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A Computational Model of Learning to Solve Problems with Diagrams

Aims

This project aims to build a computational model of learning to solve problems with diagrams, and has three main goals:

• Construct a model which learns internal representations based upon input perceptual patterns and output drawing actions. These representations will be used to drive a model of problem solving, e.g. taking the role of diagrammatic configuration schema.
• Test the EPAM-chunking theory for expert memory in a domain where visual patterns can be used as plans for problem solving behaviour.
• Provide a computer simulation of how human subjects learn to use AVOW diagrams.

Theoretical Background

Several strands of related work are being incorporated into this model.

• Actual problem solving behaviour by humans is most effective with an appropriate diagrammatic representation (Cheng & Simon, 1995).
• Studies in expertise have shown that skill differences between individuals may be accounted for by the size of their perceptual memory (Gobet, 1998).
• Models for problem solving have shown that a memory organised as diagrammatic configuration schema effectively explains the forward-chaining method of solution adopted by human experts (Koedinger & Anderson, 1990).

Constructing AVOW Diagrams

The AVOW (Amps, Volts, Ohms, Watts) diagram is one example of a class of Law Encoding Diagrams, diagrammatic representations for problem solving and learning in the sciences (Cheng, 1996, 1998). An AVOW box is a representation for an individual resistor, as shown in Figure 1.

An AVOW diagram for a circuit is formed by composing individual AVOW boxes, as shown in Figure 2. Constructing AVOW diagrams requires the subject to obey two sets of constraints simultaneously:

• first, form an accurate representation of the circuit,
• second, construct a well-formed AVOW diagram.

These constraints encourage the subject to follow a more efficient solution path, but also leave open the possibility of considerable variation in the strategies adopted by individual subjects. This last point is illustrated in Figure 4, where three different solution strategies are shown for the circuit in Figure 3(a) (these are taken from studies done here in Nottingham, see Cheng, submitted).

Learning Multiple Representations

Our implementation begins with CHREST (Gobet & Jansen, 1994), a model of chess expertise which uses an EPAM-based model of memory to explain the acquisition of perceptual chunks. In order to extend CHREST for the purpose of problem solving, mechanisms for planning and look ahead must be included. The problem of planning in constructing AVOW diagrams is in forming an overall impression of the total AVOW representation before attempting to instantiate this in a drawing. The approach taken here allows the model to include equivalence links between perceptual chunks, where each link associates a given circuit with its equivalent AVOW representation. (This is an extension of some ideas for handling multiple nets, first proposed in Gobet, 1996.)

Proposed Model

The general form of a model of problem solving with diagrams has been well established by earlier work on reasoning and inferencing with external representations, eg. Tabachneck-Schijf, Leonardo and Simon (1997). The main components are:

• an external representation: we have a computer representation of a sheet of paper, which contains line drawings of the circuit and AVOW diagrams.
• an eye for retrieving information from the external representation, and a pen for adding information.
• a Short-Term memory (STM) of visuo-spatial information, which is used, at present, for building up chunks from different visual images.
• a Long-Term memory (LTM) of visuo-spatial information, used for storing and indexing information about the chunks. In particular, as shown in Figure 5, the LTM provides an inheritance structure for the separate diagrammatic representations as well as supporting the equivalence links between the different representations.

Our current implementation uses a graphical computer environment with a directable eye for retrieving diagrammatic information from circuit and AVOW diagrams. A visual STM is used in conjunction with the extended EPAM model described above to acquire perceptual information about multiple external representations. This is presently being extended into a more comprehensive computer model of how humans learn to solve problems with diagrams.

Publications

Integrated Model

• Lane, P.C.R., Cheng, P.C-H., & Gobet, F. Learning perceptual chunks for problem decomposition. In Proceedings of the Twenty Third Annual Conference of the Cognitive Science Society, Edinburgh, Scotland, 2001.
• Lane, P.C.R., Cheng, P.C-H., and Gobet, F. CHREST+: Investigating how humans learn to solve problems using diagrams. AISB Quarterly, No.103, pp.24-30, 2000.
• Lane, P.C.R., Cheng, P.C-H., & Gobet, F. (1999). Problem solving with diagrams: Modelling the learning of perceptual information (CREDIT Technical Report No. 59, University of Nottingham). Postscript version
• Lane, P.C.R., Cheng, P.C-H., & Gobet, F. (1999). Learning perceptual schemas to avoid the utility problem. In Proceedings of the Nineteenth SGES International Conference on Knowledge Based Systems and Applied Artificial Intelligence, Cambridge, UK. Abstract

Diagrammatic Representations

• Cheng, P. C-H. (submitted). Electrifying representations for learning: An evaluation of AVOW diagrams for electricity.
• Cheng, P. C-H. (1998). A framework for scientific reasoning with law encoding diagrams: Analysing protocols to assess its utility. In M. A. Gernsbacher & S. J. Derry (Eds.) Proceedings of the Twentieth Annual Conference of the Cognitive Science Society (Mahwah, NJ: Erlbaum) pp. 232-235.
• Cheng, P. C-H. (1996). Scientific discovery with law-encoding diagrams. Creativity Research Journal, 9, 145-162.
• Cheng, P. C.-H. & Simon, H. A. (1995). Scientific Discovery and Creative Reasoning with Diagrams. In S. Smith, T. Ward, & R. Finke (Eds.), The Creative Cognition Approach (pp. 205-228). Cambridge, MA: MIT Press.

Perceptual Memory

• Gobet, F. (1998). Memory for the meaningless: How chunks help. In M. A. Gernsbacher & S. J. Derry (Eds.) Proceedings of the Twentieth Annual Conference of the Cognitive Science Society (Mahwah, NJ: Erlbaum) pp. 398-403.
• Gobet, F. (1996). Discrimination nets, production systems and semantic networks: Elements of a unified framework. Proceedings of the Second International Conference of the Learning Sciences, (Evanston Il: Northwestern University), pp. 398-403.
• Gobet, F. & Jansen, P. (1994). Towards a chess program based on a model of human memory. In H. J. van den Herik, I. S. Herschberg, & J. W. Uiterwijk (Eds.) Advances in computer chess 7, University of Limburg Press, Maastricht.

References

• Koedinger, K. R., & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14, 511-550.
• Tabachneck-Schijf, H. J. M., Leonardo, A. M., & Simon, H. A. (1997). CaMeRa: A computational model of multiple representations. Cognitive Science, 21, 305-350.