Prof. Dr. Stanislaw Schukajlow-Wasjutinski, Institut für Didaktik der Mathematik und Informatik

Open problems that can be solved by using different solution methods are an important part of the school curriculum in mathematics and science. One important type of open problem is related to the real world. Open real-world problems can be solved by constructing mathematical models and are known as open modelling problems. Prior research analyzed open problems that did not include all information that is essential for solving the problems. In this type of open problem, the initial state is open. In this grant application, we are going to analyze problems with open goal state. While solving the problems with an open goal state, different factors (e.g., temporal, and financial) can be considered to answer the question. In the project we will investigate the effects (1) of instruction that is focused on important barriers in solving problems with open goal state and (2) how the teaching of problems with open goal state affects students’ cognitive and motivational learning outcomes. We will carry out an experimental study to investigate the impact of instructions in a) how to identify solution-related factors in the goal state, b) how to set up a mathematical model, and c) how to interpret and validate mathematical results on students' solutions. Further, we will contrast two treatment programs in a quasi-experimental study and analyze the effects on interest and performance. In the first teaching program, students will solve modelling problems with an open goal state, and in the second teaching program, they will solve real-world problems with a closed goal state. The main aim of this application is to investigate students’ cognitive barriers while they solve modelling problems with an open goal state and to analyze the effects of interventions (instruction and teaching program) for problems with open goals state on students' performance. Further, we will analyze, how interventions in solving modelling problems with an open goal state will affect students’ cognitive and motivational development. This research will contribute to the overarching aim of getting new insights about dealing with open problems in instructional settings that are oriented toward self-regulation.

Empirical findings on the effectiveness of drawing instructions to promote students’ modelling competencies in the field of geometry are mixed and indicate that drawing-specific strategy knowledge (strategic knowledge about drawing, SKD) plays an important role. Results from the first phase of funding ("ViMo 1") have confirmed that declarative SKD (i.e., knowledge about the characteristics of a good drawing) is a necessary but not sufficient precondition for producing drawings with a high level of accuracy and for finding a solution to a modeling problem. The overarching goal of the follow-up project "ViMo 2" is therefore to investigate what role procedural SKD (i.e., knowledge of how to construct and use a drawing in the modeling process) plays in the effective use of drawings and in modeling performance by students in the field of geometry. In addition to procedural SKD, we are also taking into account declarative SKD as well as cognitive, metacognitive, and motivational factors. Within the framework of the follow-up project, theoretical, empirical, and practice-relevant findings can be expected in the following areas: First, the project will provide findings on the potential of strategy training for promoting mathematical modelling competencies in the field of geometry. Second, the project will contribute to research on self-generated drawings. Compared with the first phase of funding, the recording and analysis of eye movements promises additional insights into the mechanisms behind instruction in drawing at the level of strategy execution. The use of EMME (Eye Movement Modeling Examples) also promises insights into the potential of this innovative form of instruction for mathematics and science learning. Third, the project will contribute to learning strategy research. We expect to confirm and expand relationships postulated in Borkowski et al.’s (2000) theoretical model by distinguishing between declarative and procedural parts of SKD. Forth, regarding practical implications, the learning environment that we are developing in the project can be used in mathematics instruction.