Egan J. Chernoff

Research on Teaching and Learning Probability

Research in probability education is now well established and tries to improve the challenges posed in the education of students and teachers. In this survey on the state of the art, we summarise existing research in probability education before pointing to some ideas and questions that may help in framing a future research agenda.

The Fallacy of Composition: Prospective Mathematics Teachers’ Use of Logical Fallacies

The purpose of this article is to address the lack of research on teachers’ knowledge of probability. As has been the case in prior research, we asked prospective mathematics teachers to determine which of the presented sequences of coin flips was least likely to occur. However, instead of using the traditional perspectives of heuristic and informal reasoning, we have utilized logical fallacies for our analysis of the results. From this new perspective, we determined that certain individuals’—those who provided normatively incorrect responses—utilized the fallacy of composition when making comparisons of relative likelihood. In addition, we discuss how our findings impact models established in the research literature (e.g., the representativeness heuristic) and, further, we suggest that logical fallacies should supplement heuristic and informal reasoning as potential perspectives for research investigating comparisons of relative likelihood.RésuméLe but de cet article est de combler en partie le manque de recherches sur les connaissances des enseignants dans le domaine des probabilités. Comme cela avait été fait dans des études précédentes, nous avons demandé à des futurs enseignants des mathématiques quelle était la moins probable parmi les séquences présentées de tirs à pile ou face. Toutefois, au lieu d’adopter les perspectives traditionnelles du raisonnement heuristique et informel, nous nous sommes servis de sophismes logiques pour analyser les résultats. Dans cette nouvelle perspective, nous avons constaté que certaines personnes—celles qui ont fourni des réponses incorrectes sur le plan normatif—se sont servies de sophismes de composition lorsqu’elles ont comparé les différents niveaux de probabilité. Par ailleurs, nous examinons les façons dont nos résultats peuvent avoir un impact sur les modèles établis dans la recherche actuelle (par exemple l’heuristique de représentativité), et nous proposons que les sophismes logiques soient utilisés pour complémenter le raisonnement heuristique et informel dans la recherche visant à analyser les comparaisons de probabilités relatives.

Unasked But Answered: Comparing the Relative Probabilities of Coin Flip Sequence Attributes

The objective of this article is to contribute to research on teachers’ probabilistic knowledge and reasoning. To meet this objective, prospective mathematics teachers were presented coin flip sequences and were asked to determine and explain which of the sequences was least likely to occur. This research suggests that certain individuals, when presented with a particular question, answer different questions instead. More specifically, we found that participants, instead of making the intended relative probability comparison, compared the relative probability of a number of particular attributes associated with coin flip sequences. Further, we interpret participants’ attempts to reduce levels of abstraction in order to reason about probability, in a relative sense. Embracing the research literature suggesting that responses reflect individuals’ understandings of the question they were asked, this article suggests potential questions that participants have not been asked but are answering. In doing so, this article suggests that participants are providing reasonable relative probability comparisons for questions that are unasked. Finally, implications for future research are also discussed.RésuméL’objectif de cet article est d’apporter une contribution à la recherche sur les connaissances des probabilités chez les enseignants. Pour atteindre cet objectif, on a présenté à des futurs enseignants des mathématiques des séquences de tirs à pile ou face, et on leur a demandé de déterminer et de justifier quelles séquences étaient les moins probables. Cette recherche montre que certaines personnes, lorsqu’on leur pose une certaine question, répondent en fait à une question différente de celle qui est posée. Plus précisément, certains participants, au lieu de faire la comparaison attendue entre les probabilités relatives des séquences présentées, comparent plutôt les probabilités relatives d’autres caractéristiques associées aux séquences de tirs à pile ou face (par exemple l’équiprobabilité, les modèles de répétition, le hasard, les alternances, les revirements et les séries). Des implications pour d’autres recherches futures sont également abordées.

Commentary on Probabilistic Thinking: Presenting Plural Perspectives

Those of you familiar with research investigating probabilistic thinking in the field of mathematics education, might, at this point in the book, be expecting a “wish list” for future research, which has become customary (e.g., Kapadia and Borovcnik 1991; Jones et al. 2007; Shaughnessy 1992); however, we will not be adding to the list of wish lists. Instead, we have decided to, in this commentary, highlight some of the overarching themes that have emerged from the significant amount of research housed in this volume. Themes emerging from each of the four main perspectives—Mathematics and Philosophy, Psychology, Stochastics and Mathematics Education—are now commented on in turn.

Comparing the Relative Probabilities of Events

The purpose of this article is to contribute to the research investigating the use of logical fallacies, in particular the fallacy of composition, to account for normatively incorrect responses given by prospective teachers to relative probability comparisons. Our results respond to certain assumptions made regarding research on relative probability comparisons of coin flip sequences, which have suggested that participants were actually comparing events rather than sequences, and demonstrates that even when presented with events, the majority of respondents still give normatively incorrect responses. As with all research in this area, abductive reasoning is employed to substantiate our claim that the fallacy of composition is the most probable explanation of respondents reasoning.

In No Uncertain Terms: Encouraging a Critical Stance Toward Probability in School

In this chapter, we question the tendency in secondary schools to present the notion of probability exclusively through scenarios where all necessary information is readily available and appears to be wholly reliable. Such scenarios create the impression that probability is a fixed attribute of the objects that generate an event, obscuring the larger idea that the probability of an event is, rather, an enumeration of available information about the event in question, which may be subject to change. We highlight a family of problems suitable for use in the secondary school classroom where determining a solution requires not only a consideration of possible outcomes, but also the uncovering of assumptions regarding the process through which information about the event was obtained. The ambiguity provided through the presence of a middleman illustrates how differing sets of information may affect probability assessments, and encourages students to take a critical stance toward probability calculations.

Still Warring After All These Years: Obstacles to a Transdisciplinary Resolution of the Math Wars

Faced with the complex issues of modern society, a growing number of individuals and organizations have embraced a transdisciplinary approach in the attempt to resolve such issues in an ethical, socially responsible way. Such an approach may even prove to be effective in mediating (if not resolving) the math wars, a long-standing, value-laden debate about what (mathematics) children should learn in the twenty-first century and how they should learn it. However, although the math wars have evolved into a conflict involving a wide variety of individuals and groups representing various interests and disciplines, we argue that for this issue, transdisciplinarity is still out of reach. In particular, in reviewing the evolution of the math wars in the United States and in Canada through a transdisciplinary lens, we find that one major obstacle is the reluctance, and sometimes outright refusal, to step outside disciplinary constraints to engage in dialogue and collaboration with diverse stakeholders. We contend that if the attitude of opposition is maintained, we should expect a long and bitter war indeed.

Preface to Perspective I: Mathematics and Philosophy

Within the wide divergence of opinions about the philosophy of probability, there is one significant bifurcation that has been recurrently acknowledged since the emergence of probability around 1600 (Hacking, The emergence of probability: a philosophical study of early ideas about probability induction and statistical inference. Cambridge University Press, Cambridge, 1975). Hacking describes this duality of probability as the “Janus-faced nature” (p. 12) of probability, explaining “on the one side it is statistical, concerning itself with stochastic laws of chance processes” (ibid.); and “on the other side it is epistemological, dedicated to assessing reasonable degrees of belief in propositions quite devoid of statistical background” (ibid.). Although the phrase “Janus-faced” continues to be used throughout probability (related) literature, the terms used to describe the two different faces (i.e., the different theories or interpretations) of probability have not been similarly adopted.

Preface to Perspective III: Stochastics

The term “stochastics” has different meanings for different individuals. Despite a multitude of meanings, a commonly accepted “definition” is simply: probability and statistics. However, we contend the above story—for many years considered urban legend, but now actualized in introductory statistics courses around the world (e.g., Deborah Nolan and others)—best embodies the notion of stochastics in mathematics education. From this contention, by explicitly recognizing the notion of randomness within stochastics, we (re)define (for the purposes of this preface) stochastics, simply, as randomness and probability and statistics.

Preface to Perspective IV: Mathematics Education

Perspective IV: Mathematics Education presents research pertaining to two threads of investigations into probabilistic thinking: (content) topics in probability (e.g., sample space) and areas of research in mathematics education (e.g., affective domain). Within this perspective, theories, models and frameworks from mathematics education are used to investigate probabilistic thinking while other areas of mathematics education are informed by the research into probabilistic thinking.