This study investigates how the graph representation creates opportunities for young students to develop an understanding of functional relationships in pattern generalizations. The empirical data is from an educational teacher-focused classroom design research focusing on generalizations in arithmetical growing patterns in Grade 1. The results show that the students in Grade 1 are given an opportunity to reason mathematically in both recursive-and covariational thinking. The results also show how the teaching provided opportunities for the students to use multiple representations of functional thinking and how oral language is a common representation to describe relationships. However, using a well-thought-out terminology to exploit the potential of the graph representation when discussing functional relationships and generalizations appears to be important.

Pattern generalization is a key element of early algebra. However, it is also an area that causes significant problems for students as well as teachers, as it has proved challenging for elementary school students to understand the meaning of generalization. To address these problems, an intervention was done to introduce the graph and functions in relation to pattern generalizations in Grades 1 and 6. Working on graphs was new for these teachers because, in Sweden, graphs are normally not introduced in school until Grade 7. The results show that the introduction of graphs became a tool to understand and talking about a pattern generalization. As a result, their teaching on linear functions and patterns changed, and the implications of the results on mathematics education in elementary school are discussed in this paper.

Educational design research provides opportunities for both the theoretical understanding and practical explanations of teaching. In educational design research, mathematics teachers’ learning is essential. However, research shows that little consideration is given to teachers and the participation of teachers throughout the entire design process as well as in continued learning. With this in mind, educational teacher-focused design research was used to explore the challenges teachers face, and the opportunities teachers are given when they participate as actors in all the phases of educational design research - designing, teaching, and refining theoretical concepts within the teaching. In this study, the mathematics focus of the design research was generalizations in patterns with Design Principles as the theoretical frame. The results show that the participation of teachers in all the phases of a design process is central for the teachers’ learning. Moreover, challenges that the teachers encounter in the classroom provide opportunities and consequences for the continued design process and lead to changes in the teachers’ understanding of generalizations. The results also indicate that functional thinking and linear equations contributed to both the teachers’ and students’ learning about generalizations in patterns. © 2019 University of Helsinki. All rights reserved.

This paper stems from research on mathematics teachers' participation in a particular collaborative learning process that addresses the issue of mathematical communication and mathematical reasoning in relation to the teaching of algebra. Although results from the developmental research revealed changes in the working group's meaning making about mathematical communication and reasoning, whether these changes are long-term and influence the teachers' mathematics teaching over time remains unclear. The aim of this paper is to discuss possible theoretical frameworks and ways of understanding mathematics teachers' long-term learning about mathematical communication and reasoning by describing what they can learn in an organized community of practice (Wenger 1998) when working with key mathematical issues. I will use the data and results from the developmental research to design another study on long-term learning.

Mathematics teachers’ development and the understanding of what constitutes learning are an on going topic and highlighting the complexity in the processes of learning in and from practice. This study builds on the idea that mathematics teachers’ professional development needs to be based on their classroom practice (Goodchild, 2008; Kazemi & Franke, 2004). Teacher participating in a working group, a learning community, and reflect on their own teaching and students learning. Working collaborative the mathematics teacher developed understanding of mathematical communication and mathematical reasoning in their teaching algebra.

Matematiklärares profession och professionella utveckling är grundläggande för elevers lärande. Syftet med studien har varit att följa en process för att förstå vad och hur matematiklärare lär i en praktikgemenskap. En grupp matematiklärare från årskurs 1-6 har träffats regelbundet under ett år. Gruppen, reflektionsgruppen har arbetat med att utveckla och söka svar på en gemensam kärnfråga. Reflektionsgruppens gemensamma intresse var att förstå mer om kommunikation och resonemang i matematikundervisningen. Studien har sin grund i Goodchilds (2008) den utvecklande forskningscykeln som kombineras med Wengers (1998) praktikgemenskaper för att analysera hur reflektionsgruppens samtal om kommunikation och resonemang i matematikundervisningen förändras. I analysen och tolkningen används tre begrepp som analysverktyg: ömsesidigt engagemang, gemensamt intresse och delad repertoar. Resultat visar förändringar i reflektionsgruppens samtal från att förstå, till att identifiera, till att tolka och slutligen tillämpa matematiskaresonemang i matematikundervisningen. Ett resultat av studien är också förändringar i reflektionsgruppen samtal som förändras från konsensus till att problematisera kärnfrågan.

Mathematics teachers’ profession and professional development is essential to develop students’ learning. The aim of this study has been to follow a processin order to understand what and how mathematics teachers learn in a community of practice. A group of mathematics teachers grade 1-6 met regularly for a period of one year. This group, called the reflection group, was tasked with developing a common core question. The reflection group’s joint enterprise was to understand more about communication and reasoning in mathematics teaching. The study is based on Goodchild’s (2008) the developmental research cycle combined with Wenger’s (1998) communities of practice to analyse how the reflection group shifts the way of talking about communication and reasoning in mathematics teaching. In the analyses and the interpretation three concepts are used as analytical tools: mutual engagement, joint enterprise and shared repertoire. The results show shifts in the reflection group’s way of talking from to understand, to identify, to interpret and finally to apply mathematical reasoning in teaching. The result of the study also shows changes in the process as the conversation shifted from consensus to problematizing the common core question.