STUDENTS' PROBABILSTIC SIMULATION AND MODELING COMPETENCE AFTER A COMPUTER-INTENSIVE ELEMENTARY COURSE IN STATISTICS AND PROBABILITY

Modeling and simulation with the software Fathom has become an important part of an introduc- tory course on probability and statistics for future mathematics teachers at our institution. We describe our conception of modeling and simulation competence that students are supposed to acquire. We use various means such as modeling guidelines, simulation plan and a guidebook with examples for simulations to support students' learning processes. We report on results of empirical studies that made us change and extend our initial educational approach. INTRODUCTION The role of simulation in an elementary course on probability and statistics is dependent on the goals one intends to achieve in the course. Many studies address the role of simulation for developing the concept of sampling distribution and for preparing concepts and methods of infer- ential statistics (delMas, Garfield, and Chance, 1999). We focus more on the role of simulation in elementary probability. In this respect, we build on ideas developed by Gnanadesikan, Scheaffer and Swift (1987) and Konold (1994) and on ideas in the German tradition where more emphasis is given to modeling and simulation in elementary probability. Ideally the use of simulation can serve two different pedagogical purposes (Biehler, 1991). Students are to develop "modeling competence" related to elementary probability, and simulation competence can be part of it. Instead of setting up a formal model and determining unknown probabilities mathematically they can set up a model, implement it in a computer and develop estimates of unknown probabilities by means of simulation. In this sense simulation re- places mathematics. A different purpose is to use simulation for making probabilistic situations more experiential. The observed phenomenon can be used as a reference for concept develop- ment, for elaborating probabilistic intuitions, and for overcoming misconceptions. In this sense simulation replaces the real world, which is to be modeled. An appropriate software tool is essential for putting such an approach into practice. Stu- dents should work actively on analyzing data, simulating, analyzing methods and building mod- els. The software Fathom achieves best our criteria for a software tool that supports both the learning and the use of probability and statistics in problem solving (Biehler, 1997). At the department of mathematics and informatics in our University we are in the process of redesigning our introductory course on stochastics (probability and statistics) for future mathematics teachers. This is obligatory for student teachers that are going to teach in grade 5 to 10 (pupils' ages 11 to 16 years). In the last years the software Fathom was used in this course. The course starts with exploratory data analysis and descriptive statistics. In the first part of the course the students learn how to work with Fathom to analyze real data. The students can use these capabilities to analyze simulated data in the later parts of the course. Simulation as a method is introduced in parallel to the concept of probability. Situations are modeled mathematically and by simulation and results are compared. A PRIORI DIDACTICAL ANALYSIS OF FATHOM'S SIMULATION CAPABILITIES Developing Fathom as students' cognitive tool for modeling probabilistic situations im- plies that students develop a kind of mental model of the tool with regard to simulation problems. It also implies that they are to develop a certain style of working with the tool. Our a priori analy- sis aimed at supporting this process of instrumental genesis (Lagrange, 1999). As a result we con- structed several concepts and notions that we communicated to the students. We distinguish three types of simulations (simultaneous, sequential, and sampling based), we developed the notion of a step-wise "simulation plan," and we had to develop the traditional concepts of event and random variable as bridging concepts between the world of probability and the world of Fathom. In other