Literature Index

The purpose of the study reported herein was to identify important aspects of statistical knowledge needed for teaching in the middle school grades. A systematic study of the current literature, including state and national standards, was conducted to identify these important aspects and to measure the degree of emphasis or importance suggested for the content. Results show that state and national standards differ greatly in their expectations of what topics in data analysis and statistics students and teachers should master. The variation is also large in the degree of emphasis given to the content. The majority of the documents analyzed suggest giving greater emphasis to the selection and proper use of graphical data representation and measures of center and spread. Additionally teachers' standards also suggest as important the proper selection and use of teaching strategies and inference of students' understanding from their work and discourse.

Attempts to identify the development of students' knowledge in science, mathematics, and other disciplines have included proposals concerning the development of naive theories or framework theories (Carey, 1999; Ioannides & Vosniadou, 2002), abstractions or abstract structures (Fuchs et al., 2003; Hershkowitz, Schwarz, & Dreyfus, 2001), and abstract rules or schemata (Gentner & Medina, 1998; Reed, 1993). While all these ideas represent different research methods and traditions as well as attempts to explain different aspects of learning and performance, I argue that all of them suggest assumptions about the abstract nature of students' naive and developing knowledge that deserve scrutiny. This notion of abstraction, while sometimes explicit, is often hidden in researchers' broad assertions that students make use of some single idea, meaning, theory, or knowledge structure across a wide span of situations with marked contextual differences. This paper calls on researchers across theses research agendas to make more careful distinctions between claims purporting to identify apparent consistency in students' performance and claims concerning that nature or structure of the knowledge that supports that performance.

To support the newest process standard of NCTM (National Council of Teachers of Mathematics), the potential of multiple representations for teaching repertoire is explored through a real-world phenomenon for which full understanding is elusive using only the most common representation (a table of numbers). The phenomenon of "reversal of a comparison when data are grouped" can be explored in many ways, each with their own insights, including: table, platform scale, trapezoidal representation, unit square model, probability (balls in urns), and verbal form. Lesser also commented on this topic in a letter published in The American Statistician (November 2004, p.362).

The capacity for abstraction and concentration has changed. Methods of teaching Probability and Statistics must also evolve. Following is a presentation of an organized set of images, analysed using Graphs Theory, specifically adapted by syntax to each type of information. We use these, not as an illustration but as an exhaustive proof. We make this claim after having established the isomorphism between possibilities and elementary surfaces. Probability and statistics are thus unified and simplified. Combinatory Algebra and Mathematical Analysis remain tools. No more theorem will need to be learnt by heart. Fear and mathematical inferiority will be changed into an intellectual comfort. We have been using this teaching method for more than 25 years, in a basic course for non-specialized students, having written the corresponding books. Clinical and statistical tests already show its efficacy.