Knowing functions and functional thinking have recently moved from just knowledge for older students to incorporating younger students, and functional thinking has been identified as one of the core competencies for algebra. Although it is significant for mathematical understanding, there is no unified view of functional thinking and how different aspects of the concept are used as a theoretical base. In this paper, we analyse different definitions used in empirical studies. First, we did a systematic research review resulting in 19 empirical studies focusing on functional thinking with an appropriate theoretical underpinning. The definitions were analysed using an AI tool. After that, we analysed the results using intrinsic mathematical properties of how functions can be defined in mathematics to identify core aspects of the definitions. According to the analysis, two definitions capture most of the key aspects of functional thinking, and most empirical studies use these key concepts. These two definitions treat functional thinking as products or products and processes. One definition used in one empirical study stands out by theoretically operationalizing functional thinking as a process. As such, different ontological assumptions are made in the studies; however, in some cases, having the same epistemological outcome. From a methodological point of view, the cosine similarity matrix was a useful tool for an ontological analysis, but a qualitative analysis is still needed to make meaning of it.

We live in an increasingly worrying future, where researchers are looking at how we can create education for a sustainable world. In this study we investigate what such education could look like, especially for young students, reinforcing hope and an ability to act and come up with ideas of sustainable solutions with no obvious 'good' solution. The analytical tool suggested in this paper addresses the challenge of promoting and illuminating ethical reasoning in early childhood education for sustainability and mathematics. The teaching of ethics should use children's experiences as a starting point rather than theories of ethics. Using the topics of sharing as an example, and the results from a series of empirical studies, we illustrate how ethical arguments can function as backing for mathematical reasoning, and how mathematical arguments can play a similar role in ethical reasoning. We argue that the tool could assist in teaching where mathematics and education for sustainable development merge through ethical reasoning but also as an analytical tool for research.

This study investigates how collaborative, collective mathematical reasoning emerges when students jointly solve problems regarding fair sharing. Three 6-year-old children worked collaboratively in a group with a problem related to fair sharing, and their teacher was present. The data, captured in video recordings, was analysed using two frameworks: collective mathematical reasoning, and the theory of joint problem space. The results show that different things, such as physical artefacts aimed at sharing resources, challenges related to the task, and the students’ conceptions of mathematics, affect the students’ possibilities to engage in collaborative collective mathematical reasoning. 

This paper studies grade 7 (age 13) students’ expressed understanding about the equal sign/notion of equivalence in order to investigate what aspects of the concept that could be seen as an established knowledge at lower secondary school/middle school. Using items from different instruments and combining these to a new one that covered a broad spectrum of procedural and conceptual knowledge, we collected data from 159 students. The different statistical tests showed that if focusing only on separate items, it could confirm that students could be seen either having operational or relational understanding of the equal sign. However, when taking all results into account using several analyses, students’ understanding appear to be much more complex. Instead of a dichotomized view, students’ expressed knowledge of mathematical equivalence should be seen as a continuum.

The aim of this paper is to explore students mathematical and ethical reasoning in fair sharing tasks. In our study, 8-year-old students were asked to share eight biscuits between three soft toys, where one of the toys is hungry and sad. Two analytical frameworks were used, one for mathematical reasoning and one for ethical reasoning. The results show that sharing tasks can generate several solutions in which students use different mathematical arguments combined with ethical arguments. Another result is that the different arguments often interplay with, or depend on, each other, meaning that mathematical reasoning and ethical reasoning are intertwined.

L'objectif de cet article est d'explorer le raisonnement mathématique et éthique des élèves dans les tâches de partage équitable. Dans notre étude, il a été demandé à des élèves de 8 ans de partager huit biscuits entre trois peluches, l'une des peluches étant affamée et triste. Deux cadres analytiques ont été utilisés, l'un pour le raisonnement mathématique et l'autre pour le raisonnement éthique. Les résultats montrent que les tâches de partage peuvent générer plusieurs solutions dans lesquelles les élèves utilisent différents arguments mathématiques combinés à des arguments éthiques. Un autre résultat est que les différents arguments interagissent souvent entre eux ou dépendent les uns des autres, ce qui signifie que le raisonnement mathématique et le raisonnement éthique sont étroitement liés.

In this chapter, we challenge the view of mathematical modelling as a neutral practice and argue that ethical reasoning is an important part of the process. Using six cases, each describing different sharing scenarios, we analysed children’s mathematical reasoning and ethical reasoning. The cases were designed to cover a variety of aspects of mathematics and allow different types of ethics to be used. The results show that ethical reasoning plays a crucial part in task-solving, both when the children advocated for division or when they decided on sharing in unequal-sized parts.

I artikeln undersöks didaktiska val i undervisning där matematik och hållbar utveckling möts. Undervisningens syfte var att uppmuntra förskolebarn att kollektivt resonera kring fördelningsproblem. Det var barnens resonemang om möjliga lösningar som låg i fokus, inte ett rätt svar. Förskollärarens roll var att skapa möjligheter för agens: att barnen kunde göra egna bidrag i kunskapandet, där barnen kunde ställa olika konflikterande perspektiv om hållbara lösningar mot varandra och diskutera lösningarna. Utifrån metoden didaktisk modellering förde forskare och förskolläre diskussioner om undervisningen utifrån olika expertis. Dokumentationen som skapades under dessa möten analyserades, för att pröva om en didaktisk modell som hanterar de didaktisk vad- och hur-frågorna kunde hjälpa förskollärarna att skapa agens i undervisningen. Resultatet visar hur reflektion kring de didaktiska frågorna kan skapa möjligheter för agens i undervisningen. Didaktiska lösningar handlade om organisation av tid och miljö, om svårigheter att para ihop barn för kollektivt arbete samt språkutvecklande arbete.

The article examines didactic choices in teaching where mathematics and sustainable development meet. The purpose of the teaching was to encourage preschool children to reason collectively about distribution problems. It was the children's reasoning about possible solutions that was in focus, not a correct answer. The preschool teacher's role was to create opportunities for agency: enable childrens own contributions to knowledge, pit different conflicting perspectives on sustainable solutions against each other and discuss the solutions. Based on the didactic modeling method, researchers and preschool teachers discussed the teaching based on different expertise. Documentation created during these meetings was analysed, in order to investigate if the didactic what and how questions could help the preschool teachers create agency in teaching. The result shows how reflection on the didactic issues created opportunities for agency in relation to organization of time and environment, of pairing children for collective work and language development work.

This study focuses on sharing, both in equal parts (groups) or unequal parts. Children at age five attending preschool, are faced with two different tasks working in pairs. The analysis focus on the mathematical properties in the reasoning, or when mathematical arguments were replaced with an ethical reasoning. When performing division, different strategies were used, and the norm of fair share was often expressed. It was easier for the children to allocate resources when the dividend was larger than the divisor, and when dealing with a fraction, the cardinality of the number of parts appeared to be a prominent property compared to property ‘equal size’ of the parts. There were also indications of ethical reasoning where the child used different claims to convince their peer. There was a tension between the norm of equal sharing and solutions with unequal parts. One implication is that if wanting to challenge children’s mathematical reasoning in a division task, it could be fruitful to look at fractions instead of repeating tasks where the dividend is larger than the divisor.