Research

  • Calls for change in mathematics instruction and continuing professional development to foster such change is not new. Yet there have been recent calls for professional development for the topic of probability, and particularly for research to monitor and probe professional development programs (Watson, 1992). Several reports at the 1995 PME outlined the specific nature that professional development can take. For example, comparison of four teachers in the ARTISM program showed that, although the external input was the same, their individual change depended on varying local factors such as collegial support (Peter, 1995). Research was undertaken to compare two Victorian teacher professional development programs: Learning in Primary Science (LIPS) and Mathematics in Schools (MIS). As well as different discipline content, MIS and LIPS differed in their implementation. Analysis of questionnaire data showed that the needs of teachers differed for mathematics and for science, and thus the resulting professional development content reflected this difference. However, further analysis found that some mathematics topics such as Chance and Data attracted similar responses to those of science. Professional development in these schools was similar to that of science, where teachers responded favourably to specific curriculum content knowledge and activities for the classroom. This contrasts with a process approach that appeared to meet the needs of teachers involved in topics such as number and measurement. Early NUD.IST coding of interviews of teachers involved in MIS Chance projects supports this result and gives some suggestions why this might have occurred. Firstly, the topic of probability is new to most primary teachers, and secondly, probability invites a range of concepts from naive to sophisticated.

  • In this research work we study the comparison of probabilities by 10-14 year-old pupils. We consider the different levels described in research about these tasks, though we incorporate subjective distractors, which change the predicted difficulty of some items. Analysis of students' arguments serves to determine their strategies, amongst which we identify the "equiprobability bias" and the "outcome approach". Analysis of response patterns by the same pupil serves to show that the coincidence between the difficulty level of probabilistic and proportional tasks is not complete and points to the existence of different types of probabilisitic reasoning for the same proportional reasoning level.

  • The formal use of cooperative learning techniques developed originally in primary and secondary education proved effective in improving student performance and retention in a college freshman level statistics course. Lectures interspersed with group activities proved effective in increasing conceptual understanding and overall class performance.

  • 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).

  • In April of 1997 the three authors organized a conference on assessment in statistics courses for the Boston Chapter of the American Statistical Association. This conference addressed five broad areas of assessment: assessing students (e.g., objective and open-ended test questions, assessment tools besides tests including labs, projects, or cases); assessing the course (e.g., techniques for assessing students' attidudes and values and their reactions to class activities, assignments, and instructional methods); assessing textbooks (e.g., what should a teacher look for in choosing a textbook); assessing software (e.g., ease of use, accuracy, usefulness in helping students construct their own knowledge of statistics); and assessing classroom innovations (e.g., how can an instructor decide if a new classroom innovation is successful). In this paper the authors will present a summary (and their impressions) of this conference.

  • 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.

  • Although the average, or arithmetic mean, has a rich conceptual meaning, it is often simply defined as the outcome of a procedure. The purpose of this study was to compare the nature and extent of the procedural and conceptual understandings developed by two groups of students who had received different forms of instruction, one based on the traditional numerical algorithm and the other on a visual algorithm. When confronted with tasks varying along several dimensions, students adjusted or extended their basic approach for finding the arithmetic mean in ways that give insight into their understanding of this mathematical concept. While both groups of students showed a degree of understanding and flexibility with the procedure they had been taught, students who had learned the visual procedure showed a deeper conceptual understanding of the arithmetic mean.

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