The Combien? project
The “Combien?” group defined a methodology for designing
the various components of an ITS. The group built a pedagogical system
to help students to learn combinatorics using mathematical language.
From our experience of teaching combinatorics in the classroom, we have defined the mathematical bases of a solving method, which we will call “the constructive method” .
To calculate the cardinal of the set of the configurations which satisfy
the constraints given in the wording, it is possible to do so without
enumerating the configurations but simply by reasoning on a description
of the set of the configurations. Actually, the list of the elements can
be given as the result of an enumeration algorithm. So, an efficient method
for solving combinatorics problems consists in explicating this algorithm
and then analysing it in order to predict how many elements it will generate,
without having to execute it. It is this type of reasoning that we would
like the student to follow.
Observation of experts of the domain at work reveals that they know classes
of classical problems and that they know how to link to them valid schemes
of constructive one-to-one definitions. They do this by determining the
problem class through the analysis of its wording, and then instantiating
the associated scheme in order to generate the equivalent constructive
definition. The advantage of this method is that, when applicable, it
guarantees the validity of the solutions thus obtained.
A machine must build configurations according to the learner’s instructions and reason about these instructions, for example to verify that they respect the basic principle of counting. It must include a computational model of the domain that allows mathematical reasoning. Despite the anticipated variety of the machines, this model must be the same for all machines to reflect the internal consistency of the domain. It must allow to represent not only mathematical concepts, but also the problems and the learner’s activity, which is centred on a specific method (our constructive counting method).
We have defined such a model as an object-oriented conceptual model.
The higher level classes are Problem and Construction, thus reflecting
the learner’s activity: The learner is given a problem and s/he
must represent it and solve it by providing a construction (a program).
Each machine represents a certain construction type associated with a
problem type. The definition of the conceptual model is based not only
on the logical structure of the main concepts (problem, construction),
but also on a concern that students can use a representation close to
their usual language. This has been made possible by the fact that the
underlying theoretical method (the constructive counting method) has been
itself elaborated from the students’ usual answers. (see article)
In our system, we defined an internal language which we use to represent knowledge: the DESCRIPT language. It would be better to speak about a family of languages founded on a syntactic and structural common basis. Among these languages, the "description" language is used to represent the objects of the domain (including the exercises and their solutions) and has an import-export format in XML.
This language allows us to build resource editors, for example an exercise editor. Through this editor, the title and wording of the exercises can be input (in various languages) as well as a solution.Back to the top
To create the various machines of the Combien? project, we defined a formalism to specify the dialogue in an interface. This formalism, related to Petri nets, is named IREC (Interacteurs àREseaux de Contrôle: Interactors based on Control Nets). In the control net of an interactor there are variables whose values are data and controls which command the traffic of the data in the net in reaction to events on these data. IREC enables composition of interactors and communication between interactors sharing variables. AGIREC, a generic object architecture, reifies the concepts of the formalism, makes them runnable and enables to bind them with the presentation widgets. EDIREC is an interactive environment to develop interfaces which enables to edit an IREC specification and to transform it into a running code using the AGIREC architecture. EDIREC is integrated to VisualWorks and has been produced by bootstrapping. (see article1 in French, article2 in French)Back to the top
Contribution to the design of pedagogical interfaces
Inferred Lesssons for inginering
Error detection: Knowledge
In the Combien? machines, an help button is linked to each validation context. There are several kinds of help:
For each error detected, a help for understanding it is displayed. But, it is a pedagogical choice to give reflection hints instead of corrections (see article)Back to the top
Combien? proposes four "machines". They are all built as a "whole" world; they provide many contextual helps. These "machines" have been evaluated in 2002 first, in classrooms for senior high school students (Lycée carnot, Paris) , second with students in their teaching training year (in Mathematics in Paris6), third with mathematics teachers (see article). In 2003, these "machines" have been used in university Paris6 with second year students studying discrete mathematics and with senir high school students in Beauvais (Lycée Félix Faure). (see article1, article2)
Since 2004, this software is used in university René Descartes (Paris5) for second year student in a module of methodology (2004) (see article), then for second year students studying probabilities for a science degree.Back to the top
When using the software,all the events and inputs from the student are recorded in a Descript file so that the session can be re-played. This file is translated to a XML file. From XML files of all students, on one hand teachers can obtain statistics about each student production or about a group of students productions, an overview on errors which occur usually in an exercise; on the other hand students can obtain a summary about their work. (see article1, article2 in French, poster)Back to the top
Combien? software has been registered at APP(Agence pour la Protection des Programmes) in 2009
Combien? is accessible for teachers, who can choose the exercises they want to present to the students and add new exercises, and for students at Université des Sciences en Ligne (Unisciel)Back to the top