To link arithmetic with the practical and theoretical thinking of algebra, this exploratory research framed in the Modes of Thinking Theory conducted a case study to characterize modes of thinking about the set ℤ4 and its interactions in Chilean primary school teachers. For this purpose, the answers given by 30 in-service teachers to an online questionnaire were analyzed, based on a proposed cognitive model that defines the modes of thinking about the set ℤ4 with its articulators. The results show that these teachers, generally adhere to the cognitive model and evidence more articulation between synthetic-geometric and analytic-arithmetic modes than between analytic-arithmetic and analytic-structural modes, which shows less privileged theoretical thinking. In conclusion, the algebra of primary teachers can be activated by conceiving the set ℤ4 as a mathematical fragment with 4 elements constructed. Each one is considered a distinct set of congruent numbers modulo 4 that partition the set ℤ, making the concept of equivalence class contribute to the cognitive construction of the set ℤ4 as a cyclic graph of order 4.