Tools, Researchable Issues & Conjectures for Investigating What it Means to Understand Statistics (or Other Topics) Meaningfully

摘要

This paper describes a variety of sharable multi-purpose research tools which have evolved from recent studies in which models & modeling perspectives (Lesh & Doerr, 2003) were used to investigate what it means for students to develop meaningful understandings of foundation-level concepts and abilities in an introductory statistics course. Such a course typically is intended to “cover” topics ranging from basic measures of probability, centrality, and spread, to more advanced topics such as analysis of variance, hypothesis testing, or regression and correlation.? Although MMP studies have involved students from grade school through graduate school, as well as professionals in fields ranging from education to engineering, the students referred to in this paper were sophomore elementary education majors at Indiana University. This group is identified simply to give readers a frame of reference for observations made - not because it served as a treatment group or a control group of any kind… These students had sufficiently strong academic records to be admitted to a university program with high academic standards; but, compared with peers in other fields, none considered themselves to be outstanding in mathematics. The tools described here were designed mainly to investigate the following questions. (i) What are the most important “big ideas” that should be emphasized in a given mathematics topic area (e.g. statistics)? (ii) What does it mean to “understand” these ideas? (iii) How do these understandings develop? In fields like engineering, or in other “design sciences” which are more mature than mathematics education, researchers in leading research communities often devote significant portions of their time and efforts toward the development of tools and artifacts for their own use. Whereas, in mathematics education, our research community has spent relatively little time developing such tools and resources. One result of this neglect is that mathematics educators are unable to reliably observe, document, or measure either students’ or teachers’ levels of development for nearly any of the deeper or higher-order achievements that current theories hypothesize to be important. For this reason, this paper focuses on theory-based tools and tool development; and, to begin, main elements of MMP theoretical foundations are described in enough detail so that the tools will be useful to other researchers. Throughout this paper, bulleted questions or statements indicate issues about which a great deal is known – but that still should be investigated more thoroughly in order to more thoroughly understand their meanings and implications.