The purpose of this project is to analyze how the scientific concept of inverse operations can be used as a “bottom-up” approach for teaching mathematical operations with whole and rational numbers. The primary aim of this analytical review is to provide support for student teachers in their learning of scientific mathematical concepts for the purposes of teaching. The theoretical approaches applied in this project are the theories of mathematical structures, especially the theory of inverse semigroups (Abelian groups), as well as relational thinking in comparison with instrumental thinking. The methodological approach is Vygotsky’s Doctrine of Scientific Concepts. The presentation of the analytical findings is intended to illustrate a clear connection to mathematical structures, such as the concepts of inverse, as well as how these structures can support pre-service teachers’ learning with regard to teaching mathematical operations in arithmetic and algebra. The development of a theoretical approach based on this study and its analytical findings is ongoing, but it is already being implemented in the teacher education programs for Grades K-3 and 4-6 at Södertörn University in Sweden.