The aim of this article is to contribute to understanding the relationship between teaching and learning algebra at school in order to identify how schools can be supported to improve students’ learning outcomes. The students’ tests and examinations of their mathematical work and the teachers’ lessons plan and reports on the lessons’ instructions were the base data for this article. The analysis indicated that, if the teachers base their instructions on the critical aspects identified in students’ learning and open up patterns of variation in these aspects, they seem to facilitate students’ learning. The findings suggest that helping teachers develop an understanding of the students’ critical aspects can be a productive basis for helping them to make fundamental changes to their instructions and to improve their mathematical communication in the classroom.