Theory

In the Connected Learning projects, we are studying students' learning of content through exploring and constructing computer-based models of that content. This paper present a case study of a high school physic teacher's design and exploration of a computer-based model of gas molecules in a box. We follow up the case study with shorter vignettes of students' exploration and elaboration of the Gas-in-a-Box model. The cases lead us to analyze and discuss the role of model-based inquiry in science and mathematics education as well as to draw some general conclusions with respect to the design of modeling languages and the design of pedagogies and activities appropriate for model-based inquiry in classroom settings.

This chapter looks at symbolizing and mathematical learning from a social constuctivist perspective that is motivated by an interest in instructional design. The central theme is that of a concern for the way students actually use tools and symbols. Point of departure are analyses treat people's activity with symbols as an inegral aspect fo their mathematical reasoning rather than as external aids to it. As a consequence, the process of learning to use symbols in general, and conventional mathematical symbols in particular, is cast in terms of participation. Symbol use then seen not so much as somthing to be mastered, but qas a consituent part of the mathematical practices in which students come to participate. This view corresponds with the author's perspective, according to which it is essential to account for the mathematical learning not merely of individual students but of the classroom community taken as a unit of analysis in its own right. To account for this collective learning, the thoeretical construct of a classroom mathematical practice is introduced, which involves taken-as-shared ways of symbolizing.<br>Against this background an analysis is presented of the mathematical practices established duing a seventh-grade classroom teaching experiment that focused on statistical data analysis, that is based on RME theory. This analysis is supplemented with a description of the taden-as-shared ways in thish two computer-based analysis tools were used in the classroom, which is cast in terms of the emergence of a chain of signification. The chapter finishes with a reflection on the general notion of modeling. In connection with the notion of participation, a distinction is made between the use fo the term model in mathematical discourse, and an alternative formulation that relates to both semiotics and design theory.

It is not discussed that the principles and Statistical methods are necessary not only for understanding, but also for the effective exercise in any profession, especially those that are related to the health, since the variability of the clinical, biological and laboratory data, either on individuals or communities, that to come to a decision goes always accompanied by a degree of uncertainty. This is due to the undeniable probabilistic nature of the Biomedical Sciences and it is in fact the Statistics the one that provides the appropriate tools to confront the differences and that uncertainty. (Leiva, Carrera, et al., 1999). The Statistics knowledge and of their procedures allow the student and the graduate to critically appreciate the phenomena that happen to their surroundings ; it allows him to understand scientific works and to produce his own ones, besides generating data of quality and knowing about the problems that affect the population under study.<br><br>Today, any citizen needs in his daily activity certain resources of the Statistics. In the last years this has taken to radically change the teaching of Statistics in many of the countries where it is part of the Mathematics curriculum. It is necessary that the citizen learns earlier to interpret the facts that happen to his surroundings and the data that he receives permanently through any means of diffusion. Learning Statistics is nowadays unquestionably based by the instrumental contribution that this science carries out. (Gal and Garfield, 1997). Besides through statistical education research, the Statistics has been shown as a "modern discipline," useful to develop in precise form the abilities required in the global world and the information society (Ottaviani, 1999).<br><br>That had motivated the next words of Susan Starkings (1996): Mathematical education has radically changed, in many countries, over the last decade The need for mathematically literate students who can function in today's technological society has instigated a change in the content of Mathematics curriculum. From the last educational reformation our country has recently begun to introduce emphasis in Statistic with emphasis in the pre-grade curricula. Some years before only charts were given, means and standard deviations and in some cases some other position measure.

The most prominent characteristic of people's dealings with variability (that we are aware of to date) is their tendency to eliminate, or underestimate the dispersion of the data (e.g., Kareev, Arnon &amp; Horowitz-Zeliger, 2002), that is, the differences among individual observations and among means of samples from a population (Tversky &amp; Kahneman, 1971). One typically focuses on the average, and forgets about the individual differences in the material.<br><br>Shaughnessy and Pfannkuch (2002) report that when asked to analyze a set of data, many students just calculated a mean or a median. They claim that past teaching and textbooks concentrated heavily on such measures and neglected variation. Shaughnessy and Pfannkuch maintain, however, that variability is important. It exists in all processes. Understanding of variation is the central element of any definition of statistical thinking. They quote David Moore's slogan "variation matters" (p. 255).<br><br>In the history of statistics, the tendency to eliminate human variability was represented (in the first half of the 19th century) by Quetelet, who focused on regularities. According to Gigerenzer, et al. (1989), Quetelet understood variation within species as something akin to measurement- or replication-error: The average expressed the "essence" of humankind. "Variations from the average man were accidental - matters of chance - in the same sense that measurement errors were" (p. 142). Quetelet's conception of variation was diametrically different from Darwin's, who focused on variability itself and regarded variations from the mean as the crucial materials of evolution by natural selection.<br><br>One example of the tendency to ignore variability is obtained when assignment of probabilities (or weights) to a set of possible outcomes is called for. People often tend to distribute these probabilities equally over the available options (Falk, 1992; Lann &amp; Falk, 2002; Pollatsek, Lima &amp; Well, 1981; Zabell, 1988), employing what we call the uniformity heuristic. Equi-probability, or zero variability among the probabilities, is the simplest and easiest choice to fall back on.