of reasoning rationally about such data. One could therefore expect that using well-known statistical techniques protects its users against misinterpretations regarding uncertainty. Unfortunately, this does not seem to be …
Tme Volume 12, Numbers 1, 2, And 3
A Models And Modeling Approach To Risk And Uncertainty, Corey Brady, Richard Lesh
A Models And Modeling Approach To Risk And Uncertainty, Corey Brady, Richard Lesh
The Mathematics Enthusiast
In this article we describe potential contributions of a Models and Modeling Perspective to research focused on learners’ developing conceptions about uncertainty and variation. In particular, we show how a particular class of realistic problem-solving tasks can illuminate how learners develop models to identify, describe, and predict emergent patterns of regularity in the behavior of various types of systems and in the data these systems generate. We begin by situating current design work in this area within a larger project to investigate idea development in the domain of data modeling over extended (course-length) periods. We give design principles and examples …
The Role Of Probabilistic Reasoning Abilities On Adolescent Risk Taking, Maria Anna Donati, Francesca Chiesi, Caterina Primi
The Role Of Probabilistic Reasoning Abilities On Adolescent Risk Taking, Maria Anna Donati, Francesca Chiesi, Caterina Primi
The Mathematics Enthusiast
The aim of this work was to investigate the role of the cognitive system and the affective system on adolescents’ risk taking in gambling tasks characterized as different on the basis of information given to decision makers. In Study 1, we explored the role of probabilistic reasoning and sensation seeking on decision making in a non-risky context (Non-Gambling Task) and a risky context (Gambling Task) in which no preliminary information were given to participants. Results showed that adolescents referred to probabilistic reasoning only in the Non-Gambling Task. In Study 2, we explored the role of probabilistic reasoning and sensation seeking …
Calculated Risks: The Teacher As Big Data Producer And Risk Analyst, Nat Banting
Calculated Risks: The Teacher As Big Data Producer And Risk Analyst, Nat Banting
The Mathematics Enthusiast
Teachers’ work is often subjected to data analysis from outside sources in the forms of standardized examinations and media critique. This article uses the literature of risk analysis to play with two important analogies for teachers with regards to the emerging big data culture and the risk decisions therein. The complex context of the classroom facilitates the exploration of teacher as big data producer, while the multi-faceted nature of risk decisions provide the groundwork for the exploration of teacher as risk analyst. Illustrative classroom episodes portray examples of real and virtual risk faced by teachers, and a third category—curricular risk—is …
Promoting Risk Taking In Mathematics Classrooms: The Importance Of Creating A Safe Learning Environment, Sashi Sharma
Promoting Risk Taking In Mathematics Classrooms: The Importance Of Creating A Safe Learning Environment, Sashi Sharma
The Mathematics Enthusiast
Students beliefs and attitudes towards risk taking can impact on their mathematics learning and performance. However, at present, risk is not established in the field of mathematics education. The challenge for mathematics teachers in developing their students’ risk taking dispositions is to choose appropriate activities and tools that match this concept and the learning needs of the students. This paper describes some research-based ideas for promoting risk taking behaviours in a mathematics classroom. It presents interactional pedagogical strategies from a design collaborative research conducted at one secondary school. As part of the learning activities, students critically evaluated statistical investigations undertaken …
Judgment Of Association Between Potential Factors And Associated Risk In 2x2 Tables: A Study With Psychology Students, Carmen Batanero, Gustavo R. Cañadas, Carmen Díaz, Maria M. Gea
Judgment Of Association Between Potential Factors And Associated Risk In 2x2 Tables: A Study With Psychology Students, Carmen Batanero, Gustavo R. Cañadas, Carmen Díaz, Maria M. Gea
The Mathematics Enthusiast
This study was aimed to evaluate the accuracy and strategies used in the estimation of association between potential factors and associated risks when data are presented in 2x2 tables. A sample of 414 undergraduate Psychology students from three different Spanish universities was given three different tasks (direct and inverse association and perfect independence) where they had to estimate such association. Most participants judged association in the task where there was perfect independence, but the data contradicted the students’ previous expectations. The estimation of association was consistent with the perception of association and the accuracy of estimates increased with correct strategies. …
Dividing A Pizza Into Equal Parts – An Easy Job?, Hans Humenberger
Dividing A Pizza Into Equal Parts – An Easy Job?, Hans Humenberger
The Mathematics Enthusiast
Theoretically seen dividing a pizza equally is not an easy task. For instance, with a normal knife (straight cuts) one has to hit the center so that the cut is a diameter. But there are alternatives (also for dividing equally between more than two persons) which have strong connections to elementary geometry and to integral calculus. This paper deals with these alternatives elucidating the so called “pizza theorem”.
Bhaskara’S Approximation For The Sine, Karel Stroethoff
Bhaskara’S Approximation For The Sine, Karel Stroethoff
The Mathematics Enthusiast
The 7th century Indian mathematician Bhaskara (c.600 – c.680) obtained a remarkable approximation for the sine function. Many subsequent ancient authors have given versions of this rule, but none provided a proof or described how the result was obtained. Grover [1] provides a possible explanation, but I think the rule can be explained more clearly. Rather than give the rule first, we will derive it, and then discuss its accuracy, and explore some alternative approximations. Our derivation is simply an exercise in modeling.
Numerically Integrating Irregularly-Spaced (X, Y) Data, B. Cameron Reed
Numerically Integrating Irregularly-Spaced (X, Y) Data, B. Cameron Reed
The Mathematics Enthusiast
This article describes a computer program for numerically integrating under a y(x) curve of experimental data where the abscissa values are not equally-spaced. This technique is based on fitting parabolas to successive groups of three data points, and may be regarded as a generalization of Simpson’s rule.
Math As A Tool Of Anti-Semitism, Jay Egenhoff
Math As A Tool Of Anti-Semitism, Jay Egenhoff
The Mathematics Enthusiast
At Moscow State University’s Department of Mathematics during the 1970’s and 1980’s, there was rampant discrimination against Jewish and other unwanted students. The professors at the math department made a strong effort to keep Jewish students out of the department. They designed "killer" or "coffin" problems and Jewish students had to answer them during an oral exam. These problems have simple solutions, but require a clever strategy to solve them. This paper explores some of the context of this episode and provides several problems with detailed solutions.
An Examination Of Pre-Service Secondary Mathematics Teachers’ Conceptions Of Angles, Melike Yigit
An Examination Of Pre-Service Secondary Mathematics Teachers’ Conceptions Of Angles, Melike Yigit
The Mathematics Enthusiast
The concept of angles is one of the foundational concepts to develop of geometric knowledge, but it remains a difficult concept for students and teachers to grasp. Exiting studies claimed that students’ difficulties in learning of the concept of angles are based on learning of the multiple definitions of an angle, describing angles measuring the size of angles, and conceiving different types of angles such as 0-line angles, 1- line angles, and 2-line angles. This study was designed to gain better insight into pre-service secondary mathematics teachers’ (PSMTs) mental constructions of the concept of angles from the perspective of Action-Process-Object-Schema …
Common Sense About The Common Core, Alan H. Schoenfeld
Common Sense About The Common Core, Alan H. Schoenfeld
The Mathematics Enthusiast
Is the Common Core the best thing since sliced bread, or the work of the devil? Is it brand new, or a rehash of old ideas? Is it anything more than a brand name, or is there substance? Can it work, given the implementation challenges in our political and school systems? Opinions about the Common Core are everywhere, but the op-eds I’ve seen are often short on facts, and equally short on common sense. A mathematician by training, I’ve worked for nearly 40 years as an education researcher, curriculum materials developer, test developer, standards writer, and teacher. What follows is …
Book Review Of The Tower Of Hanoi: Myths And Maths (Birkhäuser), Cory Palmer
Book Review Of The Tower Of Hanoi: Myths And Maths (Birkhäuser), Cory Palmer
The Mathematics Enthusiast
As the title of the book suggests, the central topic of “The Tower of Hanoi – Myths and Maths” by Hinz, Klavˇzar, Milutinovi´c, and Petr is the famous puzzle of the same name. The classic Tower of Hanoi puzzle along with a number of its variations and related puzzles are examined in a rigorous mathematical framework.
The Development Of Calculus In The Kerala School, Phoebe Webb
The Development Of Calculus In The Kerala School, Phoebe Webb
The Mathematics Enthusiast
The Kerala School of mathematics, founded by Madhava in Southern India, produced many great works in the area of trigonometry during the fifteenth through eighteenth centuries. This paper focuses on Madhava's derivation of the power series for sine and cosine, as well as a series similar to the well-known Taylor Series. The derivations use many calculus related concepts such as summation, rate of change, and interpolation, which suggests that Indian mathematicians had a solid understanding of the basics of calculus long before it was developed in Europe. Other evidence from Indian mathematics up to this point such as interest in …
Pursuing Coherence Among Proportionality, Linearity, And Similarity: Two Pathways From Preservice Teachers’ Geometric Representations, Hyung Sook Lee, Jaehoon Yim
Pursuing Coherence Among Proportionality, Linearity, And Similarity: Two Pathways From Preservice Teachers’ Geometric Representations, Hyung Sook Lee, Jaehoon Yim
The Mathematics Enthusiast
The importance of using multiple representations of a mathematical concept and connecting the representations has been discussed in learning and teaching mathematics. The Common Core State Standards further the discussion with an emphasis on focus and coherence in teaching mathematical concepts across grades. Preservice teachers in our problem solving class were asked to use geometric representations to solve a problem that required proportional reasoning. They were also asked to sequence the works of their peers as well as their own from a developmental perspective. Sequencing geometric representations with various levels was challenging because it required showing a coherent understanding of …