Fermi Problems in Primary Mathematics Classrooms: Fostering Children's Mathematical Modelling Processes: Andrea Peter-Koop Explains How Children in Some German Classrooms Developed Problem Solving Processes through Working on Challenging Open Style Problems

Article excerpt

The difficulties which primary students experience when dealing with real-world related word problems have been discussed extensively. These difficulties are not only related to complex, non-routine problems but already occur with respect to routine problems that involve the application of a simple algorithm. Due to difficulties with the comprehension of the text and the identification of the 'mathematical core' of the problem, primary students frequently engage in a rather arbitrary and random operational combination of the numbers given in the text. In doing so, they fail to acknowledge the relationship between the given data and the real-world context. Real-world problem solving involves the 'mathematisation' of a non-mathematical situation that involves:

* the construction of a mathematical model with respect to the real-world situation,

* the finding (calculation) of the unknown, and

* the transfer of the mathematical result derived from the mathematical model to the real-world situation.

Hence, this 'mathematisation' process is frequently modelled itself in the manner shown in Figure 1.

[FIGURE 1 OMITTED]

However, while traditional word problems often do not seem to provide a suitable context for the development of mathematical modelling skills, the use of Fermi problems in the middle and upper primary mathematics classroom can help to foster students' mathematical modelling strategies.

Fermi problems

Enrico Fermi (1901-1954), who in 1938 won the Nobel Prize for physics, was known by his students for posing open problems that could only be solved by giving a reasonable estimate. Fermi problems such as, 'How many piano tuners are there in Chicago?' share the characteristic that the initial response of the problem solver is that the problem could not possibly be solved without recourse to further reference material.

However, while individuals frequently reject these problems as too difficult, Clarke and McDonough (1989) pointed out that' 'pupils, working in cooperative groups, come to see that the knowledge and processes to solve the problem already reside within the group' (p. 22). In order to stimulate collaborative modelling strategies while avoiding frustration by setting too demanding tasks, Fermi problems suitable for middle and upper primary students should:

* be based on a selection of real-world related situations that include reference contexts for primary students;

* present challenges and intrinsically motivate cooperation with peers;

* be open-beginning as well as open-ended real-world related tasks that require decision making with respect to the modelling process;

* not contain numbers in order to challenge students to engage in estimation and rough calculation and/or the collection of relevant data.

The following Fermi problems have been successfully used in grade 3 and 4 classrooms:

   How much paper does your
   school use in one month?
   (paper problem)

   How many children are
   together as heavy as a polar
   bear? (polar bear problem)

   How much water do you use in
   one week? (water problem)

   There is a 3 km tailback on the
   motorway. How many vehicles
   are caught in this traffic jam?
   (traffic problem)

Introducing the problem

Allow a brief first round of discussion before splitting the class in small groups in order to make sure that all students have understood the problem. Have equipment and literature for data collection (such as scales, information on bus/train fares, books on animals, measuring tapes, etc.) available for immediate use according to specific requests by the students. Consider that enough time is allowed for the group work. In the German classrooms the groups needed between 30 to 60 minutes in order to find a satisfactory solution. In some cases about half of this time was needed for the data collection and background research. …