M. OLIVER, T. O'SHEA,
IET, Open University, Walton Hall, Milton Keynes, MK7 6AA.
Email: M.J.Oliver@open.ac.uk
Abstract: Graphical representations have been shown to aid logic learning, especially when used in conjunction with sentential representations. After reviewing some of the problems associated with learning formal reasoning, this paper summarises two different ways in which graphical representations have been used to successfully provide this support. It then provides an example of how these two representational categories can be combined to extend existing logic learning software to support a new topic; in this case, extending Tarski's World to support the learning of Modal logic. This design is supported by adopting an object-oriented approach to the implementation, making use of Smalltalk's Model-View-Controller paradigm. This also allows three important elements to be introduced into the design which were not present in Tarski's World: an extension of the semantics to include those specific to Modal logic, the use of animation to tackle key conceptual difficulties, and the dynamic creation and support of multiple representations of the topic.
Learning logic is no trivial task. Being an abstract topic, it seems to have little relation to meaningful, real world problems (Fung, 1992; van der Pal, 1993; Oliver, 1995a). This often leads to motivational problems, which can significantly affect scores in formal language learning (Fung, 1994).
Students interviewed talked of the course FP1 as being an `absolutely new' experience, of their `panic' and `shock' when confronted with `abstract figures' and `abstract things'. They spoke of the difficulty of understanding the `formal notation', of it being like `another language... like learning Latin', of there being `too much to do in too short a time'.
(Fung, 1994, pg 12).
Even after much hard work, some students are still unable to grasp the principles of formal reasoning, although why this is so is not fully understood (Fung, 1992).
Logic can also conflict with existing beliefs (Dobson, 1994). This leads to confusion and hampers progress, since the "reasonableness" of solutions is often measured by their correlation to these same beliefs (Gill, 1994; Campbell, 1995). Such beliefs are often context dependent, inconsistent, and difficult to alter (White, 1991; Draper, 1992), making it difficult to provide convincing, general examples.
Additionally, a lack of prior training can significantly impede students' logical abilities (Fung, 1992). The most significant skills which were found to be lacking in the study by Fung and O'Shea (1992) were:
i) a familiarity with formal notation;
ii) the ability to break problems into manageable component sections
iii) being able to manipulate formulae, and
iv) the ability to abstract general principles from particular cases.
As a consequence of all these difficulties, achieving transfer for logical skills typically involves years of training (van der Pal, 1993).
Software tools which make use of external representations to support logic learning do seem to have a positive effect on learning outcomes (Fung, 1994; van der Pal, 1995; Dobson, 1996), although it should be noted that the use of graphical representations does not suit all students (Cox, 1994). There are many qualities of graphical representations which could account for this positive effect, including clarity (Campbell, 1995), the preservation of topological, geometrical, chronological and hierarchical relationships of a problem's components (Sloman, 1971; Larkin, 1987; Fung, 1994), the use of secondary notation such as adjacency, clustering, white space, labelling, etc (Larkin, 1987; Petre, 1993), and the way in which structural constraints can guide problem solving or knowledge organisation (Sloman, 1971; Barwise, 1990; Dobson, 1994; Stenning, 1993). It is also acknowledged that graphical representations have their own problems (e.g. Barwise, 1990; Parkes, 1993; Petre, 1993), but these will not be covered in this paper.
It seems unlikely, though, that each graphical representation uses all these beneficial attributes effectively. Instead, many graphical representations which have positive effects on learning seem to use only a selection of these qualities effectively.
The program Venn (produced by the Chariot Software Group), for example, provides an interface for constructing, manipulating and interpreting Venn diagrams (figure 1). These representations are used as an active part of the reasoning process, scaffolding problem solving in a very similar way to Law Encoding Diagrams (Cheng, 1996), analogical representations (Sloman, 1971), "good" diagrams (Barwise, 1990) or "vivid" diagrams (Levesque, 1986). All of these make use of structural constraints and the explicit representation of abstract relationships to direct attention to salient features and support the cognitive process.
Figure 1: The program Venn
Whilst diagrams give good support for students who know how to use them, those that find the translation between content and representation difficult often find their learning hindered. This seems to be because any perceptual cue, if used inappropriately or poorly understood, becomes a mis-cue, promoting bad inferences and incorrect rule usage (Larkin, 1987; Cox, 1994). Clearly, students must somehow acquire the ability to "read" this kind of diagram if they are to be used effectively (Petre, 1993); explicit teaching of the construction rules can be one way of achieving this.
In marked contrast is the type of interface provided by the program Tarski's World (Barwise, 1992). This consists of two windows for dealing with sentential representations of logic, together with game and world windows (figure 2). Fuller descriptions of this program can be found elsewhere (e.g. Barwise, 1992; Goldson, 1993; Oliver, 1995b).
The graphical representation provided by the world window plays a very different role to the diagrams in "Venn". Instead of actively supporting the reasoning process, this representation allows an exploration of how the rules of logic work and what they mean. It has, as a result, been described as a psychologically situated learning environment (van der Pal, 1995). The psychological nature of this description refers to the fact that this representation is neither authentic nor situated in the traditional sense (i.e. the student doesn't actually need to manipulate blocks, nor are they immersed in a real world situation); however, students describe the representation as being believable, which allows this learning environment to act as if it possessed these properties (ibid). This is closely related to the way in which graphical representations can be used as the content of reasoning and problem solving, as described by Barwise (1990).
Figure 2: A screenshot from the program Tarski's World.
Whilst Tarski's World has been demonstrated to be (in general) an effective tool for learning formal reasoning (Fung, 1994; Cox, 1994; van der Pal, 1995), one proviso has been given: learning which makes use of only this situated aspect is not effective. The program's benefits come from combining the world window with sentential representations (van der Pal, 1995). This is not surprising: taken alone, the situated representation provides no support or scaffolding for the reasoning process, playing only a passive role. Instead, it provides an opportunity for using simple examples to explore what the rules mean. The support needed for the reasoning process is then provided by the Sentence Editor window. Similar conclusions have been drawn for other topics with an abstract element, such as physics (Alessi, 1988).
To summarise, graphical representations seem to be used in at least two distinct ways: as diagrams, which scaffold the reasoning process, and as learning environments, providing content which can then be reasoned about. Since learning environments prove ineffective unless complemented by a formal representation of the topic, these two uses appear complementary. Combining them would lead to the parallel use of graphical representations, one providing a content for problems, the other actively supporting and guiding the reasoning process itself.
As noted, Tarski's World has been shown to be effective in promoting students' understanding of first order logic. Since Modal Logic is an extension of this topic, a design which extends Tarski's World would seem a good starting point for a new software tool. Tarski's World has been classified as utilising only one type of graphical representation: the learning environment. Consequently, this brief introduction to Modal logic will concentrate on the diagrams used to actively support reasoning in traditional Modal logic courses. These will provide a complementary representation which can be integrated into the design.
Modal Logic is an extension of first order logic which introduces two additional concepts: necessity and possibility. These are defined formally, but operate intuitively, so that something is necessary if it must be true, and possible if it could be true. Assigning truth values to necessary or possible statements involves evaluating an exhaustive set of situations which could occur. Each particular situation in this set is referred to as a "possible world". "Necessarily p" is true if p is true in every possible world, and "possibly p" is true if p is true in at least one of them.
Two main factors make Modal Logic difficult. The first of these lies in conceptualising the co-existence of and relationships between many possible worlds. The second is that Modal logic consists of several distinct logical systems, each describing a different kind of necessity (for example, a moral necessity would be very different from a physical necessity). Gaining insight into how these work and why each is important is a major difficulty for students.
Existing teaching texts for Modal Logic often promote the use of graphical representations to support the reasoning process (e.g. Hughes, 1972; Reeves, 1990). Although superficially dissimilar, Reeves' and Hughes' diagrams both encode the same logical rules, and are effectively equivalent. For system T, the weakest system of Modal Logic either discusses, these become tree diagrams, hierarchically mapping relationships (figure 3). Here, each rectangle represents one particular possible world, and each line, a relationship between two worlds. The sentence in the first box is the claim under investigation, 1's and 0's indicate truth values, and stars are an explicit part of the rules governing when to create a new possible world.
Figure 3: Tree diagrams for Modal Logic (system T)
The next proof (figure 4) shows a claim strong enough to satisfy the conditions of S4. Here, loops have been introduced into the diagram (e.g. between W2 and W4), which effectively introduces reduction laws into the system. A diagram for S5 involves linking each possible world to every other one, permitting further levels of cancellation. Each of these types of diagram is Law Encoding for a particular system, and comparisons between the three help to explicitly demonstrate the nature of the relationships in different Modal systems.
Figure 4: A `tree diagram' with a different set of rules (system S4)
One way of extending Tarski's World to cover Modal Logic is to add these Law Encoding diagrams into the interface. The user also needs to be able to open up more than one world window at once, so as to compare possible worlds. As has been noted (Oliver, 1995b), adapting Tarski's World would also involve placing restrictions on the situations represented, so as to prevent the disjunction of possible solutions becoming unmanageable. Removing the option of having different sized objects, reducing the size of the grid, and introducing movement relationships, is a way of maintaining the usability of the block-world model whilst introducing Modal operations. Graphical conventions to deal with groups of similar possible worlds (e.g. a set of possible worlds sharing a particular feature) are being developed.
The implementation of this design is being carried out using the programming language Smalltalk, whose Model-View-Controller paradigm is well suited to creating an interface in which many distinct windows need to interact. The underlying model comprises of a collection of objects, each of which is one possible world. These have three properties: a name, a two-dimensional array to store the objects on the world grid, and a set of relationships with other possible worlds. In turn, the objects to be displayed on the grid have two properties: a name and a shape.
This model is designed to encourage the comparison of different possible worlds by allowing separate windows to be opened on each. It also allows a new window, the "Universe" window, to be created. This would dynamically build diagrams, providing a map-like overview of the worlds under consideration. A menu allows the user to switch between different Modal systems, allowing them to see the effects this has had on the relationships between possible worlds.
Another advantage of this approach is that Modal semantics can be added to the sentential representation. This allows the program to determine whether something is necessarily true or possibly true for a given set of worlds, providing simple feedback for students. This new design is not an intelligent tutoring system, however. More meaningful, interpretative feedback will need to be provided either by the course tutor or by someone more experienced.
The third new feature supported by this design is the use of animation to enumerate the possible worlds. As mentioned, two of the major conceptual difficulties of Modal Logic are conceptualising the co-existence of possible worlds and understanding the different relationships that can exist between them. One way of overcoming these is to add a "Show Me" button, which runs through an animation of the relationships between a specified world and other possible situations. This also provides a meaningful demonstration of the link between the two different types of graphical representation (the learning environment and the law encoding diagram), promoting understanding and supporting the use of multiple representations. It also permits the option of automatically constructing a correct and complete "universe" of relationships, which could otherwise become a repetitious and uninformative exercise in book-keeping. A warning is given if the number of worlds to be described is excessively large, in which case the problem can be refined, or only a sub-section be generated.
Figure 5: The revised Tarski's World interface.
Figure 5 shows a design for the resultant interface. In this example, the user has two worlds open for inspection. World 1 shows the starting condition for a problem; world 3 shows a later case in which a Tetrahedron has moved from its starting position. The "Universe" window keeps track of the relationships between the possible worlds specified so far.
The evaluation of the software tool will follow the pattern used by Fung and O'Shea (1992). This will involve assessing the tool within the context of a Modal logic course, using it to complement existing teaching texts.
This paper has identified two uses of external representations, diagrams and learning environments, each of which has been demonstrated to enhance the learning process. It argues that further enhancement of software designs can be gained from combining these two types of representation.
By adopting an object-oriented programming approach, a software tool for Modal Logic has been devised which supports this combination of representations. This approach also permits the introduction of Modal semantics into the design. Animation promotes the understanding of the link between the two types of representation. Specifically, the Model-View-Controller mechanism of Smalltalk has been adopted as a means of dynamically creating and managing the graphical representations both of possible worlds and the relationships between them.
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