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Petri nets as a modelling tool in the development of CAL courseware
Comput. Educ. Vol. 8, No. 1, pp. 41-49, 1984 Printed in Great Britain
PETRI
NETS
0360-1315/84 $3.00 + 0.00 Pergamon Press Ltd
AS A M O D E L L I N G
DEVELOPMENT
TOOL
IN THE
OF CAL COURSEWARE
M . FERRARIS, V. MIDORO a n d G . OLIMPO Istituto Tecnologie Didattiche, Consiglio Nazionale Ricerche, Via All'Opera Pia 1l, 16145 Genova, Italy Abstract--This paper discusses a methodology for representing the subject matter for CAL dialogue. The methodology is based upon Petri Nets and is oriented to maximize the possibilities of control and individualization provided by CAL. Simple examples show how to build a content representation and how the same structure may be used as a common basis for different learning strategies. "The language of computation has become the proper dialect for discussing the basic issues of both psychology and education" (Goldstein and Papert).
INTRODUCTION Some of the most important phases in the development of CAL courseware are structuring the subject matter and defining a model of the instructional system starting from this structure. In order to accomplish these activities, a suitable language for knowledge representation is needed which meets the requirements set by the intrinsic nature of the CAL systems. This paper discusses the use of Petri Nets for structuring subject matter and the key ideas for implementing different CAL strategies (tutorial, drill and practice, problem solving, testing, simulation) on the basis of this structure.
PETRI NETS: A LANGUAGE TO STRUCTURE THE SUBJECT MATTER OF CAL SYSTEMS
Control and adaptiveness are the main features of a CAL system[l]. Effective control is possible only if there is a channel of information directed from the student to the instructional agent. This feedback mainly concerns two issues: /f and how the student can accomplish certain stated activities. To perform this evaluation, a controlled instructional process must be constructed upon well defined classes of tasks [2]; in other words classes of tasks and their relationships constitute the stucture of a CAL system. Effective adaptiveness requires that the following are satisfied: (1) The learning process should start from the actual knowledge of the student, allowing several entry points in a learning procedure. (2) A topic should be dealt with at different levels of detail allowing the student to choose the level corresponding to his knowledge. (3) The student should he able to follow his own path in learning a given subject matter according to this learning style. As a consequence the learning procedure should be composed of parallel and alternative learning activities. (4) Several learning strategies (tutorial, drill and practice, etc.) should be available to achieve a given understanding. In order to meet these requirements, the language used for structuring the subject matter should provide means to describe classes of tasks. This structure should represent parallelisms and conflicts between these classes. Furthermore, a single class should be described at different levels of abstraction. Finally, different learning strategies can be implemented starting from this representation [3]. Petri Nets seem to be a suitable tool for structuring subject matter according to these requirements [4,5]. e~E 8/1- ~
41
42
M. FERRARISet
al.
Fig. I. A Petri Net. A Petri Net[6,7] can be identified as a bipartite direct graph (Fig. 1) where the net elements denoted by O symbols are called "state elements" (s elements); those denoted by - - - - symbols are called "transition elements" (t_elements). A Petri Net models the static properties of a system. An arc from a t element to an s element indicates that the s element is an output of the t element. An arc from a n s element to a t element indicates that it i s a n input to the t element. Inaddition to this static interpretation, a Petri Net has dynamic properties which resu~ from its execution. The execution of a Petri Net is controlled by the position and movement of tokens in the net. A transition is enabled if all its input s element hold tokens. An enabled transition can fire by removing a token from each input s element and putting a token in each output s element. In Fig. 1, T1 and T2, which can be fired at the same time or in any order, are called "parallel transitions". When Petri Nets are used for representing the subject matter, t elements may be viewed as activities, input s elements as the resources used by the activity and output s elements as resources produced by theactivity. These interpretations will be refined later in t h e p a p e r . Now let us see how to use Petri Nets in the context of courseware production. Figure 2 shows the production and the operation phases of CAL courseware development. The product of this phase, called "course assembly", is a package composed of diskettes and possibly other instructional material (slides, printed material, video discs, etc.). The instructional process takes place when the student interacts with this material. This activity is called "dialogue".
author
, ~ subiect matter ~.r ~
author ~ ~ t
matter
COURSEBSSF'MBLY
~student
~
trained student
Fig. 2. Production and use of CAL courseware.
Fig. 3. "Course assembly".
Petri nets as a modelling tool in CAL courseware
43
subject matter author
~ ~
DEFINITION
SATURATION
Fig. 4. "Subject matter" structuring.
"Course assembly", as shown in Fig. 3, can be broken down into two main subactivities: (1) Subject matter structuring (which is discussed in this section); (2) Courseware design and implementation (which will be discussed in the following section). The first activity can be broken down in two subactivities: (1) definition of a "gross net"; (2) saturation of the "gross net" (Fig. 4). D E F I N I T I O N OF A " G R O S S N E T " In this phase the main steps carried out by the author are: (l) (2) (3) (4)
Informal detailed analysis of the content domain and of the aims of the instruction. Identification of the main tasks which will be dealt with by the courseware. Representation of these tasks by means of a Petri Net. Check of syntactic correctness of the Petri Net.
The product of this activity is called "gross net". Figure 5 shows a gross net referred to a very simple content domain, i.e. uniformly accelerated motion with initial velocity equal to zero. This net shows the point of view of the author with regard to a given subject and contains only the transitions considered meaningful by the author.
a=
t
I
t
2
Fig. 5. Gross net referred to "uniformly accelerated motion".
44
M. FERRARIS et al.
SATURATION However, the transitions of a gross net are a subset of all transitions existing among the s elements. In designing courseware, the author must take into account all these transitions. It is therefore necessary to make them explicit. This activity is called "saturation". The product of saturation is called a "saturated net". Figure 6 shows a saturated net. The saturated net provides a satisfactory description of the subject matter structure. To assist the author in structuring subject matter a computerized system was implemented[8]. Although saturation refers to any content domain[9], saturation carried out by this system refers only to an arithmetic domain. Starting from this content representation we are able to implement different C A L strategies.
Fig. 6. A saturated net.
I M P L E M E N T A T I O N OF D I F F E R E N T CAL S T R A T E G I E S STARTING FROM A PETRI NET REPRESENTING THE SUBJECT MATTER In this section we shall discuss the following instructional strategies: tutorial, drill and practice, problem solving, testing and simulation. Of course in a C A L process there is no rigid separation between these strategies. However, to simplify the discussion, in the following we shall deal with them one at time, considering only their distinctive features. For each strategy we discuss aims and Petri Net interpretation. As far as instruction is concerned, Petri Nets can always be interpreted in two different ways: as the cognitive procedure to be built in the student's mind by the instructional process or as the "elements" of which the instructional procedure is composed. Furthermore for each strategy key ideas for design and implementation are briefly discussed. TUTORIAL A im
The aim of a tutorial C A L strategy is to teach new topics to the student by means of a visual or verbal interaction. This statement will become more precise after having discussed Petri Net interpretation.
Petri nets as a modelling tool in CAL courseware
45
v=at
I s = __~.. at2
Fig. 7. A marked Petri Net.
Petri Net interpretation According to the first interpretation of the two cited above, in a tutorial strategy t elements represent mental operations or procedures which the student masters as a result of the inst-ructional process. The input s elements are the mental resources (images, concepts, propositions) necessary for accomplishing th~se mental activities. The output s elements are the mental resources produced by the activities. Following the second interpretation, the t elements are the units of the instructional procedure, the s elements are the topics of the given subject matter. In Fig. 7 the topics understood by the student are marked with tokens. A Petri Net in which the known topics are marked is called " m a r k e d Petri Net". A " m a r k e d Petri Net" represents the "state of the student". A tutorial process is aimed at bringing the student from an initial state in which only some topics are marked to a state in which all topics of the net are marked. A transition from one state to another takes place when the student goes through and understands a unit of dialogue (t_element in the Petri Net). The minimal condition which indicates that the student has the necessary prerequisites is that the input topics to the unit be marked. When the content of a unit has been achieved the output topic is also marked. By this mechanism, a tutorial process can be viewed as a succession of instructional units which the student "understands", as he progresses from his initial state to the desired final one. (It must be repeated that a tutorial procedure based on a Petri Net allows different initial states and different paths). The tree, shown in Fig. 8, provides a pictorial representation of all possible states and paths.
J'
II10
II00
0111
Ol I0
Ol I0
Fig. 8. The vectors "1" and "'0" represent the marking of the net. 1 indicates the presence of a token; 0 for the absence of a token; vtas = 1100 means that ~, and t have tokens.
46
M. FERRARISet al. KEY IDEAS FOR D E S I G N AND I M P L E M E N T A T I O N
Figure 3 shows that one input of this activity is the saturated net. In a saturated net there are too many t elements and, in general, it is not possible or useful to associate a unit of dialogue with each one. Therefore, the first step is to select the t elements which are associated with a learning sequence. The criteria for selecting the transition of the fundamental net should be stated by the author. "Fundamental Net" is the name for the net with the selected transitions. The net of Fig. 7 can be considered a fundamental net. The second step in designing the tutorial dialogue is to write the dialogue units for each transition of the fundamental net. The content of each unit depends on the specific instructional theory chosen by the author. In every case, a unit should contain (a) a pretest, (b) an instructional message, (c) a set of activities to be performed by the student, (d) a post test, (e) remedial material. In order to detect the initial state of the student a general pretest is required. This pretest is aimed at determining whether the student is able to enter the net and what topics the student already knows. In general, more than one path is possible starting from the same initial state. For instance in Fig. 8, starting from 0110, either T3 T2 or T2-T3 is allowed. The path chosen depends on the specific administration procedure chosen by the author. In general three alternatives are possible: (1) the student makes the selection; (2) the author makes the selection; (3) the selected unit depends on the past history of the student. Summing up, the main steps of design and implementation of a tutorial procedure are: (1) (2) (3) (4)
Selecting t elements. Writing tutorial units. Writing pretests. Defining and implementing the administration procedure. D R I L L AND P R A C T I C E
Aim " . . . Modern (psychological) theory implies very strongly that certain kinds of basic skills need not only be learned but automatized"[10]. As "tutorial" is aimed at teaching the subject matter related to a given fundamental net, so "drill and practice" is aimed at automatizing the skills (t elements) implied in that fundamental net. Petri Net interpretation In this case the two interpretations are the following: (1) The t elements represent the skills to be automatized and the s elements the knowledge or behav~our required or produced by the use of those skills. (2) The s elements represent classes of tasks or problems by means of which some skills are automatized; s elements represent the input or output data of those problems. Key ideas for design and implementation For each t element the author: (1) Writes the texts of the items. (2) Chooses the range of values of the input s elements. (3) Writes remedial units. When the student enters a drill and practice environment, he is presented with the sets of problems pertaining to those skills of the fundamental net which he or the teacher (or the author) consider worthy to be automatized. The input data for those problems can be generated automatically within a defined range. PROBLEM SOLVING Aim Problem solving is aimed at training the student to discover new procedures for solving new classes of problems, starting from operations already mastered.
Petri nets as a modelling tool in CAL courseware
47
Petri Net interpretation (1) t elements are mental operations or procedures which are not learned during the tutorial dialogue (or in general, during the instructional process), but which can be mentally built from knowledge already achieved. In our model of subject matter, those new procedures are all the t elements of the saturated net not contained in the fundamental net. (2) Like "drill and practice" t elements are classes of problems and s elements are their input or output data. The difference between drill and practice and problem solving is that the former refers to the complement of the fundamental net to the saturated net.
Key ideas for design and implementation This activity is analogous to that performed for drill and practice. The similarity is that in both cases the author writes classes of problems for the t elements; the main differences are: (1) In problem solving the number of the classes of problems is much larger than in drill and practice. (2) In problem solving it is practically impossible to write a remedial unit for each transition of the saturated net, so a remedial procedure must be implemented which shows the local structure of the subject matter to the student and gives some general hints on possible solving procedures.
TESTING
Aim Testing is aimed at supplying an accurate description of the student's understanding of a given subject, resulting from a specific instructional process. This implies that the subject matter structure used as a basis for testing must be the same as (or analogous to) the structure used as a basis of the instructional process (or, in CAL courseware, of the tutorial process). In our case this structure is provided by the fundamental net. It is possible now to restate the aim of testing in more precise, and operative, terms: testing is an instructional procedure aimed at diagnosing which nodes of a fundamental net the student has achieved, or, in other words, what is the state of the student according to a given fundamental net.
Petri Net interpretation (1) The first interpretation is that the t elements are the mental operations or skills learned in an instructional process which must be assessed by the testing process; s_elements are images, concepts or propositions necessary to execute those procedures or produced by those procedures. (2) The second interpretation is that a t element is a class of tests or problems which the student must be able to accomplish successfully to show that he has achieved a given skill, s elements are input or output data or behaviour.
Key ideas for design and implementation Petri Nets provide the basis for an operational definition of the achievement of a node T (t element): A student has achieved T when he can accomplish any task of T. For instance in our example (Fig. 7) a student has achieved T1 if he is able to calculate the acceleration in any uniformly accelerated motion with initial velocity equal to zero, given the value of v at any time t. Since it is impossible to ask the student to perform all tasks of this class we assume that a correct accomplishment of a randomly chosen task of this class indicates his skill in performing all tasks. Problems may arise when in a T procedure there are points of selection. In this case we assume the following criterion: a student is able to accomplish any task of T if he is able to perform a finite subset of tasks covering all the branches of the procedure. The problem is then to write the test items for this subset of tasks. An automatized system has been developed for assisting the author in designing test items based on these ideas[1 1]. Figure 9 shows the main functions of that system. This system assists the author in writing the test items and producing the code for computerized test delivery. It should be noticed that the test handling strategy is designed for minimizing the
48
M. FERRARIS et al.
~ROiUCE
computer code for test delivery
~ .j J-
j
PRODUCE
t
test environment Fig. 9. Design and implementation of the test environment.
workload imposed on the student: the student is first presented with the set of problems which maximize information about his learning. If he fails, the test goes deeper, trying to discover lower level gaps in his knowledge.
SIMULATION
Aims The aims of simulation are to increase the student's intuition about a given phenomenon, to train him to make predictions, to stimulate him to formulate new problems and to solve them. These aims are achieved by means of an interaction between a learner and a simulation environment.
Petri Net interpretation (1) From the cognitive point of view, t elements could be interpreted as an image of the behaviour of a procedure under given conditions (s elements). (2) From the point of view of courseware development, a fundamental Petri Net represents the model of the functions of the simulation environment. So t elements represent the algorithms which determine the behaviour of the system; s elements are the variables whose values can be set by the student or can result from the execution of those algorithms. In this case Petri Nets constitute the "specifications" of the simulation environment.
Key ideas for design and implementation Here the main point is to make the student-machine interaction as natural as possible. The student must be able to interact directly with the machine without any prerequisite in computer science. CONCLUSIONS We have tried to show that Petri Nets are a powerful tool for representing the content of a C A L system and for providing a solid guide in courseware design and implementation. However, the use of Petri Nets for structuring subject matter creates some problems which are not yet solved. When the net has m a n y elements, an enormous amount of time and space is required
Petri nets as a modelling tool in CAL courseware
49
both to saturate it a n d to draw the tree representing all possible paths. But this depends more on the complexity of h u m a n knowledge t h a n o n the tool used. These p r o b l e m s m a y be avoided in practice by i n t r o d u c i n g levels of a b s t r a c t i o n a n d stepwise refinement in the representation of a net. I n this way each level m a y c o n t a i n only a few nodes a n d even t h o u g h some generality can be lost the practical a d v a n t a g e justifies this simplification.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Landa L. N., Instructional Regulation and Control. Educational Technology Publications, New Jersey (1976). Talizina N. F., The theoretical bases of the elaboration of teaching programmes. PLET 19, 1 (1982). Pask G., Conversation Theory Applications in Education and Epistemology. Elsevier, New York (1978). Genrich H. J., The Petri Net Representation of Mathematical Knowledge. Interner Bericht ISF-76-5. Jantzen M., Structured representation of knowledge by Petri Nets as an aid for teaching and research. In Net Theory and Applications. Springer-Verlag, Berlin (1980). Petri C. A., Interpretation of Net Theory. Interner Bericht 75-07. Petri C. A., Concurrency as a Basis of Systems Thinking. Internal Report ISF-78-06. Parodi F. and Parodi M., Implementazionedi un modello di rappresentazione della conoscenza basata sulle reti di Petri. Doctoral thesis. Pask G., A proto-language (Lp the Thoughtsticker Language). Internal report (1979). Gagnd R., Implication for instructional design and effects of computer technology on interactive design and development. Educational Technology, June (1982). Ferraris M., Midoro V. and Olimpo G., A methodology for computer administered testing. Proceedings of IAAP, Edinburgh (1982).
PETRI
NETS
0360-1315/84 $3.00 + 0.00 Pergamon Press Ltd
AS A M O D E L L I N G
DEVELOPMENT
TOOL
IN THE
OF CAL COURSEWARE
M . FERRARIS, V. MIDORO a n d G . OLIMPO Istituto Tecnologie Didattiche, Consiglio Nazionale Ricerche, Via All'Opera Pia 1l, 16145 Genova, Italy Abstract--This paper discusses a methodology for representing the subject matter for CAL dialogue. The methodology is based upon Petri Nets and is oriented to maximize the possibilities of control and individualization provided by CAL. Simple examples show how to build a content representation and how the same structure may be used as a common basis for different learning strategies. "The language of computation has become the proper dialect for discussing the basic issues of both psychology and education" (Goldstein and Papert).
INTRODUCTION Some of the most important phases in the development of CAL courseware are structuring the subject matter and defining a model of the instructional system starting from this structure. In order to accomplish these activities, a suitable language for knowledge representation is needed which meets the requirements set by the intrinsic nature of the CAL systems. This paper discusses the use of Petri Nets for structuring subject matter and the key ideas for implementing different CAL strategies (tutorial, drill and practice, problem solving, testing, simulation) on the basis of this structure.
PETRI NETS: A LANGUAGE TO STRUCTURE THE SUBJECT MATTER OF CAL SYSTEMS
Control and adaptiveness are the main features of a CAL system[l]. Effective control is possible only if there is a channel of information directed from the student to the instructional agent. This feedback mainly concerns two issues: /f and how the student can accomplish certain stated activities. To perform this evaluation, a controlled instructional process must be constructed upon well defined classes of tasks [2]; in other words classes of tasks and their relationships constitute the stucture of a CAL system. Effective adaptiveness requires that the following are satisfied: (1) The learning process should start from the actual knowledge of the student, allowing several entry points in a learning procedure. (2) A topic should be dealt with at different levels of detail allowing the student to choose the level corresponding to his knowledge. (3) The student should he able to follow his own path in learning a given subject matter according to this learning style. As a consequence the learning procedure should be composed of parallel and alternative learning activities. (4) Several learning strategies (tutorial, drill and practice, etc.) should be available to achieve a given understanding. In order to meet these requirements, the language used for structuring the subject matter should provide means to describe classes of tasks. This structure should represent parallelisms and conflicts between these classes. Furthermore, a single class should be described at different levels of abstraction. Finally, different learning strategies can be implemented starting from this representation [3]. Petri Nets seem to be a suitable tool for structuring subject matter according to these requirements [4,5]. e~E 8/1- ~
41
42
M. FERRARISet
al.
Fig. I. A Petri Net. A Petri Net[6,7] can be identified as a bipartite direct graph (Fig. 1) where the net elements denoted by O symbols are called "state elements" (s elements); those denoted by - - - - symbols are called "transition elements" (t_elements). A Petri Net models the static properties of a system. An arc from a t element to an s element indicates that the s element is an output of the t element. An arc from a n s element to a t element indicates that it i s a n input to the t element. Inaddition to this static interpretation, a Petri Net has dynamic properties which resu~ from its execution. The execution of a Petri Net is controlled by the position and movement of tokens in the net. A transition is enabled if all its input s element hold tokens. An enabled transition can fire by removing a token from each input s element and putting a token in each output s element. In Fig. 1, T1 and T2, which can be fired at the same time or in any order, are called "parallel transitions". When Petri Nets are used for representing the subject matter, t elements may be viewed as activities, input s elements as the resources used by the activity and output s elements as resources produced by theactivity. These interpretations will be refined later in t h e p a p e r . Now let us see how to use Petri Nets in the context of courseware production. Figure 2 shows the production and the operation phases of CAL courseware development. The product of this phase, called "course assembly", is a package composed of diskettes and possibly other instructional material (slides, printed material, video discs, etc.). The instructional process takes place when the student interacts with this material. This activity is called "dialogue".
author
, ~ subiect matter ~.r ~
author ~ ~ t
matter
COURSEBSSF'MBLY
~student
~
trained student
Fig. 2. Production and use of CAL courseware.
Fig. 3. "Course assembly".
Petri nets as a modelling tool in CAL courseware
43
subject matter author
~ ~
DEFINITION
SATURATION
Fig. 4. "Subject matter" structuring.
"Course assembly", as shown in Fig. 3, can be broken down into two main subactivities: (1) Subject matter structuring (which is discussed in this section); (2) Courseware design and implementation (which will be discussed in the following section). The first activity can be broken down in two subactivities: (1) definition of a "gross net"; (2) saturation of the "gross net" (Fig. 4). D E F I N I T I O N OF A " G R O S S N E T " In this phase the main steps carried out by the author are: (l) (2) (3) (4)
Informal detailed analysis of the content domain and of the aims of the instruction. Identification of the main tasks which will be dealt with by the courseware. Representation of these tasks by means of a Petri Net. Check of syntactic correctness of the Petri Net.
The product of this activity is called "gross net". Figure 5 shows a gross net referred to a very simple content domain, i.e. uniformly accelerated motion with initial velocity equal to zero. This net shows the point of view of the author with regard to a given subject and contains only the transitions considered meaningful by the author.
a=
t
I
t
2
Fig. 5. Gross net referred to "uniformly accelerated motion".
44
M. FERRARIS et al.
SATURATION However, the transitions of a gross net are a subset of all transitions existing among the s elements. In designing courseware, the author must take into account all these transitions. It is therefore necessary to make them explicit. This activity is called "saturation". The product of saturation is called a "saturated net". Figure 6 shows a saturated net. The saturated net provides a satisfactory description of the subject matter structure. To assist the author in structuring subject matter a computerized system was implemented[8]. Although saturation refers to any content domain[9], saturation carried out by this system refers only to an arithmetic domain. Starting from this content representation we are able to implement different C A L strategies.
Fig. 6. A saturated net.
I M P L E M E N T A T I O N OF D I F F E R E N T CAL S T R A T E G I E S STARTING FROM A PETRI NET REPRESENTING THE SUBJECT MATTER In this section we shall discuss the following instructional strategies: tutorial, drill and practice, problem solving, testing and simulation. Of course in a C A L process there is no rigid separation between these strategies. However, to simplify the discussion, in the following we shall deal with them one at time, considering only their distinctive features. For each strategy we discuss aims and Petri Net interpretation. As far as instruction is concerned, Petri Nets can always be interpreted in two different ways: as the cognitive procedure to be built in the student's mind by the instructional process or as the "elements" of which the instructional procedure is composed. Furthermore for each strategy key ideas for design and implementation are briefly discussed. TUTORIAL A im
The aim of a tutorial C A L strategy is to teach new topics to the student by means of a visual or verbal interaction. This statement will become more precise after having discussed Petri Net interpretation.
Petri nets as a modelling tool in CAL courseware
45
v=at
I s = __~.. at2
Fig. 7. A marked Petri Net.
Petri Net interpretation According to the first interpretation of the two cited above, in a tutorial strategy t elements represent mental operations or procedures which the student masters as a result of the inst-ructional process. The input s elements are the mental resources (images, concepts, propositions) necessary for accomplishing th~se mental activities. The output s elements are the mental resources produced by the activities. Following the second interpretation, the t elements are the units of the instructional procedure, the s elements are the topics of the given subject matter. In Fig. 7 the topics understood by the student are marked with tokens. A Petri Net in which the known topics are marked is called " m a r k e d Petri Net". A " m a r k e d Petri Net" represents the "state of the student". A tutorial process is aimed at bringing the student from an initial state in which only some topics are marked to a state in which all topics of the net are marked. A transition from one state to another takes place when the student goes through and understands a unit of dialogue (t_element in the Petri Net). The minimal condition which indicates that the student has the necessary prerequisites is that the input topics to the unit be marked. When the content of a unit has been achieved the output topic is also marked. By this mechanism, a tutorial process can be viewed as a succession of instructional units which the student "understands", as he progresses from his initial state to the desired final one. (It must be repeated that a tutorial procedure based on a Petri Net allows different initial states and different paths). The tree, shown in Fig. 8, provides a pictorial representation of all possible states and paths.
J'
II10
II00
0111
Ol I0
Ol I0
Fig. 8. The vectors "1" and "'0" represent the marking of the net. 1 indicates the presence of a token; 0 for the absence of a token; vtas = 1100 means that ~, and t have tokens.
46
M. FERRARISet al. KEY IDEAS FOR D E S I G N AND I M P L E M E N T A T I O N
Figure 3 shows that one input of this activity is the saturated net. In a saturated net there are too many t elements and, in general, it is not possible or useful to associate a unit of dialogue with each one. Therefore, the first step is to select the t elements which are associated with a learning sequence. The criteria for selecting the transition of the fundamental net should be stated by the author. "Fundamental Net" is the name for the net with the selected transitions. The net of Fig. 7 can be considered a fundamental net. The second step in designing the tutorial dialogue is to write the dialogue units for each transition of the fundamental net. The content of each unit depends on the specific instructional theory chosen by the author. In every case, a unit should contain (a) a pretest, (b) an instructional message, (c) a set of activities to be performed by the student, (d) a post test, (e) remedial material. In order to detect the initial state of the student a general pretest is required. This pretest is aimed at determining whether the student is able to enter the net and what topics the student already knows. In general, more than one path is possible starting from the same initial state. For instance in Fig. 8, starting from 0110, either T3 T2 or T2-T3 is allowed. The path chosen depends on the specific administration procedure chosen by the author. In general three alternatives are possible: (1) the student makes the selection; (2) the author makes the selection; (3) the selected unit depends on the past history of the student. Summing up, the main steps of design and implementation of a tutorial procedure are: (1) (2) (3) (4)
Selecting t elements. Writing tutorial units. Writing pretests. Defining and implementing the administration procedure. D R I L L AND P R A C T I C E
Aim " . . . Modern (psychological) theory implies very strongly that certain kinds of basic skills need not only be learned but automatized"[10]. As "tutorial" is aimed at teaching the subject matter related to a given fundamental net, so "drill and practice" is aimed at automatizing the skills (t elements) implied in that fundamental net. Petri Net interpretation In this case the two interpretations are the following: (1) The t elements represent the skills to be automatized and the s elements the knowledge or behav~our required or produced by the use of those skills. (2) The s elements represent classes of tasks or problems by means of which some skills are automatized; s elements represent the input or output data of those problems. Key ideas for design and implementation For each t element the author: (1) Writes the texts of the items. (2) Chooses the range of values of the input s elements. (3) Writes remedial units. When the student enters a drill and practice environment, he is presented with the sets of problems pertaining to those skills of the fundamental net which he or the teacher (or the author) consider worthy to be automatized. The input data for those problems can be generated automatically within a defined range. PROBLEM SOLVING Aim Problem solving is aimed at training the student to discover new procedures for solving new classes of problems, starting from operations already mastered.
Petri nets as a modelling tool in CAL courseware
47
Petri Net interpretation (1) t elements are mental operations or procedures which are not learned during the tutorial dialogue (or in general, during the instructional process), but which can be mentally built from knowledge already achieved. In our model of subject matter, those new procedures are all the t elements of the saturated net not contained in the fundamental net. (2) Like "drill and practice" t elements are classes of problems and s elements are their input or output data. The difference between drill and practice and problem solving is that the former refers to the complement of the fundamental net to the saturated net.
Key ideas for design and implementation This activity is analogous to that performed for drill and practice. The similarity is that in both cases the author writes classes of problems for the t elements; the main differences are: (1) In problem solving the number of the classes of problems is much larger than in drill and practice. (2) In problem solving it is practically impossible to write a remedial unit for each transition of the saturated net, so a remedial procedure must be implemented which shows the local structure of the subject matter to the student and gives some general hints on possible solving procedures.
TESTING
Aim Testing is aimed at supplying an accurate description of the student's understanding of a given subject, resulting from a specific instructional process. This implies that the subject matter structure used as a basis for testing must be the same as (or analogous to) the structure used as a basis of the instructional process (or, in CAL courseware, of the tutorial process). In our case this structure is provided by the fundamental net. It is possible now to restate the aim of testing in more precise, and operative, terms: testing is an instructional procedure aimed at diagnosing which nodes of a fundamental net the student has achieved, or, in other words, what is the state of the student according to a given fundamental net.
Petri Net interpretation (1) The first interpretation is that the t elements are the mental operations or skills learned in an instructional process which must be assessed by the testing process; s_elements are images, concepts or propositions necessary to execute those procedures or produced by those procedures. (2) The second interpretation is that a t element is a class of tests or problems which the student must be able to accomplish successfully to show that he has achieved a given skill, s elements are input or output data or behaviour.
Key ideas for design and implementation Petri Nets provide the basis for an operational definition of the achievement of a node T (t element): A student has achieved T when he can accomplish any task of T. For instance in our example (Fig. 7) a student has achieved T1 if he is able to calculate the acceleration in any uniformly accelerated motion with initial velocity equal to zero, given the value of v at any time t. Since it is impossible to ask the student to perform all tasks of this class we assume that a correct accomplishment of a randomly chosen task of this class indicates his skill in performing all tasks. Problems may arise when in a T procedure there are points of selection. In this case we assume the following criterion: a student is able to accomplish any task of T if he is able to perform a finite subset of tasks covering all the branches of the procedure. The problem is then to write the test items for this subset of tasks. An automatized system has been developed for assisting the author in designing test items based on these ideas[1 1]. Figure 9 shows the main functions of that system. This system assists the author in writing the test items and producing the code for computerized test delivery. It should be noticed that the test handling strategy is designed for minimizing the
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M. FERRARIS et al.
~ROiUCE
computer code for test delivery
~ .j J-
j
PRODUCE
t
test environment Fig. 9. Design and implementation of the test environment.
workload imposed on the student: the student is first presented with the set of problems which maximize information about his learning. If he fails, the test goes deeper, trying to discover lower level gaps in his knowledge.
SIMULATION
Aims The aims of simulation are to increase the student's intuition about a given phenomenon, to train him to make predictions, to stimulate him to formulate new problems and to solve them. These aims are achieved by means of an interaction between a learner and a simulation environment.
Petri Net interpretation (1) From the cognitive point of view, t elements could be interpreted as an image of the behaviour of a procedure under given conditions (s elements). (2) From the point of view of courseware development, a fundamental Petri Net represents the model of the functions of the simulation environment. So t elements represent the algorithms which determine the behaviour of the system; s elements are the variables whose values can be set by the student or can result from the execution of those algorithms. In this case Petri Nets constitute the "specifications" of the simulation environment.
Key ideas for design and implementation Here the main point is to make the student-machine interaction as natural as possible. The student must be able to interact directly with the machine without any prerequisite in computer science. CONCLUSIONS We have tried to show that Petri Nets are a powerful tool for representing the content of a C A L system and for providing a solid guide in courseware design and implementation. However, the use of Petri Nets for structuring subject matter creates some problems which are not yet solved. When the net has m a n y elements, an enormous amount of time and space is required
Petri nets as a modelling tool in CAL courseware
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both to saturate it a n d to draw the tree representing all possible paths. But this depends more on the complexity of h u m a n knowledge t h a n o n the tool used. These p r o b l e m s m a y be avoided in practice by i n t r o d u c i n g levels of a b s t r a c t i o n a n d stepwise refinement in the representation of a net. I n this way each level m a y c o n t a i n only a few nodes a n d even t h o u g h some generality can be lost the practical a d v a n t a g e justifies this simplification.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Landa L. N., Instructional Regulation and Control. Educational Technology Publications, New Jersey (1976). Talizina N. F., The theoretical bases of the elaboration of teaching programmes. PLET 19, 1 (1982). Pask G., Conversation Theory Applications in Education and Epistemology. Elsevier, New York (1978). Genrich H. J., The Petri Net Representation of Mathematical Knowledge. Interner Bericht ISF-76-5. Jantzen M., Structured representation of knowledge by Petri Nets as an aid for teaching and research. In Net Theory and Applications. Springer-Verlag, Berlin (1980). Petri C. A., Interpretation of Net Theory. Interner Bericht 75-07. Petri C. A., Concurrency as a Basis of Systems Thinking. Internal Report ISF-78-06. Parodi F. and Parodi M., Implementazionedi un modello di rappresentazione della conoscenza basata sulle reti di Petri. Doctoral thesis. Pask G., A proto-language (Lp the Thoughtsticker Language). Internal report (1979). Gagnd R., Implication for instructional design and effects of computer technology on interactive design and development. Educational Technology, June (1982). Ferraris M., Midoro V. and Olimpo G., A methodology for computer administered testing. Proceedings of IAAP, Edinburgh (1982).