Representational Analysis and Design:

Law Encoding Diagrams

Overview: Pivotal role of representations in learning

The motivation for the project is claim that the representations (external symbol systems) used by a learner have critical role in conceptual learning, particularly in complex scientific and mathematical domains. The Representational Analysis and Design project is exploring the pivotal role representations can have in learning. Two main claims being investigated are:

1. representations used to support the learning process can substantially affect both what is learnt and the ease with which learning takes place; and
2. representations constrain both the nature of the conceptual structures that the learner develops and the problem solving procedures that they acquire.

The claims are being investigated by pursuing five closely related goals:

• To discover principles of effective representational systems for conceptual learning in complex scientific and mathematical domains.
• To invent or rediscover new representational systems to promote conceptual learning in selected domains - in particular Law Encoding Diagrams (LEDs).
• To analyse the difficulties of learning caused by existing poor representations.
• To understand the psychological processes underlying higher forms of cognition with such complex visual representations (e.g., problem solving, learning, discovery).
• To design and evaluate computer-based systems to support learning with LEDs.

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Theoretical Background:

The two claims about the critical role of representations for learning are based research in cognitive science, psychology, Artificial Intelligence and education that has demonstrated that representations have a fundamental role in higher forms of cognition. For example:

• Studies of human cognition have shown that:
  • different representations can dramatically affect the ease of problem solving, sometimes by over an order of magnitude;
  • experts and novices in the same domain often use alternative representations or the same representations in different ways;
  • there are often benefits of using diagrams over propositional or sentential representations;
  • using an external representation may reduce cognitive load when reasoning in complex domains.
• Computational models of problem solving and scientific discovery have demonstrated that alternative representations for the same task can require substantially different amounts computation.
• In the history of science the invention of new representations often accompanied the major discoveries made by great scientists.

The focus of the research in on a particular class of representations with some interesting properties that appear to support learning &emdash; Law Encoding Diagrams, LEDs.

Examples of how LEDs can improve conceptual learning in:

• Electricity
• Particle collisions in physics

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Principles of effective representations for learning

Various principles have been discovered and are being investigated in the project. They come in two groups.

1. Semantic transparency principles are concerned with the "static" form and meaning of the permissible expressions of the system. They identify characteristics of representational systems that aim to support learners' development of a coherent, comprehensible and memorable network of concepts, they include:
   • Integration of levels of abstraction;
   • Integration of perspectives;
   • Globally homogeneous and locally heterogeneous representation of concepts.
2. Plastic generativity (or syntactic plasticity) principles are concerned with the "dynamic" generation and manipulation of expressions. They identify characteristics of representational system aim to facilitate the use of the representation for making inferences, solving problems and exploring the conceptual structure of the domain:
   • Malleable expressions,
   • Uniform procedures,
   • Compact procedures.

The principles attempt to simultaneously satisfy the need to represent the complex knowledge of substantive domains with the nature and limitations of human information processing &emdash; a marriage of ontological requirements and cognitive epistemological constraints.

See references [1], [2] for details.

Law Encoding Diagrams, LEDs

Law Encoding Diagrams are representational systems for particular mathematical or scientific domains, which use geometric, topological or spatial constraints to capture the laws of the domain in the structure of the diagrams, such that each instantiation (drawing) of the LED representation one instance of the phenomenon and one case of the laws of the domain.

LEDs can be designed to satisfy the principles of effective representations.

LEDs for a particular domain may be considered as specialised computational device or as a specialised diagrammatic modelling language for that domain. Many LEDs are geometric diagrams. LEDs are generative in that they not only represent particular cases but can be used to model the full variety of phenomena and relations of the domain.

LEDs found in the history of science:

• Galileo's kinematics diagrams
• Newton's dynamics diagrams
• Huygen's and Wren's diagrams for one-dimensional perfectly elastic collisions
• Lewis's chemical bonding diagrams
• Somerfeld's diagrams for the electronic structure of the elements
• Nomograms in general

LEDs have been invented as part of the project:

• AVOW diagrams for electricity
• PS diagrams for probability theory
• Diagrams for particle collisions (two dimensional, variable elasticity, multiple bodies)
• AT diagrams for algebra word problems

"Interactive" examples of LEDs for particle collision in physics

See references [2], [3], [4], [5], [6] for more details.

Inventing and evaluating LEDs to discover principles of effective representations

The overall approach [see below for references]

Principles of effective representational systems [1,2] Utility and validity of principles are tested by using them to invent new LEDs for selected complex domains in science and mathematics. Principles are discovered and refined by: (1) designing new LEDs; (2) finding and studying existing LEDs; (3) analysing conventional representations &emdash; over a range of complex domains in science and mathematics. Particle collisions [5] LEDs in the history of science [3], [4] . . . . . Electricity [6], [7] Probability [2] Various lines of research are exploiting LEDs to investigate: (1) the role of representations in high level cognition; (2) how to enhance conceptual learning with effective representations; (3) how exploit such representations in software. Results of evaluations support theoretical claims about the benefits of LEDs in contrast to other representations. Experimental evaluation Classroom trials Designing and testing computer-based learning environments Modelling discovery and learning with LEDs Studies of learning processes and knowledge structure Laboratory-based experiments on problem solving and learning with various LEDs. [9], [10] Developing curricula for teaching and learning with LEDs. [-] Interactive LEDs for discovery learning. [13] Computer modelling using techniques of Artificial Intelligence. [3], [4], [14], [15] Detailed behavioural protocol analysis. [11], [12]

Designing a system of LEDs for a particular domain

The principles provide overarching constraints on the nature of a good representational systems. However, to design a particular system of LEDs for a given domain it is necessary to consider in detail (1) the structure of the domain itself (ontological constraints) and (2) the nature of the knowledge that learner will acquire (cognitive epistemological considerations). The figure lists some of the aspects to be addressed under these "external" and the "internal" perspectives.

Selected References

[1] Cheng, P. C.-H. (in press). Unlocking conceptual learning in mathematics and science with effective representational systems. Computers in Education.
[2] Cheng, P. C.-H. (1999). Representational analysis and design: What makes an effective representation for learning probability theory? (Technical No. 63). ESRC Centre for Research in Development, Instruction & Training.
[3] Cheng, P. C.-H. (1996). Scientific discovery with law encoding diagrams. Creativity Research Journal, 9(2&3), 145-162.
[4] Cheng, P. C.-H., & Simon, H. A. (1995). Scientific Discovery and Creative Reasoning with Diagrams. In S. [5] Cheng, P. C.-H. (1998). Visualizing classical particle interactions: composing diagrams to model collisions. (Technical No. 60). ESRC Centre for Research in Development, Instruction & Training.
[6] Cheng, P. C.-H. (1999). AVOW Diagrams: A Representational System for Modelling Electricity (Technical No. 61). ESRC Centre for Research in Development, Instruction & Training.
[7] Cheng, P. C.-H. (1999). Electrifying representations for learning: An evaluation of AVOW diagrams for electricity. (Technical No. 62). ESRC Centre for Research in Development, Instruction & Training.
[8] Cheng, P. C.-H. (1996). Law encoding diagrams for instructional systems. Journal of Artificial Intelligence in Education, 7(1), 33-74.
[9] Cheng, P. C.-H. (1994). An empirical investigation of law encoding diagrams for instruction. In Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society. (pp. 171-176). Hillsdale, NJ: Lawrence Erlbaum Associates.
[10] Cheng, P. C.-H. (1996). Learning Qualitative Relations in Physics with Law Encoding Diagrams. In G. W. Cottrell (Eds.), Proceedings of the Eighteenth Annual Conference of the Cognitive Science Society (pp. 512-517). Hillsdale, NJ: Lawrence Erlbaum.
[11] 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 (pp. 232-235). Hillsdale, NJ: Lawrence Erlbaum.
[12] Cheng, P. C.-H. (1999). Networks of Law Encoding Diagrams for Understanding Science. European Journal of Psychology of Education, 14(2), 167-184.
Smith, T. Ward, & R. Finke (Eds.), The Creative Cognition Approach (pp. 205-228). Cambridge, MA: MIT Press.
[13] Cheng, P. C.-H. (1999). Interactive law encoding diagrams for learning and instruction. Learning and Instruction, 9(4), 309-326.
[14] Lane, P., 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:
[15] Lane, P., Cheng, P. C.-H., & Gobet, F. (1999). Problem solving with diagrams: Modelling the learning of perceptual information. (Technical No. 59). ESRC Centre for Research in Development, Instruction & Training.