P. Arteaga Cezón1, J. García de Tomás2, C. Batanero1, R. Roa1

1University of Granada (SPAIN)
2I.E.S.O. Cinxella (SPAIN)
Combinatorics is an essential component of discrete mathematics and, as such, it has an important role to play in school mathematics. Since it does not depend on calculus it has suitable problems for different grades; usually very challenging problems can be discussed with the students, so that they discover the need for more mathematics to be learnt.

However, the teaching of combinatorics is reduced to a few lessons, on the belief that this is a topic easy for the students. Contrary to this expectation, previous research suggest that combinatorial reasoning is only fully achieved by adults when they have received specific instruction. This instruction should be based on a previous assessment of students’ intuitive competence in solving combinatorial problems.

To achieve this aim, in this paper we analyse the solution of permutation problem by a sample of 75 students in the first three compulsory secondary education courses, who had not received a formal teaching of the subject. The students were given six open-ended problems of ordinary and repetition permutations, where the type of elements (people, numbers and objects) are varied and taking into account different combinatorial models (problems of selection and location). The size of parameters in the problems were small, so that the students could use either formulas or enumeration to solve the problems.

A content analysis of the students’ written responses served to evaluate the correctness of the solution, types of errors in the solving process and strategies used to find the solution. There was a small percentage of correct solution; however the number of partially correct solution was very high. Few students used formulas, and those used were mainly incorrect. Correct or partially correct solutions to the problems were mainly based on graphical schemes, systematic enumeration or tables. Main errors included lack of discrimination between identical permutations, wrong parameter to build the permutation, confusing the type of element, repeating elements when this is not allowed, lack of recursive reasoning, failure in applying the product rule and confusing the two sets intervening in a location problem.
All these results are compared by grade, types of problem (selection or location) and type of permutation (with or without repetition) and provide useful information to teachers of combinatorics.