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Full-Text Articles in Physical Sciences and Mathematics
Developing Strategic And Mathematical Thinking Via Game Play: Programming To Investigate A Risky Strategy For Quarto, Peter Rowlett
Developing Strategic And Mathematical Thinking Via Game Play: Programming To Investigate A Risky Strategy For Quarto, Peter Rowlett
The Mathematics Enthusiast
The Maths Arcade is an extracurricular club for undergraduate students to play and analyse strategy board games, aimed at building a mathematical community of staff and students as well as improving strategic and mathematical thinking. This educational initiative, used at several universities in the U.K., will be described. Quarto is an impartial game played at the Maths Arcade, in that there is one set of common pieces used by both players, and one where stalemates are a common outcome. While some students play without apparent direction until a winning opportunity appears, others adopt a more risky strategy of building the …
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.
Levels Of Reasoning Of Middle School Students About Data Dispersion In Risk Contexts, Ernesto Sánchez, Antonio Orta
Levels Of Reasoning Of Middle School Students About Data Dispersion In Risk Contexts, Ernesto Sánchez, Antonio Orta
The Mathematics Enthusiast
The aim of this research study is to explore students’ reasoning concerning variation when they compare groups and have to interpret dispersion in terms of risk. In particular, we analyze in this paper the responses to two problems from a questionnaire administered to 82 ninth-grade students. The problems consist of choosing between two and three groups of data by comparing them. The first one composed of losses and winnings coming from a hypothetical game; the second is about medical treatments. The results show the difficulty students had in interpreting variation in a risk context. Although they identify the data group …
Risk Intuitions And Perceptions: A Case Study Of Four Year 13 (Grade 12) Students, Stephanie Budgett, Lorraine O'Carroll, Maxine Pfannkuch
Risk Intuitions And Perceptions: A Case Study Of Four Year 13 (Grade 12) Students, Stephanie Budgett, Lorraine O'Carroll, Maxine Pfannkuch
The Mathematics Enthusiast
In the New Zealand school statistics curriculum, year 12 students (aged 16-17) are required to solve problems that involve interpreting risk and relative risk within a range of meaningful contexts. In a small exploratory study we investigate the risk conceptions of four year 13 students who performed at the excellence level in their year 12 externally-assessed examination on this topic. Through questionnaires and interviews we investigate the ways in which these students perceive and express risks associated with a variety of everyday activities and also how they compare the risks of several adverse outcomes. We also explore the strategies they …
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”.
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. …
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 …
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 …
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.
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 …
Guest Editorial: Risk – Mathematical Or Otherwise, Egan J. Chernoff
Guest Editorial: Risk – Mathematical Or Otherwise, Egan J. Chernoff
The Mathematics Enthusiast
No abstract provided.
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.
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.
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: 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 …
Tme Volume 12, Numbers 1, 2, And 3
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 …
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 …
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 …
Risky Research Business: Mathematics Education Research On The Margins, Erika C. Bullock
Risky Research Business: Mathematics Education Research On The Margins, Erika C. Bullock
The Mathematics Enthusiast
Although we would like to believe that decisions about what research to conduct and how to conduct it are based solely on researcher interest and societal need, the reality is that external political and disciplinary factors do play a role. Scientifically based research (SBR) is one example of external political pressures that shape researcher choice both directly and indirectly. Additionally, disciplines like mathematics education operate under hidden curricula that have the potential to marginalize particular research foci. The purpose of this paper is to consider the implications of such a narrow focus on a young mathematics education researcher’s choices about …
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 …
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.
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 …
Teaching Risk In School, Andreas Eichler, Markus Vogel
Teaching Risk In School, Andreas Eichler, Markus Vogel
The Mathematics Enthusiast
Although risk is an important topic for society it is seldom addressed when teaching statistics and probability. In this paper we refer to this discrepancy identifying three obstacles for teaching risk in school regarding the mathematical and the situational aspect of risk. Based on two educational constructs, i.e. probability literacy and modelling, we discuss existing approaches for teaching risk in school and propose two strategies for promoting risk as a valuable issue for students based again on the distinction of the mathematical and situational aspect of risk.
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 …
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 …
Risk—A Fundamental Condition Of Doing Mathematics, Wolff-Michael Roth, Jean-François Maheux
Risk—A Fundamental Condition Of Doing Mathematics, Wolff-Michael Roth, Jean-François Maheux
The Mathematics Enthusiast
The theme of this special issue is risk. But risk is not a common topic of investigation in mathematics education, lest it be an occasional interest in “at risk” students, generally defined as those who likely will fail at school. In this study, we are not interested in this rather limited use of the risk concept. Instead, we show that risk not only is a condition of human life generally, but also a necessity for teaching and learning mathematics. To show this, we develop the concept of risk with materials from a second-grade mathematics unit on geometry. Implications are drawn …
Making Decisions About Gambling: The Influence Of Risk On Children’S Arguments, Annie Savard
Making Decisions About Gambling: The Influence Of Risk On Children’S Arguments, Annie Savard
The Mathematics Enthusiast
This article presents results from a study on decision-making towards eventual participation to gambling activities by grade 4 students. For this study, six learning situations were proposed in a fourth grade classroom. The researcher, who was also the teacher, proposed some extra activities in order to define gambling. Students learned about probability and developed, at the same time, the ability to think critically about gambling. She then proposed three fictional situations of gambling to the students, and asked them if and why they would (or would not) participate in the situation. By studying the arguments that students provided, she explored …
Students’ Language Repertoires For Prediction, David Wagner, Joseph Dicks, Paula Kristmanson
Students’ Language Repertoires For Prediction, David Wagner, Joseph Dicks, Paula Kristmanson
The Mathematics Enthusiast
Communication about prediction is complex in a number of ways. First, language is by nature recursive — language is an indicator of meaning as well as a force that shapes meaning. Second, the same language used to communicate prediction in uncertain environments is used for other purposes. In this article, we describe how the recursive nature of language impacted the choices we made in a cross-sectional longitudinal study aimed at gaining insight into children’s language repertoires relating to conjecture. We then explore some Grade 6 students’ communication about prediction to develop insight into their meaning and meaning-making with prediction language. …
Pedagogy Of Risk: Why And How Should We Teach Risk In High School Math Classes?, Nenad Radakovic
Pedagogy Of Risk: Why And How Should We Teach Risk In High School Math Classes?, Nenad Radakovic
The Mathematics Enthusiast
Risk is everywhere yet the concept of risk is seldom investigated in high school mathematics. After presenting arguments for teaching risk in the context of high school mathematics, the article describes a case study of teaching risk in two grade 11 classes in Canada- an all-boy independent school (23 boys) and a publicly funded religious school (19 girls and 4 boys). The findings suggest that the students possessed intuitive knowledge that risk of an event should be assessed by both its likelihood and its impact. Following and amending pedagogic model of risk (Levinson, R., Kent, P., Pratt, D., Kapadia, R., …