Paper session 5

Equal access for all learners to quality mathematics education


Session chair: Anette Bagger

Paper 1: Factor structure and grade-level development in the new digital Functional Numeracy Assessment (FUNA) tool for grades 3–9

Johan Korhonen, Åbo Akademi University, Vasa, Co-authors: Pekka Räsänen, Mikko-Jussi Laakso, Anu Laine, Airi Hakkarainen, Eija Väisänen, Teemu Rajala, Ulrika Ekstam, & Pirjo Aunio 


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.

Figure 1. Grade-level development in symbolic magnitude comparison. 




Aunio, P. (2019). Early Numeracy Skills Learning and Learning Difficulties—Evidence-based Assessment and Interventions. In D. Geary, D. Berch, & K.M. Koepke (Eds.), Cognitive Foundations for Improving Mathematical Learning (Vol. 5, 1st edition, pp. 195-214).

De Smedt, B., & Gilmore, C. (2011). Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties. Journal of Experimental Child Psychology, 108(2), 278-292.

Gersten, R., Clarke, B., Haymond, K., & Jordan, N. (2011). Screening for mathematics difficulties in K-3 students. 2nd edition. Portsmouth, NH: RMC Research Corporation, Center on Instruction.

Jordan, N. C., Glutting, J., & Ramineni, C. (2010). The importance of number sense to mathematics achievement in first and third grades. Learning and Individual Differences, 20, 82–88.

Skagerlund, K., & Träff, U. (2016). Number processing and heterogeneity of developmental dyscalculia: Subtypes with different cognitive profiles and deficits. Journal of Learning Disabilities, 49(1), 36-50.

Zhang, X., Räsänen, P., Koponen, T., Aunola, K., Lerkkanen, M-K., & Nurmi, J-E. (2020). Early Cognitive Precursors of Children’s Mathematics Learning Disability and Persistent Low Achievement: A 5‐Year Longitudinal Study. Child Development, 91(1), 7-27.


Paper 2: Finding Pupils’ Strengths?

Guðbjörg Pálsdóttir, School of Education, University of Iceland, Reykjavík, Iceland


The structure of teaching is based on the teacher’s knowledge of his/her pupils and the relevant matematics content. Teachers need therefore to acquire relevant information about each pupil’s competencies, as well as suitable teaching approaches, by studying the pupil’s performance in class, and in that way, build a foundation for formative assessment. It is important to analyse pupils’ strengths and help them strengthen their competences.

In spring 2019, a study was conducted as part of the course Student participation in teachers’ research which focused on the ways in which pupils engage with math problems. I supervised and collaborated with 17 elementary school student teachers in gathering data, analyzing the data and concluding from it. Eight groups from different elementary schools worked together in groups of four pupils. Three lessons were video-recorded with focus on pupils as they grappled with three math problems. The videotapes were then analyzed in view of two central themes, i.e. the pupils´ belifs and understanding of mathematics. In analyzing their views, the study looked at interest, communication, various solutions and self efficacy. Mathematical understanding was analyzed in light of their understanding of concepts, symbols and numeracy. The study also analyzed the pupils’ ability to investigate and conclude from their findings.

The findings show that most pupils are prepared to tackle new assignments in a  new learning environment. They have a good understanding of numeracy in general and have competence to explain their approaches and solutions. However, some pupils were not confident about their abilities, they were hesitant to begin and they sought affirmation from the teacher after providing satisfactory argumentation for their soluations. The pupils’ conceptual understanding was strongest with regard to numbers while they had difficulties with various concepts in geometry and algebra. It was interesting to see how the pupils engaged with their math problems and how resilience and attitude toward mathematics played a vital part in that context.

The study shows that recorded lessons can provide teachers with a beneficial basis for formative assessment. This research method can be applied for one’s own teaching with the aim of acquiring information and strengthening one’s understanding of how different teaching methods impact pupils’ learning. Teachers can use the recording to analyze how individual pupils or pupilgroups respond to differing assignments and teaching approaches, which contributes to teachers’ professional development.


Paper 3: Assessment of mathematics in Preschool-class

Anette Bagger, Örebro University, Sweden, Co-author: Helena Vennberg


Assessment in mathematics work as a gatekeeper to future education and contributes to the reproduction of disadvantage for certain groups of students. Equity is an increasing challenge for the Swedish school system (Skolverket, 2019). The government and school agencies have identified early support of students at risk as core in raising equity. Through this there is a recognition of early detection and well-designed teaching for students that fall behind (Swedish Government, 2017). One of the latest reforms in order to raise equity is mandatory assessment in mathematics in preschool-class from 2020. If a student in need of support is detected, support is supposed to be given and special educational competence has to take active part. Furthermore, teachers understanding of and knowledge about the student and support matters for the support given (Scherer, Beswick, DeBlois, Healey & Opitz, 2016).

This article reports from a project about national assessment in mathematics in preschool-class. A previous report from the project showed that there are some concerns to investigate further regarding how the student as a test-taker, the knowledge in mathematics and assessment are fabricated. Policy documents and the assessment material fabricated these in ways that creates tensions and could lead to potential pedagogical dilemmas in the carrying through of the assessment for preschool- class teachers (Bagger, Vennberg & Björklund, 2019). Therefore, the purpose of this article is to further contribute with knowledge regarding prerequisites for students in need of support and the assessment of knowledge in mathematics in preschool-class for these students. The research question that guided this purpose is: How are the students in need of support and their knowledge fabricated by teachers? This is followed by a compare regarding the fabrication that the teachers make, and the fabrication that the documents make. Finally, a discussion of the assessment of knowledge and teaching of students in need of support in mathematics is made.

The selection of preschool-classes consists of a preschool-class with a multifaceted composition regarding language, culture and socioeconomical settings. During the voluntary implementation period 2019, the preschool-class teachers were interviewed in focus groups of 4 teachers while they got acquainted with the assessment material and planned the carrying through. Hence, they talked about implications for students in need of support during and after the assessment. We derive from Popkewitz (2012) theories of fabrication of kinds of people in our analysis. The tool for analyzing is a framework already used on the steering documents and the assessment material itself (Bagger, Vennberg & Boistrup, 2019).  Fabrications are explored through categorizing the motives, values and assumptions (see Popkewitz, 2012) expressed in relation to students in need of support and their knowledge. Preliminary results are displayed in table 1.

Table 1. Examples of how the connections between students and knowledge appeared. 

Fabrication of knowledge Fabrication of student Connection 
Governed Restricted in displaying knowledge School mathematics is fostered and narrowed
Situated Fitting in or not Students’ way interacting plays out. Social aspects
Visible Responsible for displaying knowledge Test format/ content vs student’s prerequisites
Applied ability (Un)able Access to agency and capability
Fair Participating in an equal way Opportunity to participate
Usable User Activity that affords action
Too advanced Counted out Avoidance


Commonalities with the earlier performed policy analysis shows that there is a risk that knowledge indeed is limited to the items tested, and that tensions appear between learning and controlling knowledge. Also, responsibility is placed upon the individual to display knowledge which contributes to a shift from being a learner to becoming a performer.


Bagger, A., Vennberg, H. & Björklund, L, B. (2019). The politics of early assessment in mathematics education. In: Jankvist, U. T., Van den Heuvel-Panhuizen, M., & Veldhuis, M., Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. Konferensbidrag vid 11th Congress of the European Society for Research in Mathematics Education (CERME11), Utrecht, the Netherlands, February 6-10, 2019 (ss. 1831-1838). Utrecht: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.

National Agency for Education. (2019). PISA 2018. 15-åringars kunskaper i läsförståelse, matematik och naturvetenskap. Skolverket: Sverige.

Scherer, P., Beswick, K., DeBlois, L., Healy, L., & Moser Opitz, E. (2016). Assistance of students with mathematical learning difficulties: how can research support practice? – A summary. ZDM:  the international journal on mathematics education 48(5), 249-259.

Swedish Government (2017). Uppdrag att ta fram kartläggningsmaterial och revidera obligatoriska bedömningsstöd och nationella prov i grundskolan, sameskolan och specialskolan. Regeringsbeslut, Regeringen, Utbildningsdepartementet. U2017/02561/S

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

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