Elektronische Baumstrukturaufgaben zur fachdidaktischen Unterstützung in der Hochschul-Mathematiklehre für Ingenieure : Konzeption, technische Realisierung und Evaluation

• Tree-structured online exercises as a contribution of subject didactics to engineering mathematics courses at university level: concept, technical realization and evaluation

Mei, Robert Ivo; Heitzer, Johanna (Thesis advisor); Herty, Michael (Thesis advisor)

Aachen (2018, 2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2018

Abstract

This dissertation discusses the "tree-structured exercises" - a type of e-learning material developed by the Lehr- und Forschungsgebiet Didaktik der Mathematik in close cooperation with the Institut für Geometrie und Praktische Mathematik at RWTH Aachen University. The exercises are used as an optional practice element in the "Mathematics I/II" course for civil engineering and related disciplines. In a broader sense, the challenge of teaching mathematics at the transition from school to university has been the starting point for this project. That challenge has long been a topic of interest and has garnered even higher momentum over the past years - here as well as at other German universities. Among the issues addressed are rising student numbers, changes in university freshmen's mathematical background knowledge, and mathematics courses as a substantial hurdle for freshmen in STEM study programmes. Reactions include growing numbers of diverse support measures especially at universities, many of which are partly or entirely realized as e-learning. Despite frequently using modern digital methods of visualization, exercises in most of these e-learning materials still use the simple format of "task description - input of result - correctness feedback". With its emphasis on skills, this format is particularly suitable for mathematics at school level. In university level mathematics, however, tasks often become significantly more complex and consist of more interdependent steps than tasks in school level mathematics; solution processes often require a sense of overview as well as decisions that can only be made with an advanced understanding of the relevant concepts. In order to emphasize understanding and realize a more concept-oriented implementation of task solving processes, the tree-structured exercises follow a different approach: An exercise does not consist of a single question, but rather a multitude of interconnected content and question pages which cover the whole solution process - from initial trial-and-error phases to eventually writing down a neat solution. During the course of a tree-structured exercise, the "path" repeatedly splits up at question pages where the user gets asked for partial results in the solution process. In this manner, it is possible to implement additional assistance to the user after wrong answers, as well as different solutions in general. Furthermore, relevant lecture content can be revised and consolidated during the course of an exercise - with direct connection to the task at hand - on content pages. The author has taken part in the development of tree-structured exercises since autumn 2012 and has been the sole developer since spring 2014. The main intention behind this dissertation text lies in giving detailed insights into the concept, realization, usage experiences and evaluation of the tree-structured exercises to interested readers, thereby providing mathematics lecturers with a basis for deciding whether this format appears worthwhile for use in their own courses. The following aspects are featured as main parts of this dissertation: - The challenges in freshman mathematics courses - in STEM study programmes at German universities in general as well as in the specific course that the tree-structured exercises have been developed for - are described as a motivation for the tree-structured exercise project. - Subsequently, didactical basics for the tree-structure concept are explained. This includes systematic observations in the specific course on one hand and theoretical concepts from literature on the other, for instance the branching approach of programmed instruction by Norman Crowder or (local) adaptivity. Based on these concepts, this part of the text also features the specific objectives behind developing the tree-structured exercises. - As an insight into what exercises of this format can look like, the tree-structured exercises developed by the author are described in a concrete fashion. - Finally, the evaluation results of five years of tree-structured exercise usage are presented, each on the basis of anonymous data. The evaluation comprises general usage data, a student survey, a correlation analysis between exercise usage and exam performance, as well as a comparison of users and non-users based on a pretest. A substantial potential contribution of this dissertation project to research and university teaching lies in the didactical content analyses of topics in freshman mathematics, as well as the implementation of usable tree-structured exercises based on these analyses. Furthermore, the tree-structure principle can refurbish Crowder's so far little-used branching approach of programmed instruction in a modern digital fashion - providing a possible answer to the question of how to realize e-learning for (university level) mathematics that emphasizes conceptual understanding.