Student Reasoning on Probability Tasks With Coins

This article describes a subset of results from a larger study that explored middle school and high school students' probabilistic reasoning abilities across a variety of probabilistic contexts and constructs. Students in grades 5, 7, 9, and 11 at an urban, private school for boys (n = 173) completed a Probability Inventory, comprising adapted tasks from the research literature, which required students to provide answers as well as justifications of their responses. Supplemental clinical interviews were conducted with 33 students to provide further detail about their reasoning. This article focuses specifically on the probabilistic constructs of compound events and independence in the context of coin tossing. Analyses of justifications of correct and incorrect answers are provided, offering insight into students' strategies, reasoning, and underlying cognitive models. A belief framework is supported by the results of this study. Potential implications for research and instruction are also discussed.

The concepts of Probability are fundamental to the study of Mathematics, especially at the secondary school level. The main aim of this study is to investigate and identify specific students' misconceptions by secondary school students when learning Probability in Brunei Darussalam. In total, 177 Years 10 and 11 students from two schools participated in the research study. The two instruments used for this study were 'Misconception on Probability' two-tier multiple choice questionnaire and interviews. Carelessness and Incorrect method were grouped as error-typed whereas, Representativeness, Equiprobability bias, Beliefs and Human control were the four identified specific misconceptions on Probability.

Journal of Experimental Psychology: Learning, Memory, and Cognition, 2014

Mathematics Education Research Journal, 1997

Psychological Review, 2009

A long tradition of psychological research has lamented the systematic errors and biases in people's perception of the characteristics of sequences generated by a random mechanism such as a coin toss. It is proposed that once the likely nature of people's actual experience of such processes is taken into account, these "errors" and "biases" actually emerge as apt reflections of the probabilistic characteristics of sequences of random events. Specifically, seeming biases reflect the subjective experience of a finite data stream for an agent with a limited short-term memory capacity. Consequently, these biases seem testimony not to the limitations of people's intuitive statistics but rather to the extent to which the human cognitive system is finely attuned to the statistics of the environment.

By focusing on a particular alteration of the comparative likelihood task, this study contributes to research on teachers' understanding of probability. Our novel task presented prospective teachers with multinomial, contextualized sequences and asked them to identify which was least likely. Results demonstrate that determinants of representativeness (featured in research on binomial, platonic sequences) are present in the current situation as well. In identifying a variety of Context-Related features influencing teachers' choices, we suggest the context in which tasks are presented significantly influences probabilistic judgments; however, contextual consideration also provides researchers with potential difficulties for analyzing results. In addition, we identify strands for further research of contextual influence.

Applied Cognitive Psychology, 2002

This paper argues against the theory that people interpret unusual coincidences as paranormal because they misunderstand the probability of their occurring by chance. In the two studies reported here, 214 subjects were given a questionnaire on the frequency of coincidences in their lives, a series of probabilistic problems, and a scale assessing their belief in the paranormal. Believers reported more coincidences than disbelievers. Believers made more errors than disbelievers in tasks reflecting sensitivity to the relationship between expected distribution of chance events and total number of occurrences; and avoided repetitions of identical alternatives in a random sequence to a greater extent. However, the last two effects completely disappeared in a subsample of university students. It is proposed that a more frequent experience of coincidences, on the one hand, and a more biased representation of randomness, on the other, are independent consequences of a stronger propensity of believers in the paranormal to connect separate events. Copyright © 2002 John Wiley & Sons, Ltd.

2006

Data and chance are the two related topics that deal with uncertainty. On the discussions of probability and statistics in both research and instruction, the existing literature depicts an artificial separation, to which other researchers have already called attention in recognition of the inseparable nature of data and chance. Hence, this paper addresses how to integrate the discussions of distributions and probability, starting from the elementary grades. We report on a study that examines fourth-grade students' informal and intuitive conceptions of probability and distribution through a sequence of tasks for developing their understandings about probability distributions. These tasks include various random situations that students explore with a set of physical chance mechanisms and that can be modeled by a binomial probability distribution.

Subjective perception of randomness has been researched by psychologists using a variety of production and judgement tasks, resulting in a number of different descriptions for the biases that characterise people's performances. These research findings, especially those concerned with children's and adolescents' understanding of randomness, are highly pertinent to didactic practices, as new mathematics curricula for compulsory teaching levels are being proposed that incorporate increased study of random phenomena. In this article, we first analyse the complexity of the meaning of randomness from the mathematical point of view and outline philosophical controversies associated with this. Secondly, we complement previous research by comparing the meaning of randomness for 277 secondary students in two age groups (14 and 17 year-olds), through the identification of the mathematical properties they associate to random and deterministic sequences and to two-dimensional distributions. Some implications for teaching and future research are then suggested.