Teaching mathematics towards mathematical proficiency cannot easily be described since mathematical proficiency is complex and no term alone capture all aspects. The purpose of the study was to examine how mathematical proficiency in conceptual understanding can be discriminated through the concept of subtraction in grade 1 and 3 primary school. Question raised by the study was 1. What different levels in quality can be discriminated in the ways students subtracts?   2. Can progression be discriminated within subtraction operations between grade 1 and 3? Concepts like mathematical artefacts, capability and ability have been useful tools when analyzing the material. Developing mathematical thinking in students is about developing conceptual understanding anchored in the process rather than focus on procedures and outcome. When subtracting with comprehension one knows that subtraction undoes addition since subtraction is the inverse operation of addition.      The methods used are qualitative studies using observations and interviews. The material consists of notes from observations, recordings of interviews and written solutions. The study developed an analysis tool in order to compare the different ways of thinking when doing subtraction activities. Developing the tool was critical in order to compare and discriminate different levels of mathematical thinking. The results indicate that the Swedish curricula in mathematics do not support mathematical teaching towards mathematical proficiencies because it’s lack of concrete measurable goals.