Shadow education is a broad concept encompassing instructional support offered outside the regular school context. As Bray (2024) summarizes, research in this area is largely quantitative, often measuring the amount of tutoring received and its association with academic outcomes such as mathematics performance, with mixed results. Against this background, we present a systematic literature review of qualitative research that can contribute to a deeper understanding of the role of shadow education in mathematics. The following research questions were addressed: RQ1:How are the effects of the support characterized in the studies? RQ2: How is the relationship between the support and the formal schooling described?

A systematic literature search was conducted in the databases Educational Database, ERIC, Scopus, and Web of Science, resulting in 31 articles. The study was limited to studies on students aged 14–19, and data were analysed using thematic analysis.

The literature review includes digital support in asynchronous (apps, learning platforms, videos) and synchronous (online one-to-one tutoring) formats. In contrast, non-digital support was limited to synchronous formats such as tutoring programs. The effects of the support (RQ1) are characterized as affective (engagement, motivation, confidence), learning-related (understanding, student agency), social (relations, collaboration), and psychological (reduced stress, belonging). The relation between support and formal schooling (RQ2) is described as complementary, where support supplements regular classroom instruction, as contrasting, offering alternative ways to learn mathematics, and as compensatory, compensating for inadequate regular classroom instruction. In addition, the relationship is understood through more complex frameworks, such as seamless learning, which emphasizes continuity across learning settings, and transboundary learning, focusing on how skills, strategies, and identities developed with support can be transferred back into school.

This literature review will be a foundation for future qualitative research on students’ use of shadow education in mathematics. In the poster presentation, the results are further elaborated and their implications highlighted.

What happens when a Swedish mathematics teacher, who has experience of using a specific learning theory-variation theory for learning-as a tool to plan, evaluate, and revise his teaching, is asked to add and apply the Chinese framework of Bianshi (teaching with variation), which also builds on principles of variation? In this paper, we report on a case study where an elementary mathematics teacher in Sweden taught a lesson about fraction multiplication, which was video-recorded, and, in an appurtenant interview, explained the rationale behind his instructional choices. This was followed by a three-month intervention that introduced the teacher to the Bianshi framework of variation. After the intervention, a stimulated recall interview was conducted to obtain information about the way in which the teacher made sense of the Chinese framework when reflecting on his lesson about fraction multiplication. The study contributes to the discussion about whether and how the two frameworks of variation are complementary.

For many years, research on students’ understanding of the equals sign (see, e.g.,Kieran, 1981) focused on the operational (equal sign as a trigger of “to do” something) versus the relational (the equal sign describes the relation between numbers) aspect of the symbol. The conclusion has been that students need to embrace the relational understanding to manage algebraic expressions, particularly equations. In more recent research studies, students’ understanding of the relational aspect of the equals sign has been more nuanced. For instance, Jones et al. (2013) discuss the relational conception in terms of a sameness-relational conception or a substitute-relational conception, and advocate making students aware of the substitute-relational conception when teaching the concept of the equal sign. Their study was made on students in the first years of secondary school (aged 11-12 years). However, the equal sign is typically taught to students from very young age. Our question therefore is: Can students, already in the early school years have the substitute-relational conception of the equal sign?

From a previous project (Gunnarsson & Persson, 2024), we have data with 12 students in grades 1-3 (7-10 years old) when they talk about the equal sign. The students were interviewed in pairs and shown different types of expressions with equal signs and discussed these. Hence, triggered by the study of Jones et al. (2013), our data was reanalyzed, now focusing on the different relational aspects of the equal sign by using codes from Jones et al. (ibid.).

In this short presentation, the result is described in brief, concentrating on how the young students’ talk includes utterances about the different relational aspects of the equal sign. Many of the students give utterances that can be categorized as a sign of a relational conception of the equal sign. At least some of the students give utterances that can be indications of a substitute-relational conception.

Despite what many mathematics teachers may hope, students in secondary school do not seek help in their mathematics studies because they are curious about mathematics, but rather because they want to pass their tests. This is one of the conclusions of our survey of 229 students. Our study shed light on contemporary shadow education by exploring the types of support students use outside school and what motivates them to seek support in their mathematical studies. We find that regarding digital support types, students choose support where there is little or no interaction. Moreover, we find that almost half of the students claim to have used paid private tutoring, indicating a growing presence of shadow education in Sweden.

"Show all the work" we sometimes tell students when they solve tasks. But what does this mean? Is a solution better if the students have shown or not shown the component calculations? In this short report, we describe a central part of an ongoing study of conventions about what is required of a student solution to an arithmetic task to be good, or even of high quality. We made a pilot study where we asked 21 persons, upper secondary school teachers as well as researchers in mathematics education, about what aspects of a solution are important for the quality and what constitutes a high-quality solution. We show that the data can be analysed qualitatively by thematic analysis. The tentative result shows three main themes of arguments for what constitutes a high-quality solution. 

I denna presentation diskuterar vi förklaringsmodeller för elevers förståelse av likhetstecknet. Matematikdidaktisk forskning har ofta beskrivit elevers förståelse för likhetstecknet som att de antingen har en operationell (dynamisk) förståelse eller en relationell (statisk) förståelse. Vi har under intervjuer med elever i lågstadiet sett att eleverna tycks pendla fram och tillbaka mellan operationell och relationell förståelse, båda tycks finnas parallellt. Vår slutsats är att det behövs fler förklaringsmodeller än statisk och dynamisk uppfattning för att beskriva elevers svårighet med likhetstecknet.

Associativity is a central principle in mathematics. One could argue that associativity is unambiguous in a strict mathematical sense. However, in an interview study with 16 students in ages from 8 to 12years, we can show that in the interview-data there are qualitatively different types of understanding associativity. The different types of understanding are compared to previous research on mathematics education and to different definitions of associativity. Our conclusion is that the usefulness of associativity as a tool for generalised arithmetic depends on the type of understanding the students have.

Denna presentation beskriver en pågående studie om hur associativitet kan uppfattas och hanteras i matematikundervisning. Studien syftar till att beskriva elevers olika sätt att förstå associativitet och utifrån dessa designa lämpliga undervisningsinslag för att främja elevernas algebraiska förståelse. Vissa delresultat beskrivs.

Världens blickar har under de två senaste decennierna riktas mot östra Asien när lärande och undervisning i matematik står i centrum för diskussioner. En anledning till det ökade intresset är troligtvis att dessa regioner, gång efter annan, placerar sig i topp i olika internationella mätningar som TIMMS och Pisa medan Sverige och många andra OECD-länder hamnar på en mer blygsam placering. Det finns flera tänkbara förklaringar till sådana skillnader. En avgörande faktor för undervisningskvalitet och som ofta lyfts fram är den kinesiska undervisningstraditionen Bianshi Jiaoxue—undervisa utifrån variation av ämnesinnehållet. Denna tradition erbjuder flera intressanta begrepp som kan användas som verktyg för att planera och genomföra undervisning, speciellt med avseende på hur uppgifter kan sammanlänkas för att skapa effektivt och fördjupat lärande i matematik. Vi illustrerar några centrala begrepp inom traditionen genom att lyfta några exempel från en lektion om addition och subtraktion med tal i bråkform genomförd i Shanghai samt från en intervju genomförd med den undervisande läraren. Lektions- och intervjudata kommer från ett av samarbetsprojekt, finansierat av Vetenskapsrådet, mellan forskare från Jönköping University och Hong Kong University. Utgångspunkten relaterat konferensens tema är att en praktisk teori är en viktig komponent för undervisning på vetenskaplig grund.