FR2817072A1 - Computer assisted interactive teaching system has means to generate personalized teaching material according to behavior of pupil and means to evaluate degree of acquisition of material by pupil - Google Patents

Abstract

A main learning unit (M) is subdivided into irreducible elements (M') which are prerequisites for succeeding with the main learning unit and are used to generate teaching material. The mastery of a main learning unit is assessed by applying weighting factors to the acquisition percentages of the irreducible constituent elements and the acquisition percentages are found from random exercises and questions

Description

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 The present invention relates to an interactive computer-assisted teaching system.

 Document US Pat. No. 5,727,950 discloses the principle of such an interactive teaching system of the type comprising at least one computer and means for interfacing with at least one pupil, said computer having to include: means for generating pedagogical data customized according to the observed behavior of said pupil, * means for extracting evaluation data representative of the degree of acquisition of said pedagogical data by said pupil,. means for constructing a profile of said student based on said evaluation data, and. means for guiding said student according to a teaching objective and said profile.

 The teaching system of this document US Pat. No. 5,727,950 assumes, in particular, the use of intelligent agents, namely software that offers personalized assistance to a user in the consultation of information organized into a complex logical structure. The concept of intelligent agent comes from the World Wide Web and originally designates software to search, filter and route to the user the information he wishes to obtain. According to the document US 5,720,950, a smart agent can make it possible to personalize the instruction that is given to each student and to evaluate and guide it in the subject taught according to criteria that are those of the school system.

 The intelligent agents postulated by this patent use as data the answers of the student to the questions put to him. The interactive teaching system described in this US patent thus makes it possible to orient a child in a base of courses and exercises, but he can not build a personal learning program that establishes a progression based on his initial profile. .

 The invention aims at overcoming the disadvantages of the prior art by providing an interactive system of teaching assisted by

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 computer that optimizes the process of teaching a subject to a student by defining a cognitive profile of the child and by providing him at each moment work priorities corresponding to this profile.

 To this end, the subject of the invention is an interactive teaching system of the aforementioned type, in which said means for generating pedagogical data comprise a set of teaching units that are irreducible in sub-parts and hierarchized one to the other. to others according to the logical dependence links of their contents, a first unit on which a second unit depends being prerequisite for said second unit, and wherein said evaluation means comprises means for measuring the degree of acquisition of said units; irreducible teaching.

 According to this arrangement, the point of a streamlined course of the notions that are prerequisite to it is an irreducible unit of instruction. By evaluating these irreducible units linked together according to a logical structure of graph oriented to form the subject to be taught, it is possible to guide a student particularly finely in his learning path.

 Preferably, the interactive teaching system according to the invention comprises one or more of the following characteristics considered alone or in combination: said measurement means comprise means for calculating the percentage of acquisition of said irreducible teaching units, said Percentage of acquisition characterizing the probability that the student will master a notion. ; said evaluation means comprise means for random generation of exercise utterances and means of notation of said student's responses to said utterances; said evaluation means comprise means for predicting the control of a unit as a function of the percentage of acquisition of each of the prerequisite units on which it depends, as well as means for diagnosing a deficiency explaining the difficulty of controlling 'a unit ;

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said diagnostic and prognostic means comprise means for weighting the acquisition percentage of said prerequisite units C (M) as a function of their distance from said diagnosed unit; said weighting means comprise means for calculating, for each of the units M constituting a prerequisite of a unit M, a value or distance D = 1-C (M ') in which:
C (M ') =% acquisition (M') if d-min (M ') = 1,
C (M ') = min ((d-min (M ") *% acquisition (M') + d-min (M ') * C (M")) / (d-min (M') + d- min (M ")) if d-min (M ') <> 1, the min being taken on all direct successors M" of M'et d-min (M') being the minimal distance between M'et M; said diagnostic means comprise means for classifying said prerequisite units in descending order of said values of D.

 said diagnostic means comprise means for classifying said prerequisite units: in ascending order of acquisition percentage if they have the same value of D,. in ascending order of minimum distance in number of arcs to M if, moreover, they have the same percentage of acquisition, in descending order of the time elapsed since their last consultation if, moreover, they are at the same minimum distance from M ,.

 . otherwise, indifferently; said diagnostic means comprise means for detecting the deficiencies associated with a unit by selection, in order of importance, of a predetermined number of prerequisite units M'whose success rate is lower than a determined average; said diagnostic means comprise means for determining a revision program associated with a unit M by virtue of the selection, in order of importance and according to a logical sequence of functions

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 said logical dependency links, prerequisite units of said unit and determination of a minimum corrected percentage% AC-min:% AC-min = E [(3 / nb-total-prerequisites + (1-3 / nb-total -pre-required) * coeff-disp * lambda) * 100] percentile, in which, the percentiles are taken from the statistical series of% AC, lambda is a parameter between 0 and 1 that determines the size of the work program and coeff-disp is a parameter between 0 and 1 that measures the dispersion of the statistical series of pre-requisites by the formula: coeff-disp = 1-standard deviation / max-deviation.

 Other characteristics and advantages of the invention will result from the following description of an embodiment given solely by way of example illustrated by the appended drawings in which: FIGS. 1 to 6 are oriented graphs respectively illustrating the logical structure of six thematic groups relating to the teaching of calculus and algebra; FIG. 7 is an oriented graph showing the different groups of FIGS. 1 to 6 with the bridges between these groups; FIG. 8 is a graph illustrating the mechanism for presenting a lesson module to a student using the system according to the invention; FIG. 9 is an oriented graph illustrating the logical structure of the modules M1 to M7 which are the prerequisites of a module M; FIGS. 10 to 12 are oriented graphs similar to FIG. 9 illustrating the determination by the system according to the invention of the most important prerequisite modules to work for a student according to the evaluation of the acquisition percentage of each of the modules M1 to M7 by the system according to the invention; and FIG. 13 is a schematic view of a networked computer structure for implementing the interactive teaching system according to the invention.

 The interactive computer-assisted teaching system described in the following uses the logical structure of the content taught. The methods

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 Bourbaki are frequently used in modern algebra to determine the structure of objects of varying size. For example, to determine the totality of finite groups, the algebrists first sought to reduce the study of the structure of a finite group to that of other smaller groups (this is called unscrewing). Then the framework of the study being sufficiently well determined, one could repeat this operation of reduction until being reduced to the study of the irreducible objects of the theory (here the simple groups). In summary, to determine a structure, it suffices to find the reduction rules and to know all the irreducible elements.

The invention will be described in the context of a mathematics course, it being understood that this is a simple example and that it can be applied to any educational content.

Règle des signes dans de nombres-la multiplication de nombres relatifs relatifs relanfs

Le chapitre Multiplication de nombres positifs ne diffère du chapitre total que par l'explication de la règle des signes, et cette explication ne peut être réduite en sous-parties sans perdre son sens mathématique. De plus, le lien entre ces deux notions est un lien logique fort : pour comprendre la multiplication de nombres relatifs, il faut d'abord avoir compris la multiplication de nombres positifs et le différentiel d'apprentissage entre ces deux notions

n'est que d'une notion irréductible (la règle des signes). If we consider, for example, the chapter concerning the multiplication of relative numbers, this one can be reduced to two nested chapters:

Ruler signs in numbers-the multiplication of relative relative numbers relanfs

The Multiplication of Positive Numbers chapter differs from the total chapter only in the explanation of the sign rule, and this explanation can not be reduced to subparts without losing its mathematical meaning. Moreover, the link between these two notions is a strong logical link: to understand the multiplication of relative numbers, we must first have understood the multiplication of positive numbers and the learning differential between these two notions.

is only an irreducible notion (the rule of signs).

MZ'M Mc nombres positifs nombres relatifs nombres positifs &verbar; ~ ~ We will note this reduction

MZ'M Mc positive numbers relative numbers positive numbers &verbar; ~ ~

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The arrow indicates the direction of the logical relation, and one will say that the Multiplication of relative numbers admits as Prerequisite the Multiplication of positive numbers.

 This reduction operation applies to each notion of the course: one determines for each point of the course the list of the concepts which are prerequisite to it. The point of the course, purified of the notions which are prerequisites to it is then an irreducible unit of teaching. These units are called irreducible teaching units. Thus, it naturally appears an oriented graph structure whose vertices are the modules and whose oriented links correspond to the notion is a direct pre-requisite. Starting from the postulate of transitivity of the prerequisites (if A is necessary to understand B and if B is necessary to understand C, then A is necessary to understand C), the total list of concepts that are prerequisites to the understanding of a module is obtained in this graph as being the list of parent vertices of the module in question.

 As an example, we will consider that the general graph, constituted by the totality of the modules of the calculus and the algebra of a given school program, is broken down into six thematic groups: Numerical Calculation, Fractions, Literal Calculation, Functions & Data management, Equations & systems, Order relationship & inequations.

 Each module belongs naturally to a single group but sometimes depends on modules that are in other groups. These links between modules of different groups are bridges between the different topics taught.

 Each group can be broken down into sub-groups of fixed grade level, formed by the modules concerning a point of the program of this or that class (6th, 5th, etc.). All of these groups and subgroups, with their interconnections, represent a leveled thematic table of contents for the college program. In fact, the nonlinear structure of the network makes this table of contents more like a map of subjects giving a more exploded vision but closer to the logic of the school curriculum.

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 The six thematic groups (Numerical Computation, Fractions, Literal Calculation, etc.) are illustrated respectively in Figures 1 to 6 by sub-graphs formed, for each group, by the modules belonging to it as well as the bridges between these modules and the other groups. Each module is identified by a code: in the notation a. b. c, a corresponds to the group number, b to the school level where this concept is addressed and c an identification number. For example, module 2. 5. 7 is module number 7 of group 2 (fractions) and grade level of Sem. Figure 7 illustrates the different groups and bridges between these groups.

 In general, the modules describe a definition, a property, a theorem or a method. They fall into two categories. The first consists of lesson modules; it corresponds to the explanation of an original concept or a new idea. The second consists of the methodology modules; we do not learn a new concept but rather a new way of using our knowledge to solve a classic type of problem.

 The treatment of these two types of modules differs because the student may appear to be more passive in one case than in the other. In fact, in the first case, the student is asked for more understanding (and therefore reflection) whereas in the second case, the student is required to have a greater ability to adapt and repeat a method. This distinction would correspond to that which is made between a lecture and a session of Directed Works.

 The lesson modules are used to present then define a notion, or introduce and then state a property or theorem.

exposé à l'élève, suit le plus souvent possible ce que l'on appelle l'arche de la connaissance illustré par le graphique de la figure 8 : - on commence en 10 par introduire la notion à partir d'un problème ou d'un constat logique accessible à un élève qui domine les bases nécessaires : on donne ainsi le but du module. The logical plan of the contents of the module, that is to say, its way of being

exposed to the pupil, follows as often as possible what is called the ark of knowledge illustrated by the graph in figure 8: - we begin by introducing the notion from a problem or from a logical statement accessible to a student who dominates the necessary bases: one thus gives the goal of the module.

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 - we study in 11 a very simple example which allows to describe the notion or the result without having to formalize it: one thus makes work the intuition of the pupil.

 - we formalize and mathematize in 12 the concept by stating the definition or the theorem concerned: we give the mathematical heart of the module.

 - we give in 13 more complex examples of use of the property or understanding of the definition: we show how we can apply the notion.

 - 14 different exercises are proposed to the student: one evaluates his understanding.

 As shown in Figure 8, this arch process is based on a student's acquired base and returns through evaluation to a new estimate of the student's level, and its peak corresponds to the mathematization of the concept. that is, at the most complicated and important point of the module.

 Thus constructed, the module describes the different phases necessary for the assimilation of a notion by a student.

 Evaluations are based on original exercises of various levels, corresponding exactly to the module (so few exercises synthesis ...).

Their statement is based on that of standard exercises, for example 6 in each module, in which the variables are set randomly. We therefore have for each module a very large stock of different exercises (sometimes several thousand).

 The exercises can be of type QCM, sentences with holes, questions with alphanumeric entry of the answer to the keyboard, or graphical operation with the mouse (action on a sensitive part of the screen, drag-and-drop of objects etc. ). Some will also be realized in limited time.

 Each exercise is thus composed of a routine of generation of the random constants that it uses, of a statement and a correction (text and sometimes images) generated by these constants, of an interface of capture of the answer and of correction of the veracity of this answer. A computer format is pre-defined in the software model.

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The evaluation will be described in more detail later.

The purpose of the methodology modules is to help a student assimilate an exercise resolution method after understanding and understanding the lesson.

 A methodology module begins with the data of an exercise statement, such that the student could meet in a book. The screen is then split into two windows. In the first, is the statement of the exercise that the computer will solve in front of the student. In the second, there is a similar statement (with the same logic as the previous one but with different numerical values); the student is responsible for solving by following the method used by the computer step by step.

 Thus, at each elementary stage of the resolution, the computer explains what it will do, then realizes it and finally displays in the window reserved for the example of the pupil a sentence with holes similar to the one that comes from display the computer (holes that can be filled with words, numbers, or symbols). The student must then fill it correctly, using the numerical values of his own statement.

 If the student fails to complete his sentence correctly, he has the opportunity to force the computer to answer in his place (and therefore to see a correction of his mistakes and / or misunderstandings). However, the computer blocks the exercise and allows the student to start at the beginning with new numerical values, and so on until the student succeeds in reproducing the method without any fault.

 An evaluation similar to that of the lesson modules is planned at the end of a methodology module. However, here the complete reading of the module implies a minimum knowledge of the level of the student: the latter has been able to reproduce the method in a case certainly close but not exactly identical. In general, we will have more information on the level of the student that the interactivity of the module will be large.

 In order to locate in this logical structure that is the oriented graph of the modules, the user needs first of all a visual representation and a map. This visual representation must be simple enough and

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 Intuitive to be accessible to all, as well as rely on strong logical rules to allow to set up a narrative code that is the test of time and the variation of the contents.

 Thus the modules and their links, illustrated in Figures 1 to 6, are grouped, no longer by themes, but this time by school year, and by distributing the modules in such a way that, if we superimpose the different performances per year, two modules of two distinct years, one of which is prerequisite to the other, are located one above the other. This leads to a three-dimensional representation of the module graph.

 In pictorial language, the oriented graph put in three dimensions can be likened to a city or a palace in which there are six buildings, corresponding to the six thematic groups described above. Each building is divided for example into four floors, one per grade level. These floors consist of rooms. Each room is associated with a module and contains doors and hatches, which lead to corridors, walkways or stairs, themselves leading to other rooms. These corridors, walkways or other stairs are in correspondence with the logical links of the oriented graph.

 This image offers a topological representation of the knowledge taught and a simple visual identity to the complex graph of the modules, offering many mnemonic means of learning to the student. Faced with this city, the user behaves like an explorer who, to learn mathematics, must discover the secrets contained in each room. The logical principle that one can only understand a notion if one has read and understood all those that are prerequisites to it, translates into an obvious principle of everyday life: one explores a house that 'floors after floors, rooms after rooms, making sure that the previous explorations have been well done.

 In order to control and guide the hypermedia multidimensional reading of the user, the computer makes a permanent evaluation of the degree of acquisition of each concept and of each competency involved by the student.

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 This evaluation is proposed automatically after consulting the content of a module, but can be started at the request of the student itself, during a new passage in the room of the module. The student is offered exercises of three different levels: - Direct application of the module: here, the exercises are simple and fast (Q. C. M. most often); they allow to evaluate the elementary understanding of the module (the vocabulary, the exposed examples etc.); in the case of the methodology modules, these exercises are integrated in the course consultation.

 - Comprehension of the module in depth: here, the exercises are more complicated and allow to evaluate the understanding of the subtleties presented in this module (more complex use cases etc.).

 - Appropriation of knowledge: here, the resolution of exercises requires an effort of imagination and synthesis that allows the student to understand how to use this module in cases that require more autonomy on his part; he integrates here the new notion in his mathematical baggage.

 The mechanisms for determining a percentage of acquisition of a module from the results of a series of exercises will now be described. To avoid the definitive and sometimes traumatic aspect of the traditional school evaluation, the percentage of acquisition of a module must be a weighted average of the statistical percentages of success in the proposed exercises.

 For each level of difficulty, there are two typical exercises that can be declined a very large number of times. The student can evaluate himself as many times as necessary on each of the levels without risk of learning the answers by heart. For each typical exercise, the computer takes into account the results of the last five attempts (which may not have all occurred in the same session) and gets a percentage of success in this exercise. This percentage, on the other hand, becomes significant only after a certain minimum number of attempts.

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 Each level of difficulty is composed of two typical exercises which each represent a part of the success rate of the exercises of the level of difficulty in question (this part is variable and depends on the exercises).

 The percentage of acquisition of the module is a weighted average of these three percentages of success per level. The relative weights and calculation methods for exercises performed as part of a methodology module and as part of a lesson module are given in Tables 1 and 2, respectively. There are p1, p2 and p3. Percentages of success at Level 1,2 and Level 3 exercises. A trusted student is allowed not to evaluate on Level 1 or Level 1 and Level 2 exercises for Lesson Level 2 or Level 2 modules. the case of the methodology modules (it is impossible to avoid the level 1 evaluation that takes place when reading the module).

Table 1

<Tb>
<tb> Exercises <SEP> performed <SEP> weight <SEP> relative <SEP>% <SEP> of acquired
<tb> module <SEP> of <SEP> method. <SEP> level <SEP> 1 <SEP> level <SEP> 2 <SEP> level <SEP> 3
<tb> levels <SEP> 1,2 <SEP> and <SEP> 3 <SEP> 30% <SEP> 30% <SEP> 40% <SEP> 0, <SEP> 3 * p1 + 0, <SEP> 3 * p2 + 0, <SEP> 4 * p3
<tb> levels <SEP> 1 <SEP> and <SEP> 3 <SEP> 43% <SEP> xxx <SEP> 57% <SEP> 0.43 * p2 + 0, <SEP> 57 * p3
<Tb>
Table 2

<Tb>
<tb> Exercises <SEP> performed <SEP> weight <SEP> relative
<tb>% <SEP> d <SEP> Acquisition
<tb> module <SEP> of <SEP> lesson <SEP> level <SEP> 1 <SEP> level <SEP> 2 <SEP> level <SEP> 3
<tb> levels <SEP> 1,2 <SEP> and <SEP> 3 <SEP> 20% <SEP> 40% <SEP> 40% <SEP> 0, <SEP> 2 * p1 + 0, <SEP> 4 * 0.4 * p2 + p3
<tb> levels <SEP> 2 <SEP> and <SEP> 3 <SEP> xxx <SEP> 50% <SEP> 50% <SEP> 0.5 * p2 + 0, <SEP> 5 * p3
<tb> level <SEP> 3 <SEP> xxx <SEP> xxx <SEP> 100% <SEP> p3
<Tb>

L'ordinateur qualifiera le module de non évalué si aucun exercice n'a été tenté. Si par contre trop peu d'exercices ont été tentés (soit moins de deux exercices différents ont été tentés, soit moins de la moitié des exercices tentés ont une pourcentage de réussite non significatif), le pourcentage d'acquisition ne sera pas affiché et sera qualifié de non significatif.

The computer will qualify the module as unassessed if no exercise has been attempted. If, on the other hand, too few exercises have been attempted (ie less than two different exercises have been attempted, ie less than half of the exercises attempted have a non-significant success rate), the acquisition percentage will not be displayed and will be qualified as not significant.

 Using dynamic global data about a student and his or her use of the system and dynamic local data for each module that the student accessed, the system builds an electronic profile of this student.

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 This electronic profile is updated automatically and continuously: each student's operation results in an infinitesimal modification of his electronic profile. The consultation of the module is any reading of its course, followed or not evaluation but also any evaluation session without reading. A session is defined as any uninterrupted sequence of consultations, that is, any continuous session of use of the system.

The data constituting the electronic profile are as follows:
Global data: - User name, login and password
Current school level of the user (possibly school, class name, options ...) - Number of modules consulted at least once, percentage of the total number - Total number of consultations made (here several per module can be counted ) - Average duration of a module consultation - Total number of exercises attempted (respectively successful) - Average number of exercises performed (respectively successful) per module.

 - Average number of exercises performed (respectively successful) per consultation.

- For each session: duration of use of MATH S, number of modules consulted, number of exercises performed (respectively successful) and consultation history
Local data:
The list of all previous consultations (date, duration,% acquisition at the end of the consultation, number of exercises attempted (respectively successful) - Number of consultations - Total consultation time - Percentage of acquisition of the module - For each exercise of this module:

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- Value of the weighting coefficient (ie of difficulty) in the calculation of the acquisition percentage (this currently fixed figure can be brought to vary with the level of the student) - Total number of attempts - Total number of successes - Table of last five results
The electronic profile is a computer file whose format is prepared for processing within the system, for export to a spreadsheet and / or a database, for transmission over a network (possibly secure) and finally for a synchronization between different places of use (school and house for example).

 The electronic profile is a true mathematical assessment of the student in mathematics, which takes as much into account his results at the time of the consultation as the evolution of these results.

 A first use of the electronic profile is statistical.

 There are many histograms, pie charts, and so on. describing the student's progress as well as a series of data describing the exact situation of the student at a given moment. We then have a powerful tool of monitoring and control at several levels: - Self-control of the user: the student can by consulting, printing and archiving these statistical data to realize and monitor its progress.

 - Control and monitoring of parents / teachers: the computer keeping in memory the behavior of the student, one can easily check his attendance to work, the evolution of his performances etc. These reports play the role of both the quarterly report card for the comprehensiveness of the information obtained and the liaison book for the permanent aspect of the link that is created between parents and teachers.

 - Control of progress by the computer: hypermedia functions guide the student in reading backwards or downwards)), to search for bases not understood. However, it is unlikely that a student will be totally free in his navigation within the module graph,

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 know how to move appropriately and can decide how to progress.

Thus the standard user will want the computer to guide him in the order of chapters and modules to read. A module is validated if its acquisition percentage is greater than 40%. Navigation control follows the following principle:
It is impossible (because pedagogically useless) to access a module whose prerequisites are not validated.

 Optimally, the pupil must start the use of the system by the first chapter of the basic course (6th): integers and decimals (module 1.6.1). Indeed, this module, real gateway to the city, is the only one to have no other module prerequisites and to be prerequisites of all other modules. The computer will block the student on this module until it is validated. This validation will be the equivalent of a key that will open the logical doors to the following room (s). Thus, the pupil is like an explorer lost in a labyrinth of which he knows only the pieces he has already visited. The tests to discover the names and the contents of the following parts are the exercises that the computer proposes to him. However, following the principle highlighted previously, the student can only enter a module if all the doors or stairs that access it are open, which implies that the validation of a module is not always enough to open all the accesses connected to it. Thus, this mode of navigation, which we will describe as controlled, allows a student to freely and quickly access the basics that he does not dominate enough (although validated) via hypermedia links, on the other hand he does not authorizes to progress in its reading only if it possesses the minimum of necessary bases validated.

 If the pupil starts his use at a level other than the basic one (the 6emu), the computer makes accessible all the entries of the floor corresponding to his level (an entry is here a module which has all its prerequisites in the lower levels) and uses the temporary percentages obtained by the synthesis exercises to define whether the modules of the previous years are a priori validated. All the basics at the student level are

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 therefore accessible via hypermedia links but will not necessarily validated.

 The previous navigation models have the disadvantage of sometimes requiring students to a long job before accessing the desired module. Even if, pedagogically, it is always better to go back to unfulfilled bases before progressing in a class, the rhythm of the student's class could compel him to wish to work on a specific chapter (for a homework assignment for example). that the computer would prevent it. Finally, the student would not be able, in case of total control of the computer, to give free rein to his curiosity in the exploration of the mathematics course.

 It is therefore offered to the student a free navigation mode, in which all access to all modules are open. In this case, the student can move in any direction of the buildings and explore according to his curiosity or his school requirements the different modules.

 A student in free navigation mode can also self-evaluate the different modules he consults. However, only evaluations of the modules accessible to him in controlled navigation mode are counted in his electronic profile. To move from one module to another, the student has several possibilities hypermedia type. It can use hypertext links, to go from a module to a prerequisite module. It can also use a building plan to go from one module to another module which is prerequisite. He can simply navigate on sight using a 3D representation of the module and its doors and stairs to other modules.

 Finally, the student can use a search engine to locate the module or modules that are related to the part of the course he wants to use. Each module is therefore associated with a series of keywords that are specific to it. A student typing a word like fraction sees appearing the list formed by all the modules associated with the keyword fraction. He just has to click on the module of his choice to access it.

 There is a second use of the electronic profile of a user, based on an algorithmic approach of type graph and

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 to provide, via intelligent agents, a system for detecting gaps and determining work program.

 The intelligent agent concept comes from the World Wide Web. An intelligent agent is software that provides personalized assistance to a user in the consultation of information organized into a complex logical structure. This help is based on a precise knowledge of the user's needs and prior behavior and a detailed analysis of the navigation structure and the information it contains.

In the system described here, the complex data structure is the oriented graph of the modules. The user is certainly offered simple navigation and search systems via multidimensional hypermedia reading and is even assisted by the computer in its progression in the controlled navigation mode. However, many pedagogical questions remain unresolved: - When the student fails to validate a module, how to be sure that the source of the errors is related to the very concept of the module and not to a distant gap whose pupil is not necessarily conscious? - How in general to identify the deficiencies related to a module or set of modules and propose a program of revisions that allow the student to fill these gaps? - If we want to understand a notion that is explained in a module that is not yet accessible (many modules not yet acquired or consulted separate us from our goal), what is the optimal work program (in terms of time and chances of success) to access it without skipping pre-requisites but without getting lost by reading notions incidental to our problem?
To meet these pedagogical needs, the cognitive model of mathematical knowledge that is the oriented graph of the modules and the exhaustive knowledge of the prior behavior of the user in relation to this knowledge, contained in his electronic profile, are used. These parameters are integrated into an intelligent agent whose goal is to answer these pedagogical questions as independently as possible.

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The answers given by the intelligent agent are essentially based on the algorithm described below.

 Consider an oriented graph of modules, each module of which is evaluated by the percentage of acquisition of the electronic profile and point out a particular module, called M. The algorithm will give the list of the prerequisites of M, whose acquisition percentage is less than a given percentage pmin and within this subgraph, the algorithm will rank the prerequisites to M in order of importance with respect to the understanding of M, in a sense that takes into account the distance in number of arcs between the prerequisites and M as well as the percentage of acquisition of these prerequisites or the time spent since the last reading.

 The exploration algorithm itself will not be detailed: it is of the Breadth-First Search type and quite fast since the graph is small and has no circuit.

min ( (d-min (M") *% acquisition (M') +d-min (M') *C (M"))/ (d-min (M') +d-min (M")) ; si d-min (M') < > 1 le min étant pris sur tous les successeurs directs M"de M'. La moyenne utilisée ci dessus est prise en pondérant le % d'acquisition de M'par l'inverse de d-min (M') et C (M") par l'inverse de d-min (M"). Comme la distance minimale représente l'éloignement logique entre M et son pré-requis, la valeur 1-C (M') apparaît comme étant le pourcentage d'importance de M'par rapport à M qui tient compte du niveau de l'utilisateur. Cette valeur est notée D (M'). Ainsi plus la notion est éloignée, plus ce nombre risque de diminuer mais un pré-requis On the other hand, the technique of weighting paths that makes it possible to determine the classification of the pre-requisites sought will be explained below. With each pre-requisite I associate a minimum arc distance between M'et M, noted d-min (M ') (which is easily calculated during the implementation of the BFS algorithm) and a coefficient C (M ') defined recurrently as follows:
C (M ') =% acquisition (M') if d-min (M ') = 1,
C (M ') =

min ((d-min (M ") *% acquisition (M ') + d-min (M') * C (M")) / (d-min (M ') + d-min (M ")) if d-min (M ') <> 1 the min being taken on all the direct successors M "of M" The average used above is taken by weighting the acquisition% of M'by the inverse of d -min (M ') and C (M ") by the inverse of d-min (M") Since the minimum distance represents the logical distance between M and its prerequisite, the value 1-C (M') appears as the percentage of importance of M'compared to M which takes into account the level of the user.This value is denoted by D (M '), so the further away the notion is, the more likely it is to decrease but a pre-requisite

<Desc / Clms Page number 19>

 more distant than another may be considered more important if it is much less dominated than the other.

The prerequisites can therefore be ranked in descending order of importance by following the lexicographic order applied to the triplet (D,% acquisition, d-min, time-out):
Two prerequisites are classified:
1) In descending order of the values of D;
2) If they have the same value of D, in ascending order of the acquisition percentage;
3) If, moreover, they have the same percentage of acquisition, in ascending order of minimum distance in number of arcs to M;
4) If, moreover, they are at the same minimum distance from M, in descending order of the time elapsed since their last consultation.

 5) Otherwise, indifferently.

The passage of time (measured by the time-elapsed variable) can be integrated more finely in the ranking of prerequisites. Indeed, suppose that we have two pre-requisites that have almost the same importance with respect to M (via the coefficient D) but such that the slightly higher pre-requisite was consulted the same day, whereas the pre-requisite - slightly less important issue was last consulted several months ago. The classification explained above does not take into account the passage of time which, intuitively, indicates to us that it is the prerequisite which has not been consulted for several months which should be more important. The passage of time is integrated by introducing the concept of corrected acquisition percentage (% AC) defined by:
If time-lapse <30 days, then% AC =% acquisition;
Otherwise% AC =% acquisition / log (elapsed time), the elapsed time being counted in numbers of days and the logarithm being taken in base 30.

 Table 3 gives values of the percentage of reduction allowing to go from the percentage of acquisition to its corrected value:

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Table 3

<Tb>
<tb> time <SEP> elapsed <SEP>! <SEP> to <SEP> 30 <SEP> 60 <SEP> 120 <SEP> 180 <SEP> 365 <SEP> 730 <SEP> 1460
<tb>% <SEP> of <SEP> Reduction <SEP> 100, <SEP> 00% <SEP> 94, <SEP> 61% <SEP> 87, <SEP> 77% <SEP> 83, <SEP> 40% <SEP> 75, <SEP> 80% <SEP> 68, <SEP> 89% <SEP> 62, <SEP> 80%
<Tb>

It is then useful to replace in the above description the acquisition percentage by its corrected value.

 These notions-except that of the passage of time-will be illustrated hereinafter by examples related to the oriented graph of FIG. 9 on which the modules M1 to M7 are prerequisites of the module M.

 Several cases describing the calculations of C and D, for each of the modules M1 to M7 as a function of its distance d-min to M and its acquisition percentage (% acquisition) are given in the following tables, then the response of the algorithm in the case where p-min is fixed at 60% (one starts directly from the graph of the prerequisites of M) is determined. The influence of time is not taken into account: for this it suffices to replace the acquisition percentage with the% AC.

Example 1
The values of C and D for each of the modules M1 to M7 are given in Table 4.

Table 4

<Tb>
<tb> Module <SEP> M1 <SEP> M2 <SEP> M3 <SEP> M4 <SEP> M5 <SEP> M6 <SEP> M7
<tb> d-min <SEP> 1 <SEP> 1 <SEP> 2 <SEP> 2 <SEP> 2 <SEP> 2 <SEP> 3
<tb>% <SEP> acquisition <SEP> 20% <SEP> 50% <SEP> 60% <SEP> 80% <SEP> 40% <SEP> 50% <SEP> 60%
<tb> C <SEP> (%) <SEP> 20% <SEP> 50% <SEP> 53% <SEP> 60% <SEP> 27% <SEP> 30% <SEP> 42%
<tb> D = 1-C <SEP> (%) <SEP> 80% <SEP> 50% <SEP> 47% <SEP> 40% <SEP> 73% <SEP> 70% <SEP> 58%
<Tb>

la mesure où il est le plus près et le moins bien acquis), que la branche M1, M5 est la plus faiblement acquise, suivie par les branches M1, M6, M7 (principalement la partie M1, M6) et que ces deux branches son nettement moins acquises que les deux autres. In this first example, if we intuitively try to determine the most sensitive prerequisites for understanding M, we conclude that M1 is probably the most important module with respect to M (in

the extent to which it is nearest and least well acquired), that the branch M1, M5 is the weakest acquired, followed by the branches M1, M6, M7 (mainly the part M1, M6) and that these two branches are significantly less than the other two.

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 We find these diagnoses by ordering the vertices according to the values of D: the three least well-acquired modules with respect to M are very clearly M1, M5 and M6.

 We note that M6 is more important if we count it in the branch that passes through M1 than if we count it in the branch that passes through M2. This is why we retain the lowest value, which better describes the role of M6.

 The response of the algorithm is illustrated in Figure 10 which shows that the most important modules are in descending order of importance: M1, M5, M6, M7.

Example 2
The values C and D for each of the modules M1 to M7 are given in Table 5.

Table 5

<Tb>
<tb> Module <SEP> M1 <SEP> M2 <SEP> M3 <SEP> M4 <SEP> M5 <SEP> M6 <SEP> M7
<tb> d-min <SEP> 1 <SEP> 1 <SEP> 2 <SEP> 2 <SEP> 2 <SEP> 2 <SEP> 3
<tb> ~ <SEP>% <SEP> acquisition <SEP> 70% <SEP> 50% <SEP> 80% <SEP> 30% <SEP> 50% <SEP> 20% <SEP> 90%
<tb> C <SEP> (%) <SEP> 70% <SEP> 50% <SEP> 60% <SEP> 43% <SEP> 63% <SEP> 40% <SEP> 60%
<tb> D = 1-C <SEP> (%) <SEP> 30% <SEP> 50% <SEP> 40% <SEP> 57% <SEP> 37% <SEP> 60% <SEP> 40%
<Tb>

Here, the distribution of the percentages of acquisition is more irregular but remains rather homogeneous. The main trends are that the problems are in the M2 branch, M4 and the M6 module mainly. We find again these diagnoses by comparing the different values of D, which are here more homogeneous. It should nevertheless be noted that this weighting system gives M3 and M7 a relatively high relative importance with respect to the M5 module. This is why the complete algorithm couples this weighting system with a pruning of the pre-requisite graph (which is here a tree!): After the determination of the coefficients D, we remove from the graph of the pre-requisites those which have a high percentage of acquisition.

 The response of the algorithm is illustrated in Figure 11: the most important root is M6 and the most important modules are in descending order of importance M6, M4, M2.

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Example 3
The values of C and D for each of the modules M1 to M7 are given in Table 6.

Table 6

<Tb>
<tb> Module <SEP> M1 <SEP> M2 <SEP> M3 <SEP> M4 <SEP> M5 <SEP> M6 <SEP> M7
<tb> d-min <SEP> 1 <SEP> 1 <SEP> 2 <SEP> 2 <SEP> 2 <SEP> 2 <SEP> 3
<tb>% <SEP> acqulsltlon <SEP> 0% <SEP> 0% <SEP> 0% <SEP> 40% <SEP> 70% <SEP> 70% <SEP> 90%
<tb> C <SEP> (%) <SEP> 0% <SEP> 0% <SEP> 0% <SEP> 13% <SEP> 23% <SEP> 23% <SEP> 50%
<tb> D = 1-C <SEP> (%) <SEP> 100% <SEP> 100% <SEP> 100% <SEP> 87% <SEP> 77% <SEP> 77% <SEP> 50%
<Tb>

Dans ce dernier exemple, trois modules n'ont pas encore été consultés et le module M4 est de loin le plus important. Le calcul des coefficients D traduit à nouveau bien la situation.

In this last example, three modules have not yet been consulted and the M4 module is by far the most important. The calculation of the coefficients D again reflects the situation well.

The response of the algorithm is illustrated in Figure 12: the most important root is M3 and the most important modules are in descending order of importance: M1, M2, M3 and M4, only the last one having already been consulted.

The detection of gaps (DL) and the determination of a revision program (DPR) A pupil may have difficulty increasing his percentage of acquisition of a certain module M. This may be due to the notion explained in M but this can also be related to a prerequisite of M, validated but poorly dominated. Any deficiencies of the student with respect to this module must be studied to ensure that it is the new notion that poses a problem.

This difficulty may be voluntarily reported to the computer by the student; It can also be diagnosed with a high number of exercises performed (ie above average) associated with a very low success rate below the average (far below average means below average) standard deviation) or at a high reading time with a very low acquisition percentage or any other negative combination of time criteria, number of years and percentage of acquisition.

Whenever such a difficulty arises, the computer offers the student the choice between looking for the gaps that seem the most urgent

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 to fill or the exhaustive determination of all the prerequisites that it would be good to upgrade.

 In the first case, the computer starts the DL. It determines through the key algorithm, the list ranked deficiencies in order of importance. It calculates the mean and the standard deviation of this series and proposes the ranked list of the first three deficiencies in order of importance which it then completes by the following, as long as their importance is greater than the average plus the preceding standard deviation . Students can access the corresponding modules, read them and evaluate themselves again. Once his or her electronic profile has been modified, the student can return to the evaluation of the problem module at any time or request an update of the search for gaps and the identification of those that seem most important (the determination of D takes into account all the acquisition percentages of all the successors of the module, modifying a single percentage can therefore cause a variation of many coefficients D and change the composition and order of the list of gaps to be filled).

 In the second case or if after consultation of three modules that represent significant gaps (determined by the DL), the student has still not validated the module M, the computer begins the determination of a work program DPR .

 These two procedures are quite similar but there is a fundamental difference between DL and DPR. In the case of DL, it is suggested to the student to first review the most important gaps, which are often also the closest ones in the prerequisite tree, hoping to solve the problem quickly. This rapid method, however, may fail if the understanding of these gaps themselves is based on the prior understanding of other, more distant gaps, and so on.

 The DPR has been designed to overcome this defect. In this case, the computer offers the student to resume, in the logical order, all the gaps, from the farthest to the nearest. This work is much longer for the student but in this case, we are sure to rely on basics understood or at least read.

<Desc / Clms Page number 24>

 The DPR begins just like the DL by determining, through the key algorithm, the ranked list of prerequisites in order of importance.

The computer then determines a minimum percentage (corrected)% AC-min, which will allow it to detect the prerequisites that seem good to upgrade, by the following formula:% Arc-mon = E [(3 / nb-total-prerequisite + (1-3 / nb-total-prerequisite) * coeff-disp * lambda) * 100] percentile.

 In this formula, the percentiles are taken with respect to the statistical series of% AC, lambda is a parameter between 0 and 1 which is chosen by the user in order to control the size of the work program (by limiting the number of modules selected ) and coeff-disp is a parameter between 0 and 1 that measures the dispersion of the statistical series of pre-requisites by the formula: coeff-disp = 1-standard deviation / max-deviation.

 The coeff-disp values for different lambda values are given in Table 7 in the case of Examples 1 to 3.

Table 7

<Tb>
<tb> one hundred <SEP> selected <SEP>;<SEP>%<SEP> AC-min
<tb> lambda = O <SEP> lambda = 0.25 <SEP> lambda = 0.5 <SEP> lambda = 0, <SEP> 75 <SEP> lambda = 1
<tb> Ex <SEP> 1 <SEP> 0.4508 <SEP> 42 <SEP>;<SEP> 50% <SEP> 49 <SEP>;<SEP> 50% <SEP> 55 <SEP>;<SEP> 53% <SEP> 62 <SEP>;<SEP> 57.2% <SEP> 68 <SEP>;<SEP> 60%
<tb> Ex <SEP> 2 <SEP> 0.3331 <SEP> 42 <SEP>;<SEP> 50% <SEP> 47.50% <SEP> 52.52, <SEP> 4% <SEP> 57.58, <SEP> 4% <SEP> 61 <SEP>;<SEP> 50%
<tb> Ex <SEP> 3 <SEP> 0, <SEP> 2995 <SEP> 42, <SEP> 20.8% <SEP> 47.32, <SEP> 8% <SEP> 51 <SEP>;<SEP> 41.8% <SEP> 55.49% <SEP> 59 <SEP>;<SEP> 56, <SEP> 2%
<Tb>

Then, the prerequisite tree is pruned by removing the vertices that have a% AC greater than% AC-min. The tree obtained is that of the prerequisites that the computer suggests to revise. This choice is motivated to take into account the dispersion of the series while imposing that the number of pre-requisites selected is at least three. The more scattered the series are (the values are far from the average), the more coeff-disp is close to 0 and the lower% AC-min is low, which corresponds to the fact that the pre-requisites being statistically either very dominated or very little dominated, it is good to suggest revision only to those who are very weakly dominated. Conversely, the more the series is grouped, the more coeff-disp is close to 1 and the more% AC-min is high. In

<Desc / Clms Page number 25>

 In this case, the prerequisites are essentially all the less or less well controlled and the computer will not be able to remove as much in its pruning.

 Once the pruning is done, all modules not in the final tree are blocked and any shortcuts between the selected prerequisites are created to describe all branches of the tree: from one point of logical view, we replace the oriented graph of all modules by this sub-graph of the selected pre-requisites. The reading and evaluation rules in this subgraph are always the same, however only the help bubbles available while the underlying hypertext links are blocked.

 In addition, the computer ranks in descending order of importance the list of possible starting points for reading (which are the leaves of our tree). He thus suggests the starting point of the most important way to understand the final notion.

 Finally, the computer guides the student in his reading as follows: after the consultation of a pre-requisite, the student can continue to progress in the path via the rules of multidimensional reading in the sub-graph, the access to modules that open as soon as all their prerequisites have been consulted (so we can continue until we reach a Y-type branch in reverse, ie a successor who has different prerequisites in the trees). If it can not continue this reading (or do not want to continue it alone), the computer eliminates from the tree the module that has just been consulted; new entries will appear. Then it takes again the graph of all the prerequisites and starts again these calculations taking into account the modification brought by the consultation (even if the pupil does not evaluate himself, the reading can modify the time-elapsed parameter and thus the value % AC). He then suggests the starting point of the new, most important path to understanding the final notion.

 Once the final notion is reached, if the user still can not improve, the computer offers a new revision program, the determination rules are the same but% AC-min is increased by 20 points. If after completion of this new revision program, the student

<Desc / Clms Page number 26>

 still can not validate the module, it is clear that what blocks it is contained in the module itself and not in its prerequisites.

Determining a work program (TPD)
As in any system that allows access to a large amount of information, it is important to include a search engine that locates information about a keyword. A library-style indexing system of the Library of Congress is planned (it can be semi-automated by the presence of summaries for each module).

However, a search engine just locates information but does not consider the user requesting the information. This problem is at the root of the creation of intelligent agents on the WWW.

Given the reading modes in the system described, it is also in this case, an interesting problem. Indeed, if the student is looking for a module that he has already consulted, it is not absurd to allow him to access it via a search engine. On the other hand, if he looks for information that he has not validated all the prerequisites, he can understand nothing by accessing it directly.

 This scenario is very common: a student may wish to prepare a classroom assignment on equations; it locates the associated modules in the graph but may not have the basics necessary to understand them. It is therefore not pedagogically desirable for him to have direct access to these notions.

 In general, the ideal pedagogical work from the most effective point of view is one that begins where our knowledge ends and whose reading leads us to an understanding of the notion (s) of interest to us as soon as possible ( so avoiding unnecessary digressions ...). Thus, if the student asks the search engine to locate a module, the computer should give him not only the position of the module sought but also determine if necessary a work program allowing him to move from his current level of knowledge to level necessary to understand the desired module. The algorithm that undertakes such a determination is the DPT. It will be described as part of its use via a

<Desc / Clms Page number 27>

 search engine. A student has entered one or more keywords into the computer.

 First suppose that a single module corresponds to these keywords. The user is offered the choice between two reading options: do not consult validated modules, or check again the validated modules that the computer has considered the least well acquired. In the first case, the algorithm used is almost identical to the DPR but without taking into account the passage of time (so we do not use the% AC) and by removing from the tree the prerequisites that have a percentage of acquisition greater than the validation percentage.

 In the second case, or if the student fails to reach and validate the desired module using only unvalidated modules, the computer launches the complete DPR already explained.

 If now, several modules correspond to these keywords, the computer will determine the sub-graph of the prerequisites of these modules (which no longer has any reason to be a tree) and the list of leaves (outgoing vertices). and roots (incoming vertices) of this subgraph. For each sheet, the computer will associate a coefficient D with each prerequisite of this sheet. Then, for each element of this subgraph, it keeps as final value which measures its importance, the greatest value of the coefficients D which have been calculated to it. We then obtain a lexicographic classification of the roots in descending order of importance (the parameter d-min of the lexicographic order is here the minimum distance from a pre-requisite to any sheet of the subgraph). At this point, the user must choose between the two playback options again. In the first case, the computer will just remove the already validated modules from the subgraph and then propose the starting point of reading which seems to him the most important. The reading rules are then the same as for the DPR.

 In the second case or if after consultation of the previous work program, at least one module is not validated, then the computer analyzes the previous analysis as follows:

<Desc / Clms Page number 28>

requis sont celles de la DPR mais appliquées à l'ensemble des modules du sous-graphe.

required are those of the DPR but applied to all modules of the sub-graph.

What really distinguishes TPD and DPR is time management.

In the case of the DPR, the temporal constraint is of the type as fast as possible. Indeed, the DPR, like the DL, intervenes by interrupting a normal reading. Thus, the diagnosed revisions must take place as quickly as possible. On the other hand, in the case of the TPD, there is a voluntary action and out of normal consultation of the user. This can therefore have different time constraints (for example, the class duty on the desired notion takes place in 10 days).

 Once the deadline is entered, the computer displays a calendar in which the student enters the number of hours he has per day to use the system. From there, the computer deduces the average number of modules that must be viewed per session, and then more specifically, it suggests a program of work day after day. He proposes to start with the most important root, then to go back to find a Y-branch upside down. He then moves on to the next sheet and then goes back up and so on. Once all the sheets have been exhausted and all the modules that have been programmed for reading have been deleted, the computer starts the process again from the new roots (which are the branches mentioned above).

When the order is determined, the modules are distributed per day avoiding where possible the separation of paths on two sessions.

 Once the program of work has begun, the computer controls its realization, modifies it according to the work of the pupil, etc.

 In order to be able to take into account in real time changes and updates to the content and structure, the addition of new buildings or floors and improvements to navigation tools and secondly , to make all the possible uses for the same user compatible with one another (in the classroom, at home and in different subjects), the system described is preferably implemented in the form of a client / server architecture comprising:

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 a server containing pedagogical data, user metadata (use clipping, common general objectives, etc.) and managing their electronic profiles; the server responds to client machine calls via a protocol, TCP / IP for example; . client machines with browsers that can display content, manage multi-dimensional hypermedia, perform assessments, update electronic profiles, and manage smart agents.

 As represented in FIG. 13, the server S may consist of a central server accessible from client machines such as personal computers C1, C2, Cn via a network such as the Internet or an input.

 Of course, alternatively, the system according to the invention may be constituted by a local machine such as a PC filling 1 times the server function and the client machine.

 It goes without saying that the embodiments described are only examples and they could be modified, in particular by substitution of equivalents without departing from the scope of the invention.

Claims (10)

 CLAIMS 1. Interactive teaching system comprising at least one computer and interface means with at least one student, said computer comprising:. means for generating personalized educational data based on the observed behavior of said student,. means for extracting evaluation data representative of the degree of acquisition of said instructional data by said student, and. means for constructing a profile of said student based on said evaluation data, and. means for guiding said pupil according to a teaching objective and said profile, characterized in that: said means for generating pedagogical data comprise a set of teaching units (M, M1-M7) which are irreducible in subparts and hierarchized with respect to one another according to the logical dependence links of their content, a first unit on which a second unit is pre-required for said second unit, and said evaluation means comprise means for measuring the degree C (M) of acquisition of said irreducible teaching units.  2. System according to claim 1, characterized in that said measuring means comprise means for calculating the acquisition percentage C (M) of said irreducible teaching units (M, M1-M7), said acquisition percentage characterizing the probability that the student will master a notion.  3. System according to claim 2, characterized in that said evaluation means comprise means for random generation of exercise statements and means for rating responses of said student to said statements. <Desc / Clms Page number 31>  4. System according to any one of claims 2 and 3, charac- terized in that said evaluation means comprise prognostic means for controlling a unit as a function of the acquisition percentage of each of the prerequisite units on which it depends. , as well as means of diagnosing a deficiency explaining the difficulty of controlling a unit.  5. System according to claim 4, characterized in that said diagnostic and prognostic means comprise means for weighting the acquisition percentage of said prerequisite units C (M) according to their distance from said diagnosed unit.  6. System according to claim 5, characterized in that said weighting means comprise calculation means, for each of the units M'constituant a prerequisite of a unit M, a value or distance D = 1-C (M ') in which :

 C (M ') =% acquisition (M') if d-min (M ') = 1, C (M') = min ((d-min (M ") *% acquisition (M ') + d-min (M ') * C (M ")) / (d-min (M') + d-min (M")) if d-min (M ') <> 1, the min being taken on all direct successors M "de M'et d-min (M ') being the minimum distance between M'et M.  7. System according to claim 6, characterized in that said diagnostic means comprise means for classifying said prerequisite units in descending order of said values of D.  8. System according to claim 7, characterized in that said diagnostic means comprise means for classifying said prerequisite units: in increasing order of the acquisition percentage if they have the same value of D, in ascending order of minimum distance in number of arcs to M if, moreover, they have the same percentage of acquisition, <Desc / Clms Page number 32>  - in descending order of the time elapsed since their last consultation if, moreover, they are at the same minimum distance from M ,.

 - otherwise, indifferently.  9. System according to any one of claims 6 to 8, characterized in that said diagnostic means comprise means for detecting gaps associated with a unit by selection, in order of importance, of a predetermined number of units. prerequisites Me whose success rate is below a certain average.  10. System according to any one of claims 6 to 8, characterized in that said diagnostic means comprise means for determining a revision program associated with a unit M through the selection, in order of importance and according to a logical sequence according to said logical dependency links, prerequisite units of said unit and determination of a minimum percentage corrected% AC-min:% AC-min = E [(3 / nb-total-prerequisites + (1-3 / nb-total-pre-requisite) * coeff-disp * lambda) * 100] percentile, in which, the percentiles are taken from the statistical series of% AC, lambda is a parameter between 0 and 1 that determines the size of the work program and coeff-disp is a parameter between 0 and 1 which measures the dispersion of the statistical series of pre-requisites by the formula: coeff-disp = 1-standard deviation / max-deviation.