Mathematics Education (Secondary Levels)

Dissertation Project of Prof. Dr. Tobias Rolfes

Materials for the Doctoral Thesis Completed in 2017

Title: Functional Reasoning - Empirical Results on the Influence of Static and Dynamic Representations. [Funktionales Denken − Empirische Ergebnisse zum Einfluss von statischen und dynamischen Repräsentationen]

Digital version of the dissertation [german]


  •  Dissertation Award of the Department 7: Natural and Environmental Sciences 2018. [won]

The influence of static and dynamic representations on functional reasoning

The thesis focuses on the influence of external representations on the application and learning of functional reasoning. To this end, the theoretical foundations first clarify and elaborate the concept of functional thinking and discuss the role of representations for functional thinking. Based on this, the Integrated Model of Text and Picture Comprehension and the Adaptive Control of Thought-Rational, two information-theoretical models for the cognitive processes involved in applying and learning functional thinking, are considered from the perspective of cognitive psychology. The theoretical foundations conclude with a definition of the learning efficiency of external representations as distinct from the use efficiency of external representations.

In the quantitative-empirical section, several experiments are reported. Experiment E1a and Experiment E1b focused on the usage efficiency of tables and graphs in functional reasoning. The empirical data showed that for quantitative functional reasoning, a table was more usage efficient than a graph. In contrast, for qualitative functional reasoning, the efficiency of use depended on whether the graph had to be inspected only roughly or in detail. Thus, for qualitative functional reasoning, a graph was more use-efficient than a table only if the graph had to be inspected only roughly to solve the task. However, if detailed inspection of the shape of the graph was necessary, the table was more utility efficient. The results of Experiment E1b provided evidence that for functional reasoning, the efficiency of use of external representations is relatively independent of the language, cultural background, and age of the subjects.

In quasi-experiment QE2, multirepresentational learning of functional reasoning with both tables and graphs was compared to monorepresentational learning with either tables only or graphs only. The results of the quasi-experiment showed that multirepresentational learning produced advantages in learning efficiency for qualitative functional reasoning. However, in quantitative functional reasoning, no higher learning efficiency could be demonstrated from multirepresentational learning compared to monorepresentational learning with graphs only. However, the results also indicated that multirepresentational learning requires more time than monorepresentational learning, but can lead to higher learning gains.

Experiment E3 compared the learning efficiency of static, linear-dynamic, and interactive-dynamic representations for functional reasoning. The results showed that both forms of dynamic representations were more efficient for learning than static representations. However, no difference in learning efficiency was demonstrated between interactive-dynamic and linear-dynamic representations.

[This dissertation was written and is only available in german. Our summary was translated by our website team. Possible quotations have been translated in accordance with scientific regularia.]

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