This study investigates how collaborative, collective mathematical reasoning emerges when students jointly solve problems regarding fair sharing. Three 6-year-old students worked collaboratively with three problems 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 are presented through four themes, each contributing to the problem-solving space and students’ possibilities to engage in collaborative collective mathematical reasoning. The themes are physical artefacts aimed at sharing resources, challenges related to the nature of the task, students’ conceptions of mathematics, and the teacher’s contributions, each playing different roles depending on where in the reasoning process they appear.

This article reports an empirical study investigating how first-grade students in multilingual classes in Sweden experience number lines in an algebraic teaching aligned with the El'konin-Davydov curriculum. A phenomenographic analysis revealed that the students (n = 150) experienced number lines in terms of three qualitatively distinct categories: (a) Mathematical properties, (b) Relationships between the properties, and (c) Operations on a number line. These categories involved different types of algebraic thinking, identified in the students’ joint analytical work. The higher categories encompass the lower ones, and a greater variety of algebraic thinking was identified in the higher categories. The first, qualitatively lowest, category includes students’ experiences with points on a number line and the distances between them. The second category includes experiences about relationships between the properties in category one (the iterated unit). The third and highest category includes aspects of value (that the value is from the starting position to a specific position, distinguishing between positions and values), the direction of the number line (relationships between unknown quantities depending on their locations on the number line), and the relationships between operations (e.g., addition and multiplication). Suppose the number lines had been introduced in a ready-made form consisting of numerical positions; the first and second categories identified in this algebraic teaching might not have been possible. The results indicate that these students might use properties and relationships on a number line to enable their mathematical reasoning.

Den här artikeln handlar om möjligheter och utmaningar för barn i åldrarna fyra till åtta år. De fick arbeta med olika fördelningsdilemman och visade då förmåga till både matematiska och etiska resonemang i sina lösningsprocesser.

In this paper, we focus on the expansion of a task developed in one educational context and moved into another educational context. The working process was conducted as a collaboration within a research team consisting of researchers and teachers. The aim of this paper is to describe how a task designed for students to work in base four, was collaboratively expanded into a task that deals with both base four and base ten. The overarching idea is the importance of early exposure to the structural aspects of the positional system and to advocate for exploring different bases to enhance learning. The theoretical framework of the project is based on learning activity, inspired by the El’konin and Davydov mathematics curriculum. The paper includes an analysis of the task staged in a Swedish classroom, highlighting collaborative design work, task features, and outcomes. The results indicate challenges and insights in students’ understanding of base ten as one of many possible base numbers.

This study examines the collective mathematical reasoning when students and teachers in grades 3, 4, and 5 explore fractions derived from length comparisons, in a task inspired by the El´konin and Davydov curriculum. The analysis showed that the mathematical reasoning was mainly anchored in mathematical properties related to fractional or algebraic thinking. Further analysis showed that these arguments were characterised by interplay between fractional and algebraic thinking except in the conclusion stage. In the conclusion and the evaluative arguments, these two types of thinking appeared to be intertwined. Another result is the discovery of a new type of argument, identifying arguments, which deals with the first step in task solving. Here, the different types of arguments, including the identifying arguments, were not initiated only by the teachers but also by the students. This in a multilingual classroom with a large proportion of students newly arrived. Compared to earlier research, this study offers a more detailed analysis of algebraic and fractional thinking including possible patterns within the collective mathematical reasoning. An implication of this is that algebraic and fractional thinking appear to be more intertwined than previous suggested.

The aim of this thesis is to develop and explore teaching possible to promote algebraic thinking together with young, multilingual students six to twelve years old. One underlying assumption for the aim is that algebraic thinking can be developed by students participating in learning activities that are characterized by collective mathematical reasoning on relations between quantities of positive whole and rational numbers. Two overall research questions support this work: (1) What in students work indicate algebraic thinking identified in learning activities and as experiences of algebraic thinking? (2) How can learning models manifest in learning activity, in what ways do learning models change and enhance, and which characteristics of learning actions are enabled?   

Data was produced by interviews and from research lessons with students in lower grades in a multilingual Swedish school. The research lessons were focused on learning activity as suggested by Davydov (1990, 2008/1986), aimed at developing theoretical thinking – here algebraic thinking. They were staged in two research projects conducted as networks of learning studies. In these learning studies, the group of teachers iteratively designed and revised learning activities whereby the students could identify mathematical knowledge and collectively solve mathematical problems. 

The findings in the articles signal that learning models were developed as rudimentary, preliminary, prototypical and finally symbolic. Rudimentary models were grounded in algebraic thinking when the students analysed problem situations and identified the problem. Preliminary and prototypical models were developed by initiating and formalising actions understood as algebraic thinking. Different tools were initiated by the students and the teachers. These tools were formalised by the students. The students used algebraic symbols and line-segments to think together when comparing different quantities (Article 2). They carried out operations using unknown quantities when reflecting on additive and multiplicative relationships (Article 3). The students also used algebraic symbols to reflect on subtraction as non-commutative (Article 3). The different tools they used interacted on different levels of generalisation (Article 1). Algebraic thinking grounded the students reflections but interacted with, for example, fractional thinking in their arguments during the development of their learning models (Article 4). The different ways of thinking interacted in arguments when developing the rudimentary, the preliminary and the prototypical models. However, in the conclusion of their collective reasoning and in the development of the symbolic learning models, these different ways of thinking were intertwined in the same arguments (Article 4).

As a conclusion, the four articles signal that learning models including algebraic symbols developed in a learning activity can be used by newly-arrived immigrant students to reflect on structures of numbers.

This article discusses algebraic thinking regarding positive integers and rational numbers when students, 6 to 9 years old in multilingual classrooms, are engaged in an algebraic learning activity proposed by the El’konin and Davydov curriculum. The main results of this study indicate that young, newly arrived students, through tool-mediated joint reflective actions as suggested in the ED curriculum, succeeded in analysing arithmetical structures of positive integers and rational numbers. When the students participated in this type of learning activity, they were able to reflect on the general structures of numbers established as additive relationships (addition and subtraction) as well as multiplicative relationships (multiplication and division) and mixtures thereof, thus a core foundation of algebraic thinking. The students then used algebraic symbols, line segments, verbal, written, and gesture language to elaborate and construct models related to these relationships. This is in spite of the fact that most of the students were second language learners. Elaborated in common experiences staged in the lessons, the learning models appeared to bridge the lack of common verbal language as the models visualized aspects of the relationships among numbers in a public manner on the whiteboard. These learning actions created rich opportunities for bridging tensions in relation to language demands in the multilingual classroom.

Kan multiplikation förstås på något annat sätt än som upprepad addition? Här prövar författarna ett nytt sätt att undervisa om multiplikation. Genom att atbeta med indirekt mätning skapas ett behov av multiplikation.

This article explores grade 1 students’ different ways of experiencing quantity comparisons after participating in teaching designed as a learning activity using tasks from the Davydov curriculum. A phenomenographic analysis generated three hierarchical ways of experiencing comparisons: counting numerically, relating quantities, and conserving relationships. The first category comprises arithmetic ways of thinking, whereas the second and third categories comprise algebraic ways of thinking. Algebraic thinking was identified as reflections on relationships between quantities at different levels of generalisation. The implications of these results in relation to learning activity theory are discussed.

The issue for this paper is to identify algebraic reasoning through students´sense-making actions, during a lesson, where students and a teacher develop learning models for mixed numbers. The analysis focuses the students’ work, trying to make sense of the unknown fractional part of the number. This unknown part was elaborated when the students suggested to “add a little bit more” to construct equality. The un-known part developed to a fractional part with help of an emerging learning model containing algebraic symbols: B=W+p/a. In this activity. The potentialities in the students’ algebraic reasoning were identifyed as: an additive relationship between the integer and the fractional part of the number, and a multiplicative relationship between the numerator and the denominator in this fractional part.