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8

Structure and Design of an

Intelligent Tutorial System

Based on Skill Assignments

Dietrich Albert

Karl-Franzens-Universit¨

at Graz

Martin Schrepp

Ruprecht-Karls-Universit¨

at Heidelberg

Since the beginning of research in artiﬁcial intelligence several attempts have

been made to construct intelligent tutorial systems (ITS). Such an ITS consists

in general of a representation of the knowledge in a special domain, a diagnos-

tic procedure to determine the knowledge of a student working with the system,

teaching material, and a procedure for adaptive teaching. This chapter demon-

strates how an extension of the theory of knowledge spaces can be used for the

design of a domain-independent ITS. The main components of this ITS are a

representation of the skills necessary in the knowledge domain and their depen-

dencies as a surmise system, a set of questions related through a skill assignment

to the skill states used for knowledge diagnosis, and a rule that relates skill states

to teaching operations. The ITS is adaptive with respect to the consideration of

the prior knowledge a student possess and with respect to the learning speed of

a student. The strict formalized description of the systems components and their

interactions during the teaching process guarantees that an ITS with the described

properties can be implemented easily.

179

Albert, D., & Schrepp, M. (1999). Structure and Design of an Intelligent Tutoring System Based on Skill

Assignments. In D. Albert & J. Lukas (Eds.), Knowledge Spaces: Theories, Empirical Research, and

Applications (pp. 179–196). Mahwah, NJ: Lawrence Erlbaum Associates.

180 ALBERT AND SCHREPP

INTRODUCTION

One of the research topics discussed in artiﬁcial intelligence is the approach to use

computers in education for diagnosing knowledge (for example Brown & Burton,

1978) or teaching (e.g., Mandl & Hron, 1985; Okamoto & Matsuda, 1993; Slee-

man & Brown, 1982).

Intelligent tutorial systems (ITS) differ from other forms of computer-assisted-

instruction (e.g., drill-programs, electronic textbooks, or simulation programs) in

being “intelligent” with respect to two criteria. First, the system must be able to

diagnose the status of the student’s knowledge in the underlying knowledge do-

main. Second, the tutorial strategy must be adaptive to the prior knowledge of the

student and must be able to improve the learner’sknowledge.

There exists a variety of architecturesfor ITS in different knowledge domains.

Most of the constructed systems work in highly formalized domains, like sub-

disciplines of mathematics, for example, symbolic logic (Matsuda & Okamoto,

1992), arithmetic (Takeuchi & Otsuki, 1994), fraction calculation (Kondo, Watan-

abe, Takeuchi, & Otsuki, 1990), integration (Kimball, 1982), physics (Ploetzner

& Spada, 1993), language acquisition (Kunichika, Takeuchi, & Otsuki, 1994),

or the usage of software systems or programming languages (Heines & O’Shea,

1985; Yasuda & Okamoto, 1991).

Most of these architectures are highly specialized in their speciﬁc knowledge

domain and it is, therefore, not possible to use them in other domains.

There is a wide agreement in literature that an ITS must contain at least four

basic components, that interact during the teaching process. These components

are a knowledge base, a student model, a teaching component, and a diagnostic

component. Existing ITS differ in the way in which these components are con-

ceptualized, especially in their in general domain speciﬁc student models.

The knowledge base contains all relevant knowledgeof the domain for which

the ITS is constructed. This component represents the knowledge of an expert in

the domain and is often realized as an expert system.

The student model is a representation of the cognitive abilities or skills of

students in the knowledge domain. It consists, in general, of a set of states. Each

state represents a special state of knowledge in which a student may possibly be

during the learning process, that is during his or her interaction with the ITS. The

learning process is represented by a sequence of states of the student model.

The teaching component consists of teaching materials (instructions, exer-

cises, or demonstrations) and a rule that determines for each state of the student

model, which part of this material is relevant for a student who is in that particular

state.

The state of knowledge of a student is, in general, not directly observable be-

cause the student model contains hypothetical assumptions about necessary cog-

nitive abilities or cognitive procedures. The diagnostic component is used to infer

this state from the interaction of thestudent withthe ITS.

8. INTELLIGENT TUTORIAL SYSTEM 181

The goal of the teaching process is to enable the student to learn all relevant

information of the knowledge domain (or,at least, a sufﬁcient large subset of this

information), so that her or his knowledge after the teaching process terminates is

more or less equal to the knowledge contained in the knowledge base of the ITS.

We want to show how the theory of knowledge spaces (Doignon & Falmagne,

1985, 1998) and skill assignments (Doignon, 1994a; Lukas & Albert, 1993; Ko-

rossy, 1993) can be used to develop a general architecture of an ITS. This general

architecture is domain independent. Therefore it can be used for the concrete

construction of ITS’s in different knowledge domains.

Two properties of the theory of knowledge spaces and skill assignments are

especially important for this approach. First, the theory of knowledge spaces

allows the formulation of effective and adaptive procedures for the diagnosis of

knowledge (see Falmagne & Doignon, 1988; Doignon, 1994b). Second, skill

assignments allow a precise description of the connection between observable

behavior in problem solving and underlying cognitive abilities or skills (Doignon,

1994a; Korossy, 1993, 1997).

In the following section, we describe the basic ideas of an ITS based on the

theory of knowledge spaces and skill assignments. Then, we show how this basic

system can be generalized by incorporating some important new components.

In the last section possibilities for further developments are sketched and open

questions leading to further research are discussed.

BASIC IDEAS OF AN ITS BASED ON SKILL

ASSIGNMENTS

In this section, we develop the basic architectureof an ITS based on skill assign-

ments. This basic architecture is very simple and therefore limited in its applica-

bility because the main goal of this section is to clarify the basic ideas underlying

our approach.

For the formal description of our basic architecture we have to introduce some

notations. We write in the following Pow for the power set, the set of all

subsets of a set . Seq denotes the set of all tuples of elements of a set

with Seq

IN

.

We divide the components of our ITS into statical and dynamicalcomponents.

A component is called dynamical if it is updated during the teaching process, thus

being adaptive to the student’s behavior, and statical if this is not the case. We

begin with the description of the statical components.

182 ALBERT AND SCHREPP

Knowledge Base

The knowledge necessary in the domain is represented through a set of skills.

Dependencies between these skills are described by a surmise function

Pow Pow . A surmise function assigns to each skill the different sets of pre-

requisites of this skill. We interpret as “Every student has

to master all skills from at least one of the subsets of to be able to

reach mastery of ”. The surmise function is used during the teaching process to

determine an optimal learning path. The surmise system can be considered

as the knowledge base of the system.

Student Model

Given and , we can deﬁne the student model by:

is the set of all subsets of that contain with every skill at least one set of

prerequisites of . is the set of all subsets of consistent with the dependencies

of skills described by . Therefore, the knowledge of every student working

with the system can be described by an element of , if we presuppose that

contains all the skills relevant1in the knowledge domain and that describes

the dependencies between these skills correctly. If the knowledge of a student is

described by we call the competence state of that student.

Diagnostic Component

The skills in are hypothetical and are not directly observable constructs. There-

fore, our system must contain a possibility to infer the skills a student possess, the

student’s competency, from his or her solving-behavior.

Let be a set of questions or problems in the underlying knowledge domain.

We assume in the following that the questions in are especially constructed for

the diagnosis of the skills in . Thus, we can assume that the mastery of allskills

in is sufﬁcient for the ability to solve all questions in . None of the questions

in require a skill . The systematical construction of questions for a given

set of skills is discussed for example in Lukas and Albert (1993), Held (1993),

Albert, Schrepp, and Held (1994), or Korossy (1993, 1997).

Because we want to diagnose the skills a student possesses from her or his

solving behavior, we have to relate the skills in to the ability of students to solve

the questions in . This is done by a skill assignment Pow . For a

1Relevant skills are the skills that form the speciﬁc bodyof knowledge in the underlying knowledge

domain. The ability to add two integers is an example of such a relevant skill in the domain of

elementary arithmetic. General skills that are necessary for understanding the instructions or problems

of the domain, such as the ability to read, are presupposed and not considered as members of .

8. INTELLIGENT TUTORIAL SYSTEM 183

question , we interpret as “A student answering correctly

to question must master all skills from at least one of the sets ”.

Because of this interpretation we can assume that the states of the

student model contained in are minimal with respect to , for all

we have . For a more detailed discussion of the dependency between

the skills possessed by a student and her or his solving behavior, see Korossy

(1993, 1997).

The skill assignment determines for each state of the student model

the answer behavior consistent with this state by

is the set of all questions in that can be answered correctly by a student

with competency .

Teaching Component

To allow teaching operations, the system must contain materials that can be pre-

sented to students in order to improve their knowledge. Let be a set of instruc-

tions, like examples, facts, informational texts, or exercises. We represent as a

union of pairwise disjoint sets , with if and only if

. Each of these sets represents a special type of instruction.

The teaching operations should be adaptive to the prior knowledge of a stu-

dent. Therefore, we have to relate them to the skills in . This relation is estab-

lished by a function Pow Seq , which relates skills and their

prerequisites to an instructional sequence. We interpret

for

IN

as “The instructional sequence should be presented to a stu-

dent who masters all skills in and who does not master ”. can be

considered as the teaching component of the system.

Hypothetical Student States

To enable an adaptive teaching strategy, the system must also contain dynamical

components, that is components that change during the interactions of a student

with the system.

The actual information of the system about the knowledge of a student work-

ing with it can be represented as a subset of . This subset consists of

all states of that are consistent with the previous answers of the student on

the presented questions. We call the set of hypothetical student states. If

are the questions already answered correctly by a student and

the questions answered incorrectly by that student we have:

184 ALBERT AND SCHREPP

Storage of Information

The interactions between student and system must be stored to use them for teach-

ing decisions. For example, which questions the student has already answered

must be stored and also which information was already presented to him or her.

This is done in the history of the teaching process. is a list of all previ-

ously presented questions, obtained answers, and presented instructions. Thus,

is an element of Seq , where correct false . For ex-

ample means that ﬁrst question

was presented and answer was obtained, second question was presented

and answer was obtained. Afterwards, instructions were presented to

the student, followed by a presentation of question , on which answer was

obtained, and so on.

Teaching Process

Up to this point, we have described only the components of the systems archi-

tecture. Now, we describe how these components interact during the teaching

process. The teaching process can be considered as an interaction between ﬁve

modes of the system.

The ﬁrst mode is the START mode. Here the system sets because, up

to this point, no information about the students knowledge is obtained. Then, the

system changes into the second mode, called DIAGNOSIS mode.

The DIAGNOSIS mode consists of three steps. In the ﬁrst step, the system

tries to determine a question , for which an obtained answer guarantees an opti-

mal reduction of the set of hypothetical student states consistent with the obtained

answer of the student. This can be done by the half-split procedure in the diag-

nostic algorithm described in Falmagne and Doignon (1988).

For the set of hypothetical student states and a question , we deﬁne

as the set of all elements of that are consistent with the fact that

the student solves question and as the set of all elements of that

are consistent with the fact that the student fails in solving . Formally, we have

and . The sets

and are pairwise disjoint, that is we have and we

have .

We deﬁne a question as being optimal for diagnosis if the value of

card card is minimal over all . For an optimal question

the cardinality of the sets and is optimal balanced, that is the sets

and differ in cardinality as little as possible.

The described procedure can be regarded as the diagnostic component of the

system. Notice that a question already answered (correct or incorrect) by the

student can not be optimal as long as the set of hypothetical student states

contains at least two elements.

8. INTELLIGENT TUTORIAL SYSTEM 185

In the second step, an optimal question , determined by the ﬁrst step, is

presented to the student. If the answer of the student is (where 1

represents “correct” and 0 “false”) then is replaced by .

The third step tests if there exists a skill not mastered by the student, (for-

mally: ), for which a set of prerequisites is completly mastered,

(formally: there exists a with ). If this is

the case, then the system changes to the TEACH mode. If this is not the case, the

system continues with the ﬁrst step of the DIAGNOSIS mode.

The TEACH mode starts if the DIAGNOSIS mode has detected a skill not

mastered by a student who masters for a set of prerequisites of all skills from

. Formally, this means that and .

It is possible that the DIAGNOSIS mode has detected more than one skill with

this properties. If this is the case, the system has to decide which of these skills

should be taught ﬁrst. We discuss this problem in the following section. For the

moment, it is sufﬁcient to assume that the decission is made by chance.

The TEACH mode then presents the instructional sequence to the

student. After the presentation of this instructional sequence the system changes

to the TEST mode.

The TEST mode consists of three steps. In the ﬁrst step the system checks if

the student has reached mastery of the last skill taught by the TEACH mode.

Therefore, a question is presented for which skill is necessary and only skills

from are required, where denotes the set of prerequisites to detected in

the third step of the DIAGNOSIS mode. Formally, this means that the system

presents a question with .

In the second step, the system checks if the presented question is answered

correctly by the student, indicating that he or she had learned the required skill. If

the student fails in answering correct, that is if , the system goes back to the

TEACH mode and starts the presentation of again. If the student gives

the correct answer to the presented question is replaced by

and the third step of the TEST mode checks if ,

indicating that the student masters all skills required in the domain. If this is the

case the system changes to the END mode, if not the system changes to the third

step of the DIAGNOSIS mode.

The END mode informs the student that he or she had reached the learning

goal and ﬁnishes the process.

The teaching process is depicted in Fig. 1 as a ﬂow chart.

GENERALIZATIONS OF THE BASIC SYSTEM

In the previous section, we introduced an architecture for an ITS based on the

theory of knowledge spaces. Because our main goal in this section is to work out

the general ideas underlying an application of knowledge space theory in the ﬁeld

START

Set

DIAGNOSIS

Select optimal question

Present and replace through

for obtained reaction on

Test if exists, with

Yes

TEACH

Present instructional sequence

TEST

Present

Is obtained reaction correct ?

Yes

Is ?

Yes

END

No

No

No

FIG. 1. Teaching process described as an interaction between the ﬁve modes of the system.

186

8. INTELLIGENT TUTORIAL SYSTEM 187

of computerized teaching, this basic architecture is very simple and is therefore

restricted in its applicability. In this section, we try to clarify the limitations of the

basic architecture and work out methods to overcome them. So, at the end of this

section, we are able to formulate a generalization of our basic architecture.

In our basic architecture, we have considered only two possible answer types

of a student to a presented question by assuming that such an answer is either cor-

rect or incorrect. This approach has the advantage of simplicity, but does not use

the whole information about the student’s competency contained in the obtained

answer.

To use this information completely we have to introduce the answer in the

skill assignment and clarify the role of the answer in the update of the set of

hypothetical student states within the ﬁrst and second step of the DIAGNOSIS

mode.

Therefore, we introduce the set of all possible answers (possible inputs to

the system) a student can give to the questions in . We relate the skills in to

the questions in and possible answers in by a skill assignment2

Pow . The interpretation of is “A student who answers

to question must at least master all skills from one of the sets ”. As

in the previous section, we can assume that the sets in are

minimal with respect to , i.e. .

As in the previous section, the skill assignment determines for each state

the answer behavior consistent with this state by

is the set of all tuples of questions and obtained answers that can be pro-

duced by a student with competency .

Now, we have to generalize the half-split procedure in the ﬁrst step of the

DIAGNOSIS mode. For the set of hypothetical student states and a question

, we deﬁne as the set of all elements of , that are consistent

with the fact that the student gives answer on question ,

. We deﬁne a question as being optimal for diagnosis if

the value card card is minimal over all .

This means that for the optimal question the cardinality of the sets is for

all possible answers optimal balanced, that is the sets differ in cardinality

as little as possible. The sets must not be pairwise disjoint, we can have

for , and they can be empty, that is we can have

for some .

As before in the second step of the DIAGNOSISmode, an optimal question is

presented to the student and if the answer is , then is replaced by .

2In the following we denote concepts that are proper generalizations of concepts introduced earlier

by the same symbols as in the previous section. Since, for example, the generalized concept of a

skill assignment is for correct incorrect identical with the concept of a skill assignment as

introduced in the previous section, it is denoted by the same greek letter .

188 ALBERT AND SCHREPP

To make things easy, in the formulation of the basic architecture we have

drawn implicitly the unrealistic assumption that a student answers always accord-

ingly to her or his competency described by a state of the student model. This

assumption ensures that our rules governing the update of in the DIAGNOSIS

mode do not end with an empty . Remember that the teaching process starts

with and that after each obtained answer of a student to a question ,

the set is replaced by .

But if, as a result of a lucky guess or a careless error, the students answer to

question is not consistent with one of the states in , that is with the students

previous interactions with the system, we can receive an empty . In this case,

we have to ﬁnd a new rule for the update of in step two of the DIAGNOSIS

mode.

We distinguish between two cases. If we replace just as before

by . If conversely we replace by

, enlarging by considering all states

of the student model that differ only in one skill from the states in . The en-

largement of is necessary because we have to consider the possibility that the

obtained answer on is in accordance with the competency of the student and

is empty as a result of a lucky guess or careless error to one of the previous

questions, that leaded to an elimination of the correct state of the student.

To ensure that a wrong assumption concerning the students’ competency, for

example, resulting from a careless error, can be corrected, we have to assume that

is well–graded (Falmagne & Doignon, 1988). This means that for two arbitrary

states , there exists a chain of states

in with card for . Each contains exactly one

skill more than his predecessor in the chain.

This assumption is necessary because we enlarge in the case of an error in

diagnosis only by the states that differ in only one element from the states in .

If in a not well–graded one state differs in at least two skills from all other

states, this state can not be reached by the described rule if it is once eliminated

from , for example as a result of a careless error.

One important restriction of the basic architecture is that, given a skill not

mastered by a student who masters all skills from a set , where is a

set of prerequisites of , there is only one sequence of

instructions that lead to the mastery of by the student. Generally one would ex-

pect many such instructional sequences having all the same intended effect on the

students’ knowledge. For example, a missing skill may be taught alternatively by

presenting some examples, some instructional texts, or a mixture of both. Which

of these alternatives is best for a particular student may depend on the student’s

personality.

We include the possibility of alternative instructional sequences by assigning

to a skill and a set of prerequisites of a set of instructional sequences through

a function . So we regard as a function Pow Pow Seq . The

8. INTELLIGENT TUTORIAL SYSTEM 189

interpretation of is “To a student mastering all skills from

and not mastering , one of the instructional sequences should be

presented to reach mastery of ”.

Because we had assumed in our basic architecturethat contains only

one instructional sequence, we must assume that this instructional sequence is

repeated until the student masters skill . If the TEST mode ﬁnds that a student

has not learned the last skill taught by the TEACH mode by presenting the se-

quence , this sequence is presented again. This may be problematic for

two reasons. First, the instructional sequence may be inappropriate for

the student, so that a repetition of this sequence makes no sense. Second, a repe-

tition of informations already presented may lead to a motivationaldecrease.

We can overcome this limitation by assuming that solely such instructional se-

quences can be presented to a student that have not been presented before as long

as contains at least one informational sequence not already presented.

Formally, this means that to a student, who had already received the instructional

sequences (these sequences are stored in the history of the teaching

process), only instructional sequences from can be pre-

sented if this set is not empty. If is empty, then an arbitrary

element of determined by chance should be presented again to the stu-

dent.

Now the problem arises which of these sequences should

be presented to a student during the TEACH mode of the system. This can be

solved by a choice rule which chooses one of these sequences with respect to

the previous interactions between system and student.

Such a choice rule can be implemented in several ways. In the following,

we discuss one such possibility. We divide the set Seq of instructional se-

quences into a ﬁnite number of pairwise disjoint subsets , that is we

have Seq and for .

These subsets represent instructional types, for example instruction by ex-

amples, or instruction based on texts. For the classiﬁcation of sequences into

types, the description of as a union of pairwise disjoint sets of different instruc-

tions introduced in the previous section can be used.

We can expect that students differ in the type of instruction they prefer and

with which they learn best. So the choice rule should ensure that a student

receives the type of instruction which is most effective for her or him, that is

should be adaptive to the success of a special instructional type in the previous

interactions between system and student.

We deﬁne the effectivity of an instructional type as the relative fre-

quency of already presented instructional sequences from leading to a mastery

of the intended skill. The effectivity can be measured by the TEST mode.

So the effectivity is a function .

We use the effectivity for the construction of an adaptive choice rule by as-

suming that the probability for the presentation of a sequence

190 ALBERT AND SCHREPP

increases with the effectivity of the instructional type to which belongs.

If we should have if and only if

. The decision between members of the same instructional type is

drawn by chance as before.

Another limitation of the basic architecture is that the goal of the teaching

process is mastery of all skills in by the student, the teaching process terminates

only if . In a practical application it may be for two reasons necessary

to deﬁne subgoals of the teaching process.

First, for some students the mastery of a subset of may be sufﬁcient, for

example, if they need only basic knowledge in the domain and do not want to

become experts there. Second, the teaching process may have taken too much

time for one session, so we have to formulate points where an interuption makes

sense.

Such subgoals can be introduced by marking a subset of . An element of

is called a subgoal of the teaching process.

Which elements of are suitable as subgoals? A subgoal should be a closed

piece of knowledge, all skills in it should be connected. If we interpret the surmise

function on as a description of the contentual dependency of the skills the

optimal candidates for subgoals are the basis elements of . The reason is that

a basis element includes with each skill only such skills, which are either

elements of a set of prerequisites of or for which itself is an element of one of

their sets of prerequisites. Therefore we chose as a subset of the basis of .

To make use of the subgoals for the control of the teaching process, we only

have to change the third step of the test mode of our system. This step tests if

, if the student masters all skills from , and changes to the END mode

if this is the case. This step can be easily generalized by assuming that it tests if all

hypothetical student states include a basis element , that is if the student masters

at least all skills from . Formally, this means that it is tested if the condition

is fulﬁlled. If this is the case the student is asked

if he or she wants to interrupt the process. If the student decides to interrupt, the

history of the teaching process is stored and can be used in the next session

to continue the process properly. If the learning goal of a particular student is a

proper subset of , for example, if the student only needs basic knowledge in the

domain, the process can be terminated if this special goal is fulﬁlled. In this case,

we have also to ensure that skills not contained in the learning goal of the

student are not taught, even if the diagnostic procedure ﬁnds out that the student

does not master these skills. This can be done easily by replacing by in all

components of the system and in the teaching process. This means that the system

“forget” all information about skills in and reacts just as if the set of skills

necessary in the domain would be .

The learning goal of a particular student may be given explicitly, for example,

if the system is used by students to fulﬁll a special course requirement, or asked

interactively at thebeginningof the teaching process from the student.

8. INTELLIGENT TUTORIAL SYSTEM 191

The set of subgoals of the teaching process can also be used to solve a

problem already mentioned in the previous section.

The problem arises if the DIAGNOSIS mode has detected at least two skills

and not mastered by a student and sets of prerequisites and for

respectively for which all skills from and are mastered

by that student. The system has then to decide which of these skills should be

taught ﬁrst. In the previous section, we had assumed that this decision is drawn

by chance.

Given a set of subgoals we can improve the decision strategy by assuming

that skills are taught ﬁrst that ensure that a subgoal is reached. Only if this rule

is not sufﬁcient to decide between two skills, for example if none of these skills

leads to a subgoal, the decision is drawn by chance by an implemented random

decision strategy. This procedure ensures that the teaching strategy tries to teach

ﬁrst closed pieces of knowledge, i.e., subsets of in which all skills are contentual

connected by , instead of unconnected pieces of knowledge.

Another possible extension of our basic architecture is the introduction of the

learning speed or learning ability of students into the teaching strategy. Students

clearly differ in their ability to learn new material. This difference between stu-

dents should be considered in an ITS to ensure that the strategy the system uses for

the presentation of instructions reﬂects the learning ability of a student working

with the system.

For example, for a student who is able to learn fast it may be optimal to present

a number of instructional sequences during one step of the teaching process in or-

der to allow the student to aquire a number of skills during that step. For a student

who learns slowly, it seems more appropriate to present only the instructions nec-

essary to learn one missing skill during one step of the teaching process.

The teaching process described in the previous section considers only the sec-

ond case of the example. This can be seen from the interaction of the TEACH

and the TEST mode of the system. Assume a student who does not master skill

but who masters all skills from , where is a set of prerequisites of

. We assumed in the previous section that in such a case, the TEACH mode

presents the instructional sequence and afterwards the system changes to

the TEST mode to ensure that skill was aquired by the student. Therefore, the

procedure described in the previous section seems to be optimal only for students

who learn slowly.

Because the learning speed of a student is not known to the ITS before the

teaching process starts, it must be measured during this process. This can be done

adaptively by the TEST mode. The central idea is that the number of cases in

which the student had aquired the skill taught in the previous TEACH step is an

indicator for the learning speed of that student.

We measure the learning speed of a particular student by an integer greater

than , that is we can have . At the beginning of the teaching

process we set for every student. If the TEST mode detects in successive

192 ALBERT AND SCHREPP

steps, where is a ﬁxed level, that all skills3taught in the previous step are aquired

by the student, then is replaced by . If the TEST mode detects conversly

that a student has not aquired the skills taught in the previous TEACH step and

is greater than , then is replaced by .

Now, we have to describe how the learning speed inﬂuences the choice

of skills that will be taught during the TEACH mode. Assume that the DI-

AGNOSIS mode had detected a skill not mastered by a student who masters

all skills from , where is a set of prerequisites of . If ,

then the TEACH mode presents just as before an element and changes

to the TEST mode. If then the TEACH mode checks if there

exists skills with for

. This means that is a set of prerequisites of the

skill . If such skills exists, then the system presents instructional sequences

from , i.e. gives the

student the possibility to aquire all the skills in one step. If such skills

does not exist, the system choses the maximal number of skills with the

described properties and presents instructional sequences corresponding to these

skills. We have to ensure in this case that an aquisition of this skills by

the student does not lead to an increase of the value . So only such cases are

considered in the update of , where the number of skills presented during the

TEACH step is equal to the actual value of the learning speed .

DISCUSSION

The ITS described in this article consists of several structures, for example

, which interact during the teaching process. The design of these structures

inﬂuences the performance and effectivity of the system. We illustrate this by an

example.

The effectivity of the system depends on its ability to diagnose which skills a

student does not master. Especially the speed of the diagnosis is very important,

since it may be demotivating for a student if a lot of questions will be presented to

her or him until a lacking skill is diagnosed and teaching materials are presented.

The speed of the diagnosis depends on the functions and . For example a

stricter results in a faster diagnosis.

Thus, it may be adequate to chose as strict as possible, even if this may result

in an elimination of some true states from which occur with low frequency in

the intended population. This approach increases the speed of the diagnosis while

it decreases its accuracy. Such manipulations of the componentsmust be handled

carefully. Their efﬁciency can be evaluated only in a concrete application.

3We assume here that the TEACH mode can teach more than one skill in each step. How this can

be realized is described later.

8. INTELLIGENT TUTORIAL SYSTEM 193

The described ITS contains no procedure for the repetition of already mas-

tered skills. We have not introduced such a procedure since we assumed that the

skills are teached in increasing sequence concerning their difﬁculty4. Therefore

we can assume that the skills which are teached ﬁrst, i.e. the easy ones, are prac-

ticed again during the aquisition of the skills which require them as prerequisites.

But if we assume that the teaching process may be interupted, for example after

a subgoal is reached, for a long period of time it seems necessary to introduce a

procedure for the repetition of skills. Such a procedure requires a rule for select-

ing the skills mastered in the previous session which should be repeated at the

beginning of the new session. How such a rule can be formulated and integrated

into the teaching process is at the moment an open question.

We have formulated our ITS domain independent. For a concrete application

of the ITS in a special knowledge domain several steps are necessary. First, the

skills relevant in the domain, i.e. the set , must be found and the dependencies

between these skills, i.e. the surmise function , must be formulated. Second,

questions must be constructed which can be used for diagnosis of the skills pos-

sessed by students. Then their connection to the skills, i.e. the skill assignment

, must be formulated. Third teaching materials, i.e. the set , must be devel-

oped and it must be stated which instructions should be presented to a student in

a special state of the student model by constructing the function .

These steps should be left to experienced teachers in the domain. But even if

we assume that a group of such experienced teachers enforced such a necessary

analysis of the domain we have to be aware of the risk that some of the structures

may contain errors which may inﬂuence the performance of the system

negatively.

For example an indadequate construction of may lead to a student model

containing many states which will result in a very slow knowledge diagnosis.

This risk can be reduced with two different approaches. One approach is to

test the assumptions contained in empirically. How this can be done is for

example described in Albert and Held (1994), Albert, Schrepp and Held (1994),

Schrepp (1993), Held (1993), or Korossy (1993, 1997). But if the number of

skills in or the number of questions in is high this method may be problem-

atic, since to many empirical data will be necessary to draw conclusions on the

correctness of and .

A second approach to reduce the inﬂuence of an incorrect formulation of the

basic structures is to make the functions adaptive. We will illustrate this

idea by an example. Assume that we have , i.e. the experts assumed

that should be presented to a student mastering all skills from and

not mastering . Assume further that the instructional sequence has not the

intended effect, i.e. only a few students show mastery of after was presented

to them. Remember that this can be recognized by the TEST mode. We can made

4This is realized within the system since the teaching procedure makes use of the surmise function

to chose the skills which should be teached next.

194 ALBERT AND SCHREPP

easily adaptive if we assume that is eliminated if less than a percentage of

students reaches mastery of after is presented to them. We have to ensure

here that will not become empty. Another possibility is to chose the

instructional sequences accordingly to the probability of their success.

This example shows that it is relatively simple to make the system adaptive

concerning , since we have a possibility to measure the success of an instruc-

tional sequence directly. For and this is more complicated, because the cor-

rectness of these functions can not be observed directly. If, for example, a hypo-

thetical skill state is never observed by any student this may be due to the

fact that contains to much states, i.e. is formulated not strictly enough, or due

to the fact that is not constructed adequately, i.e. there is no combination of

solved and unsolved questions from which implies state . It is at the moment

not clear how and can be changed adaptively due to their success to explain

the behavior of students.

We have formulated in this paper the structure of an ITS based on the theory

of knowledge spaces and skill assignments. The next step in the development of

our work should be the implementation of the ITS and its application in a special

knowledge domain. Since we have described the components of our system as

well as their interaction strictly formalized, the implementation of the system

in a language of logical programming such as Prolog or Lisp should cause no

problems.

A concrete application of this ITS can be used to evaluate the systems efﬁ-

ciency and can giveraise to new insights in the dynamic of the teaching process,

which may lead to further improvement of the theoretical structures.

ACKNOWLEDGEMENTS

The research reported in this paper was supported by Grant Lu 385/1 of the

Deutsche Forschungsgemeinschaft to J. Lukas and D. Albert at the University

of Heidelberg.

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