Relate before calculate: Students’ ways of experiencing relationships between quantities
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The aim of this article is to contribute with findings concerning students’ ways of experiencing general mathematical structures and, in particular, relationships in additive structures.When students discern relationships in additive structures, it may lead to positive consequences for students’ future ability to perform calculations in addition and subtraction tasks. In the study, semi-structured interviews were conducted with students in grades 3, 8, and 9. An illustration showing a set of different quantities was the starting point during the interviews, together with an opening question regarding how the diverse quantities could be equalised. After the students’ discussions, they were asked if this could be described mathematically using written symbols. The students’ expressions concerning the phenomenon “relationships between quantities” were analyzed using phenomenography as an analytical tool. According to phenomenography, there are a limited number of ways in which a phenomenon can be experienced. Further, it is not about exploring how many individuals hold a specific experience that is of interest. In the case of this article, it is about capturing qualitatively different ways of experiencing the phenomenon relationships between quantities. Despite no specific numbers being presented, many students attributed specific numbers and values when expressing relationships between quantities. The students expressed general mathematical structures only to a limited extent and, in those cases, mostly only after encouragement from the interviewer. Following the phenomenographical analysis, the students’ ways of experiencing “relationships between quantities” are: as something that has to be calculated, or as something that has to be related. The first of these was most common in all grades. In this study, one critical aspect was identified, namely, that quantities are related to each other, additively. Instead of introducing mathematics with a focus on answer-oriented tasks, it is essential to introduce mathematics based on general structures such as additive structures. Even if the students are not familiar with such a mathematical “culture”, it is worth it. This was confirmed in our study.