COMBINING AND TRANSLATING BETWEEN REPRESENTATIONS

Shaaron Ainsworth, Peter Bibby and David Wood

Abstract

Diagrams are typically presented with other representations. Traditionally these representations have been textual, but new technology is allowing diagrams to be iombined with many other representations. Research is presented that suggests that multiple (graphical and propositional) representations can improve children’s understanding of estimation. Using a measure of representational co-ordination that predicts learning outcomes, we have found that improvement in estimation depends on the extent that learners recognise similarity between representations.

Introduction

The advent of modem computer technology has allowed both new types of graphical representation such as animation and new dynamic and interactive uses of traditional graphical representations. Computers provide the opportunity to simultaneously represent information in multiple ways, which can be dynamically linked. It has been suggested that by using such multiple representations, learners will come to see and apply complex ideas in novel ways and that the learning that results from using multiple representations will not be shallow and procedural, but flexible and insightful (Kaput 1989).

Two types of claims can be distinguished concerning the advantages of learning with multiple representations. The first is that they support different ideas and processes, allowing specific emphasis to be placed on aspects of complex ideas. The second is that if co-ordinated they promote a deeper understanding of the domain by supporting the development of referential meaning. There is increasing empirical support for many of the proposed benefits of multiple representations (e.g. Tabachneck et a1 1994, Cox & Bma in press) especially that they promote different processes.

In order to design effective learning environments that employ multiple representations, we must consider how they affect learning in two separate but related ways. Firstly, students must come to understand what each representation tells them about the domain to be learnt. Each individual representation within the multi-representational system has associated properties which may affect what and how the students learn. (e.g. Larkin & Simon 1987, Stenning & Oberlander 1995). Secondly, in a multi-representational system, students must learn how the representations relate to each other. Hence, translation and co-ordination of representations should also affect what students learn (Kaput 1989).

Empirical Work

CENTS is a learning environment that uses multiple representations to teach the skills and concepts of computational estimation. Children are taught a number of different strategies for estimating and are given information about the accuracy of their answer through multiple representations. Knowledge of how to transform numbers in order to achieve accurate results is very difficult for children. (Case & Sowder 1990).

This research was conducted at the ESRC Centre for Research in Development, Instruction and Training.

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Two representations are used to express the accuracy of the estimates. Both are based on the percentage deviation of the estimate from the exact answer but emphasise different aspects of accuracy (e.g. direction in addition to magnitude). They give feedback on the accuracy of answers and users must also act upon the representations in order to predict the accuracy of their estimates.

In order to test predictions about how to combine multiple representations, accuracy can be displayed in different ways. There are a number of alternative ways of classifying representations. The most relevant to this paper are the distinction between informational and computational equivalence, mathematical and pictorial (where pictorial is being used to denote a real world image) and graphical and propositional, although many others exist (e.g. Green 1989, Stenning & Oberlander 1995). Three different combinations of representations were created for the current experiment. They were informationally equivalent but varied whether they were mathematical (a histogram and a numeric display) or pictorial (a splat wall and archery target) or a mixture (target & numeric display). Both pictorial representations were graphical, but the mathematical and mixed displays include one graphical and one propositional representation.

This allowed us to analyse to two primary questions: what factors influence how users come to understand how the multiple representations relate to the domain to be learnt and; what factors influence how learners come to understand how the representations relate to each other.

An evaluation with 48 9-10 year old children (three experimental groups with the different representations and one non-intervention control) has been conducted in a local primary school. Briefly the experimental procedure consisted of pre-testing children’s mental mathematics ability and skill both at estimating and at knowing the accuracy of your estimate. Then each child used CENTS twice over a month with the experimenter present to provide interface support but not direct teaching. Finally, children completed estimation post-tests.

Results

A number of significant results were found.

e Relative to the control group, all three experimental groups improved significantly in their performance of estimation. They became more accurate and used more appropriate strategies.

Two of the experimental groups (mathematical and pictorial) became significantly better at predicting the accuracy of their experiments, the mixed group did not.

Traces of children’s representation use whilst using the computer were examined and comparisons between their first and second computer trial were analysed.

0 The maths group became significantly better at predicting the accuracy of their estimates over time. The pictures group did not improve, but at time one there was a strong trend for them to be better than the other groups and at time two they demonstrated almost identical performance to the maths groups. The mixed group did not improve and were significantly worse than the maths group at time two. Relative to the other groups, the mixed group were worse at predicting accuracy using the representations.

* If users were recognising that both representations told them essentially the same information, then we would expect their behaviour to be effectively the same on both representations. Correlating users’ performance across the different representations suggests that the maths groups and the pictures groups were converging their behaviour on the different representations over time but the mixed group were not. Hence, it would seem that the mixed group did not come to understand the relation between the different representations.

Discussion

In order to examine the properties of the representations that account for these results, it is insufficient to look at each representation in isolation. Both representations in the mixed condition were present in the either of the other conditions and were used successfully in those groups. Children who used the mixed representations do not appear to have recognised the similarities between them and as a result were unable to translate their actions from one representation to another.

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Examining the properties of each representational system in turn, we can ask why translation occurs in pictures and maths but not in the mixed conditions. The pictorial representations are based on the same metaphor and are acted upon and read off in similar manners. In other words, both the format and the operators for these representations are almost identical. In addition to these similarities, pictures are 'ambient symbol systems' (Kaput 1989). Children of this age will have had considerable opportunity to interpret language and pictures, but relatively little experience with other representations. Translation between the different mathematical representations also occurs successfully. Although action and read off involve very different processes and one representation is graphical and one is propositional, mapping between the representation may be facilitated as both representations use numbers. DuFour- Janvier et a1 (1987) suggested that children only believed that two representations were of the same thing if they both used the same numbers. The mixed representations differ in terms of modality (propositional and graphical) and reading and action procedures. In addition they also mix mathematical and non-mathematical representations. Failure of overlap therefore occurs at all levels.

Conclusion

Kaput (1987) has argued that the co-ordination of multiple representations is fundamental to a deep understanding of mathematics. This claim is generalisable to many complex domains. Although co- ordination appears analytically necessary it has received little empirical support. Looking at the convergence of users' behaviour across time provides one such measure and in our study is associated with learning outcomes. As children's co-ordination of representations increases their ability to reason about the accuracy of their estimations also increases. Our measure of representational co-ordination allows us to examine the integration of representations of different kinds and allows us to move beyond looking at single kinds of representations in isolation, be they diagrammatic or propositional.

Multiple representations benefit users to the extent to which users recognise the similarities between the representations. Simply providing multiple representations is no guarantee of this occurring. When designing learning environments, particularly those that combine diagrams with other representations, we must consider how to support the users co-ordination of multiple representations. This should be done both in terms of the format of the representations and the operators that act upon it.

References

Case, R., & Sowder, J. T. (1990). The development of computational estimation: A neo-piagetian analysis. Comition and Instruction, 1(2), 79-104.

Cox, R., & Brna, P. (in press). Supporting the use of external representations in problem solving: The need for flexible learning environments. Journal of Artific ial Intelligence in Education.

Dufour-Janvier, B., Bednarz, N., Belanger, M. & (1 987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Eds.), Problems of Representation in the Teachine and Learning of Mathematics Hillsdale, NJ: LEA.

Green, T. R. G. (1989). Cognitive Dimensions of Notations. In A. Sutcliffe & L. Macaulay (Ed.), PeoDle and ComDuters V, Cambridge University Press.

Kaput, J. J. ( I 989). Linking representations in the symbol systems of algebra. In S. Wagner & C. Kieran (Eds.), Research Issues in the Learning and Teaching of Algebra Hillsdale, NJ: LEA.

Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11, 65-99.

Stenning, K.. & Oberlander, J. (1 995). A Cognitive Theory of Graphical and Linguistic Reasoning: Logic and Implementation. Cognitive Science, 97-140.

Tabachneck, H. J. M., Leonardo, A. M., & Simon, H. A. (1994). How does an expert use a graph ? A model of visual & verbal inferencing in economics. In A. Ram & K. Eiselt (Ed.), 16 Annual Conference of the Cognitive Science Socieh. Georgia Institute of Technology, Atlanta, Georgia: E A .

0 1996 The Institution of Electrical Engineers. Printed and published by the IEE, Savoy Place, London WCPR OBL, UK.

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