## FractAl - Fraction competence and algebra competence: measuring, modeling, and explaining their predictive relationships.

This is an interdisciplinary project in cooperation with the Department of Educational Psychology (Prof. Dr. Michael Schneider) at the University of Trier.

*Brief description:*

Algebraic skills, such as solving equations, are an important component of higher mathematical competencies. Recent studies show that competencies in dealing with fractions predict algebraic competencies over several years, even when controlling for potentially confounded variables. Fraction competence, mediated by algebra competence, could therefore influence the development of higher mathematical competencies. However, empirical findings on the relationships between fractional and algebraic competence are so far scarce, and the underlying causal mechanisms are not yet known. Fractional competence exemplifies Numerical Cognition, while Algebra requires primarily non-numerical mathematical reasoning. Therefore, exploring the relationships between fractional and algebraic competence also increases understanding of the connections between numerical and non-numerical mathematical thinking in learning. However, neither fraction nor algebra literacy are homogeneous constructs. They include, for example, conceptual and procedural knowledge, some of which are independent facets of competence.

The goals of the project are to (a) find more sophisticated empirical evidence about which facets of fraction and algebra competence predict each other, (b) test alternative hypotheses about mediating processes of predictive relations, (c) examine the direction of predictive relations, and (d) test the causal effect of interventions that explicitly address and link fraction competence, algebra competence, and the mediating processes.

To accomplish these goals, four studies are planned with fifth through ninth graders. In Study 1, conceptual and procedural facets of fraction and algebra literacy will be examined using confirmatory factor analyses. Longitudinal path analyses will be used to test alternative hypotheses about possible mediators between fraction and algebra literacy in Study 2. Study 3 tests whether the predictive relationships are bidirectional, such that early algebra competence predicts learning success in school-based instruction about fractions. Study 4 is a field experiment to examine the extent to which two instructional interventions that link fraction literacy and algebra literacy by strengthening one of two mediators have a positive impact on algebra learning success.

Overall, the project aims to (a) better understand the relationships between numerical and non-numerical mathematical reasoning, (b) better predict the development of mathematical competence in students, (c) explain how students' widespread problems in fraction learning subsequently also cause deficits in algebraic reasoning, and (d) make instruction on algebra more effective.

### Team

Prof. Dr. Andreas Obersteiner

Prof. Dr. Michael Schneider, University of Trier

Michael D'Erchie

Claire Forsmann, University of Trier

### Project-Related Publications and Presentations

Forsmann, C., **D'Erchie, M.**, Obersteiner, A., & Schneider, M. (2023). Is the Correlation Between Fraction and Algebra Knowledge Based More on Concepts Than Procedures? *20th Biennial Conference of the European Association for Research on Learning and Instruction (EARLI)*. Thessaloniki, Greece.

Forsmann, C., **D'Erchie, M.**, Obersteiner, A., & Schneider, M. (2022). *Developing Measures of Conceptual and Procedural Fraction and Algebra Knowledge in Grade 9: A Pilot Study* [Poster presentation]. Meeting of the scientific network “Developing and stimulating competencies: Methodological challenges and opportunities for research”*,* Leuven, Belgium.