How does statistical thinking differ from mathematical thinking? What is the role of mathematics in statistics? If you purge statistics of its mathematical content, what intellectual substance remains? In what follows, we offer some answers to these questions and relate them to a sequence of examples that provide an overview of current statistical practice. Along the way, and especially toward the end, we point to some implications for the teaching of statistics.
This paper will describe the initial phases of a three-year project whose purpose is to devise assessment instruments and follow the implementation of the Chance and Data content of A National Statement on Mathematics for Australian Schools. A discussion of the rationale used in devising items will be followed by descriptions of the four types of instruments developed. Finally, the results of pilot trials carried out in recent months will be presented.
Curriculum recommendations in mathematics at international and national levels have advocated increased attention to probability instruction K-8 (Australian Education Curriculum Corporation, 1994; Department of Education and Science and the Welsh Office, 1991; National Council of Teachers of Mathematics, 1989). In response to these recommendations, current curriculum materials have placed increased emphasis on the teaching and learning of probability (Berle-Carman, Economopoulos, Rubin, Russell, & Corwin, 1995; Chandler & Brosnon, 1994). With respect to teaching and learning, numerous studies (Fennema, Franke, Carptenter, & Carey, 1993; Fuys, Geddes, & Tischler, 1988; Lamon, 1996; Mack, 1995), advocate the use of research-based knowledge of students' thinking to inform instruction. Although there has been considerable research on students' probabilistic reasoning (e.g., Falk, 1983; Fischbein, Nello, & Marino, 1991; Hawkins & Kapadia, 1984; Piaget & Inhelder, 1975; Shaughnessy, 1992), none of this research has generated a framework for systematically describing and predicting students thinking in probability. Moreover, research has not generated or evaluated instructional programs at the elementary and middle school levels that are guided by research-based knowledge of students probabilistic thinking (Shaughnessy, 1992). This paper reports on a program of four research studies on probability in the elementary and middle grades. In particular, it examines: a) a research-based framework for describing and predicting how elementary and middle grades' students think in probability; b) an instructional program in probability for the elementary level that was informed by the research-based framework on students probabilistic thinking; and c) two instructional programs in the middle grades, one emphasizing conditional probability and independence, the other focusing on probabilistic thinking and writing in the context of probability.
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.
While important efforts have been undertaken to advancing understanding of probability using technology, the research herein reported is distinct in its focus on model building by learners. The work draws on theories of Constructionism and Connected Mathematics. The research builds from the conjecture that both the learner's own sense making and the cognitive researchers, investigations of this sense-making are best advanced by having the learner build computational models of probabilistic phenomena. Through building these models, learners come to make sense of core concepts in probability. Through studying this model building process, and what learners do with their models, researchers can better understand the development of probabilistic learning. This report briefly describes two case studies of learners engaged in building computational models of probabilistic phenomena.
It is my philosophical position that post graduate success is highly associated with autonomous learning. The student needs practice in (1) teaching herself statistical theory and application, (2) self-diagnosis of conceptual strength and weakness, and (3) the process of transcribing a well-defined researchable problem into readable prose. As parents, we have lovingly attempted to instill the sense of pride and accomplishment which can only come from independent success. We don't advocate abandonment, that is, a sink or swim correspondence course approach. The instructor who clarifies, guides, challenges, and supports is worth her salt and has only fully completed her charge when--like the parent--she's obsolete. The purpose of this paper is to explore motivational strategies for motivating student commitment to this autonomous learning objective.
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 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.
There are benefits of teaching inference via confidence intervals (CIs), rather than null<br>hypothesis significance testing (NHST). However, CIs are not without misconceptions.<br>First, we provide empirical evidence that CI presentations of data can help alleviate some<br>typical misinterpretations of results, leading to more accurate conclusions and more justified<br>decisions. Second, we demonstrate that CIs are also prone to particular types of<br>misconceptions. Finally, we present interactive figures and simulations that, when used with<br>guidelines for CI interpretation, should lead to more insightful interpretations of research<br>results and fewer misconceptions.
Fifty-six high school mathematics teachers participated in a four-day technology-intensive professional development experience designed to support their understanding of the big statistical idea of "comparing distributions." Content pretests and teacher interviews informed the hypothetical learning trajectory and design of professional development beyond that from the research literature. Additional data sources included content post-tests and interviews, video-tape of the professional development experience, and teachers' constructed responses to written reflection prompts during the session. Retrospective analyses surfaced a striking phenomenon I have chosen to call dynamic technology scaffolding (DTS) which involves coordination of increasingly-sophisticated technological tools during statistical investigation with the purpose of supporting learners' conceptual understanding of an important statistical big idea.