1. Advisor: Ming-Puu Chen
The Use of Symbols, Words, and Diagrams as Indicators of Mathematical Cognition
Pyke, C. L. (2003). The use of symbols, words, and diagrams as indicators of mathematical cognition: A causal model.
Advisor: Ming-Puu Chen
Reporter: Lee Chun-Yi
Doctoral Student at Department of Information and Computer Education, National Taiwan Normal University.

2. Introduction
A model that describes and explains thinking ought to provide explanatory, predictive, or heuristic functions (Paivio, 1990).
In a few areas of school mathematics, such as the development of proficiency with number concepts and operations, there are well-developed models and theories about memory organization and storage (e.g., Behr, Harel, Post, & Lesh, 1992) and models that explain additive and multiplicative thinking (e.g., Fuson, 1992; Greer, 1992).

3. Introduction
Representation is one area where recent research activity has suggested the need for new and more unified models to guide research and practice (Kaput, 1998).
A useful model in this domain might begin to
(a) articulate relations among the variables involved in strategic representation, revealing details about the processes that are responsible for internal and external representation of mathematical situations (Paivio, 1971, 1990);
(b) relate strategic representation proficiency to existing knowledge and skills that have been established as predictors of problem-solving outcomes (e.g., reading and spatial ability); and
(c) inform practitioners about how students use and create internal and external representations to make sense of and solve problems (Cuoco, 2001).

4. Introduction
A goal of the study was to present educators with a valid and useful alternative model as they seek to restructure their understanding of how students approach representational activity during problem solving.

5. Dual Coding Theory: DCT (Paivio, 1990)

6. Individual, Context, and Strategic Representation Variables

7. Strategic Representation
Representation strategies, are used to reproduce written text from memory of a text and draw a diagram that represents a nontext information presentation (Goldin & Janvier, 1998).
.Referential strategies, are responsible for how nontext images are formed from words and how text images are formed from pictures (Paivio, 1971, 1990).
Transformation strategies are defined as composed of the organizational processes and transformational processes identified in DCT (Paivio, 1990).

8. Task Context
Text-based exam
Graphic-based exam

9. The Path Model Tested

10. Sample Task

11. Analytic Strategy
Path analysis (Pedhazur, 1982), a special case of more general applications of structural equations modeling (Hoyle & Smith, 1994), was used to empirically evaluate the relationships among the variables as predicted by the theoretical model proposed for this study.
The method selected employed ordinary least squares regression procedures to estimate parameters and evaluate the model.
Consistent with the path analytic approach outlined by Pedhazur (1982), multiple regression procedures were used to estimate model parameters and calculate a goodness of fit index derived for this type of analysis by Specht (1975).
The goodness of fit index was used to quantify the ability of the model to reproduce the correlation matrix among the variables.
A transformation of the index was tested against a chi-square distribution for statistical significance.

12. Descriptive Data
An analysis of skewness and kurtosis for each variable showed that the distributions of scores approximated normality and were suitable for further analysis.

13. Regression Results

14. Evaluating the Model
The calculated goodness of fit value (GFI =.938) did not reject the fit of the data.
The goodness of fit index was transformed into a statistic with an approximate 2 distribution (Pedhazur, 1982; Specht, 1975) and tested for statistical significance.
The critical value (w > 11.07) would reject the model. The calculated value (w = 10.82, df = 5, p < .05) did not reach the critical value and the model remained tenable, implying that the variance that would be added by including the omitted paths in the model would not significantly improve prediction of the outcome.

15. Effects among Variables

16. The End
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