The trade-off between complexity and accuracy. Preparing for computer-based adaptive instruction on fractions

ABSTRACT

PS-I learning approaches with an initial problem-solving phase (PS) followed by an instruction phase (I) can be optimised by providing adaptive (individualised) instruction that builds on students’ problem-solving products. A computer-based implementation of both phases offers the possibility of adaptivity at the transition from the problem-solving to the instruction phase. However, it also poses two challenges for the problem-solving phase, which should allow for (1) complex problem-solving processes (process validity) and (2) accurate diagnosis of problem-solving products. Increased openness of the learning environment can increase the process validity and decrease diagnostic accuracy and vice-versa (complexity-accuracy trade-off). Using two variants of a computer-based learning environment for fraction comparison with different extent of openness (O⁺ and O^-), we investigate whether both requirements (complex problem-solving processes and accurate diagnosis of the problem-solving products) can be met simultaneously. To check process validity (complexity of problem-solving processes), we compare the problem-solving products with those from previous research on an analogous paper-based version. Our results show that valid and comparable problem-solving processes can be achieved in both computer-based learning environments (O⁺ and O^-). Moreover, only the learning environment O^- enables an accurate computer-based interpretation of students’ products and thus resolving the complexity-accuracy trade-off.

KEYWORDS:

• Problem-based learning
• learning from errors
• productive failure
• fractions
• diagnosticity
• learner model

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In their work on complex problem solving, Dörner and Funke (2017) describe several phenomena that are specific to complex problem solving, including the potential for failure. The potential for failure increases with the complexity of the problem (Dörner & Funke, 2017). Thus, the process validity in the present study is analogous to the situation in research on complex problem solving (Dörner & Funke, 2017).

2 In this paper, we distinguish between assessment and diagnosis. By assessment, we refer to a specific phase in the instructional process (assessment phase as opposed to learning phase). By diagnosis, we refer to the process of data interpretation that can take place in each of these phases (learning phase and assessment phase).

Additional information

Notes on contributors

Antje Boomgaarden

Antje Boomgaarden is a PhD student in didactics of mathematics at the University of Education Freiburg, Germany. Her research focuses on the computer-based implementation of multiphase learning scenarios and the analysis of eye-movements within students’ comparison of typical solutions.

Katharina Loibl

Dr. Katharina Loibl is professor for interdisciplinary research on learning and instruction at the University of Education Freiburg, Germany. Her research focuses on instructional designs (such as PS-I), cognition, and teachers’ diagnostic competence. In her research projects, she combines the psychological perspective with the subject-specific perspective.

Timo Leuders

Dr. Timo Leuders is professor of mathematics education at the University of Education Freiburg, Germany. His research comprises studies in mathematical learning and cognition, mathematics teaching, and teachers’ diagnostic competence. He also engages in teacher professional development, publications for teachers, and the design of textbooks.