Ola Helenius

• Professor
• Researcher and developer at University of Gothenburg

Mathematical communication, encompassing writing in, about, and for mathematics , is a critical competency. Defining excellent mathematical writing standards, however, remains challenging. To address this, we conducted a systematic review of 48 scholarly works on quality in mathematical writing. Our findings reveal mathematical writing for differen...

We present a two-dimensional framework capable of characterizing the algebra content in textbook series spanning over at least all years of comprehensive schooling. Our framework extends well-known previous work and subdivides school algebra into algebra classes such as structure, operations on non-numerical symbols, functional thinking, and patter...

The fraction representation can symbolize different mathematical concepts because the symbol a/b is polysemic. Since students' difficulties in acquiring the different conceptual meanings that the fraction representation denotes are well documented in research, we examined two commonly used textbook series in Sweden concerning how the polysemic aspe...

We use a key case from implementation research in mathematics education, where results involve a claim of teachers' beliefs influencing the implementation outcome, to argue that alternative interpretations are possible. The case represents how argumentations about teachers' beliefs as an influential factor are carried out in implementation research...

Fractions are core content of elementary school mathematics, and conceptual knowledge of fractions is essential when developing a comprehensive understanding of fractions. Previous research, however, has indicated limitations in student teachers' fraction knowledge. This study investigated 57 Swedish elementary school student teachers' conceptions...

We use a single key case to understand whether a research-based innovation to improve students' formal written mathematical communication has implementability and achieves desired outcomes for the target population. We investigate how and in what way students' communication competency can develop when teachers use an innovative teaching model targe...

In a teacher-researcher collaboration, a teaching model designed for developing students' competence in producing formal written mathematical communications of solutions to problems has been piloted. The sequence was implemented in a mathematics course in the International Baccalaureate program. By taking advantage of a standardized thesis writing...

From an implementation perspective, there is a need to research the effects of large-scale projects. We focus on a project that engages over 1,000 teachers who implement a teaching model via highly scripted teacher guides. In this single case study, we analyzed video recordings of a third-grade teacher's innovation enactment of a teaching cycle. We...

Articles in mathematics education that report on the implementation of innovations do not always report models for how the innovation program is supposed to attain its effects. We analyze an article with no such explicit theory of change with the aim of providing a background case for discussing the role of theories of change in implementation rese...

We report on a study investigating how the notions of large-scale and small-scale projects are interpreted. Some of the methodological procedures of the Delphi technique were used, including posing a couple of scale-related questions to a group of 14 educational researchers who had experience in leading or investigating the large-scale implementati...

We present the results of a systematic literature review of research articles that self-identify as describing the implementation of innovations from mathematics education research in educational practice. We classified 103 articles according to study type, the type of teaching/learning phenomenon the implementation targeted, and whether the innova...

Mathematical writing in school is difficult to teach. Reasons include that it is a domain in which there are rules for some parts of the writing like for example mathematical notation while others must be negotiated locally in the classroom. Writing is also seldom separated from the process of solving mathematical problems, creating a situation whe...

Scale and scaling are central concepts in projects aimed at implementing innovations for improving mathematics education. This study aims to provide conceptual clarity on the notions of scale and scaling in mathematics education research. The meanings of these terms are handled tacitly, and there is diversity in how they are used in the literature....

Theoretical aspects of symbol systems and the theories of conceptual fields and representation by Vergnaud are used to unpack the symbols and formalism competency. We illustrate how mathematical symbols are initially used as labels for phenomena in concrete situations or through iconic representations. Later, however, developing mathematical symbol...

We examine the feasibility of using the Documentational Approach to Didactics to gauge the fidelity and character of teachers' implementation of scripted teaching sequences for the teaching of arithmetic in primary school. By analyzing a video-recorded lesson, we found that the teacher adapted the scripted teaching sequence by merging it with other...

The number of books at home is commonly used as a proxy for socioeconomic status in educational studies. While both parents’ and students’ reports of the number of books at home are relatively strong predictors of student achievement, they often disagree with each other. When interpreting findings of analyses that measure socioeconomic status using...

In this paper we consider implications of the current worldwide 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...

Understanding how children acquire mathematical abilities is fundamental to planning mathematical schooling. This study focuses on the relationships between mathematical cognition, cognition in general and neural foundation in 8 to 9-year-old children. We used additive mathematics tests, cognitive tests determining the tendency for proactive and re...

We examine the feasibility of combining elements from the framework of program theory with the Documentational Approach to Didactics to gauge the fidelity and character of teachers' implementation of scripted teaching sequences for primary school. By analyzing two video-recorded lessons from two teachers' implementation of the program Thinking, Rea...

We examine the feasibility of combining elements from the framework of program theory with the Documentational Approach to Didactics to gauge the fidelity and character of teachers’ implementation of scripted teaching sequences for primary school. By analyzing two video-recorded lessons from two teachers’ implementation of the program Thinking, Rea...

With this essay, we pay a tribute to Gérald Vergnaud's legacy.

We reassess the relation between students’ socioeconomic status (SES) and their achievement by treating SES as multidimensional instead of unidimensional. We use data from almost 600,000 students in 77 countries participating in the 2018 PISA assessment of student achievement in math, science, and reading. The composite measure of SES that PISA use...

The development of a large-scale professional development project for Swedish mathematics teachers is retrospectively examined. By referring to documentation produced by stakeholders in the development process, the stakeholder's design recommendations and underlying assumptions on teacher development are described. Seeing the development as a co-de...

Mathematics is a polysemic enterprise, where the same name is given to different things. Polysemy is present whenever mathematical patterns identified in different circumstances share the same structure. This structure will be subsumed under the same mathematical symbolism. Therefore, it is of interest to theorize upon progress in concept knowledge...

By examining the effects of a national-scale teacher professional development (PD) program on instructional practices and student mathematics achievement, we contribute to calls for empirical studies investigating the impacts of such programs conducted at scale. The program corresponds well with core critical features of high-quality teacher PD and...

In this paper we report on a framework for analyzing progress in concept knowledge as represented in textbooks. The framework is based on theories of Vergnaud, combined with a theory of Bruner. Two Swedish textbook series are analyzed with respect to how they introduce mathematical content and how it supports the progress of concept knowledge throu...

While the importance of handling the mathematical key concept proportion is well documented for children and adolescents, studies on adults are rare. We report three tasks, investigating if adults' experiences from everyday life support their abilities to deal with proportional situations concerning volume scaling, speed, and cost-effectiveness. Op...

Iszák and Beckmann recently argued for a coherent approach to multiplication, based on a specific model for thinking about multiplication. We argue theoretically against this variant of coherence. We base our arguments on an exemplification of how mathematical thinking at four different cognitive levels involves an element of polysemy and exemplify...

This paper reports on a case study of a student’s rationales for learning mathematics. We operationalize Stieg Mellin-Olsen’s educational concept of rationales for learning and apply the concept on data consisting of three semi-structured interviews with a student in the Swedish prison education program. Our analysis shows that the student’s ration...

One of the many things teachers do is to give feedback on their students’ work. Feedback pointing out mistakes may be a key to learning, but it may also backfire. We hypothesized that feedback based on students’ mistakes may have more positive effects in cultures where teachers have greater authority over students, which we assume to be cultures th...

I den här texten ges en kortfattad översikt till den vetenskapliga bakgrunden till kompetensutvecklingsprojektet som är en bärande del i SKL's och NCM's projekt Ledning och styrning, matematik. Utifrån ett professionsperspektiv innehåller denna satsning två komponenter. Den ena är en fortbildningskomponent. Det andra är en undervisningskomponent. D...

This study reports on a case study of a student’s rationales for studying mathematics. We operationalize Stieg Mellin-Olsen’s educational concept of rationales for learning and apply the concept on data consisting of three semi-structured interviews with a student in the Swedish prison education program. Our analysis shows that the student’s ration...

Students' peer collaboration efforts in mathematics and statistics is a topic that has increasingly gained attention in research. In any collaboration, authority relations play a role for how meaning is constituted: Whenever things are discussed and decisions are made, authority is involved in a sense that some arguments or persons may be more conv...

Up until today, research has gathered extensive knowledge on students’ understanding of multiplication (e.g. Clark & Kamii, 1996; Fischbein, Deri, Nello, & Marino, 1985; Harel & Confrey, 1994; Larsson, 2016; Thompson & Saldanha, 2003). We know that concepts preferably develop over long periods, through experience of a large number of situations (Ve...

The outcomes of a longitudinal randomized number sense intervention including pre-, post- and follow-up tests are reported. The intervention was conducted in Grade 0 when children were six years old. Using Concrete-Representational-Abstract principles, a structured explicit 10- week program focusing on numbers and collective reasoning about represe...

The present paper describes the principles behind a large scale professional development project in Sweden that is presently designed and implemented. The project involves five municipalities and around 500 teachers in grades 0-3. In contrast to most contemporary programs, this professional development program builds on implementing pre-designed te...

In this paper, we investigate mathematics teachers' conception of the relationship between mathematics and programming. The context of the investigation is a recent curriculum reform in Sweden that makes programming a compulsory element of the national mathematics standards. Following up on an in-service training initiative, we conducted a pilot su...

Can instructional quality be measured using TIMSS items on how often certain instructional practices are used in the mathematics classroom? We focused on three instructional practices that have been the topics of longstanding debates in the educational literature: memorizing formulas, listening to the teacher, and relating mathematics to daily life...

Data generation in statistics education is often conducted by the students them- selves; however, the question of what learning opportunities the data generation process offers has only been studied to a small extent. This paper investigates to what extent data generation is an observational and procedural vs. a conceptual activity. We inquire into...

In this chapter, an adapted version of Dowling’s distributive strategies is used to show how two children, Klara and Teo, are provided with differential enhancement in the mathematical learning opportunities that they are offered. The analysis shows that the use of everyday settings of mathematics problems, including expectations about the social r...

This paper puts the state of mathematics education research in Sweden in a historical context. The periods 1950-1984, 1985-1999, 2000-2008 and 2008 to present are discussed. An analysis of the content of PhD dissertations from 1918 to 2017 is presented.

Speed may seem like a simple concept, but similar to other rate-based concepts, tasks involving speed can be significantly challenging for students. We examine adult and 16-year-old students' work with a task involving the average speed over two equal distances. Solutions are analyzed by applying scheme theory and the distinction between the predic...

Speed may seem like a simple concept, but similar to other rate-based concepts, tasks involving speed can be significantly challenging for students. We examine adult and 16-year-old students' work with a task involving the average speed over two equal distances. Solutions are analyzed by applying scheme theory and the distinction between the predic...

Play-based preschool pedagogy usually relies on informal teaching while policy trends and some research call for increased formalisation of the pedagogy. Using Bernstein’s concepts of classification and framing, this article characterises mechanisms that link evaluation of preschool to the push towards the formalisation of teaching in preschool. Mo...

Vergnaud's theory of conceptual fields combine Piaget's work on schemes with a theory of semiotics and enable epistemological analysis of both operative and predicative forms of knowledge. We examine the possibility of combining Vergnaud's theory with Inferentialism as developed by Brandom, which is a pragmatic semantic philosophy of language and m...

Schemes were Piaget's most important concept. Through work of Vergnaud, schemes were connected to representations and theoretical models from Piaget were connected to principal insights from Vygotsky. We suggest that the scheme concept can be elaborated further by detailing the relationship between schemes and semio-tics. We consider a case of an a...

The use of freely-available web-based materials in professional development has rarely been investigated in mathematics education research. In this article, the responses to a survey by 267 preschool teachers about their use of online professional development materials are described. The web materials were based on a design model and the survey res...

By comparing two large-scale professional development programs' content and impact on student achievement, we contribute to research on critical features of high quality professional development, especially content focus. Even though the programs are conducted in the same context and are highly similar if characterized according to established rese...

Denna systematiska översikt sammanställer forskning om digitala lärresurser för att utveckla barns och elevers kunskaper i matematik. Forskningen spänner över försko­ lan till och med gymnasieskolan och anknyter speci kt till matematikundervisning­ en. I denna rapport redovisas översiktens resultat för förskolan. De digitala lärresurser som studera...

Denna systematiska översikt sammanställer forskning om digitala lärresurser för att ut­ veckla barns och elevers kunskaper i matematik. Forskningen spänner över förskolan till och med gymnasieskolan och anknyter speci kt till matematikundervisningen. I denna rapport redovisas översiktens resultat för grund­ och gymnasieskolan. De digitala lärresurs...

By combining linguistic analysis with metaphors for mediating between a body of knowledge and teaching practice, we exemplify curriculum developers’ voices in teacher guides. We discuss two types of voices, that of the Expert, who talks through the teacher and restricts the pedagogical agency of the teacher, and that of the Philosopher, who speaks...

Nordic preschool practice includes little formal teaching but both policy trends and some research challenge this model. Using Bernstein's concepts classification and framing, mechanisms that link mathematical objectives in preschool to the push towards formalization of teaching are characterized. A conceptualization of preschool teacher's work of...

This paper reports on a study on students' rationales for studying mathematics. The question, how you can characterize students' motivation for studying mathematics in relation to their social and educational context, is theory driven, based on a sociopolitical theoretical perspective. We operationalize Stieg Mellin-Olsen's educational concepts on...

This volume contains the proceedings of MADIF 10, the tenth Swedish Mathe- matics Education Research Seminar, held in Karlstad, January 26–27, 2016. The theme for this seminar was ICT in mathematics education: the future and the realities. The MADIF seminars are organised by the Swedish Society for Research in Mathematics Education (SMDF). MADIF ai...

In the Swedish mathematics prison education program, students with a history of failure in the regular school system get a second chance to recapture lost years of education. Experience from teaching paint a picture of motivated, goal-oriented students that perform very well in this extraordinary context. How can that be? Our study aims to investig...

In the Swedish mathematics prison education program, students with a history of failure in the regular school system get a second chance to recapture lost years of education. Experience from teaching paint a picture of motivated, goal-oriented students that perform very well in this extraordinary context. How can that be? Our study aims to investig...

In this paper, three Swedish studies focusing on student teachers in transition from university to teacher practice are analyzed with respect to similarities and differences in how the teacher students describe the mathematics teaching they want to do as well as how they relate to teaching they already see carried out. Despite the different theoret...

This case study investigates how a group of 12-year-old pupils contextualizes a task formulated as an equation expressed in a word problem. Of special interest is to explore in detail the phenomenon of pupils working with manipulative-based equation-solving methods in a task involving another real world context. The pupils’ small group discussions...

The informal character of preschool mathematics, engaged in during children’s play, places complex requirements on preschool teachers. It also leads to challenges in developing appropriate analytical tools for researching teacher work. In this paper a framework, the ”didaktic space”, is described and used to analyse interactions between preschool t...

Mathematical performance is highly correlated with several general cognitive abilities, including working memory (WM) capacity. Here we investigated the effect of numerical training using a number-line (NLT), WM training (WMT), or the combination of the two on a composite score of mathematical ability. The aim was to investigate if the combination...

This small-case study combines a content related and a dialogical approach, in an in-depth analysis of how three 12-year-old pupils in a video recorded small group discussion construe the meaning of the letter n in an algebraic expression. The findings indicate that the pupils used a rich variety of contextual resources in their sense-making attemp...

One of Bishop’s six mathematical activities is playing which includes modelling, hypothetical thinking and abstraction. These can be in young children’s play, but do they by their presence make this play mathematical? In this chapter, we explore this question by first defining play and then comparing its features with what is known about mathematic...

Preschool teachers' expectations about what mathematics they should engage children in are generally centred about numbers and counting. However, the Swedish preschool curriculum and research into young children's development of mathematical understanding suggest that children can be offered a much richer set of ideas. In this paper, we examine mat...

Researchers rarely discuss methodological issues in regard to preschool mathematics education and if they do, they do not take their starting point from reconceptualisations of what mathematics might be for preschool children. This paper presents as an analytical tool the " didaktic space " that arose when responding to issues related to the analys...

We describe the design process for an intervention program in the domain of number in Swedish preschool class. A consequence of the design-feedback cycle was that the initial idea of combining a learning trajectory based approach with a socially driven teaching based on collective reasoning was revised. The resulting design keeps the emphasis on st...

This paper theorises the design of Skolverket's preschool and preschool class professional development web modules. By contrasting different models of teacher change, components are identified that designers of professional development materials may need to consider. Data from the decisions taken in designing the Skolverket project were analysed in...

This book presents chapters based on papers presented at the second POEM conference on early mathematics learning. Chapters broaden the discussion about mathematics education in early childhood, exploring the debate about empasizing construction versus instruction. Sections investigate the teaching and learning of specific mathematical content, ear...

In this paper, the distinction between Bernstein’s horizontal and vertical discourse is used to show how two children are restricted in their possibilities to learn mathematics. The social relationships set up within contexts, both of the problems being solved, and between participants, contributed to the horizontal or vertical discourse being empl...

This volume contains the proceedings of MADIF 9, the Ninth Swedish Mathe- matics Education Research Seminar, held in Umeå, February 4-2, 2014. The MADIF seminars are organised by the Swedish Society for Research in Mathematics Education (SMDF). MADIF aims to enhance the opportunities for discussion of research and exchange of perspectives, amongst...

From 2012 to 2016 all teachers of mathematics, in primary through to upper secondary and adult education, in Sweden are to be given the opportunity of receiving state-coordinated professional development (PD), generally involving around one meeting per week for a year. We examine the ways in which this programme and its content are research-based b...

This paper presents the results of a pilot study using questionnaires on the role of video case studies in professional development material for preschool and preschool class teachers in Sweden. Although the analysis of the results show that some questions in the questionnaire need adapting, generally it seems that it is possible to find out inform...

This study investigates the impact of a national reform in Sweden introducing mathematical competency goals. Data were gathered through interviews, classroom observations, and online surveys with nearly 200 teachers. Contrasting to most studies of this size, qualitative analyses were conducted. The results show that teachers are positive to the mes...

This article presents the Swedish action plan for mathematics: the reasons for launching it, its main objectives, and how it was carried out. The authors put forward a fresh view of what mathematics can contribute at school. The action plan aims to generate interest in mathematics among both pupils and the general public. The authors present the ac...

C. Bergsten, B. Grevholm, H. Strømskag Måsøval & F. Rønning (red.) (2007). Relating practices and research in mathematics education. Proceedings of NORMA 05, fourth Nordic conference on mathematics education. Trondheim: Tapir Academic Press. ISBN 978-82-519-2212-8

In 1977 Kervaire and Murthy presented three conjectures regarding $K_0 \mathbb{Z} C_{p^n}$, where $C_{p^n}$ is the cyclic group of order $p^n$ and $p$ is a semi-regular prime. The Mayer-Vietoris exact sequence provides the following short exact sequence $$0\to V_n\to \pic (\mathbb{Z} C_{p^n})\to \cl \mathbb{Q} (\zeta_{n-1})\times \pic (\mathbb{Z} C...

There is an Mayer-Vietoris exact sequence involving the Picard group of the integer group ring $\Z C_{p^n}$ where $C_{p^n}$ is the cyclic group of order $p^n$ and $\zeta_{n-1}$ is a primitive $p^n$-th root of unity. The "unknown" part of the sequence is a group. $V_n$. $V_n$ splits as $V_n\cong V_n^+\oplus V_n^-$ and $V_n^-$ is explicitly known. $V...

In 1977 Kervaire and Murthy presented three conjectures regarding K 0 ZC p n, where C p n is the cyclic group of order p and p is a semi-regular prime that is p does not divide h (regular p does not divide the class number h = h h ). The Mayer-Vietoris exact sequence provides the following short exact sequence 0 ! Vn ! Pic ZC p n ! Cl Q(inGamma1 )...

Thesis (Ph. D.)--Chalmers University of Technology, 2002.

We re-prove a result by M. A. Kervaire and M. P. Murthy [Comment. Math. Helv. 52, 415–452 (1977; Zbl 0355.12009)] concerning Picard groups of integer group rings ℤC, where C is a cyclic group of prime power order. Our method is more elementary than the one used by Kervaire and Murthy and relies on the construction of certain multiplicative maps by...

In this paper we prove two different generalizations of Kummer's Lemma that describes when a unit of a cyclotomic field is a p-th power of another unit. One of these results is then used to prove a theorem about the Picard group of the integer group ring ZC, where C is a cyclic group of prime power order. The theorem was first proved by Kervaire an...