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### Session chair: Bjarnheiður Kristinsdóttir

## Paper 1: Counting in year one as predictor for achievement in year four in Danish students

**Pernille Bødtker Sunde, VIA University College, Aarhus. ****Co-authors: Pernille Ladegaard Pedersen and Peter Sunde**

### Abstract

Knowledge of early predictors of later achievement is important to detect students at risk of developing mathematical difficulties in order to initiate early intervention (Gersten, Jordan, & Flojo, 2005). Early differences in strategy use in arithmetic has been linked to later mathematic performance in general and arithmetic ability specifically (e.g. Gersten et al., 2005; Vanbinst, Ghesquière, & De Smedt, 2014). However, knowledge of the relationship between early strategy use and later achievement within specific mathematical knowledge areas are scarce.

In 63 Danish students, we investigated the extent to which assessment results of general mathematical achievement and strategy use in single-digit addition in early year one predicted achievement in more advanced mathematics (Number and arithmetic (AR), proficiency with fractions (FR), equations (EQ), word problems (WP)) in year four.

Frequency of specific strategy use year one was calculated from student’s responses to 36 single-digit addition problems. Achievement scores in year one was number of correct answers on a standardized mathematics test with items on number (18), arithmetic in context (4), symbolic addition (20) and symbolic subtraction (20). Achievement scores in year four was number of correct answers in tests of AR, FR, EQ and WP.

In total, student information attainable in year one (strategy use, mathematics achievement and cognitive functioning) explained significant amounts of the variance in the four categories of year-four achievement (WP: 28%, AR: 37%, FR: 21%, EQ: 47%) of which strategy use predictors alone explained 20% (WP), 8% (AR), 12% (FR) and 33% (EQ). For WP and EQ, strategy use measures (i.e. the proportion of problems solved with ‘count all’) explained significant additional variation in year four achievement that could not be accounted for by general mathematical achievement scores year one (WP 9 %, EQ 21 %). Hence, for WP and EQ, information on how a student do single-digit addition (strategy use) in year one was a better predictor than how well the student do arithmetic (number of correct answers) in year one.

From these results, we hypothesie that systematically obtained measures of how young students solve simple single-digit addition problems might provide useful information about their foundational number knowledge that in turn may predict risks of low achievement later in school.

### References

Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38(4), 293–304. https://doi.org/10.1177/00222194050380040301

Vanbinst, K., Ghesquière, P., & De Smedt, B. (2014). Arithmetic strategy development and its domain-specific and domain-general cognitive correlates: A longitudinal study in children with persistent mathematical learning difficulties. Research in Developmental Disabilities, 35, 3001–3013. https://doi.org/10.1016/j.ridd.2014.06.023

## Paper 2: The use of formative assessment in special education to enhance mathematical equity, access, and empowerment

**Catarina Andersson, Senior lecturer and researcher at the Department of Science and Mathematics Education, Umeå University, catarina.andersson@umu.se **

#### PRESENTATION (PDF)

### Abstract

Taking the idea of inclusion seriously implies to recognize differences in students’ learning processes as contributing to the mathematics classroom practice rather than treating some learning strategies and trajectories as abnormal and problematic (see Scherer, et al., 2016). Long ago, Fuchs and Fuchs (1986) argued for using formative assessment instead of categorizing learners. Because formative assessment is based on the assumption that students’ needs are diverse and by definition requires that information about students’ learning is used to adjust teaching and learning processes (e.g. Black & Wiliam, 2009), this classroom practice allows the use of students’ success as well as misunderstandings to bring learning forward. In addition, making the students involved in assessment and learning processes may enhance access to the learning community and the same mathematics when working as resources for each other.

Thus, formative assessment has the potential to approach diverse learning needs as normal and as a resource. The forthcoming study aims at finding ways of using formative assessment to provide all students qualitative mathematics education without marginalizing some of them.

A previous study showed that a group of special education teachers in mathematics (SETMs) who had learnt about formative assessment in their education later found this knowledge beneficial and useful (Andersson, 2020). However, also challenges appeared regarding how to implement formative assessment in inclusive terms (see also Watkins, 2007) and how to design cooperation with regular mathematics teachers. Using collaborative research methodology, the researcher and a group of SETMs will systematically design, use, evaluate and redesign tools and principles for how to use formative assessment to enhance mathematical equity, access, and empowerment.

The author wishes to discuss methodological issues, more specifically how to perform high-quality research when the research process is under constant development.

### References

Andersson, C. (2020). *Formative Assessment – from the view of Special Education Teachers in Mathematics. *Manuscript in preparation

Black, P., & Wiliam, D. (2009). Developing the Theory of Formative Assessment. *Educational Assessment, Evaluation and Accountability, 21*(1), 5–31.

Fuchs, L. S., & Fuchs, D. (1986). Effects of Systematic Formative Evaluation: A Meta-Analysis. *Exceptional Children, 53*(3), 199–208.

Scherer, P, Beswick, K., DeBlois, L., Healy, L., & Moser Opitz, E. (2016). Assistance of students with mathematical learning difficulties: how can research support practice? *ZDM, Mathematics education, 48*(5), 633–649

Watkins, A. (Ed.) (2007). *Assessment in Inclusive Settings: Key Issues for Policy and Practice.* Odense, Denmark: European Agency for Development in Special Needs Education.

## Paper 3: Creative mathematics for the diverse learning group

**Ósk Dagsdóttir, University of Iceland, oskdags@hi.is**

### Absract

Creativity is a human trait related to our evolution and crucial to our survival. Yet there has historically been little focus on it in research. However, there has been a growing interest in the phenomena in recent decades. Many school experts have started to focus on creativity as a pivotal part of all learning. In Iceland this is seen in the National Curriculum and the recent educational policies and creativity should be fostered in all education and across subjects.

Creative mathematics is a concept that raises a lot of eyebrows. The work of mathematicians is highly creative dealing with uncertainty, problem solving, divergent thinking and searching for answers through trial and error. This creativity is however rarely seen in school mathematics where students are expected to work alone following given algorithms and are rewarded for being quiet and quick. Although many teachers want to emphasize creativity in their mathematics classroom, they often lack the means and ability to put that into practice. Teachers need to have had creative experiences themselves in order to support their students with it.

This educational action research takes place within an Icelandic compulsory school. All the teachers at the participating school take part in a two-year PD (professional development) program on creative mathematics. The program includes seminars on creative mathematics learning, discussion and opportunities to work on creative projects such as problem solving, working like real mathematicians, experimenting with patterns, hands-on and interdisciplinary learning. The teachers then work towards bringing these ideas into their classrooms.

I have been privileged to be the facilitator of this PD program. I host the seminars and work with the teachers to foster creativity inside and outside their mathematics classroom. Collaborative data was gathered from seminars and work with teachers. The data included interviews, surveys, videos, field notes, and was analyzed using qualitative methods. Case studies were chosen to shed light on how the teachers experience the effect of the program on their views, pedagogy, teaching practices and student learning.

This presentation focuses on the case of Anna. Anna teaches younger pupils and although she did not take any mathematics education as part of her teacher training, she claims she appreciates mathematics and thinks that it is inherently fun to solve problems and think about the mathematical reality. She says that she has previously based her teaching mostly on her intuition and the materials provided by the school. When faced with a class of 3^{rd} graders that scored low on a standardized evaluation test for the four basic mathematical operations, I embarked on a journey with her and her co-teacher to do a continuous two-month workshop working with creative mathematics. We met frequently to plan lessons and had sessions with the students of two to three hours twice a week.

Anna explains that what she got out of the workshop was finally having the time and the group size to embark on a conversation about mathematics, to include hands-on learning and to work with play and games. She states that by discussing the meaning of the concepts they are using in mathematics and getting them to express their thoughts about numbers and operations, the students develop a deeper understanding in mathematics and discuss the meaning behind mathematical concepts. She feels that the hands-on learning connects concepts to hands and senses and that games brought joy and life into the lessons.

She took a post test on the students with the basic operations and found that their understanding and ability improved greatly and although she cannot exclude that they would have done so without the workshops she feels that they took a jump with this work and that it prepared them for further learning. This can be reflected in theory of creative mathematics and shows how the diverse learning group can benefit from a teacher that focuses on creative learning and how important it is to offer longer lessons with fewer students and hands-on projects.

## Paper 4: Creating students’ access to the classroom discussion with silent video tasks

**Bjarnheiður Kristinsdóttir University of Iceland bjarnhek@hi.is, Co-authors: Freyja Hreinsdóttir and Zsolt Lavicza **

### Abstract

Silent videos are short, animated video clips without sound or text, that focus on one mathematical concept each. In a silent video task, students watch one silent video on a previously studied topic selected by the teacher and in turn work in pairs to prepare and record their voice-over for the video. After handing in, the whole group listens to and discusses students’ responses to the silent video task. These responses can include different narratives, explanations, and descriptions. The teacher leads the group discussion with the goal to clear misunderstandings and approach a common understanding of the mathematical concepts shown in the video. In this paper presentation I will report on findings from a design-based research study (e.g. Bakker, 2018; Cobb, 2000) where silent video tasks were designed and developed in collaboration with upper secondary school teachers and their students in Iceland. The aim was to develop the silent video tasks’ instructional sequence and to clarify in what sense silent video tasks might be useful in the mathematics classroom. In the first phase of the study, four teachers in four schools assigned one silent video task to their 17-year-old students. Results from the first phase indicated that silent video tasks might be useful as a part of formative assessment (Kristinsdóttir, Hreinsdóttir, & Lavicza, 2020). Thus, in the second phase of the study, I (the first author) worked with three teachers in two schools that used formative assessment. Two of the teachers assigned one silent video task to their 16-year-old students and one teacher assigned three different silent video tasks, distributed over one term, to his 16-year-old-students. Existing classroom norms (Yackel & Cobb, 1996) in all three classrooms were such that students normally only expressed themselves verbally about mathematics when asking for support regarding tasks from their textbook. The research questions were: How can silent video tasks be used for formative assessment and how do they affect teaching and learning in the mathematics classroom? Data consisted mainly of interviews with teachers and classroom observations. I used questions along the five dimensions from the Teaching for Robust Understanding framework (Schoenfeld, 2018) as a theoretical framework during data analysis. The task created a new “talking about mathematics” situation. Teachers noticed that students who normally were inactive in groupwork, among them two students with autism spectrum disorders, took part in writing a script for a voice-over and/or recording a voice-over. This indicated that the new situation supported students in getting access to and taking part in the classroom discussion. Results also indicate that, during the classroom discussion, teachers get an opportunity to recognise and create connections between their students’ and their discipline’s discourses connected to the mathematics shown in the video.