# Theory

• ### A rationale for an instructional sequence for analyzing one- and two-dimensional data sets

We present an instructional sequence on analyzing one- and two-dimensional data sets. This instructional sequence is based on teaching experiments in a 7th-, and 8th-grade classroom in Nashville, TN. The sequence we present here, however is not just a smooth version of the sequence that we have tried out. What we learned during these teaching experiments made us reconsider a lot of the original instructional activities. The aim of this paper is not to present the result of these considerations as a ready-made instructional sequence. Instread, out min objective is to offer a rationale for a revised instructional sequence. This rationale is construed from the following ingredients: original considerations that formed the point of departure for our teaching experiment, experiences in the classroom, re-considerations, and new insights.

• ### Promoting Active Learning in Introductory Statistics Course Using the PACE Strategy

In the recent years, statistics educators have been actively rethinking how students learn statistics and how to teach introductory statistics. Furthermore, the current technology continues to open new opportunities for developing innovative teaching strategies. This article presents a paradigm, the PACE approach, for teaching the introductory statistics. PACE stands for projects, activities, cooperative learning using computer and exercises. The approach begins with in-class hands-on activities and cooperative team work. The class lectures are organized to provide the basic concepts and guide students through the activities using team work and computer to help students understand the concepts and problem-solving strategies. Exercises are designed to reinforce the basic concepts and to practice solving real world problems. Projects are self-selected by students under some guidance provided by the instructor. Report writing and oral presentation are emphasized. It is believed that self-selected projects reflect student's interest, and hence better motivate them to be active learners. The paradigm of integrating these components together in a structured system motivates students to be actively involve with their learning.

• ### Statistical mechanics for secondary school: The GASLAB multi-agent modeling toolkit

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.

• ### In A World of Data, Statisticians Count

Someone needs to bring reason and logic to this mass movement to solve all problems with data, and that task should fall to the statistician. Now, I realize that data sets are collected and analyzed by practitioners in many fields, but statisticians are the only professional group educated specifically to ask (and, hopefully, answer) the deep questions about data quality, reliability, and validity, and to seek optimal solutions to data production and analysis that can apply across a wide range of applications. At present, it may be true that statistics is more in demand than are statisticians, but there are plenty of opportunities for the latter. It is high time we expand our numbers so that we can meet the ever-increasing need for statisticians in the information age.

• ### Modeling, symbolizing, and tool use in statistical data analysis

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.

• ### Multimedia and Multimedia databases for teaching Statistics

The breakneck advance of multimedia capabilities and internet technologies offers an unprecedented opportunity to improve the quality of teaching and learning. Nowadays the use of multimedia resources and WWW-supported learning environments is a crucial issue in education and further education. Integrating visualization, animation, interactive experiments, sound and hotlinks to relevant internet sites opens completely new dimensions of learning. Modern multimedia may also incorporate new communication channels and could be part of emerging virtual educational networks.<br><br>Statistics seems to be particularily suitable for illustrating the benefits of multimedia-based teaching. On the one hand, Statistics connects quite different fields of application. This interdisciplinary character of the science can be well demonstrated by suitable videos and motivating examples closely related to people's life. On the other hand, multimedia represents an ideal platform for visualizing statistical concepts and for discovering basic statistical principles by self-driven experiments. Multimedia software for Statistics can go beyond closed instructional microworlds by offering properly maintained subject-specific gateways to recent statistical data and supplementary information from the rapidly growing internet.

• ### Teaching Statistics in the First Years of the University with the Emphasis on the Solution of Problems

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.

• ### Demonstrative Prototype of the Statistical Methods

This paper deals with the introduction of hypermedia technologies in statistical methods as well as their applications in the training of students. A hypermedia prototype to show statistical methods was developed and the toolkits are Toolbook and Kappa. Several screens were developed in a electronic book. The screens show the calculations modules. Some statistical graphs are shown through the screens. The electronic book shows about theoretical methods to calculate means, medians and variance for samples and it provides a deep knowledge about statistical methods. The consultation sessions are quite complete, allowing the student to learn the theory and practice to solve statistical problems. An hypertext system represents the information, in a different way from others usually employed because it is presented as a non-linear mode and, therefore, allowing to take advice with information in agreement with user interests.<br><br>Hypertext and hypermedia systems allow us to reach the following objectives:<br><br>* The structure of the classical text files;<br>* Non-Linear navigation on any selected order of the stored text;<br>* Cooperation, i. e., information with different formats, text, graphs, images, video and voice;<br>* Interaction, what means, sophisticate access tools, as graphical interfaces.<br><br>The paper debates about publications that suggest the benefits of hypermedia systems applications in personal training models.

• ### What are the Clues for Intuitive Assessment of Variability?

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.

• ### Industrial training in quality improvement Part IV: An instrucional model implications for secondary and college instruction

Now that you have heard about the evolution of GAPS, as well as a description of the content and the instruction al approach, I would like to discuss at a more theoretical level an instructional model which underlies the GAPS course. In addition, I will touch on implications for teaching at the high school and university levels.