Unlocking conceptual learning in mathematics and science with effective representational systems

Abstract

The representational analysis and design project is investigating the critical role that representations have on conceptual learning in complex scientific and mathematical domains. The fundamental ideas are that the representations used for learning can substantially determine what is learnt and how easily this occurs, and that to improve conceptual learning effective representations should be found or invented. Through the conceptual analysis and empirical evaluation of a class of representations that appear to be particularly beneficial for conceptual learning, Law Encoding Diagrams (LEDs), the project has identified certain general characteristics of effective representations. In this paper a descriptive model of the components and processes of conceptual learning is presented and used for several purposes: to explain why the nature of representation used for learning is critical; to demonstrate how representations possessing the identified characteristics of effective representations appear to support the major processes of conceptual learning; to consider how computers may further enhance the potential benefit of LEDs for conceptual learning.

Introduction

One of the major challenges now facing the use of computers in education is to develop systems that effectively support conceptual learning in substantial mathematical and scientific domains. Computer-based learning has had some success at promoting procedural or skills-based learning (e.g. Anderson et al., 1995, Lesgold et al., 1992; see also papers by Wood and Wood, and Wood, Underwood and Avis in this volume), but it is by no means obvious that the design of these systems can be extended to support conceptual learning.

This paper summarizes the theoretical and empirical aspects of a programme of research that is investigating how best to support conceptual learning. The basic claim of this work is that the nature of the representational systems used for such learning, to a significant extent, determine what is learnt and how easy it is to obtain a good conceptual understanding. The keys to conceptual learning are effective representations; the programme of research will be referred to as ‘the representational analysis and design project’.

Work in cognitive science on problem solving, expertise and the nature of representations has demonstrated the pivotal role that representations have in human cognition, especially in reasoning and problem solving. Following Skemp, 1971, Kaput, 1992 and others, the representational analysis and design project is extending this work into the arena of learning and in particular to conceptual learning. Motivation for the approach of the present project comes, in part, from studies of the nature of scientific discovery which have demonstrated that the role of representations are fundamental to the success of the scientific enterprise. Major figures in the history of science made their discoveries by carefully selected representations, or by inventing new representations. These representations had informational and computational properties that facilitated the discoveries. Thus, an appealing notion is that learners may also benefit substantially in similar ways by being given carefully selected representations, or even specially invented representations, for use in conceptual learning.

To introduce some of the issues to be covered in this paper, consider two learners’ attempts at solving a particular physics problem: one using a conventional algebraic approach and the second an approach using a diagrammatic representation. Both solutions were produced by participants in the empirical evaluations that have been conducted on the effectiveness of different representations for learning as part of the present project. The problem involves finding the velocities of two elastic bodies which have collided head-on, given the masses of the bodies and their initial velocities.

A picture illustrating this class of problems is shown as the first step of the solution attempt in Fig. 1. (The two pages have been accurately redrawn for clarity.) This solution was produced by experimental participant PB, who was a graduate physicist working for his doctorate in Physics. This attempt was generated in the pre-test of an experiment. PB would have been well schooled with the relevant laws and algebra during his undergraduate degree. Given the initial state of the problem, in which body-A impacts body-B coming in the opposite direction with a different speed, PB drew the diagram showing the initial situation, Step 1 in Fig. 1. The solution involves writing down the algebraic laws for momentum and energy conservation as applied to this class of phenomena, Steps 2 and 4, and substituting in the given values, Steps 3 and 5. At Step 6, PB begins to think about substituting one equation into the other to obtain an equation with just one unknown variable, but he realizes that he has made a mistake in the substitution of values into the momentum equation (at Step 3), so he corrects this before proceeding (Step 7). The full substitution and elimination procedure follows, Steps 8 and 9. PB obtains a quadratic equation but does not solve it. Instead, he reverts to the earlier equation derived from the energy conservation laws (at Step 8) and simply finds a numeric solution by considering squares of integers (Steps 10 and 11). The equation derived from the momentum law (at Step 7) is then used to successfully check this numeric solution (Step 12). PB is happy to state the final solution (Step 13). However, he does not realize that the values are the same as the initial velocity values given in the problem statement, which he had earlier used to label the diagram at Step 1.

Despite being a graduate physicist it is clearly not the case that PB has a good understanding of this basic topic of physics. Unfortunately, PB’s performance is not unusual compared to the other graduate physicists and engineers that have been studied doing similar problems (Cheng, 1996c). This solution attempt is a good example of the difficulties that can arise from the use of a poor representation. Mistakes have been made in the manipulation and interpretation of the expressions generated during the solution, some of which PB has done well to spot and correct, others he has missed completely. Effort has been wasted pursuing a series of inferences in a particular direction which was then abandoned. Knowledge about techniques for the manipulation of the representation itself is necessary for problem solving but is not directly relevant to understanding the nature of the domain, in this case methods to solve pairs of simultaneous equations and the solution to quadratic equations. It is easy to lose track of the goal during problem solving when so much algorithmic work has to the done, as shown by PB not realizing that the solution values were identical to the given values.

One of the claims of the representational analysis and design project is that such difficulties can be avoided by giving learners better representations. Fig. 2 shows an alternative approach to the solution using a similar, but more difficult, problem. It is more difficult because the masses of the bodies are not equal. Fig. 2 is an example of a class of representations that was invented by the scientists, Huygens and Wren, who discovered the laws governing this domain (Cheng & Simon, 1995). The magnitude of the masses are shown by the lengths of lines m₁ and m₂ in the middle of Fig. 2. The given initial velocities are shown by the arrows at the top of Fig. 2, u₁ and u₂. The final velocities have been found by construction of the diagram according to certain geometric rules that encode the laws of momentum and energy conservation for this domain. After impact, the bodies depart in the same direction but v₁ is much greater because its mass is much smaller.

The diagram in Fig. 2 was actually produced in the post-test of an experiment, by participant JT who was a graduate psychologist with little knowledge of physics (in contrast to PB, above). JT learnt about the domain for 40 min using a computer-based learning environment that exploits these diagrams and was developed as part of the present project. No direct instruction on the structure or use of the diagrams was given to JT who simply learnt about the domain by manipulating the diagrams on screen and comparing them with an animated simulation of the collisions. Despite the limited amount of instruction and the relatively short time on the system, JT rapidly acquired the rules governing the structure of the diagrams. The post-test was the first time she drew the diagrams for herself, but she successfully applied them to problems she could only guess at before using the system. The experiments conducted in the present project, with a variety of representations in computer-based and conventional media, show that such dramatic changes in approaches to problem solving with new representations is not unusual.

The diagram in Fig. 2 is clearly simpler than the numerous complex expressions in Fig. 1, so the chances of making errors when generating or modifying such representations are likely to be lower. The form of the diagram also reflects the structure of the domain, the pairs of arrows for initial and final velocities giving a simple image of what is happening, so the interpretation of the representation is supported naturally. The diagram uses simple geometric constraints to encode the laws of the domain that are directly applied to elements representing particular properties, whereas the syntactic algebraic rules operate on quite arbitrary expressions remote from the phenomenon itself.

The contrast between Fig. 1, Fig. 2 exemplifies how the representation a learner uses can determine what and how easily they learn. Thus, an approach which appears to have great potential for improving conceptual learning is to judiciously select, or invent, appropriate and effective representations. This paper will summarize the theoretical and empirical work conducted to articulate and test this claim. The paper has three main sections.

The first section considers why it is that representations can exert such an influence on conceptual learning, or to put it another way, to explain why the processes of conceptual learning are so sensitive to the choice of representations. To achieve this, it is necessary to consider the nature of conceptual understanding and the processes of conceptual learning. This is done by presenting a general descriptive model of the components and processes of such learning. This model is used to identify the major barriers that an ineffective representation may raise in the path of the learner; to show that previous computer-based approaches to promote conceptual learning have been largely piecemeal in their attempts to support learners overcoming such barriers, and to argue that the choice of an effective representation can, in a systemic fashion, minimize these barriers.

The second section of the paper also uses this model to provide a context in which to consider the nature of effective representations. Various characteristics of effective representations are identified and described. A particular class of representations, Law Encoding Diagrams (LEDs — illustrated in Fig. 2), has been the main focus of the empirical studies and conceptual analysis in the project, because they possess these characteristics and appear to be effective in supporting conceptual learning. Empirical evaluations of these LEDs have shown that they can improve learning compared with traditional algebraic approaches (Cheng, 1996d). Other systems of LEDs have also been specially designed for various domains (Cheng, 1999a, Cheng, 1999b, Cheng, 1999c) and further empirical evaluations have demonstrated that they can promote learning (Cheng, 1999e). An example with LEDs for electricity is presented later.

The third and final section of the paper considers general educational issues and implications concerning the introduction of novel representations.