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### Session chair: Helena Roos

## Access in mathematics education

**Organiser: Helena Roos, Linnaeus University, helena.roos@lnu.se**

#### PRESENTATION (PDF)

### Summary

The focus of this symposium is on access for every student in inclusive mathematics education. Based on four papers, access is explored and problematized from different perspectives. Firstly, we address the question of who the student is, that should get access in inclusive classrooms. Most often, when discussing special educational needs in mathematics and inclusion, students struggling to get access are emphasized. However, also students that are in excessive access to mathematics need something other than the regular mathematics education. If considering also these students, the notions of accessibility and inclusion become even more challenging as it widens the definition of diversity in inclusive classrooms. Secondly, written representations of accessibility and inclusions in Swedish policy documents are focused on. The selected policy documents are on municipality level, as those documents are closest to the everyday work of mathematics teachers and classrooms. Explicitly, the expressions and interpretations of equity in these policy documents are focused on. Thirdly, we move into an inclusive mathematics classroom at one Swedish primary school. An educational design research study was conducted in this classroom where co-teaching is used in strive towards inclusion. Examples are presented of designed inclusive mathematics lessons realized by the means of teachers’ mathematical and relational competencies. Finally, we focus on assessment in relation to access. Access in situations of assessment implies not only that students have access to an immediate test situation but that their access is sustainable also in future situations of assessment. Thus, challenges and opportunities that teachers face in moments of sustainable assessment for *all *students are focused on.

## Paper 1: What is special about special educational needs in mathematics?

**Helena Roos, Linnaeus University, Sweden**

### Abstract

Often when discussing special educational needs in mathematics (SEM) and inclusion, students struggling to get access to the mathematics are in focus (Roos, 2019a). Here different labels are used to describe the struggling students, such as for example low achievers (Scherer, 2020), students with mathematical learning difficulties (Scherer, Beswick, DeBlois, Healey & Opitz, 2016) and students with mathematics learning disabilities (Lewis & Fisher, 2016). Research has also shown a difference in the use of lables depending on the epistemologcal research field where it is used (Bagger & Roos, 2015). Here, the field of mathematics education tend to be more focused on relational aspects of SEM than the field of special education. Even though there is a difference in the use of labels, none of the above mentioned epistemological fields tend to focus on students that are in excessive access to mathematics, but need something other than the regular mathematics education to *get access to learning* when discussing inclusion and SEM. Though, there are a few scholars who discuss inclusion in relation to these students, sometimes referred to as “gifted students”, (e.g. Leikin, 2011), or mathematically talented students (e.g. Shayshon, Gal, Tesler & Ko, 2014). Because of the difference in the use of labels there is a risk of having multiple different research directions using different vocabulary, though it is the “same” issues that needs to be taken into consideration in the mathematics education in relation to inclusion (Roos, 2019b). This makes it hard for both research and practice to come to solid conclusions to provide access to learning in mathematics for all.

Roos (2019b) argues for including both students in struggle to gain access, and students in access to mathematics education when discussing SEM and inclusion in order to focus access to learning. This way to view inclusion in relation to SEM helps to provide an education so that everybody has the possibility to “attain their maximum potential” (Oktaç, Fuentes & Rodriguez Andrade, 2011, p. 362). Here, the notion of inclusion becomes both challenging and interesting, as it requires work in different directions, depending on the situation and the students, to cover the diversity of SEM in order to obtain access to learning for every student in an inclusive classroom.

To conclude, what is special about SEM is that it focuses the special need in the learning situation to provide *access to learning for every student*, highlighting the need of taking both students in struggle to gain access, and students in access to mathematics education into consideration when planning and implementing inclusive mathematics teaching. Hence, SEM is a notion situated in space and time.

### References

Bagger, A., & Roos, H. (2015). How Research Conceptualises the Student in Need of Special Education in Mathematics. In O. Helenius, A. Engström, T. Meaney, P. Nilsson, E. Norén, J. Sayers, M. Österholm. *Development of Mathematics Teaching: Design, Scale, Effects. Proceeding of MADIF 9. The Ninth Swedish Mathematics Education Research Seminar* Linköping: Svensk förening för MatematikDidaktisk Forskning – SMDF, 2015, p. 27-36.

Leikin, R. (2011). The education of mathematically gifted students: Some complexities and questions. The Mathematics Enthusiast: 8 (1) article 9, p. 167-188.

Lewis, K. E., & Fisher, M. B. (2016). Taking stock of 40 years of research on mathematical learning disability: Methodological issues and future directions. Journal for Research in Mathematics Education, 47(4), 338–371.

Oktaç, A., Fuentes, S. R., & Rodriguez Andrade, M. A. (2011). Equity issues concerning gifted children in mathematics: A perspective from Mexico. In B. Atweh, M. Graven, W. Secada, & P. Valero (Eds.). Mapping Equity and Quality in Mathematics Education. (pp. 351–364). Dordrecht: Springer.

Roos (2019a). Inclusion in mathematics education: an ideology, a way of teaching, or both? *Educational Studies in Mathematics, 100*(1), 25–41.

Roos (2019b). The meaning of inclusion in student talk: Inclusion as a topic when students talk about learning and teaching in mathematics. Ph. D. Thesis. Linnaeus University: Växjö.

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.

Scherer P. (2020). Low Achievers in Mathematics—Ideas from the Netherlands for Developing a Competence-Oriented View. In: van den Heuvel-Panhuizen M. (eds) International Reflections on the Netherlands Didactics of Mathematics. ICME-13 Monographs. Springer, Cham. pp. 113.132.

Shayshon, B., Gal,H., Tesler, B., & Ko, E. (2014). Educational Studies in

Mathematics Education 87, 409–438.

## Paper 2: Meaning and operationalization of equity in municipality mathematical action plans.

**Åsa Maria Johansson, Cecilia Lindegren-Österholm, Helena Roos, Linnaeus University, Sweden**

### Abstract

In the Swedish school, as well as internationally, the notion of equity has in recent decades gained a prominent position (Atweh 2011; Gutierrez 2009; Pais & Valero 2011; Swedish National Agency for Education 2006, 2012). Simultaneously, the governing of the Swedish school has moved from the state to the municipalities and the focus has moved towards goals and results (Swedish School Inspectorate 2014). One way for the municipalities to govern the results is by written action plans, also in action plans in mathematics (Swedish Agency for Public Management 2018). In this study, the concept of action plan refers to a plan developed by the municipality for mathematics development in compulsory school.

As Critical Discourse Analysis (CDA) states, by investigating discourses and how they operate in a social practice, and by realizing how power-relations shapes assumptions and actions of equity in a totality of politics, ideologies, pedagogy and economic and political strategies, changes can be achieved (Fairclough 2010). But how is the notion of equity interpreted and expressed in action plans in relation to mathematics learning? By the means of CDA, meanings and operationalizations of equity take place in a social reality, where a chain of power-relations based on politically stated goals and requirements, determines how action plans in mathematics is realized at a school level. In this way, municipality mathematical action plans, may play a crucial role of influencing the meaning of equity as concept as well as the way it operationalizes.

The aim of this study is to investigate equity in action plans in mathematics focusing meaning and operationalization. The research questions of the study are:

- What are the different meanings of equity described in the mathematical action plans?
- How is the operationalization of the concept of equity expressed in mathematics action plans?

By using CDA, text from five action plans in mathematics was analyzed. The focus of the analysis was partly to investigate texts and identifying discourse(s) and actions associated with discourse(s), and partly to identify social practice(s) in which the discourses and actions are found.

The result ended up in the construction of three discourses of equity, where equity is described and operationalized; *Equity as Achievement, Equity as Access *and *Equity as Power and Identity.*

The most prominent discourse is the discourse of *Equity as Achievement*. Here, equity is reached by results at tests in relation to quality. This discourse is visible in a social context of systematic quality work in schools as well as in a steering process from governmental level to municipality and school level.

A second discourse identified is *Equity as Access*. This discourse is defined by social, physical and pedagogical learning environment and access to different actions of support. By the development of the teaching, by collegial cooperation and special support for students in need, equity is attained by increasing access.

A third discourse identified is *Equity as Power *and* Identity*. This discourse implies that teaching should give opportunities for a good future in terms of work and participation in society. This at the same time as the student develop an identity, an interest, motivation and self-efficiency.

Thus, in the chain of power-relations of schools, the imprints of equity in the organisation depend ultimately on the competence and the knowledge possessed by officials, how that knowledge has been internalized and related to laws and regulations. Hence, how this knowledge is processed and translated into the action plans affects the distribution of equity in the municipality. The result of this study highlights opportunities as well as responsibilities in the use of language to display equity in action plans.

### References

Atweh, B. (2011) (red.). Quality and Equity in Mathematics Education as Ethical Issues. I Atweh, B., Graven, Me., Secada, W. & Valero, P. *Mapping Equity and Quality in Mathematics Education*. Dordrecht: Springer Verlag, ss. 63–75.

Fairclough, N. (2010). *Critical discourse analysis: the critical study of language*. Harlow: Longman.

Gutierrez, R. (2009). Embracing the inherent tensions in teaching mathematics from an equity stance. *Democracy and Education*, 18(3), ss. 9–16.

Pais A. & Valero P. (2011). Beyond Disavowing the Politics of Equity and Quality in Mathematics Education. I Atweh B., Graven M., Secada W. & Valero P. (red). *Mapping Equity and Quality in Mathematics Education, *ss. 35–48. Springer, Dordrecht.

Swedish Agency for Public Management (2018). *Strategier och handlingsplaner: ett sätt för regeringen att styra? *Stockholm: Statskontoret.

Swedish National Agency for Education (2006).* Vad händer med likvärdigheten i svensk skola? En kvantitativ analys av variation och likvärdighet över tid*. Rapport 275. Stockholm: Fritzes.

Swedish National Agency for Education (2012). *Likvärdig utbildning i svensk grundskola? En kvantitativ analys av likvärdigheten över tid. *Skolverkets rapport 374. Stockholm: Skolverket.

Swedish School Inspectorate (2014). *Från huvudmannen till klassrummet *Skolinspektionen.

## Paper 3: Access to displaying knowledge during assessment – a matter of sustainability

**Anette Bagger, Örebro University, Anette.Bagger@oru.se**

### Abstract

Success to access tests, predestinates access to success in various ways for students. Success to learn and to progress in further education, among other. A challenge for teachers and researchers in the field is to afford all students access to display knowledge. This connects to sustainability in the assessment situation for all, and not only those who already have access.

**Research topic**

Education of today is highly driven by assessment in different forms. The emphasis on and understanding of knowledge as something that needs to be visual or seen in order to count, is core in this (Bagger, Roos & Engvall, 2020). For some students, the displaying of knowledge is a struggle. An urgent task for the teachers to manage is therefore to support students in making their knowledge visual and thereby possible to assess. Access is through this, a matter of sustainability in the context of assessment. This proposal aims at discussing what sustainable assessment is, from a special educational perspective. With sustainable assessment I draw on Bound (2000) and refer to assessment that provides the student the means to both access the test situation and visualize their knowledge, but also stretches beyond the immediate assessment situation. This could for example mean that the situation helps the student to learn test taking strategies that are sustainable now and in future situations of assessment, and that the student is comfortable with being assessed.

**Theoretical framework**

The data used to exemplify and explore this topic, consists of moments of assessment, seen from the experience of teachers who work with students with disabilities. The theory used is the philosophical work of Bornemark (2018a, 2018b) and her writing of intellectus and ratio. I will also display how concepts regarding the assessment of knowledge and learning for students with disabilities are colonialized and how concept imperalism has taken place. Also, the concept of “förpappring” of caring practices will be used. For students with disabilities participation and the caring part of education is important. Förpappring means that activites and thoughts are put down on documents and the ratio practice is over used, whilst other important aspects that is accessible in the intellectus practice, remains cloaked.

**Material and expected findings**

Results from this exploration displays challenges and opportunities that the teachers face in moments of assessment. The role of adaptions in order for both the teacher and student to have access to the students knowledge and the assessment procedure is highlighted. Finally, I conclude how social epistemology works with the practice of intellectus and ratio and how these affects the possibilities to promote sustainable assessment.

**Relevance to the Nordic educational research**

The neoliberal logic in schooling and the testing regime of childhood is prevalent in the Nordic school systems, although at different degrees. I claim that this affects students with disabilities, especially if their particular needs of adaptions are not fulfilled. In addition, there is a risk that their knowledge is not considered as important in the comparative discourse with high emphasis on test results as valuable when quality of countries and schools is assessed.

### References

Bagger, A., Roos, H. & Engvall, M. Directions of intentionalities in special needs education in

mathematics. *Educational Studies in Mathematics* (2020). https://doi.org/10.1007/s10649-020-09945-4

Bornemark, J. (2018a). Det omätbaras renässans: En uppgörelse med pedanternas världsherravälde (Första upplagan. ed.).

Bornemark, J. (2018b). The Limits of Ratio: An Analysis of NPM in Sweden Using Nicholas of Cusa’s Understanding of Reason. In Btihaj Ajana (Ed.) Metric Culture, (pp.235 – 253).

## Paper 4: Sustained students’ participation in mathematics by means of teachers’ mathematical knowledge and relational abilities

**Malin Gardesten, Linnaeus University, malin.gardesten@lnu.se**

### Abstract

Recent studies in the intersection between mathematics education and special education show that students’ access to mathematics is an area that needs further exploration. Previous research has shown that the teacher needs to know the students for them to get access to mathematics. In turn, this requires abilities to identify a student’s different prerequisites in mathematics as well as seeing the student as a person (Roos, 2019). Thus, the required abilities are founded in teachers’ mathematical knowledge when teaching mathematics and teachers’ relational abilities. This study aims to explore students’ participation in mathematics the by means of teachers’ mathematical knowledge and relational abilities.

This study has a social perspective on learning, which implies learning as social participation (Wenger, 1998). The concepts communities of practice, negotiation of meaning, participation and reification from Wenger (1998) are used to understand students’ participation in mathematics. The community of the mathematical classroom where the participants negotiate the meaning of the mathematical content is of interest. Participation refers to students taking part, engaging in mathematics, and contributing to the practice of mathematics in the community of the mathematical classroom. Reification implies processes or products congealing mathematical concepts and affects what is experienced. Learning as participation occurs through engaging in actions and interactions with students’ peers and teachers concerning mathematics

An intervention was carried out where the researcher, together with the two participating teachers, identified the existing and the desired mathematics teaching in one grade 5. The setting was a co-taught organized mathematics classroom that strived towards inclusion. Together, the general mathematics teacher and the special education teacher in mathematics identified a need to develop the whole-class discussions in mathematics. The existing situation implied that the same 4-5 students always were active during mathematics lessons, sharing their solutions and reasoning in whole group discussions, while the other students remained quiet. Goals were set regarding developing every students’ possibilities to discuss and reason in small groups as well as in whole class.

The intervention was conducted through an iterative cyclic process during two semesters where a series of teaching activities were designed, implemented and revised. The teaching activities were grounded in theories on whole-class discussions (Stein et al., 2008), exploratory talk in small groups (Mercer & Sams, 2006), and the framework of Universal Design for Learning that composes principles for making learning accessible for every student (Rose & Meyer, 2002).

Through the intervention, the interactions between the teachers and the students were explored to investigate the coordination of teachers’ mathematical knowledge and relational abilities. Meanwhile, students’ participation in mathematics during the whole-class discussion as well in small groups were evaluated. The intervention was documented in several ways; the planning and teaching activities were documented by video-recorded observations, interviews with the teachers and the students were audio-recorded, and the students’ written work was collected.

For analysis of the video recorded observations, a methodological framework built on two theoretical approaches was used. A deductive content analysis was carried out using theory network strategy by coordination (Bikner-Ahsbahs & Prediger, 2010). Firstly, the analysis focused on teachers’ mathematical knowledge using the categories of foundation, transformation, connection and contingency from The Knowledge Quartet (Rowland, 2013; Rowland et al., 2005). Then, on the same data material, the second layer of deductive analysis was conducted using the categories of tact and inclusive stance from the Pedagogical Relational Teachership (Ljungblad, 2019; Ljungblad, 2021). Instances of the data material based on video-recorded teaching activities where the theoretical approaches overlapped were inductively analyzed regarding students’ access to learning mathematics through participation in mathematics, framed by social practice theory (Wenger, 1998).

Tentative results indicate that the reifications, such as the organization of the mathematics lessons, is reflected in teacher-student interactions, which may sustain student participation in mathematics. Expected contributions give a deeper understanding of students’ (un)sustained participation in mathematics.

### References

Bikner-Ahsbahs, A., & Prediger, S. (2010). *Networking of theories—an approach for exploiting the diversity of theoretical approaches*. Springer.

Ljungblad, A.-L. (2019). Pedagogical Relational Teachership (PeRT) – a multi-relational perspective. *International Journal of Inclusive Education*,* 23*, 1-17. https://doi.org/10.1080/13603116.2019.1581280

Ljungblad, A. (2021). A Taxonomy for Developing the Relational Dimension of the Teaching Profession. In (Manuscript submitted for publication ed.): University of Gothenburg, Department of education and special education.

Mercer, N., & Sams, C. (2006). Teaching children how to use language to solve maths problems. *Language and Education*,* 20*(6), 507-528.

Roos, H. (2019). *The meaning (s) of inclusion in mathematics in student talk: Inclusion as a topic when students talk about learning and teaching in mathematics* [Doctoral dissertation, Linnaeus University Press].

Rose, D. H., & Meyer, A. (2002). *Teaching every student in the digital age: Universal design for learning*. Association for Supervision and Curriculum Development.

Rowland, T. (2013). The Knowledge Quartet: the genesis and application of a framework for analysing mathematics teaching and deepening teachers’ mathematics knowledge. *Sisyphus-Journal of Education*,* 1*(3), 15-43.

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. *Journal of Mathematics Teacher Education*,* 8*(3), 255-281.

Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. *Mathematical Thinking and Learning*,* 10*(4), 313-340.

Wenger, E. (1998). *Communities of practice : learning, meaning, and identity*. Cambridge University Press.