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This study contributes to understanding mathematics teachers’ noticing based on teacher-selected moments through a case comparison of two third-grade mathematics teachers in China and two in Sweden. The data includes written and oral reflections, and students’ worksheets. A four-dimensional (mathematics, mathematics teaching, mathematics learning, and general pedagogy) coding framework was used to analyse commonalities and differences between the teachers’ noticing. The results revealed both commonalities but also important differences. The biggest differences were shown in mathematics teaching and general pedagogy. The study brings some first insights into Chinese and Swedish teachers’ noticing and provides practical implications on how teaching in the East and West can learn from each other.

This study explores which methods Swedish students use for multi-digit addition and subtraction after being taught the simplification methods regroup and move the difference. The data consist of 30 students’ solutions to one subtest of the Grade 3 national mathematics test. Results show that students used a variety of methods, with splitting numbers by place value being the most common, despite having lower success rates. The simplification methods were used less frequently, but more successfully. The findings suggest that conceptually focused teaching of different methods can support students in selecting and using efficient methods, although additional support may be needed to increase actual use. The study also confirms earlier research showing that Stepwise methods outperform Split methods, especially in subtraction.

This article reports on a study of mandatory assessment in mathematics for preschool-class children (aged six) and teachers' opportunities to advocate Care for the learning of mathematics. Care for learners and Care for mathematics intersected in the teachers' Grouping of students and their ability to Know the student. These play out as teachers promote availability and justice to students, and as teachers strive to make students' knowledge visible, promote accuracy and gain legitimacy. It is concluded that teachers' opportunities to advocate care for the learning of mathematics needs to be present also in the administration of assessment and to be put in the context of the school's culture and student population, for teachers to be able to promote fair assessment for each student.

Den här presentationen beskriver en pågående studie om lärares arbete matematikklassrummet. Tillsammans med två verksamma lågstadielärare utforskar vi matematiska situationer i matematikundervisningen. Syftet med studien är att bidra med kunskap om vad som villkorar lärares möjligheter att utveckla tillgängliga matematiska situationer. Studiens datamaterial består av videoupptagningar och lärarnas reflektionsloggar. Preliminära resultat kommer att kunna presenteras på konferensen. Målsättningen är, förutom att uppnå studiens syfte, även att utveckla ett verktyg som kan utgöra stöd för lärares utveckling av matematikundervisning.

In Sweden, there has been a development of continuously more and earlier mandatory assessments in schools in the last decades. One of the latest is mandatory assessment in preschool classes from 2019, with one of the purposes to identify students in need of Special Education in Mathematics (SEM students). Preschool class is a compulsory school year for 6-year-olds which precedes compulsory school. The study at hand explores how teachers create knowledge of mathematics and the SEM student during their work with this mandatory assessment. Results show that mandatory assessment might affect how teachers assess knowledge, develop their teaching, and (re)produce knowledge of the six-year-olds as knowers and of knowledge in mathematics. We join the vast amount of research that has shown that mandatory assessment is a potentially sensitive situation for students who are already unprivileged, who face obstacles in their learning or who struggle with mathematics. Awareness of processes of inclusion, exclusion and (in)equity during assessment and in the following education is central to not count (SEM) students out as competent learners with valuable knowledge.

This chapter is an analysis and comparison of students' subject knowledge in mathematics in Grade 3 (Age 9) after an extensive whole-class intervention using the teaching material think, reason, and count (TRC) during the preschool-class year (Age 6). The national test (NT) reveals that the TRC-group (n = 79; 4 classes) performed better than the control group (n = 55; 3 classes), and a significant difference is shown on two of the subtests. This chapter shows that inclusive mathematics education in preschool-class focusing on reasoning about representations of numbers develops good mathematical knowledge and is especially supportive for students at-risk. Correlation of the tested areas of mathematics to TRC reveal that at-risk-students perform better in all areas of TRC.

Early identification of children at risk is essential to prevent and remediate mathematical learning difficulties (Gersten, Clarke, Haymond, & Jordan, 2011), but it requires valid and reliable assessment tools (Aunio, 2019). Measures developed for younger students (<9y) have usually covered areas like number sense, relational skills and counting skills, because of their predictive value for later mathematical skills (Jordan, Glutting, & Ramineni, 2010; Zhang et al., 2020). Studies on mathematical learning difficulties across different grade levels show that number processing (De Smedt & Gilmore, 2011; Skagerlund & Träff, 2016) and basic arithmetic skills (Zhang et al., 2020) seem to differentiate students with MLD from their peers. Consequently, the aim of this study was to investigate the factor structure and grade-level development in the new digital Functional Numeracy Assessment (FUNA) tool for grades 3–9. Students from both Swedish and Finnish speaking schools in Finland participated in the study (N = 4070). FUNA is a digital assessment tool embedded in the VILLE collaborative learning platform developed by the Centre of Learning Analytics of the University of Turku. FUNA consists of seven different subtasks; symbolic magnitude comparison, number-dot matching, number sequence, numerical ordering, single-digit addition, single-digit subtraction, multi-digit addition and subtraction. In the first four tasks, reaction time (RT) and accuracy was used to form an efficiency measure (number of correct items/medianRT of the correctly solved items). Students had 2 minutes to solve as many tasks as possible for single-digit addition and subtraction respectively, and 3 minutes for the multi-digit addition and subtraction tasks. Preliminary results show that there is grade-level development in symbolic magnitude comparison, F (6, 4063) = 213.34, p < .001, 𝜂2𝑝𝜂p2 =.24 (Figure 1). Students in higher grades process numbers more efficiently compared to students in lower grades. In NORSMA we will also present the grade-level development in the other subskills and the factor structure of FUNA.

Förskoleklassen har en speciell position mellan två skilda kulturer där utveckling och lärande har olika betydelse. Avhandlingens syfte är att bidra med kunskap om förskoleklasslärares möjligheter att följa och främja alla elevers utveckling och lärande i matematik. Avhandlingen består av fyra studier. Studie I utgjordes av en intervention med en intervention- och en kontrollgrupp som både följdes genom observationer och utvärderades genom för- och eftertest. Interventionen bestod av undervisningsmaterialet Tänka, resonera och räkna i förskoleklass (TRR), samt tillhörande fortbildning. Två uppföljningsstudier genomfördes. I Studie II undersöktes vilka effekter TRR hade på matematikutvecklingen på kort och lång sikt. Studie III undersökte eventuella skillnader på nationella provet gällande prestation på olika matematikområden. Slutligen i studie IV undersöktes genom en textanalys, hur policydokument har potential att styra införandet av bedömningsstödet Hitta matematiken (HM) i förskoleklass utifrån framskrivandet av eleven, matematiken och bedömning. Genom studie I, II och IV besvaras hur stöd- och undervisningsmaterial formar förskoleklasslärares föreställningar och förmågor relativt matematik, bedömning och elever som riskerar att hamna i matematiksvårigheter. Studie I, II och III besvarar hur TRR påverkar kunskapsutveckling i matematik hos elever som riskerar att hamna i matematiksvårigheter (i-risk-elever) på såväl kort som lång sikt.

Resultaten i avhandlingen visar att förskoleklasslärares förmåga att följa i-risk-elevers kunskapsutveckling i TRR ökade med förskoleklasslärarnas medvetenhet om sina egna ämneskunskaper. Denna ökning ägde rum trots att bedömning var en perifer aspekt i TRR. Resultaten visar vidare att med TRR blev tillfällen för bedömning en naturligt integrerad process i undervisningen. I vilken förskoleklasslärarna i sina bedömningar identifierade innehåll i undervisningen och svårigheter i lärandet av ett matematiskt innehåll snarare än elever. Detta skiljer sig från bedömningsstödet, för vilket resultatet visar att bedömnings-tillfällen framställs som en avgränsad planerad aktivitet där målet är just att kontrollera elevernas kunskaper. Bedömning är i bedömningsstödet, i motsats till i TRR, framställd som ett behov att fånga elevens inre föreställningar, tänkande och brister. 

Resultaten visar också att elevernas kunskapsutveckling var större i TRR-gruppen än i kontrollgruppen, särskilt för i-risk-elever. Detta indikerar att inkluderande matematikundervisning med fokus på resonemang om representationer av tal ger möjligheter att utveckla god räkneförmåga för alla elever och är särskilt stödjande för i-risk-elevers lärande. Avhandlingens resultat visar alltså en positiv effekt för i-risk-elever, och kanske lika viktigt, inga indikationer på att TRR skulle vara sämre för elever som inte är en i-risk-elev.

The Swedish preschool-class is a unique type of schooling between preschool and school, intended as a bridge between two cultures, where development and learning have different meanings. The aim of this thesis is to contribute to the knowledge of preschool-class teachers’ opportunities to track and promote the development and learning of all students in mathematics. The thesis consists of four studies. Study I was a controlled intervention. The intervention consisted of the teaching material Think, reason and count in preschool-class (TRC) and an auxiliary professional development programme. Both the TRC group and the control group were followed by observations and evaluated by pre- and post-tests. Two follow-up studies were carried out. Study II investigated the effect of TRC on mathematics development in the short and long term. Study III looked further into differences in results on the subtests of the national test in grade 3. Study IV examined how the student, mathematics and assessment are fabricated in policy documents regarding the imposition of the assessment material for preschool-class, Find the mathematics (FM). Together, Study I, II and IV answer how support and teaching materials shape preschool-class teachers’ ideas and abilities regarding mathematics, assessment and students at risk of mathematical difficulties. Study I, II and III answer how TRC affects the mathematical development of students who are at risk of mathematics difficulties (at-risk-students) in the short and long term.

The results show that, with TRC, the preschool-class teachers’ ability to track the knowledge development of at-risk-students increased with awareness of the teachers’ own subject knowledge, despite the fact that assessment was a peripheral aspect of TRC. The results show that, with TRC, assessment became a naturally integrated process in teaching, where preschool-class teachers identified difficult content and obstacles in learning mathematical content rather than identifying students. In contrast, the results show that in FM, assessment opportunities are presented as defined, planned activities where the goal is to check students’ knowledge. In FM, assessment is fabricated as a need to capture the student’s inner beliefs, thinking and shortcomings. In addition, the results show that students’ mathematical development was greater in the TRC group than in the control group, especially for at-risk-students. This indicates that inclusive mathematics teaching with a focus on reasoning about representations of numbers provides opportunities to develop good numeracy for all students and is particularly supportive to at-risk-students. The results show a positive effect for at-risk-students, and perhaps just as important: no indication that TRC hamper students who are not at-risk-students.

One of the latest reforms to increase the equity and quality in mathematics education is mandatory national assessment in Preschool-class. It is supposed to govern the action, focus and professional language of the pedagogues. In an earlier study of preparatory work, policy decisions and assessment material, we discovered that there is a risk for schoolification of the Preschool-class, through this reform. A second risk is that students could be locked into "levels" of achievement and a third risk lies in a potential narrowing down of the curriculum (Bagger, Björklund, Vennberg, Accepted). At the same time, earlier research show that early assessment can lead to positive development for students in need of support (Vennberg & Norqvist, 2018). Deriving from our initial study of national assessment in preschool class, we have planned for a follow up project with the aim to contribute to knowledge of and developing the practice about the preschool class teacher’s work with national assessment in mathematics. During the conference we will present and discuss ethical and methodological challenges in this project. More specifically, we want to discuss what is required of practice close approaches in these ethically demanding situations of vulnerability and assessment in mathematics with young students. A key issue is the opportunities for teachers and researchers to manage the information collected during the  national assessment and handling of sensitive data, how they jointly can contribute to the development and management of the knowledge that generated about students' opportunities to learn and to demonstrate their knowledge. And finally, in what way this might contribute to assessment and teaching in mathmatics continues to have its starting points in the unique assignment of the preschool-class.

References

Bagger, A., Björklund, L. & Vennberg, H. (accepted). The politics of early assessment in mathematics education. CERME11 in Utrecht, January 2019.

Vennberg, H. & Norqvist, M. (2018). Counting on – long term effects of an early intervention program. In Bergqvist, E., Österholm, M., G, C & Sumpter, L. (Eds.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, (pp. 355-362). Umeå, Sweden: PME.

One of the latest reforms in Sweden in order to increase equity and quality in education is making national assessment compulsory in preschool-class (age 6). The claimed political volition is all students’ best possible mathematical development. In this paper, we examine the preparatory work, the assignment to the National Agency of Education, and the assessment material itself, searching for what meaning is inscribed regarding the student, mathematics and assessment. The results imply that the politics of the assessment might exaggerate a search for flaws and control instead of promoting all students learning and in addition contributing to the schoolification of preschool-class.