Journal Article

  • In this report we present the results of an international research study on the training of researchers in mathematics education. The study was carried out by some members of The International Study Group on Theory of Mathematics Education. The research involved developing a questionnaire which was mailed to numerous institutions all over the world and the analysis of the answers which were received. The main objective of the study was to collect international data about the training of reserachers in mathematics education and to establish an information network about graduate programs in the field. A total of about 150 questionnaires were sent out and 78 answers received. Fifteen of these answers came from universities that wish to participate in the network although they do not have a program at present.

  • Formal methods abound in the teaching of probability and statistics. In the Connected Probability project, we explore ways for learners to develop their intuitive conceptions of core probability concepts. This article presents a case study of a learner engaged with a probability paradox. Through engaging with this paradoxical problem, she develops stronger intuitions about notions of randomness and distribution and the connections between them. The case illustrates a Connected Mathematics approach: that primary obstacles to learning probability are conceptual and epistemological; that engagement with paradox can be a powerful means of motivating learners to overcome these obstacles; that overcoming these obstacles involves learners making mathematics--not learning a "received" mathematics and that, through programming computational models, learners can more powerfully express and refine their mathematical understandings.

  • A series of studies was conducted to elucidate a phenomenon here referred to as the "illusion of control". An illusion of control was defined as expectancy of a personal success probability inappropriately higher than the objective probability would warrant. It was predicted that factors from skill situations (competition, choice, familiarity, involvement) introduced into chance situations cause individuals to feel inapproriately confident. In Study 1 subejcts cut cards against either a confident or a nervous competitor: in Study 2 lottery participants were or were not given a choice of ticket; in Study 3 lottery participants were or were not given a choice of either familiar or unfamiliar lottery tickets; in Study 4, in a novel chance game, subjects either had or did not have practice and responded either themselves or by proxy; in Study 5 lottery participants at a racetrack were asked their confidence at different times; finally, in Study 6 lottery participants either received a single three-digit ticket or one digit on each of 3 days. Indicators of confidence in all six studies supported the prediction.

  • This study examined students' reasoning about simple repeated choices. Each choice involved "betting" on two events, differing in probability. We asked subjects to generate or evaluate alternative strategies such as betting on the most likely event on every trial, betting on it on almost every trial, or employing a "probability matching" strategy. Almost half of the college students did not generate or rank strategies according to their expected value, but few subjects preferred a strategy of strict probability matching. High-school students showed greater deviations from expected value than college students. Similar misunderstandings were observed in a choice task involving real (not hypothetical) repeated trials. Large gender differences in prediction strategies and in related computational skills were observed. Subjects who understand the optimal strategy usually do so in terms of independence of successive trials rather than calculation. Some subjects understand the concept of independence but fail to bring it to bear, thinking it can be overridden by intuition or local balancing (representativeness).

  • Although numerous research studies have focused on issues related to the teaching of statistics, few studies have focused on the training of people who may become statistics teachers. The purpose of this study was to examine doctoral students' preparation in statistics in the field of education. A national survey was conducted of twenty-seven quantitative methods (QM) programs. One QM professor from each program was identified and asked to describe and evaluate the training of QM and non-QM doctoral students at his or her institution. The vast majority of professors indicated that most or all of the students in their QM programs received training in the "old standard" procedures--ANOVA, multiple regression, and traditional multivariate procedures, whereas fewer than half of the professors indicated that most or all of their QM students received training in more recent procedures such as bootstrapping and multilevel models. Professors were also asked to rate the skills of their QM students in areas such as mathematical statistics and computing on a scale from "Weak" to "Strong". Most professors gave high ratings to their QM students' skills with statistical packages, but gave much more mixed ratings of their QM students' training in mathematical statistics. Nearly half of the professors thought that most of their QM students could have benefited from one or two additional statistics courses. Results are discussed in terms of training future doctoral students.

  • How can ideas, techniques, and applications taken from Exploratory Data Analysis (EDA) enrich mathematics instruction? How do students and teachers respond to ideas of EDA? How must EDA be transformed in order to reach a pedagogically useful position in the mathematics curriculum within general education? The paper describes some results of a teaching experiment concerning ideas of EDA. It was explored how basic new displays such as stem-and-leaf and boxplots can be taught and learned and how they have to be regarded in the context of more traditional statistical displays and newer computer supported displays. A new structuring of the cognitive tool kit for elementary data analysis is sketched. EDA is communicated to teachers and students as detective work. The paper describes ways and problems to do this and how such ideas were transformed in the classroom. Difficulties that arose with using open material and complex data sets in the classroom are discussed with an example concerning deaths in traffic accidents in west Germany from 1953-1987.

  • Proper interpretation of standardized test scores is a crucial skill for K-12 teachers and school personnel; however, many do not have sufficient knowledge of measurement concepts to appropriately interpret and communicate test results. In a recent four-year project funded by the National Science Foundation, three web-based instructional presentations in educational measurement and statistics were developed and evaluated (Zwick et al., 2008). These modules were found to be particularly effective for pre-service K-12 teachers. The primary challenge of the project was to deliver the material in three short 25-minute web-based presentations. In this paper, we discuss the design principles, technical considerations, and specific instructional approaches implemented in the modules, invoking principles from cognitive psychology research. Based on evidence gathered from our project and previous research in teacher education and multimedia learning, we offer suggestions for presenting educational measurement and statistics concepts in a multimedia learning environment.

  • Concepts in probability can be more readily understood if students are first exposed to probability via experiment. Performing probability experiments encourages students to develop understandings of probability grounded in real events, as opposed to merely computing answers based on formulae.

  • We consider how probability theory can be used to increase one's chances of winning in the new television game show The Colour of Money.

  • When two sealed envelopes contain money, one twice as much as the other, a player should be indifferent between them. But when one envelope is opened, one's decision should vary as a function of the observed value and one's subjective probabilities.

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