Personalised Web-Based Learning Systems Klaus-Dieter Schewe1 , Bernhard Thalheim2 , Alexei Tretiakov1 1) Massey University, Department of Information Systems & Information Science Research Centre Private Bag 11 222, Palmerston North, New Zealand [k.d.schewe|a.tretiakov]@massey.ac.nz 2) Christian Albrechts University Department of Computer Science and Applied Mathematics Olshausenstr. 40, D-24098 Kiel, Germany [email protected]

Abstract

A similar approach to learner modelling was followed in [2].

For web-based e-learning systems personalisation, i.e. the self-adaptation of a system to the preferences and goals of the learners is a highly desirable feature. In this article we start from learner profiles to model preferences and goals. Furthermore, we refine course outlines that are first modelled by directed graphs by Kleene algebras with tests. Then we show how systems can be personalized to different learner profiles simply by reasoning with equations.

Learner profiles identify various characteristics that impact on the learning behaviour. The possible values for these characteristics define the learner space, and learner profiles can then be formally defined by (meaningful) subsets of this space. In [4] convex and aggregate learner types have been defined this way. Then the problem to solve is the self-adaptation of the system to the profiles, i.e. the personalisation of the learning system.

1

In this paper we adopt the view that the purpose of modelling learner profiles is merely to associate preferences and goals with them, which will direct the personalisation. Goals can be expressed by conditions that must be reached. Preferences can be the conditional or unconditional selection of a particular learning path in case the learner is offered a choice.

Introduction

Web-based learning systems are systems in which the system is supposed to act in the role of a teacher. The design of such systems usually starts with setting up a course outline, which can be formally described by a directed graph, called outline graph in [1]. In an outline graph the vertices represent learning units, and the edges navigation links between them. The learning units include assessments and tests. Very often, the outline graph is just a straight sequence of learning units.

In order to use such preferences and goals for personalisation we first formalise course outlines following [7], i.e. we add actions and Boolean conditions, then concentrate on the possible flow of these actions. For this we allow sequencing, parallel execution, choice and iteration, which leads to the algebraic structure of a Kleene algebra with tests [5]. In doing so the course outline However, learning success is not primarily due to the will correspond to an algebraic term involving actions right organisation of the learning material. It even more and conditions, goals become postconditions, and preferdepends on the right approach to meet the learners’ ences can be formalised by equations. As a consequence needs. In particular, in [3] the advantages of letting the we can apply term rewriting to realise personalisation. learner drive the learning process have been emphasised. This will constitute the major contribution of this paper. In order to meet the needs of a particular learner it becomes necessary to model the learners. In [4] learner We proceed as follows. In Section 2 we briefly describe profiles and types have been defined adapting the whole- couse outlines going beyond directed graphs, i.e. we inperson approach from [6], which provides a rather com- troduce actions, conditions and various ways to combine prehensive set of human factors that impact on learning. them. In Section 3 we briefly discuss learner profiles em1

• If p is a process and ϕ is a Boolean condition, then the guarded process ϕ·p and the post-guarded process p · ϕ are processes.

phasising preferences and goals and there formalisation in our new framework. Section 4 then describes the personalisation process. We conclude with a brief summary.

2

Note that we did not introduce an operator for parallel execution, say p1 kp2 . As actions are selected by a human learner, we do not need such an operator, as we can replace p1 kp2 by the sequence p1 · p2 and claim commutativity, i.e. p1 · p2 = p2 · p1 must hold.

Course Outlines

Course outlines deal with the problem to describe the navigation of learners through an e-learning system. As shown in [1] we may exploit finite, directed graphs for We usually write 1 instead of skip. We also used an this purpose. additional process 0 that can never be executed. Thus, an outline graph is a finite, directed graph G = Furthermore, we use · (often omitted) both for sequences (V, E), i.e. V and E are finite sets with E ⊆ V × V . The and for logical AND, and + both for choice and logical vertices, i.e. the elements of V , are called learning units, OR. This overloaded use of symbols, however, does not and the edges, i.e. the elements of E are links between cause problems [5]. For logical negation we use ¯, so we these units. Furthermore, with each learning unit u ∈ V write ϕ¯ for the negation of ϕ. we associate a view C u describing the data content of So, we can represent each course outline by a single althis learning unit. gebraic term. Each link ` ∈ E from u1 to u2 corresponds to a possible transition from the source learning unit u1 to the target Example 1. Consider part of a course outline that learning unit u2 . Such a transition should be triggered deals with linear equations. Let us assume we allow by an action initiated by the learner. This action can learners to learn just about how to do some practical simply be a navigation, but in general we may think of calculations, which they may complete with a practical more complicated actions. Therefore, each link is asso- test. Learners may also learn about the theory behind ciated with the name of an action that can be issued in these calculations, in which case they first have to pass a that learning unit. In addition, it is also associated with test about their knowledge in basic linear algebra. The a data type expressing the information communicated theory unit will be followed by an examination. from learning unit u1 to learning unit u2 . So, we get six learning units: u0 = introduction, u1 = Actions on a learning unit may depend on the success- linear algebra test, u2 = practical calculations, u3 = ful completion of the learning unit by the learner, i.e. theory, u4 = practical test and u5 = examination, and we obtain pre- and postconditions. Though each action the following nine actions: may be complex in itself, we will treat actions as beα0 = enter system ing atomic, i.e. we ignore updates on any underlying α 1 = choose theory (between u0 (or u2 ) and u1 ) database, in which learning material is stored. α2 = choose practice (between u0 (or u3 ) and u2 ) Now concentrate on the flow of actions. Actions can be α3 = sit entry test (between u1 and u1 ) executed sequentially or parallel, iterated, and we may allow choice between actions. We may also add an action α4 = continue theory (between u1 and u3 ) skip, which does nothing, so we can then also express α5 = continue exam (between u3 and u5 ) optionality. These possibilities to combine actions lead α6 = continue test (between u2 and u4 ) to operators of an algebra. α7 = sit exam (between u5 and u5 ) Formally, let A = {α1 , . . . , αk } be a set of actions. Then α8 = sit test (between u4 and u4 ) define inductively the set of processes P determined by A as follows: Then we need the equations α2 α4 = α4 α2 , α6 α4 = α4 α6 and α8 α4 = α4 α8 to express the desired parallelism. Fur• Each action α ∈ A is also a process, and skip is a thermore, we consider the following three atomic condiprocess. tions: • If p1 and p2 are processes, then the sequence p1 · p2 ϕ1 = entry test passed and the choice p1 + p2 are also processes. ϕ2 = examination passed • If p is a process, then also the iteration p∗ is a proϕ3 = practical test passed cess. 2

With these actions and conditions at hand we can represent the course outline by the following algebraic term:

Example 2. Let us continue our previous example. For a learner who will not learn the mathematical theory, but only obtain some practical skills in solving linear equations we obtain the unconditional preference rule α1 = 0, i.e. the learner will never choose the theory part.

∗

α0 (α2 (α6 ((ϕ¯3 α8 ) ϕ3 + α8 ) + 1) + 1) (α1 (ϕ¯1 α3 )∗ ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)

3

For a learner who will learn the mathematical theory, but prefers to get some practical skills first, we obtain the preference rule α1 x + α2 y = α2 y (x and y are arbitrary expressions).

Learner Profiles

Let us briefly review what we understand under learner profiles or types. For this we identify general and subject-specific characteristics of users such as the ability to think abstractly, the ability to memorise the learnt material, the need for guidance, the background knowledge, etc. So formally, we obtain a non-empty set C of learner characteristics.

For a learner who will learn the mathematical theory and prefers to go straight into it, we obtain the preference rule α1 x+α2 y = α1 x (x and y are arbitrary expressions). Let these three learner types be denoted by LT1 , LT2 and LT3 , respectively. Furthermore, we may use any Boolean condition built from the conditions that occur in the course outline as a goal that is associated with a learner type.

In addition, for each of these learner characteristics we obtain a set of values called the scale of the characteristic and denoted as S(c). In most case we can assume a linear order ≤ defined on a scale. Then the learner space LS is the cartesian product of these scales, i.e. Y LS = S(c).

Example 3. Let us continue our previous examples. For a learner of the type LT1 the goal might be ϕ3 , i.e. the learner wants to pass the practical test. For learners of type LT2 or LT3 the goal is ϕ2 , i.e. the learners want to pass the examination.

c∈C

Thus, each element of the learner space is a tuple, and each component of this tuple indicates the value of a certain learner characteristic. That is, the learner space captures our knowledge about the different combinations of learner characteristics.

4

Personalisation

A learner type LT corresponds to several points in the In [7] it was shown that course outline as defined in this learner space, i.e. LT ⊆ LS. Learner profiles and types paper give rise to Kleene algebras with tests (KATs), i.e. have been discussed intensively in [4]. Now we may link the following axioms are satisfied: the user types and the course outline based on preference rules. Such preference rules can be expressed by • + and · are associative, i.e. for all p, q, r ∈ K we equations as follows: must have p+(q +r) = (p+q)+r and p(qr) = (pq)r; • + is commutative and idempotent with 0 as neutral element, i.e. for all p, q ∈ K we must have p + q = q + p, p + p = p and p + 0 = p;

• An equation p1 +p2 = p1 expresses an unconditional preference of activity (or process) p1 over p2 . • An equation ϕ(p1 + p2 ) = ϕp1 expresses a conditional preference of activity (or process) p1 over p2 if the condition ϕ is satisfied.

• 1 is a neutral element for ·, i.e. for all p ∈ K we must have p1 = 1p = p;

• Similarly, an equation p(p1 +p2 ) = pp1 expresses another conditional preference of activity (or process) p1 over p2 after the activity (or process) p.

• for all p ∈ K we have p0 = 0p = 0; • · is distributive over +, i.e. for all p, q, r ∈ K we must have p(q + r) = pq + pr and (p + q)r = pr + qr;

• An equation p1 p2 + p2 p1 = p1 p2 expresses a preference of order. ∗

• p∗ q is the least solution x of q + px ≤ x and qp∗ is the least solution of q + xp ≤ x, using the partial order x ≤ y ≡ x + y = y.

∗

• An equation p = pp expresses that in case of an iteration p will at least be executed once. 3

In addition to the axioms of KATs and the equations as follows: that are defined by preference rules there are general α0 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) equations on the course outline that are independent from learner types, e.g. conditions as the following ones: (α1 (ϕ¯1 α3 )∗ ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)ϕ3 = • If an action p has a pre-condition ϕ, then we obtain α0 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1)ϕ3 = the equation ϕp ¯ = 0. α0 ϕ¯3 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1)ϕ3 = (α0 α2 α6 ((ϕ¯3 α8 )∗ ϕ3 )ϕ3 • If an action p has a post-condition ψ, we obtain the equation p = pψ. So the desired personalised course outline in this case is • If an action p is triggered by a condition ϕ, we obtain the equation ϕ = ϕp.

α0 α2 α6 ((ϕ¯3 α8 )∗ ϕ3 ,

• In addition we obtain the exclusion condition ϕψ = i.e. the learner will choose the practical part, then proceed to the practical test and sit this test until s/he has 0 and the tautology ϕ + ψ = 1. passed it. This is exactly what we would have expected intuitively. So we obtain a set of equations ΣLT for each learner type LT . If p is the algebraic expression representing For users of type LT3 with the goal ϕ2 we calculate as the course outline and ψ represents the goal associated follows: with LT , the personalisation problem can be restated as α0 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) follows: (α1 (ϕ¯1 α3 )∗ ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)ϕ3 = Find a minimal p0 such that pψ = p0 ψ holds. ∗ α0 (α1 (ϕ¯1 α3 ) ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)ϕ3 In order to solve this formal version of the problem we simply start with pψ and apply all equations in ΣLT until we obtain an expression p0 ψ, in which p0 is minimal. The So we only discard the initial choice between theory and decidability results in [5] ensure that such a p0 will always practice, and the desired personalised course outline in this case is be obtained. α0 (α1 (ϕ¯1 α3 )∗ ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)

Example 4. Let us continue our previous examples.

First we should add some more general equations. For instance, when entering the system the usual assumption would be that the learner has not yet passed the entry Finally, for users of type LT2 with the goal ϕ2 we cannot test nor the practical test nor the examination. This discard any part of the course outline, so there will be can be expressed by postconditions to α0 , so we get the no change in this case. equations: α0 = α0 ϕ¯1

α0 = α0 ϕ¯2

5

α0 = α0 ϕ¯3

Conclusion

Similarly, only sitting the corresponding test or examina- In this paper we presented an algebraic approach to retion may change the assumption on the three conditions. alise the personalisation of web-based learning systems. For this we formalised course outlines by algebraic terms So we obtain more equations as follows: that involve the actions of the learners, Boolean conditions and junctors that express sequential or parallel ϕ¯1 αi = αi ϕ¯1 for all i 6= 3 composition, iteration and choice. In doing so, the gist ϕ¯2 αi = αi ϕ¯2 for all i 6= 7 of learner profiles, which we see in preferences and goals, ϕ¯3 αi = αi ϕ¯3 for all i 6= 8 can be formalised by equations and postconditions, respectively. As the underlying algebraic structure is that The for users of type LT1 with the goal ϕ3 we calculate of a Kleene algebra with tests, we showed how term 4

rewriting can be exploited to solve the personalisation problem. This approach allows us to model learner profiles, preferences and goals in a purely declarative way leaving it to a rather simple algebraic mechanism to do the personalisation. This shifts the emphasis from the technical solution to the design of content to be learnt and learners to be understood and correctly represented.

References [1] Binemann-Zdanowicz, A., Schewe, K.-D., and Thalheim, B. Adaptation to learning styles. In Proceedings of the IEEE International Conference on Advanced Learning Technologies – ICALT 2004 (2004), Kinshuk, C.-K. Looi, E. Sutinen, D. G. Sampson, I. Aedo, L. Uden, and E. K¨ahk¨onen, Eds., IEEE Computer Society, pp. 121–125. ¨bscher, R. What’s in a prerequisite? In Pro[2] Hu ceedings ICALT 2001 (2001), T. Okamoto, R. Hartley, Kinshuk, and J. P. Klus, Eds., IEEE Computer Society, pp. 365–368. [3] Jantke, K.-P., Memmel, M., Rostanin, O., Thalheim, B., and Tschiedel, B. Decision support by learning-on-demand. In Proceedings DSE 2003, CAISE Workshops (2003), pp. 317–328. [4] Kaschek, R., Schewe, K.-D., Thalheim, B., Kuss, T., and Tschiedel, B. Learner typing for electronic learning systems. In Proceedings of the IEEE International Conference on Advanced Learning Technologies – ICALT 2004 (2004), Kinshuk, C.-K. Looi, E. Sutinen, D. G. Sampson, I. Aedo, L. Uden, and E. K¨ ahk¨ onen, Eds., IEEE Computer Society, pp. 375–379. [5] Kozen, D. Kleene algebra with tests. ACM Transactions on Programming Languages and Systems 19, 3 (1997), 427–443. [6] Martinez, M. Key design considerations for personalized learning on the web. Educational Technology and Society 4, 1 (2001). [7] Schewe, K.-D., and Thalheim, B. Reasoning about web information systems using story algebras. In Proceedings ADBIS 2004 (2004), A. Benczur, J. Demetrovics, and G. Gottlob, Eds., vol. 3255 of LNCS, pp. 54–66.

5

Abstract

A similar approach to learner modelling was followed in [2].

For web-based e-learning systems personalisation, i.e. the self-adaptation of a system to the preferences and goals of the learners is a highly desirable feature. In this article we start from learner profiles to model preferences and goals. Furthermore, we refine course outlines that are first modelled by directed graphs by Kleene algebras with tests. Then we show how systems can be personalized to different learner profiles simply by reasoning with equations.

Learner profiles identify various characteristics that impact on the learning behaviour. The possible values for these characteristics define the learner space, and learner profiles can then be formally defined by (meaningful) subsets of this space. In [4] convex and aggregate learner types have been defined this way. Then the problem to solve is the self-adaptation of the system to the profiles, i.e. the personalisation of the learning system.

1

In this paper we adopt the view that the purpose of modelling learner profiles is merely to associate preferences and goals with them, which will direct the personalisation. Goals can be expressed by conditions that must be reached. Preferences can be the conditional or unconditional selection of a particular learning path in case the learner is offered a choice.

Introduction

Web-based learning systems are systems in which the system is supposed to act in the role of a teacher. The design of such systems usually starts with setting up a course outline, which can be formally described by a directed graph, called outline graph in [1]. In an outline graph the vertices represent learning units, and the edges navigation links between them. The learning units include assessments and tests. Very often, the outline graph is just a straight sequence of learning units.

In order to use such preferences and goals for personalisation we first formalise course outlines following [7], i.e. we add actions and Boolean conditions, then concentrate on the possible flow of these actions. For this we allow sequencing, parallel execution, choice and iteration, which leads to the algebraic structure of a Kleene algebra with tests [5]. In doing so the course outline However, learning success is not primarily due to the will correspond to an algebraic term involving actions right organisation of the learning material. It even more and conditions, goals become postconditions, and preferdepends on the right approach to meet the learners’ ences can be formalised by equations. As a consequence needs. In particular, in [3] the advantages of letting the we can apply term rewriting to realise personalisation. learner drive the learning process have been emphasised. This will constitute the major contribution of this paper. In order to meet the needs of a particular learner it becomes necessary to model the learners. In [4] learner We proceed as follows. In Section 2 we briefly describe profiles and types have been defined adapting the whole- couse outlines going beyond directed graphs, i.e. we inperson approach from [6], which provides a rather com- troduce actions, conditions and various ways to combine prehensive set of human factors that impact on learning. them. In Section 3 we briefly discuss learner profiles em1

• If p is a process and ϕ is a Boolean condition, then the guarded process ϕ·p and the post-guarded process p · ϕ are processes.

phasising preferences and goals and there formalisation in our new framework. Section 4 then describes the personalisation process. We conclude with a brief summary.

2

Note that we did not introduce an operator for parallel execution, say p1 kp2 . As actions are selected by a human learner, we do not need such an operator, as we can replace p1 kp2 by the sequence p1 · p2 and claim commutativity, i.e. p1 · p2 = p2 · p1 must hold.

Course Outlines

Course outlines deal with the problem to describe the navigation of learners through an e-learning system. As shown in [1] we may exploit finite, directed graphs for We usually write 1 instead of skip. We also used an this purpose. additional process 0 that can never be executed. Thus, an outline graph is a finite, directed graph G = Furthermore, we use · (often omitted) both for sequences (V, E), i.e. V and E are finite sets with E ⊆ V × V . The and for logical AND, and + both for choice and logical vertices, i.e. the elements of V , are called learning units, OR. This overloaded use of symbols, however, does not and the edges, i.e. the elements of E are links between cause problems [5]. For logical negation we use ¯, so we these units. Furthermore, with each learning unit u ∈ V write ϕ¯ for the negation of ϕ. we associate a view C u describing the data content of So, we can represent each course outline by a single althis learning unit. gebraic term. Each link ` ∈ E from u1 to u2 corresponds to a possible transition from the source learning unit u1 to the target Example 1. Consider part of a course outline that learning unit u2 . Such a transition should be triggered deals with linear equations. Let us assume we allow by an action initiated by the learner. This action can learners to learn just about how to do some practical simply be a navigation, but in general we may think of calculations, which they may complete with a practical more complicated actions. Therefore, each link is asso- test. Learners may also learn about the theory behind ciated with the name of an action that can be issued in these calculations, in which case they first have to pass a that learning unit. In addition, it is also associated with test about their knowledge in basic linear algebra. The a data type expressing the information communicated theory unit will be followed by an examination. from learning unit u1 to learning unit u2 . So, we get six learning units: u0 = introduction, u1 = Actions on a learning unit may depend on the success- linear algebra test, u2 = practical calculations, u3 = ful completion of the learning unit by the learner, i.e. theory, u4 = practical test and u5 = examination, and we obtain pre- and postconditions. Though each action the following nine actions: may be complex in itself, we will treat actions as beα0 = enter system ing atomic, i.e. we ignore updates on any underlying α 1 = choose theory (between u0 (or u2 ) and u1 ) database, in which learning material is stored. α2 = choose practice (between u0 (or u3 ) and u2 ) Now concentrate on the flow of actions. Actions can be α3 = sit entry test (between u1 and u1 ) executed sequentially or parallel, iterated, and we may allow choice between actions. We may also add an action α4 = continue theory (between u1 and u3 ) skip, which does nothing, so we can then also express α5 = continue exam (between u3 and u5 ) optionality. These possibilities to combine actions lead α6 = continue test (between u2 and u4 ) to operators of an algebra. α7 = sit exam (between u5 and u5 ) Formally, let A = {α1 , . . . , αk } be a set of actions. Then α8 = sit test (between u4 and u4 ) define inductively the set of processes P determined by A as follows: Then we need the equations α2 α4 = α4 α2 , α6 α4 = α4 α6 and α8 α4 = α4 α8 to express the desired parallelism. Fur• Each action α ∈ A is also a process, and skip is a thermore, we consider the following three atomic condiprocess. tions: • If p1 and p2 are processes, then the sequence p1 · p2 ϕ1 = entry test passed and the choice p1 + p2 are also processes. ϕ2 = examination passed • If p is a process, then also the iteration p∗ is a proϕ3 = practical test passed cess. 2

With these actions and conditions at hand we can represent the course outline by the following algebraic term:

Example 2. Let us continue our previous example. For a learner who will not learn the mathematical theory, but only obtain some practical skills in solving linear equations we obtain the unconditional preference rule α1 = 0, i.e. the learner will never choose the theory part.

∗

α0 (α2 (α6 ((ϕ¯3 α8 ) ϕ3 + α8 ) + 1) + 1) (α1 (ϕ¯1 α3 )∗ ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)

3

For a learner who will learn the mathematical theory, but prefers to get some practical skills first, we obtain the preference rule α1 x + α2 y = α2 y (x and y are arbitrary expressions).

Learner Profiles

Let us briefly review what we understand under learner profiles or types. For this we identify general and subject-specific characteristics of users such as the ability to think abstractly, the ability to memorise the learnt material, the need for guidance, the background knowledge, etc. So formally, we obtain a non-empty set C of learner characteristics.

For a learner who will learn the mathematical theory and prefers to go straight into it, we obtain the preference rule α1 x+α2 y = α1 x (x and y are arbitrary expressions). Let these three learner types be denoted by LT1 , LT2 and LT3 , respectively. Furthermore, we may use any Boolean condition built from the conditions that occur in the course outline as a goal that is associated with a learner type.

In addition, for each of these learner characteristics we obtain a set of values called the scale of the characteristic and denoted as S(c). In most case we can assume a linear order ≤ defined on a scale. Then the learner space LS is the cartesian product of these scales, i.e. Y LS = S(c).

Example 3. Let us continue our previous examples. For a learner of the type LT1 the goal might be ϕ3 , i.e. the learner wants to pass the practical test. For learners of type LT2 or LT3 the goal is ϕ2 , i.e. the learners want to pass the examination.

c∈C

Thus, each element of the learner space is a tuple, and each component of this tuple indicates the value of a certain learner characteristic. That is, the learner space captures our knowledge about the different combinations of learner characteristics.

4

Personalisation

A learner type LT corresponds to several points in the In [7] it was shown that course outline as defined in this learner space, i.e. LT ⊆ LS. Learner profiles and types paper give rise to Kleene algebras with tests (KATs), i.e. have been discussed intensively in [4]. Now we may link the following axioms are satisfied: the user types and the course outline based on preference rules. Such preference rules can be expressed by • + and · are associative, i.e. for all p, q, r ∈ K we equations as follows: must have p+(q +r) = (p+q)+r and p(qr) = (pq)r; • + is commutative and idempotent with 0 as neutral element, i.e. for all p, q ∈ K we must have p + q = q + p, p + p = p and p + 0 = p;

• An equation p1 +p2 = p1 expresses an unconditional preference of activity (or process) p1 over p2 . • An equation ϕ(p1 + p2 ) = ϕp1 expresses a conditional preference of activity (or process) p1 over p2 if the condition ϕ is satisfied.

• 1 is a neutral element for ·, i.e. for all p ∈ K we must have p1 = 1p = p;

• Similarly, an equation p(p1 +p2 ) = pp1 expresses another conditional preference of activity (or process) p1 over p2 after the activity (or process) p.

• for all p ∈ K we have p0 = 0p = 0; • · is distributive over +, i.e. for all p, q, r ∈ K we must have p(q + r) = pq + pr and (p + q)r = pr + qr;

• An equation p1 p2 + p2 p1 = p1 p2 expresses a preference of order. ∗

• p∗ q is the least solution x of q + px ≤ x and qp∗ is the least solution of q + xp ≤ x, using the partial order x ≤ y ≡ x + y = y.

∗

• An equation p = pp expresses that in case of an iteration p will at least be executed once. 3

In addition to the axioms of KATs and the equations as follows: that are defined by preference rules there are general α0 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) equations on the course outline that are independent from learner types, e.g. conditions as the following ones: (α1 (ϕ¯1 α3 )∗ ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)ϕ3 = • If an action p has a pre-condition ϕ, then we obtain α0 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1)ϕ3 = the equation ϕp ¯ = 0. α0 ϕ¯3 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1)ϕ3 = (α0 α2 α6 ((ϕ¯3 α8 )∗ ϕ3 )ϕ3 • If an action p has a post-condition ψ, we obtain the equation p = pψ. So the desired personalised course outline in this case is • If an action p is triggered by a condition ϕ, we obtain the equation ϕ = ϕp.

α0 α2 α6 ((ϕ¯3 α8 )∗ ϕ3 ,

• In addition we obtain the exclusion condition ϕψ = i.e. the learner will choose the practical part, then proceed to the practical test and sit this test until s/he has 0 and the tautology ϕ + ψ = 1. passed it. This is exactly what we would have expected intuitively. So we obtain a set of equations ΣLT for each learner type LT . If p is the algebraic expression representing For users of type LT3 with the goal ϕ2 we calculate as the course outline and ψ represents the goal associated follows: with LT , the personalisation problem can be restated as α0 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) follows: (α1 (ϕ¯1 α3 )∗ ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)ϕ3 = Find a minimal p0 such that pψ = p0 ψ holds. ∗ α0 (α1 (ϕ¯1 α3 ) ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)ϕ3 In order to solve this formal version of the problem we simply start with pψ and apply all equations in ΣLT until we obtain an expression p0 ψ, in which p0 is minimal. The So we only discard the initial choice between theory and decidability results in [5] ensure that such a p0 will always practice, and the desired personalised course outline in this case is be obtained. α0 (α1 (ϕ¯1 α3 )∗ ϕ1 α4 (α2 (α6 ((ϕ¯3 α8 )∗ ϕ3 + α8 ) + 1) + 1) α5 (ϕ¯2 α7 )∗ ϕ2 + 1)

Example 4. Let us continue our previous examples.

First we should add some more general equations. For instance, when entering the system the usual assumption would be that the learner has not yet passed the entry Finally, for users of type LT2 with the goal ϕ2 we cannot test nor the practical test nor the examination. This discard any part of the course outline, so there will be can be expressed by postconditions to α0 , so we get the no change in this case. equations: α0 = α0 ϕ¯1

α0 = α0 ϕ¯2

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α0 = α0 ϕ¯3

Conclusion

Similarly, only sitting the corresponding test or examina- In this paper we presented an algebraic approach to retion may change the assumption on the three conditions. alise the personalisation of web-based learning systems. For this we formalised course outlines by algebraic terms So we obtain more equations as follows: that involve the actions of the learners, Boolean conditions and junctors that express sequential or parallel ϕ¯1 αi = αi ϕ¯1 for all i 6= 3 composition, iteration and choice. In doing so, the gist ϕ¯2 αi = αi ϕ¯2 for all i 6= 7 of learner profiles, which we see in preferences and goals, ϕ¯3 αi = αi ϕ¯3 for all i 6= 8 can be formalised by equations and postconditions, respectively. As the underlying algebraic structure is that The for users of type LT1 with the goal ϕ3 we calculate of a Kleene algebra with tests, we showed how term 4

rewriting can be exploited to solve the personalisation problem. This approach allows us to model learner profiles, preferences and goals in a purely declarative way leaving it to a rather simple algebraic mechanism to do the personalisation. This shifts the emphasis from the technical solution to the design of content to be learnt and learners to be understood and correctly represented.

References [1] Binemann-Zdanowicz, A., Schewe, K.-D., and Thalheim, B. Adaptation to learning styles. In Proceedings of the IEEE International Conference on Advanced Learning Technologies – ICALT 2004 (2004), Kinshuk, C.-K. Looi, E. Sutinen, D. G. Sampson, I. Aedo, L. Uden, and E. K¨ahk¨onen, Eds., IEEE Computer Society, pp. 121–125. ¨bscher, R. What’s in a prerequisite? In Pro[2] Hu ceedings ICALT 2001 (2001), T. Okamoto, R. Hartley, Kinshuk, and J. P. Klus, Eds., IEEE Computer Society, pp. 365–368. [3] Jantke, K.-P., Memmel, M., Rostanin, O., Thalheim, B., and Tschiedel, B. Decision support by learning-on-demand. In Proceedings DSE 2003, CAISE Workshops (2003), pp. 317–328. [4] Kaschek, R., Schewe, K.-D., Thalheim, B., Kuss, T., and Tschiedel, B. Learner typing for electronic learning systems. In Proceedings of the IEEE International Conference on Advanced Learning Technologies – ICALT 2004 (2004), Kinshuk, C.-K. Looi, E. Sutinen, D. G. Sampson, I. Aedo, L. Uden, and E. K¨ ahk¨ onen, Eds., IEEE Computer Society, pp. 375–379. [5] Kozen, D. Kleene algebra with tests. ACM Transactions on Programming Languages and Systems 19, 3 (1997), 427–443. [6] Martinez, M. Key design considerations for personalized learning on the web. Educational Technology and Society 4, 1 (2001). [7] Schewe, K.-D., and Thalheim, B. Reasoning about web information systems using story algebras. In Proceedings ADBIS 2004 (2004), A. Benczur, J. Demetrovics, and G. Gottlob, Eds., vol. 3255 of LNCS, pp. 54–66.

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