Who Built the Egyptian Pyramids? the Use of Computational Tools in Solving Problems to Make Students Think about Randomness

ABSTRACT When learning probability, one of greatest difficulties for students is, quite simply, concept of chance. The reasoning which students use to decide if a sample has been obtained by chance often opposes formal probabilistic ideas. However, usual teaching model usually shows itself to be inefficient when helping students to overcome these learning difficulties. In this work, we describe an experiment based on an alternative teaching and learning model evaluated in a group of students in first year of an engineering degree at University of Basque Country (Spain). Computers are incorporated into problem solving process as model construction tools helping students overcome their difficulties. Results obtained show that computers, with right activities, can be effective intervention instruments to help students to look in greater depth at concepts and procedures. 1. INTRODUCTION The ideas and intuitions which students generally present on probability are associated with their everyday experience with chance. These intuitions are usually vague and confused: will probably rain tomorrow, the sequence 1, 2, 3, 4, 5, 6 cannot win etc. It is difficult to operate with these insubstantial intuitions which are so different from those related to natural numbers for example. However, these intuitions have a practical value for individuals, as they are useful to make judgements in chance-related situations. The scientific response to problem of chance is theory of probability. However, solutions which this theory provides are often contrary to informal intuitions. As a result, former are displaced by latter [2]. There has been a lot of research showing that individuals' probabilistic judgements are often guided by systematic use of a few inference patterns. These patterns are very intuitive yet incorrect [16]. One of these patterns consists of believing that fragments of outcome sequences which appear ordered and symmetrical cannot be caused by chance as they do not look random. These ideas lead, for example, to considering it not particularly probable that 12345 could be a winning lottery, compared with any other number which has disordered digits [5, 19, 12]. In this work, we are going to explain activity plan which we have developed so that students can experiment on how random sequences should look. In order to design our activities plan, we have taken student's learning difficulties into account, as recommended in research into teaching mathematics [11]. 2. THE ROLE OF COMPUTERS IN CONSTRUCTING PROBABILISTIC MODELS AND TEACHING MATHEMATICS Currently, computers play an important role in constructing models which are used to explain experimental evidence and to obtain new properties for phenomenon being studied. For this reason, as indicated by Santos-Trigo [18], recent curriculum proposals recognized importance of using computational tools to comprehend mathematical ideas and solve problems. De Guzman [7] indicates that influence of computers is visibly modifying very nature of our mathematics. Dynamic systems, differential equations, statistics, algebra, geometry, topology are some of many fields where this influence is clear. However, as indicated by Biehler [3], it is difficult to generalize on merits of computers in education. These merits on whole depend on software and didactic strategy used. In same respect, Balachef & Kaput [1] mentioned desperate need to research all aspects related to teaching and learning with computers. In activity plan which we have designed, computer simulations provide visualizations for students to compare their intuitions on chance and scientific ideas. Simple probabilistic models are constructed involving roulette wheels and drums and then these models are combined to form other more complex models. …