Investigation of routes on various grids

Problems in the curriculum have existed since ancient times, if not earlier. For example, an Egyptian mathematical manuscript (the Ahmes Papyrus) consisting of a collection of problems, was written about 1650 B.C. This has not changed in nearly 4000 years. The central idea of problems has been a common to school mathematics historically and to school mathematics today. Traditionally, when we have spoken about problems we have meant word problems and especially so-called story problems. From our contemporary view, these are only one particular kind of problem. These are problems with the help of which we present pupils with a pseudo - real world. Through this type of problem we present students with a situation or a task and in the form of a question with a goal that the student must achieve. We can say that these are „goal - aimed" problems, or in contemporary terminology closed problems. We require that the student choose some previously learned algorithm (if he/she knows more than one), and with this perform a calculation. They are often placed at the end of a set of algorithmic exercises and with help of them we practise what we have just learned. In many countries today, as in our own, these problems are of central importance, and it is by solving such problems that we can assess the student. Here is a typical example of a story problem when we must calculate with fractions: Story problem: On a trip at the end of school year there were 25 pupils. 3/5 of them are girls. How many girls and how many boys participated in the trip? A great change occurred in the concept of the problem when we began to consider the relationship between the student and a problem. Reys et al. (1984) argued that „a problem involves a situation in which a person wants something and does not know immediately what to do to get it". In this explanation of a problem, it is very important that the pupil wants to achieve the goal. If for example, we give to our student the task of making a present for his/her mother for Mothers' Day, it is problem for him/her only if they want to do it (in this new approach).