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Generalization is one of the core aspects of algebraic thinking, and research literature points to the importance of objectification when generalizing mathematical discourses. To increase the field’s understanding of the objectification process in various algebra discourses, our study analyses Grade 6 teachers’ use of words, symbols, routines and narratives when they introduce various algebraic concepts. Our results show that there exist instances of objectification in the teachers’ discourses, but these are all implicit, which can make them difficult for students to notice. Thus, teachers need to be made more aware of the importance of making the objectification process explicit. Furthermore, our results show that instances of objectification are tightly connected to the use of formal algebraic symbols. This highlights the importance of the transition from concrete/iconic to formal/arbitrary representations.

Computational thinking has become part of the mathematics curriculum in several countries. This has led recently available teaching resources to explicitly integrate computational thinking (CT). In this paper, we investigate and compare how curriculum resources developed in Denmark — digital teaching modules — and Sweden — printed mathematics textbooks — have incorporated CT in mathematics for grades 1–6 (age 7–12). Specifically, we identify and compare the CT and mathematical concepts, actions, and combinations in tasks within these resources. Our analysis reveals that Danish tasks are oriented toward CT concepts related to data, actions related to programming, and mathematical concepts within statistics. This is different from Swedish tasks, which are oriented toward CT concepts related to instructions and commands, actions related to following stepwise procedures, and mathematical concepts related to patterns. Moreover, what is most dominant in one country is almost or completely absent in the other. We conclude the paper by contrasting these two approaches with existing knowledge on computational thinking in school mathematics.

In the last decade, programming and computational thinking (CT) have been introduced on a large scale in school curricula and standards all over the world. In countries such as the UK, a new school subject—computing—was created, whereas in countries such as Sweden, programming was included in existing subjects, notably mathematics and technology education. The introduction of programming and CT in technology education implies a particular relationship between programming and technology. Programming is usually performed with technological artefacts—various types of computers—and it can also be seen as a specific branch of engineering. This book analyses the background to and current implementation of programming and computational thinking in a Swedish school technology context, in relation to international developments. The various chapters deal with pertinent issues in technology education and its relation to computers and computing, for example, computational thinking and literacy, teachers’ programming competence, and computational thinking, programming, and learning in technology education. The book includes examples from educational research that could also be used as inspiration for school teaching, teacher education and curriculum development

Här redovisas resultat från ett fyraårigt forskningsprojekt om implementeringen av programmering i skolans matematikundervisning. Både lärare och läromedel har studerats och resultaten väcker frågor om satsningen varit så genomtänkt som man hade kunnat önska.

In this chapter, we consider opportunities and challenges of integrating programming in school mathematics by looking closer at the Swedish case. Especially, we aim to unpack how the integration of programming in mathematics can affect students’ opportunities to develop mathematical competencies. Three examples taken from a Swedish research project are described and discussed in detail. In the first example, we focus on differences regarding syntax and semantics in programming and mathematics, relevant in a learning perspective. In the second and third examples, we highlight the teachers’ views of how programming may enhance the traditional learning of school mathematics, and describes one particular lesson where students solve pattern problems in Python. Each example is discussed in relation to different mathematical competencies.

I ett pågående forskningsprojekt tittar författarna närmare på programmering som innehåll i skolmatematiken sedan införandet i läroplanen 2018. Här redovisar de hur lärare tolkat uppdraget och vilken roll programmeringen har i de matematiklektioner med programmering som lärarna planerat och genomfört.

In this symposium, we will discuss different ways of analysing mathematics textbooks from a methodological point of view. The discussion will be based on examples from five separate ongoing analyses of Swedish textbooks divided into two methodological approaches; one where analysis is conducted within an established theoretical framework, and one where analytical tools are constructed through combining aspects of different theories. The symposium will be held at the conference MADIF13.

In this paper we consider implications of the current world-wide inclusion of computational thinking in relation to children’s development of algebraic thinking. Little is known about how newly developed visual programming environments such as Scratch could enhance early algebra learning. The study is based on examples of programming activities used by mathematics teachers in Sweden, teaching students aged 10–12 years during the first two years of implementing programming in the mathematics curriculum. Informed by Chevallard’s praxeology in terms of praxis and logos, we describe, unpack, discuss and expand these activities. Core issues related to algebra found in the three activities are as follows: making implicit variables explicit; using a counter variable; and identifying parameters as a specific type of variable. Our findings show that, in addition to already identified uses of variables in early algebra, programming activities in the early years bring in new aspects and new ways of treating variables that could, potentially, enhance students’ understanding of variables and generalization, provided that programming praxis is embedded in an appropriate algebra logos.

To get an overview of the characteristics of the studies in mathematics education research that explicitly state that they deal with implementation, we have conducted a systematic review. In this paper, we report on a subset of the identified studies from the review, dealing with large-scale professional development for teachers. For the subset of the 11 identified papers, we ask the question: What designs are used to support teachers to adopt new ideas in their practice and what dimensions of scaling are considered in the studies? To articulate design and dimensions of scaling, we draw on theoretical constructs from both mathematics education research and more general implementation research. Results indicate that the choice of facilitating strategy impacts the dimensions of scaling considered in the implementation.

Med utgångspunkt i den senaste forskningen diskuterar bokens författare programmeringens inträde i skolmatematiken. Vilken roll har programmeringen i skolmatematiken? Kan elever lära sig matematik genom programmering? Vilka möjliga broar mellan matematik och programmering kan skapas? Vilka är utmaningarna? Diskussionen utgår från elevernas, lärarnas och lärarutbildarnas perspektiv. Innehållet i boken innefattar skolans alla nivåer, från förskola till gymnasiet. Programmering i skolmatematiken vänder sig till blivande och verksamma lärare, och till lärarutbildare, men även till personer med övergripande ansvar för skolans digitalisering på lokal, kommunal eller nationell nivå.

Det här kapitlet jämför programmering och matematik ur ett språkligt och begreppsligt perspektiv. Genom att betrakta ett programmeringsspråk som en ny form av matematisk representation menar författarna att elevers erfarenhet av programmering kommer att påverka deras förståelse av matematiska begrepp. Eftersom programmeirng i den svenska matematikkursplanen för grundskolan har skrivits in som en del av det centrala innehållet i algebra, undersöker författarna specifikt begreppen variabel och funktion, samt användningen av algebraiska symboler, framförallt likhetstecknet. Två undervisningsexempel analyseras i detalj, ett program i Scratch och ett program i JavaScript. Författarna argumenterar för att lärare behvöer känna till närr symboler har olika betydelser i programmering respektive matematik, och hurbetydelsen av matematiska begrepp, såsom variable och funktion, får delvis andra innebörder i programmering.

Det avslutande kapitlet i boken sammanfattar de utmaningar och möjligheter som följer av att programmering integreras i matematikämnet. Utifrån Chevallards teori om didaktisk transposition av kunskap beskriver författarna tre olika sätt som lärareatt arbetar med programmering och matematik. Det blir tydligt att programmeringsundervisning i matematik kan komma att skilja sigvariera ganska mycket åt beroende på hur lärarna tolkar relationen mellan matematik och programmering, vilka uppgifter och programmeringsmiljöer de väljer, och vad de ser som syftet med undervisningen. AvI slutningsviset av kapitlet återkopplar författarna till tidigare kapitel för att summera de utmaningar och möjligheter somvad det nya kunskapsinnehållet medför., Vetenskapsrådet, 2018-03865

I ett pågående forskningsprojekt tittar författarna närmare på programmering som innehåll i skolmatematiken sedan införandet i läroplanen 2018. Här redovisar de hur lärare tolkat uppdraget och vilken roll programmeringen har i de matematiklektioner med programmering som lärarna planerat och genomfört.

As with any programs in teacher education, Swedish mathematics teacher education is influenced by changing political winds, developments in Information and Communication Technology (ICT), culture, history, PISA results, research-based program designs, and a fair amount of passion. Content and outcomes are nationally determined and include the requirement of a strong research foundation, but this is often not how practcing techers work, which exerts its own pull on teacher education. The specific implementations of programs take different forms at the universities that offer mathematics teacher education. In order to provide a comprehensive yet meaningful ntroduction to both the current system and current practices, we describe the overall organization of Swedish mathematics teacher education, and then offer short cases of implemented programs. To ensure inclusivity, the various parts are written by mathematics educators from the respective institutions. In this way, both variation across mathematicas teacher education for diffrent grade levels and variation across different institutions working with the same national directives can be distinguished. Issues such as the academization of teacher education are problematized, as are other forces that constitute the crosscurrents in Swedish mathematics teacher education.

När programmering skulle inkorporeras i skolans arbete valde Sverige i sin läroplansrevidering 2017 att skriva in det i matematikämnet, med stark koppling till algebra. Samtliga matematiklärare ställdes då inför utmaningen att undervisa i programmering. Vi undersöker här resultatet av 32 lärargruppers gemensamma arbete med att planera och genomföra lektioner i programmering i matematik i grundskolan. För att få insikt i hur lärare tolkar uppdraget och transponerar läroplananes beskrivning av programmering till klassrumspraktiken analyserar vi det matematiska innehållet i dessa lektioner samt vilken syn på relationen mellan matematik och programmering som framträder i lärarnas beskrivning av syfte, lärandemål, aktivitet och reflektion. Vi finner att programmeringsaktiviteter i 1/3 av lektionerna inte kopplas till något traditionellt matematiskt innehåll. I övriga lektioner är det främst aritmetik eller geometri som utgör det matematiska innehållet. Få explicita kopplingar görs till algebra förutom till begreppet variabler, men då är det främst variabler inom programmering som avses. I materialet framträder fyra olika relationer mellan matematik och programmering: 1) enbart programmering; 2) matematik som en kontext för programmering; 3) programmering som ett verktyg för att effektivisera beräkningar; 4) programmering som ett verktyg för att utforska matematik. Resultaten diskuteras i relation till matematikämnets syfte och innehåll i den svenska läroplanen., When incorporating programming in the school curricula, Sweden decided to integrate it with mathematics, and specifically within the core content of algebra. As a result, all mathematics teachers at all levels were faced with the challenge of teaching programming. In this study we analyse documentation from 32 lessons studies where groups pf teachers have planned, conducted, and revised lessons on programming within the school subject mathematics. To gain insight into how the teachers interpret the new task and thus contribute to the transposition of knowledge from the curriculum level to the classroom level, we analyse the mathematical content in these lessons and the relations between mathematics and programming that emerge in the way the teachers describe the aim, the activites and the learning outcomes of the lessons. We find that 1/3 of the lessons do not connect to any traditional mathematics content, and the rest of the lessons mostly focus on arithmetic or geometry. Few explicit connectiions are made to algebra, execpt for the variable concept, but when variables are treated the focus is on variables in the programming sense rather than algebra. Four different relationships between mathematics and programming emerge in the data: 1) programming without connecting to mathematics; 2) mathematics as a context for programming, 3) programming as a tool for efficient calculations; 4) programming as a tool for exploring mathematics. The results are discussed relation to the transposition of knowledge of mathematics as a school subject.

As with any programs in teacher education, Swedish mathematics teacher education is influenced by changing political winds, developments in Information and Communication Technology (ICT), culture, history, PISA results, research-based program designs, and a fair amount of passion. Content and outcomes are nationally determined and include the requirement of a strong research foundation, but this is often not how practcing techers work, which exerts its own pull on teacher education. The specific implementations of programs take different forms at the universities that offer mathematics teacher education. In order to provide a comprehensive yet meaningful ntroduction to both the current system and current practices, we describe the overall organization of Swedish mathematics teacher education, and then offer short cases of implemented programs. To ensure inclusivity, the various parts are written by mathematics educators from the respective institutions. In this way, both variation across mathematicas teacher education for diffrent grade levels and variation across different institutions working with the same national directives can be distinguished. Issues such as the academization of teacher education are problematized, as are other forces that constitute the crosscurrents in Swedish mathematics teacher education.

In tackling the well-known challenges of learning algebra, Sfard (2008) highlights the need for objectification of mathematical discourse. Teachers can support students in this objectification by modelling the discourse they want students to develop (Sfard, 2016). Our study uses Sfard’s (2008) definition of objectification in order to analyse grade 6 teachers’ algebra discourse. The purpose is to understand if and how objectification occurs in the discourse when algebra is introduced, and how this might influence students’ understanding of algebra. Data consist of video recordings from three Swedish teachers’ introductory lessons in algebra with 12-year old students, with four consecutive introductory algebra lessons for each teacher. The recordings are from a larger international video study. Teachers’ discourse about algebraic entities was analysed concerning word-use, visual mediators, routines, and endorsed narratives (Sfard, 2008). Preliminary results show that there were mainly three algebraic entities that were in focus in the lessons, described as equations, expressions and variables. Generally, we note that as more formal algebraic symbols were introduced and used by two of the teachers, the more objectified became their discourse. This objectification did not occur in the third teacher’s discourse, which lacked an early introduction of formal symbols. This could imply that symbols reinforce objectification. Particularly, the progression of the objectification process concerning expressions was very prominent over the course of one teacher’s four lessons. The word use clearly changed from treating the introduced symbols as processes, where an expression was “to describe something with the help of a variable”; to objects, where an expression was “a system of symbols” that “could be used in an equation”. A more comprehensive discussion of the results, including routines and endorsed narratives, will be included in our presentation.

The teaching of algebra has over the years descended from courses in higher education to become an integrated part of compulsory school education. Subsequently, the teaching and learning of algebra in the early grades has become an internationally established field of research (c.f. Cai & Knuth, 2011; Kieran, Pang, Schifter & Ng, 2016; Kaput, Carra- her & Blanton, 2008). Also in the Nordic and Baltic region mathematics educators have developed an interest in this field. In this special issue, some of the recent Nordic research is presented.In the autumn of 2017 a call for papers was announced. It resulted in fourteen submitted papers, and finally the seven articles in this issue of NOMAD. Combined, they provide one picture of the contemporary research. In addition to this we would like to comment on other research in this field we are aware of. However, bear in mind, this is not a structured literature review. Our aim is merely to outline the millieu of this thematic issue. We mainly describe larger research groups and projects, and apologise in advance if we have omitted other important research. [...]