# Exploring the role of conceptual knowledge, task complexity, and affect in secondary school students’ geometry proof and geometry calculation skills in two educational systems

A central feature of mathematics as a scientific discipline is its focus on axiomatic theories and deductive proof as a means of constituting scientific knowledge. Even though generating mathematical arguments is described as a goal of standard and curriculum documents around the world, studies have repeatedly shown that students struggle to understand the function of a proof, validate proofs against mathematical standards, and also to construct proofs for given mathematical statements. In this context, the specific epistemological complexity of proof problems is often contrasted against other problem types, which require the same conceptual knowledge, e.g. multi-step geometry proofs and multi-step geometry calculation tasks. Different factors have been discussed that influence students’ performance on these kinds of proof and calculation tasks, for example students’ knowledge of geometric theorems and concepts, their affect, but also task characteristics such as the complexity of the geometric diagram involved.

Apart from other, more general socio-cultural differences, the role of proof in geometry instruction in German and Taiwanese curricula has been described by textbook analyses in the past, and particular differences in the role of proof in the curriculum were found. For example, while proofs serve mostly a validation function to establish the truth of mathematical statements in German textbooks, proofs serve as one task type to practice the application of geometric theorems and properties in a problem-solving context in Taiwanese textbooks.

The main goal of the project is to further our understanding how the factors mentioned above influence students’ performance on geometry proof and calculation tasks, and how this influence is moderated by the socio-cultural context. Knowledge of these effects is vital to grasp the specific challenges students in both educational systems face when learning geometric problem solving in general, but also when facing the specific epistemological complexity of proof problems.

*Goal 1: Investigate the specific complexity of geometry proof tasks in comparison to geometry calculation tasks.*

To this end, the project team has developed pairs of geometry problems. Each pair consists of a geometric proof problem and a geometric calculation problem, which are based on the same geometric figure and which require the application of the same theorems and properties in the same sequence.

*Goal 2: Investigate the effects of the socio-cultural context, in particular the specific role of proofs in the geometry curriculum, on students’ proof construction skills.*

To isolate the effects of the socio-cultural and curriculum differences on geometry proof, specifically, the project controls for students’ geometry calculation skills in the parallel tasks when comparing students’ geometry proof skills.

*Goal 3:* *Investigate the effects of diagram complexity on students’ geometry proof and calculation performance.*

Based on a new model of diagram complexity, the project investigates how different aspects of diagram complexity impact students’ performance on geometry proof and calculation tasks. Moreover, the comparison of two task types and two socio-cultural contexts allows to investigate the stability of the identified effects.

*Goal 4: Investigate the effects of students conceptual geometry knowledge, their general and task-related affect on students’ geometry proof and calculation skills.*

Beyond establishing well-known results from expectancy-value-theory, the project aims to investigate how the impact of these factors on students’ skills is moderated by task type (proof vs. calculation) and socio-cultural context (Germany vs. Taiwan).

Stefan Ufer (Ludwig-Maximilians-Universität München, Germany)

Hui-Yu Hsu (National Tsing Hua University, Taiwan)

Markus Vogel (University of Education, Heidelberg, Germany)

Ying-Hao Cheng (University of Taipei, Taiwan)

Jiang-Chen Cheng (National Taipei University of Education, Taiwan)

Sarah Ottinger (Ludwig-Maximilians-Universität München, Germany)