ABSTRACT VIEW
SOLVING COMBINATORICS PROBLEMS: FROM THE DESIGN OF A SOLVING METHOD TO A TRAINING SYSTEM
F. Le Calvez, H. Giroire
University Pierre and Marie Curie - Paris 6 (FRANCE)
Solving combinatorics exercises is a difficult task, because of the gap between the statement and the mathematical objects involved. In France, it is now taught in the first year of university and no much time is devoted to it. Teachers are eager to enhance the students' competency in this domain.
Let us take an example of the combinatorics problems on which we work: “How many 5-letter words are there with exactly one occurrence of the letter a?”. We will name SeC the set of these elements. The general form of the exercises is: "given a set or sets, count within some universe of configurations the elements that satisfy some constraints". It is not sufficient to give the cardinal of SeC, the student must be able to justify it and this is the very difficulty for them.
With mathematics teachers we have elaborated a solving method "the constructive method" which is based on the multiplicative principle and is close to the students' usual solving. The student has to define the universe and to construct by steps one element of SeC. Each step corresponds to give a constraint which defines a subset; the student has to calculate at each step the cardinal of this subset (number of possibilities). In the example, two steps occurs:
I choose 1 position for letter 'a' n: 5*1
I choose 4 positions for letter not 'a' n: 1*25*25*25*25
The cardinal of SeC is 5*1*1*25*25*25*25 (multiplicative principle)
Using this method, it is possible to make students more familiar with the underlying mathematical concepts and reasoning, and to enhance their modelling competency.

We have designed learning solving activities based on the method to enable students to train themselves. Combinatorics exercises have been classified in accordance of their solving schema. For each class of exercise we have designed an interface (a "machine") with which the student can express his/her solving using the constructive method. Special attention has been made to offer interfaces which are close to the students' natural ways and enable them to be more familiar with the underlying concepts.
The system (named Combien?, How Many?), at each step of the solving calculates if the on-going solution of the student can conduct to a right one (indeed, there are several possible solving ways for solving an exercise); if it is not, the system gives hints to enable the student to understand his/her error and to proceed. Thus, the system is designed for training and not for assessment.

The system has been used for 6 years in university. Students are satisfied with having an exercise solving method that enables them to start the solving process easily and that is easy to use. Teachers praise the fact that Combien? exhibits the mathematical objects and that, thus, students can become familiar with them. The use of Combien? encourages students not only to make more and more exercises but also to justify their numerical solutions.

Combien? is multilingual (English, French, Spanish and soon German and Portuguese). Furthermore, it offers author possibilities; teachers can add exercises and analyze the tracks of students' solving. See http://combien.lip6.fr for more information. At this time we are working on the integration of Combien? in ActiveMath system.

Le Calvez, F., Giroire, H., & Tisseau, G., (2008) Design of a Learning Environment in Combinatorics based on Problem Solving: Modeling Activities, Problems and Errors, IJ of Artificial Iintelligence in EDucation, (18)