Mathematics Education (Secondary Levels)

Dissertation Project of Dr. Michaela Lichti

Materials for the doctorate completed in 2018

Title: Encouraging Functional Thinking - Experimenting with Representational Materials or Computer Simulations [Funktionales Denken fördern − Experimentieren mit gegenständlichen Materialien oder Computer-Simulationen]

Digital version of the dissertation [german]
Functional reasoning test (english version)
Functional reasoning test [german version]

Here the student workbooks for the group that experimented with representational materials and the group that experimented with computer simulations are provided, each as PDF and as WORD files, as well as the GeoGebra simulations used:

Materials Group Workbook, PDF file [german]
Workbook material group, WORD file [german]
Workbook simulation group, PDF file [german]
Workbook simulation group, WORD file [german]
GeoGebra simulations [german]


  • Dissertation Award of the Department 7: Natural and Environmental Sciences 2019 [won]
  • Dissertation Award of the University of Koblenz-Landau, Landau Campus 2019. [won]

Real experiments or simulations - What function(s)?

Functional relations are part of the mathematics lessons of every grade level, they are relevant for subjects like biology, chemistry or social studies and equally in everyday life, e.g. when a cup of coffee cools down. However, students rarely realize that they are dealing with functional relationships. Their functional understanding often shows weaknesses (Leinhardt et al. 1990) and needs encouragement. This study addresses the question of whether this promotion should occur with real experiments or simulations.

Theoretical Background

Functional understanding is divided into three basic aspects (Vollrath 1989). The assignment aspect involves that each x (the independent variable) is assigned exactly one y (the dependent variable). The covariation aspect describes the change behavior of a function. It refers to the question in which way the dependent variable changes when the independent variable is varied. The object aspect characterizes that a function is considered as a whole and treated as an independent object. In order to be able to promote the functional understanding of students, one has to take these aspects into account. Especially the covariation aspect usually causes difficulties (Rolfes et al. 2013) and needs special attention.



Experiments represent one way to promote functional thinking. Both working with materials (real experiments) and experimenting with simulations (GeoGebra) can be implemented in the classroom. This project will empirically investigate whether the two approaches produce different levels of learning progress in functional understanding among grade 7 students. After developing a test to measure functional understanding, the appropriate real-world experiments with the corresponding simulations will first be selected in a preliminary study, and then possible differences in student learning gains will be determined in the main study using different interventions and pretest/posttest designs.


Leinhardt, G., Zaslavsky, O. & Stein, M. K. (1990). Functions, Graphs, and Graphing. Tasks, Learning, and Teaching. Review of Educational Research 60 (1), 1–64. doi:10.3102/00346543060001001 

Rolfes, T., Roth, J. & Schnotz, W. (2013). Der Kovariationsaspekt von Funktionen in der Sekundarstufe I. In G. Greefrath, F. Käpnick & M. Stein (Hrsg.), Beiträge zum Mathematikunterricht 2013. (S. 834–837). Münster: WTM Verlag.

Vollrath, H.-J. (1989). Funktionales Denken. Journal für Mathematikdidaktik (10), 3–37.


[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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