An exploration of Scottish teachers’ perceptions of equitable teaching practices in mathematics

Abstract

Equitable teaching practices encompass evidence-based strategies utilised by teachers to support the learning of all pupils, while recognising each pupil’s unique background. This study delves into the perspectives of Scottish teachers regarding equity in mathematics, as evidenced by their self-reported practices. By analysing semi-structured interviews with 29 teachers from various school levels (early years, primary, and secondary) through Variation Theory, we were able to uncover common practices, such as ability grouping versus mixed-ability grouping, as well as practices specific to each level. We also identified overlaps between early-years/primary and primary/secondary, highlighting the importance of transitioning from one school level to another. It is imperative for teacher education programmes to address these transitions more explicitly in order to ensure greater continuity and consistency in equitable teaching practices.

Keywords:

• Equitable teaching practices
• Variation Theory
• early years
• primary
• secondary
• Scotland

REVIEWING EDITOR:

• Serafina Pastore, Education, Psychology, Communication, Università degli Studi di Bari Aldo Moro, Bari, Italy

Subjects:

• Sociology of Education
• Teachers & Teacher Education
• Classroom Practice
• Curriculum Studies

Introduction

The sociopolitical nature of school mathematics has been discussed by scholars for some time now (for example, Appelbaum, 1995; Popkewitz, 1988; Skovsmose, 1994). Nevertheless, following the relatively few examples of previous decades, the field of mathematics education took a more explicit sociopolitical turn around 2000 (Gutiérrez, 2013), by focusing on questions such as: who decides what is included in school curricula (Appelbaum & Davila, 2007); who is excluded from school mathematics (Xenofontos, 2015); how concepts like equity and social justice influence discussions of curricula and the teaching/learning of the subject (Xenofontos et al., 2021). This sociopolitical turn is also reflected in several policy initiatives around the world. For example, in the United States we see a shift in focus by the National Council of Teachers of Mathematics from conceptual understanding, as expressed in from the Curriculum and Evaluation Standards (NCTM, 1989), toward equity and worthwhile learning opportunities for all learners in the Principles documents of the current century (NCTM, 2000, 2014). Similar observations can be made for the constituent countries of the United Kingdom (Moscardini, 2014). For example, Scotland, from where this paper draws, currently places a similar emphasis on equity in policy documents such as the Delivering Excellence and Equity in Scottish Education: A Delivery Plan for Scotland (Scottish Government, 2016), the Scottish Attainment Challenge (Education Scotland, 2018), and the revised Professional Standards 2021 for Scotland’s Teachers (General Teaching Council for Scotland, 2021).

This paper is built on several fundamental values and views its co-authors share. First, we strongly believe that policy initiatives alone are not capable of addressing the needs of all children (Aguirre et al., 2017; Celedón-Pattichis et al., 2018), especially when policies pathologise marginalised learners’ identities (Popkewitz, 2004; Walshaw, 2010). Second, we recognise the central role of teachers in promoting equity in and through school mathematics at personal, interpersonal, institutional, and cultural levels (Willey & Drake, 2013). Third, we are aware that perceptions of equity and equitable teaching practices vary significantly across educational systems (Graven, 2014; Lee et al., 2018; Xenofontos, 2019). This is consistent with an ecological view of agency, which positions teachers as actors always acting ‘by means of their environment rather than simply in their environment’ (Biesta & Tedder, 2007, p. 137). Finally, we identify a need for a more general study of teachers regardless of specific common characteristics. Many individual studies on equitable mathematics teaching practices focus exclusively on early-years (Buchheister et al., 2019; Lee & Ginsburg, 2009), or on primary (Felton-Koestler, 2017; Wager, 2014) or secondary teachers considered to be ‘exceptionally equitable,’ or on teachers having a specific interest in learning more about and incorporating related practices in their teaching (Boaler & Staples, 2008; van Es et al., 2017). We draw on data from a wider project in Scotland, focusing on mathematics/numeracy teachers at the three school levels who were not necessarily identified as exceptionally equitable, and explore the following research question:

What are the similarities and differences of teaching practices at early-year, primary, and secondary education, perceived as equitable by the teachers themselves?

We first turn our attention to the relevant mathematics education literature, in order to frame and locate our study. Next, we present our methodology, followed by our main findings. In closing, we discuss implications of our work, as well as avenues for future studies.

Theoretical considerations

Understanding equity in mathematics education

Defining equity is not a straightforward endeavour, not least because it is highly contested term, with various meanings (Secada et al., 1995; Wagner et al., 2012). A common misconception is to confuse equality for equity (Taylor et al., 2019; Walshaw, 2010; Yolcu, 2019). While the two are seemingly similar, due to their underlying assumptions about fairness, we treat them as different since equality is concerned with giving all students the same amount of a ‘good’ (e.g. resources, support, etc.), while equity assumes and acknowledges that different individuals have different learning needs; therefore, differentiated support is provided so that each student can reach a goal (Levitan, 2015). Equity might, in the latter sense, be more closely associated with equal outcomes. In some accounts, equity is used interchangeably with social justice (Esmonde & Caswell, 2010; Planas & Civil, 2009), while in others the two terms are distinguished; nonetheless, their relationships are explicitly presented and discussed (Gregson, 2013; Leonard, 2018; Xenofontos et al., 2021). Gutiérrez (2002, p. 153) aspires to a notion of equity that is achieved when we are ‘unable to predict student patterns (e.g. achievement, participation, the ability to critically analyse data or society) based solely on characteristics such as race, class, ethnicity, sex, beliefs and creeds, and proficiency in the dominant language’. In subsequent work, Gutiérrez (2008) introduces four dimensions of equity: access (e.g. resources available to engage with quality mathematics), achievement (e.g. standardised test scores, participation rates), identity (maintaining cultural, linguistic, familial connections), and power (agency to affect change in school or society). Access and achievement are part of what she calls the dominant axis, while identity and power are part of the critical axis. Boaler (2006, 2008), in turn, proposes another dimension, relational equity, which moves the focus away from school outcomes and achievement in tests, by drawing attention to the ways students learn to treat peers and the respect they learn to have for people from different circumstances to their own. Gutiérrez’s first two dimensions might fall under Lee et al.’s (2018) equity as a standard of excellence (measurement of outcomes), while the latter two, along with Boaler’s proposition, relate to equity as a moral imperative (warranted by the ethical and moral standards of a democratic society, where every individual has the right to learn). In our discussion of what Scottish teachers report about their perceptions of equity and its pursuit, we recognise both the utopian characteristics of Gutiérrez’s aspirations and, as we describe below, an unsurprising lack of theoretical nuances in the teachers’ spontaneous, implicit definitions of equity. Our study was designed to elicit the everyday sensibilities that the teachers self-describe more than to probe analytic interpretations of outcomes versus opportunities and resources.

Equitable teaching practices in mathematics

We define equitable teaching practices as evidence-based classroom actions employed by teachers to support all students’ learning, as a response to an acknowledgement of each student’s background (Berry & Wickett, 2009). We emphasise the evidence-based character of such practices, as teachers may hold beliefs about practices they consider equitable, which, in reality, may be contributing to the maintenance or reproduction of inequalities and social exclusion (Swanson et al., 2017; Xenofontos, 2016). To provide concrete examples of equitable practices, some authors use Gutiérrez’s four equity dimensions and break each down to clear sample activities (e.g. Buchheister et al., 2019; Myers et al., 2015). Others illustrate emergent frameworks, each of which have a different focus or point of origin. Some related examples are mentioned here: Banes et al. (2018) on the importance of discussion, Moschkovich (2013) on the role of language use in the mathematics classroom, and its importance for other-language learners; Jamar and Pitts (2005) on the importance of teachers exhibiting high expectations, especially for students from ethnically marginalised backgrounds; Hand (2012) on teachers’ dispositions towards equity play an important role in the enactment of equitable mathematics instruction; Boaler (2006, 2008, also Boaler & Staples, 2008) on the importance of mixed-ability grouping; and van Es et al. (2017) on teachers’ noticing for equity.

Various frameworks with different foci share similarities while conceiving equitable practices in divergent ways. Nevertheless, the studies reported above are based on primary empirical data. For this reason, we consider the work of Bartell et al. (2017), based on a metasynthesis of several empirical studies, as more comprehensive. Bartell and her colleagues report nine specific equitable practices for teaching mathematics: (1) draw on students’ funds of knowledge; (2) establish classroom norms for participation; (3) position students as capable; (4) monitor how students position each other; (5) attend explicitly to race and culture; (6) recognise multiple forms of discourse and language as a resource; (7) press for academic success; (8) attend to students’ mathematical thinking; and (9) support development of a sociopolitical disposition. (For more details about each practice, readers should refer to the original article of Bartell et al. (2017), in which each practice is linked to a set of other research articles, as evidence of its effectiveness.) These practices echo the work of Ladson-Billings (1995), who highlighted six habits of highly effective mathematics teachers for urban/African-American students: (1) students are helped to become intellectual leaders in the classroom; (2) students are apprenticed in a learning community rather than taught in an isolated and unrelated way; (3) students’ real-life experiences are legitimised as they become part of the official curriculum; (4) teachers and students participate in a broad conception of literacy that incorporates both literature and oratory; (5) teacher and students engage in a collective struggle against the status quo; and (6) teachers are cognisant of themselves as political beings.

In a recent review, Yolcu (2019) identified three convergent themes in studies examining equitable practices. Firstly, the importance of school mathematics and achievement of equity are typically justified by a perceived ‘gatekeeping’ role of mathematics for access to other disciplines of study and for future educational and economic opportunities. Secondly, perhaps unintentionally, these studies regenerate psychological distinctions of studying mathematics without providing information about the social and cultural identities of students involved. Finally, they paradoxically consider equitable teachers at once as both agents of change and objects of research. We emphasise yet another convergent theme among such studies: their exclusive focus on a specific school level (early years, primary, secondary). To the best of our knowledge, examining actual equitable practices or teachers’ perceptions of what constitutes an equitable teaching practice across school levels is largely ignored by research. This article addresses the gap through our focus on teachers’ perceptions of teaching practices, in particular, those practices understood as equitable by teachers across the early years, primary, and secondary levels. A deeper understanding of teachers’ self-reported practices is a proxy for their actual classroom practices (Clunies‐Ross et al., 2008). This generates starting points to flag continuities and discontinuities in children’s school experiences, and to consider how to better celebrate or address them. Attention to these continuities and discontinuities is particularly important given research findings indicating a persistent general decline in engagement with mathematics as learners transition from one school level to another (Martin et al., 2015). Declines in self-efficacy beliefs, motivation, and performance (Deieso & Fraser, 2019), accompany a reinforcement of stereotypes regarding gender-based mathematics performance (Denner et al., 2016). These observed differences are typically attributed to factors such as teachers’ self-efficacy beliefs (Midgley et al., 1989) and teachers’ and parents’ emphases on goal (Friedel et al., 2010), as well as teachers’ different approaches in using instructional materials (Fan et al., 2013). Nonetheless, while we consider teachers’ perceptions important for the reasons explained above, we are fully aware that beliefs often operate subconsciously and beyond articulation (Leatham, 2006; Raymond, 1997; Xenofontos, 2016), so that teachers’ self-reported practices do not necessarily reflect what happens in reality.

The study and its methods

Some information about the Scottish context

Within the Scottish education system, the term equity is discussed in various policy documents, and explicitly associated with poverty and children’s socioeconomic status (Ellis & Sosu, 2015; Scottish Government, 2016, 2017; Education Scotland, 2018). The Scottish Government calculates a relative measure of deprivation, called the Scottish Index of Multiple Deprivation (SIMD), based on data across seven domains: income, employment, education, health, access to services, crime, housing). Each area in Scotland is then assigned an SIMD value, from 1 (most deprived) to 10 (least deprived). Closing the so-called poverty-related attainment gap (gap between performances of pupils from affluent and deprived areas) has become a top national priority with the introduction of the Scottish Attainment Challenge (Education Scotland, 2018). Current actions related to this national strategy include the Pupil Equity Funding programme (additional funding allocated directly to schools, based on the number of students from low SIMDs attending them) and Getting It Right For Every Child (a national approach to improving outcomes and supporting the wellbeing of children, young people, and their parents). Furthermore, the Scottish Government (2016) makes clear that ‘[e]nsuring effective transitions between primary and secondary education is particularly important’ (p. 14), especially since ‘[t]he Scottish Attainment Challenge has been set up to improve educational outcomes in communities with a high concentration of children living in poverty’ (p. 25).

Participants

The General Teaching Council for Scotland (GTCS) registers teachers either as primary or secondary (www.gtcs.org.uk). Primary teachers are generalists who work with students aged 3-12 in nurseries and primary schools. Anecdotally, some primary teachers self-identify as early-years teachers, due to their preference for working with children aged 3-7. Those using the primary teacher label prefer working with children 7-12 years old. Secondary teachers teach their specialist subject area and work with 12-18 year-old pupils.

Study participants are teachers working in the Central Belt of Scotland. Even though it is a large region with areas of different affluence levels, it also includes areas with the lowest SIMDs in the whole country¹. Volunteer teachers were sought via the networks of local authorities, our own professional networks, and on social media. Also, teachers who voluntarily expressed interest in the project passed the details onto other potential participants. We are aware that this type of snowball sampling may limit diversity in the sample, and consequently the type of data collected; however, at the time this study was conducted we were primarily interested in the school level-related differences in an exploratory manner. In total, 29 teachers were recruited, 8 of whom self-identified as early-year teachers and 11 as primary teachers, while 10 were secondary mathematics teachers. Other than one teacher in early-years, two primary and one secondary, all other participants had more than five years of professional experience. The five-year threshold is often seen as a critical time frame during which teachers have had sufficient time to develop a wide range of professional experiences (Tricarico et al., 2015). In previous studies (e.g. Buchheister et al., 2019; Felton-Koestler, 2017; van Es et al., 2017), participating teachers were recruited because researchers had used specific criteria a priori to label them as ‘exceptionally equitable’; therefore, the research intention was to further analyse the knowledge, beliefs, and practices of those teachers. In contrast, our study teachers expressed interest in participation themselves. Of course, one might claim that our participants could potentially be ‘exceptionally equitable’. Nevertheless, it was not our intention to use any predetermined recruitment criteria to evaluate a participant’s level of excellence in employing equitable teaching practices.

In the presentation of our findings, we referred to each of the 29 participants by their school level and assigned number (e.g. EY3 – 3^rd early-year teacher, PT2 – 2^nd primary teacher, ST7 – 7^th secondary teacher). We maintain UK English in the actual quotes by teachers, to remain faithful to participants’ use of language.

Through the lens of Variation Theory

In a previous paper based on data from the same project (Xenofontos & Hizli Alkan, 2022), we discussed teachers’ perceptions on marginalisation in school mathematics. As concluded, all 29 participants mainly reproduced the social-class/poverty discourse of policymakers, while very few recognised other marginalising variables (for example, gender or English language competence). Yet, none of the teachers described how such variables might be interlinked. In this report, we explore these teachers’ self-reported practices for addressing what they themselves named as causes of marginalisation. In doing so, we refrain from employing any predetermined theoretical framework regarding equitable practices. Such an approach would have narrowed our work to an examination of whether specific equitable practices, as identified in the academic literature, were reported by our participants. Rather, in acknowledging that mathematics teachers’ beliefs, knowledge, and practices are culturally situated (Andrews 2009; Xenofontos 2018), we are interested in documenting and understanding the range of practices teachers at the three school levels in this specific cultural context identify as responses to the causes of marginalisation. In other words, while in Xenofontos and Hizli Alkan (2022) our focus was on what teachers pointed out as causes of marginalisation in school mathematics, here we explore the variation of strategies they claimed to employ as a response to such causes in their mathematics classrooms. We locate the work presented here in the tradition of Variation Theory (VT). As a conceptual tool, VT draws from and extends the ideas of phenomenography, a theoretical and methodological approach to mapping people’s experiences that strives to articulate the world through the standpoint of those experiencing it (Marton, 1981). Taking a second-order perspective, phenomenography explores variation in the ways a phenomenon is perceived within a group of people, including possible misconceptions of reality (Marton, 1986). Phenomenography had at one point in time been heavily criticised as merely descriptive and undertheorised, and as exclusively focused on methodological issues (despite this not being the intention of its founder, Ference Maron). It has evolved to what is now known as VT (Marton & Booth, 1997). VT and phenomenography are both built on the idea of variation: describing variation in human experiences and explaining why it exists (Orgill, 2012). As an interlinked theoretical and methodological approach, VT is gaining increasing appreciation in mathematics education research (see Björklund et al., 2021; Essien, 2021; Melhuish et al., 2020). Nevertheless, this approach has, hitherto, mostly been used from the perspective of mathematics learners. A novel aspect of our work here lies with the fact that VT is explicitly employed to examine and explain variation from teachers’ perspective, and specifically in the ways they claim to respond to marginalisation in school mathematics through what they believe to be equitable teaching practices.

Data collection and analysis

All participants were invited to an individual semi-structured interview. Each interview was audio-recorded, lasted approximately 45-55 minutes, was held at each participant’s school, and was conducted by either one of the first two authors or a research assistant. As part of a wider project, the interview protocol included questions about teachers’ perceptions of and experiences related to (a) marginalisation and the attainment gap in mathematics, (b) equitable mathematics teaching practices, and (c) concepts like equity, inclusion, diversity, and social justice. In Table 1 below, we present the complete interview protocol. Since the interviews were semi-structured, not all questions were posed in the exact same order or phrasing. In fact, several questions from the interview protocol were omitted in cases where they were covered by responses to other questions. The protocol was prepared in advance to support researchers in case the discussion got stuck or lost its flow.

Table 1. The complete interview protocol.

General topic	Questions	Prompts (in case informants get stuck)
Teacher’s professional profile – Icebreaking questions	Could you share a few words about yourself, your role in this school, years of experience, etc.?	Years of experience Qualifications Which subjects/years do you teach? Professional dispositions
	How was your own experience with learning numeracy/mathematics in your own schooling?	Self-efficacy Teaching methods used by your teachers
	How would you describe your numeracy/mathematics experiences through what you learned during your own teacher education (as prospective teacher) and, later, professional development (as a practising teacher)?	Potential changes Content Instructional methods used
Perception of the attainment gap in numeracy/mathematics in Scotland	As you may be aware, here in Scotland there is extensive discussion of the attainment gap, especially in mathematics/numeracy. How do you understand this gap? Could you give any examples from your own professional experiences where you observed gap(s)? How does it impact your day-to-day life as a mathematics teacher? Why do you think some children do not perform as well as others in school mathematics? Why do some children find themselves on the margins?	Potential reasons: Poverty Language Migration Gender Social class Ethnicity Disability Sexual orientation
Conceptualisations of concepts: equity, inclusion, diversity, social justice, and their links to numeracy/mathematics	Card activity Give teachers four cards with the terms ‘equity’, ‘inclusion’, social justice’, ‘diversity’. These are some of the concepts found in the academic literature in relation to attainment gaps. They can also be found in Scottish policy documents. What does each of these mean to you, in general and in relation to numeracy/mathematics? How do you understand/define these concepts? Give teachers an A3 piece of paper and some pens. Could you use these cards and illustrate the relationships between these concepts and the attainment gap? Feel free to move these cards around, remove any card you find not so relevant, and add any other concept you think is relevant.	Illustrations can be of any form, as for example: Venn diagram Charts Drawings
Equitable teaching practices for bridging the attainment gap	What specific practices do you use and consider effective in addressing all children’s learning needs, especially underperforming children? Can you give specific examples?

Each interview was transcribed verbatim soon after it had been conducted. Due to the exploratory nature of our work, readers should keep in mind that our aim was not to confront teachers with their misconceptions, but rather to document these if they emerged during an interview. On reflection, we are aware that some of our interview questions could be followed up with a ‘why?’ more explicitly, so that teachers’ (mis)conceptions could be explored deeper. Retrospectively, we admit that this might not have taken place, as we – the interviewers – were particularly cautious not to ‘upset’ informants, especially as they were sharing with us personal-professional stories of a sensitive matter.

Even though this paper focuses on self-reported practices that are deemed equitable by teachers, our analyses were not based solely on teachers’ responses to those specific questions (see last general topic from our interview protocol). On the contrary, we examined interview transcripts holistically, as teachers could be reporting relevant teaching practices in their responses to other questions. Although the differences between self-reported and actual practices will remain unknown for this research, some literature suggests that teachers’ self-reported practices reflect their actual classroom practice (Clunies‐Ross et al., 2008). We consider teachers’ self-reported practices as part of their pedagogy, which plays a key role in promoting equity, and therefore as critical to investigate.

As noted above, the VT perspective helps us to document, understand, and interpret the variation among teachers’ reporting of practices they see as equitable. While ‘conception’ is the main unit of analysis in phenomenography and VT (Marton & Pong, 2005), there is no standard approach to how the variation of conceptions should be identified and described (Marton & Booth, 1997). This can also be seen in other studies in mathematics education that examine phenomena through the lens of VT; a variety of diverse analytical approaches are documented (see for example, Björklund et al., 2021; Essien, 2021; Melhuish et al., 2020). For the purposes of this study, reflexive thematic analysis (Braun & Clarke, 2021) was chosen for its ability to highlight how perspectives and beliefs are constructed within their unique socio-cultural contexts (Xenofontos, 2018). The first two authors worked independently to assign coding and data categories, and then together to collate, discuss and combine the codes into broader categories related to our questions. This method is similar to how grounded theorists move from open to axial coding (Strauss & Corbin, 1998). The third author subsequently served as a critical reader for reviewing categories from an outsider perspective to the data (Baskerville & Goldblatt, 2009). Following Braun and Clarke’s (2021) advice on moving away from a need to achieve qualitative data saturation, we do not focus on quantitative measures such as frequency or percentages.

Ethical issues

The study complies with the ethical guidelines of the British Educational Research Association (BERA, 2018). Specifically, all participants were informed both verbally and in writing about the aims of the project and signed informed consent forms. The project received ethical approval from the General University Ethics Panel, University of Stirling (number of approval GUEP 742), as the first two authors worked at that university during the time the project was conducted. Furthermore, Lincoln and Guba (1985) criteria for trustworthiness, which involve credibility, transferability, dependability, confirmability and authenticity, were applied to ensure the rigour and quality of our research. For credibility, we provide details about the demographics of our participants, as well as the Scottish context, while not divulging the names of teachers, students, or schools to safeguard confidentiality and traceability issues. To address dependability, which refers to the stability of data over time and under different conditions, we shared our findings with critical friends (former teachers in Scottish schools and now colleagues in academia), who assured us that our findings reflected the nature of Scottish context. Confirmability was achieved through researcher triangulation as explained in the data analysis. For transferability, we explicitly acknowledged the sociocultural context of our study so that other researchers can make decisions about the extent to which other settings are similar to ours. Nevertheless, we are interested in exploring possible convergences and/or divergences between the perceptions of teachers working at the three school levels, rather than generalising our data to the wider teacher population of Scotland. Finally, for authenticity purposes, we provide representative quotes from our data, which were shared and discussed with our Scottish colleagues/critical friends to ensure the veracity of their representativeness.

Findings: teaching practices perceived as equitable

Ability vs mixed-ability grouping across school levels

With ability grouping, we refer to all types of grouping based on perceptions of children’s ability (i.e. multi-level classrooms, cross-level grouping, within-class grouping). We are aware that there are different labels in the Anglo-Saxon literature (Taylor et al., 2019). For example, in the US literature, the term ‘tracking’ is sometimes used as synonymous to what in the UK literature is called ‘setting’ (the practice of separating children into ability groups for each school subject) and sometimes to ‘streaming’ (segregating children by setting the same groups for all subjects). In this study, teachers’ perceptions about and preferences of ability and/or mixed-ability grouping appear to be polarised. Across school levels, we see teachers discussing the benefits of a specific grouping type, in attempts to justify their choices.

In the first cluster, teachers favoured ability grouping as a pragmatic choice. According to them, ability grouping gives teachers the flexibility to work on children’s individual needs and to manage time and the classroom better. Examples are presented below from each school level. In these examples, teachers mainly base their arguments on the support of ‘low’ attainers and those with additional learning needs:

You are quite quick to becoming aware of who are getting it and who aren’t. I also send children out with an SLA [Support for Learning Assistant], so they [the SLAs] would be used to help push up the attainment for targeted groups. I have a group of just three for numeracy who have a quite big attainment gap. The rest of the class are working on addition and subtraction. (EY8)

‘I think, if you use ability groups, if you’ve got one group to focus on, it would be really good if you want to practice something and if you want to make sure they understand. In a class where you’ve got a wide range of abilities and there’s only one of you, it could be very hard to manage so I think on the managing side of things if you’ve got one group, yeh that’s great’. (PT7)

I really struggle to imagine how mixed ability could work for us. Even within set classrooms you still have quite a range of ability. Within our top sets you’ll have pupils who will struggle quite a lot and some who will be absolute highflyers. Our bottom sets are still working mostly at early and first level in secondary classrooms right up to third year. I can’t imagine how pupils who need targeted support can possibly keep up with the work of a mixed ability class. (ST9)

Conversely, in the other cluster, teachers expressed their preferences towards mixed-ability grouping, a practice which they perceived as scaffolding/leverage of the quality of learning. EY1, for example, talked about the support children who face difficulties can receive from peers: ‘they need that scaffolding of someone actually leading them into the next things. It doesn’t need to be an educator, it could be their best friend but you need those activities to be going on’. Similarly, PT10 explains:

Sometimes I might work with the whole group as a collective but it just depends on what we’re doing. Sometimes they’ll work with a partner, their own choice of partner and more often a friend who’s at the same ability level as them. Sometimes I’ll put them with a partner of mixed-ability pairings so that people with a better understanding of maths concepts are working with the ones that aren’t ready, but where you would ideally want them to be so they can help explain things and push each other a little. So, if they are working in pairs, they will help push each other along (the ones who need that assistance, the 1-1 assistance) cause the person they are working with should be able to explain it. (PT10)

For ST10, mixed-ability grouping serves the additional purpose of leveraging the aspirational targets of pupils:

I think by starting a complex instruction of mixed ability there, you’ve got aspirational targets and members of classes. If you’re mediocre at maths then you know you’re mediocre at maths. ‘I’m not bad at it, but I’m not as good as the person over there’. If you look around the class, ‘I’m in the same class with X, Y and Z and they’re excellent at maths, I want to be like them’. There’s aspiration for these kids, if you take that away then something changes, there’s nothing to aspire to. (ST10)

Practices specific to each school level

As presented above, teachers across school levels expressed polarised views about student grouping (ability vs mixed-ability). While this issue was common across school levels, other responses about equitable teaching practices were particularly homogeneous within school levels. We remind readers that we did not analyse data separately for each school level; nevertheless, our analyses pointed out strong within-level similarities. These similarities also included reflections on participants’ teacher education, which was perceived by almost everyone to be inadequate as preparation for addressing pupils’ diverse needs in mathematics. For example, EY3 saw their mathematics teacher education programme ‘not as comprehensive as I would have liked’, while others, like PT2 and ST1, thought that placement experiences were more useful than their theoretical coursework. In ST1’s words, ‘[i]f I’m honest I don’t know that there was anything [in the teacher education programme] that was useful. I think what was useful was the classroom experience, the placement experience. I certainly didn’t find that the training we were given was of benefit to me’.

Tables 2–4 illustrate practices perceived as equitable by the early-year, the primary, and the secondary teachers, respectively. These tables demonstrate the variations of practices reported at each level. In their individual interviews, teachers discussed combinations of these practices. In addition, while these practices are presented here as distinct, they inevitably involve some overlaps. For example, the practice play in Table 2 involves the use of language and communication that can also be found in the practice whole-class sharing and talk. Furthermore, as mentioned earlier, why each practice was perceived as equitable was not explicitly asked during the interviews (an acknowledged shortcoming of our data collection). Nonetheless, in the three Tables 2–4 we included a column alongside each representative quote with our interpretation of why each practice seems to be perceived as equitable by teachers. Our interpretations do not rely exclusively on particular quotes but derive from our readings and discussions of all similar quotes addressing each practice.

Table 2. Practices perceived as equitable by early-years teachers.

Practices	Representative responses	Why teachers perceive this as equitable
Play	EY4: We do a lot of experiencing through play, you’ve got those kids that come in straight away and are talking about bigger and smaller, ‘I got the bigger blocks and you got the smaller blocks’. Or when they play with the water tray, and they’re pouring and measuring. They naturally do it a lot more whereas there’s some children you really have to coax that conversation out of them, like ‘Who’s got the biggest one? Who’s got the smallest one?’	Play as a rich learning opportunity for children who do not otherwise actively participate
Scaffolding	EY1: If I wanted to see wee Johnny today, I would be thinking about numeracy because I want to know what he can do for numeracy. I might do what we call an observation which is, I might have something that I bring in myself which would be an adult-led activity or I might today just decide to watch his play and interact with him doing the play but in terms of numeracy. So he may be playing with the cars in the garage and I say to him, ‘park 4 cars for me’ to see whether he can park, if he knows what 4 is. And ‘if another one comes up, how many have you got now?’	Exploring individual pupils’ prior knowledge and skills to inform instructional planning (differentiated learning)
Hands-on activities and concrete/visual materials	EY6: Numeracy in my classroom doesn’t have limits. So it’s not, like, ‘right, this week we’re going to learn about time’ and then you put all the clocks away and they don’t see them again. In my classroom I’ve got shapes, clocks, different materials to count with because it’s important that children get that realness and understanding that’s not just something that they put away and that’s how they learn by engaging and activities.	Expanding the boundaries of mathematics teaching, mostly through hands-on activities and making connections between or among concepts
Whole-class sharing and talk	EY5: We need to make everyone aware that not everybody’s going to get the answer the same way, and that’s why you share strategies. There’s always lots of different ways to get to the same answer, but also giving them that help and support and being equitable for everyone and understanding their backgrounds, understanding what they need maybe different to what someone else needs. For me, that’s equity.	Providing pupils with opportunities to publicly share ideas, and to be exposed to others’ ideas
Outdoor learning	EY3: I think the addition of lots of outside experiences for children is having a huge impact. Children are really gaining those real-world skills. You’re seeing amazing problem-solving, you’re seeing the very early stages of numeracy and development, shape sorting, size, measure. These lovely organic experiences that come from it and the role that I’m doing, I’ve been really fortunate to see some fantastic outdoor play experiences that have really run with children’s interests, where they are at … and the really rich discussions around measure and capacity, especially around water trays and things like that. These are the experiences that are gonna spark that interest.	Outdoor learning as powerful curricular choice, indirectly increasing pupils’ interest in and motivation for learning mathematics Open-ended (formative) assessment as supporting differentiated learning

Table 3. Practices perceived as equitable by primary teachers.

Practices	Representative responses	Why teachers perceive this as equitable
Whole-class discussion	PT4: I ask questions to the whole class. ‘How would you do this?’ or ‘Is that right?’, ‘How can you prove it’s right’. It gets them thinking much more. And then find very quickly if there are other children under the radar, and pick them out much faster.	Using simple formative assessment to inform support and remediation
Differentiation by task	PT6: Option 1, cold, is the easiest task, option 2, medium, is the middle, and option 3, hot, is more challenging, and then depending on the ability of the class you can have an extra hot challenge, and children choose themselves which one they would like to do.	Encouraging pupil autonomy and personal choice
Whole-class sharing of different strategies	PT2: You can get 10 children to do one sum and when you ask them how they got to the answer, they’ll tell you different ways of how they’ve done it because they’ve used different strategies. I usually try and get children to share their strategies with the rest of the class. […] Because I find that the way I might sometimes teach some children will get it, but others are doing it a different way, so being aware of diverse approaches to get them to do it.	Sharing of diverse strategies can support learning for all
Standardised testing	PT7: We have just done mock numeracy testing. Now, if I had looked at my class baselines and class work, I could tell you exactly where I thought children were, which ones I think are going to achieve first level, and which are going to struggle, who may get it by Christmas time.	Standardised approaches provide data regarding who might be falling behind
Encouraging the use of concrete materials and visuals	PT1: I’ve heard children being told not to count on their fingers when they’re a certain age. Not always, but I’m talking about things that I’ve heard and I always make a point of saying ‘adults count on fingers, that’s what we do’. So I think when children are at the upper stages of primary school, some teachers don’t expect them to use concrete materials. I don’t agree with this. I tell my pupils that it is valid for them to solve problems and work out things using concrete materials in a variety of ways. In my class, I encourage them to use materials.	Physical models and representations support learning even at upper primary grades, which is perceived as supporting equity because it supports learning of all pupils
Technology-aided practices	PT4: I’ve used some websites. I know some teachers don’t use them to their full potential, but I can assign specific targets for what [students] need to work on. I can use the internet as an assessment tool. This isn’t just playing a game. That repeated practice that some children need, so game-based learning is quite good as well. […] I’m planning a blog for next term, using digital technology, Spheros, like robots that you programme, to teach angles and direction. The children struggle with 270° or 180°. I’m hoping, cause I’m trying something new, that this will be the thing.	Creating opportunities for pupils to practice their learning flexibly in digital environments
Outdoor learning	PT11: Outdoor learning, when it’s not freezing, yeh, it helps. You can do anything outside. Things we’ve done outside, like using the chalks and metre sticks to create right angles, lines of symmetry, creating symmetrical shapes and everything. So outdoor has its place, absolutely.	Applying mathematical ideas outside the classroom

Table 4. Practices perceived as equitable by secondary teachers.

Practices	Representative responses	Why teachers perceive this as equitable
Differentiation by task	ST9: Differentiation by task is probably the one I was using more. For example, when I last taught my absolute lowest set which was a first-year set, they were still working on adding double digits, and we would quite often have a sheet of 60 add questions. I knew that the weakest pupil was just working on adding 2 single digits together, so I made sure they got that sheet. I know other pupils were working on maybe a double digit and a single digit, bridging the tens barrier so they got sheets that matched. Our more able pupils might have two double digits, with or without bridging the tens barrier.	Encouraging pupil autonomy and personal choice
Provision of additional sessions for exams	ST5: There’s seven of us (note: mathematics teachers) that give up an hour every single week to run three sessions. I mean it’s not going to be one-to-one with the tutor. We’ll never gonna be able to match that but we can give up an hour. I mean I’ve seen it from 10 all the way up to an entirely full room, so we can give those pupils the opportunity there that they may not be able to afford at home. Their parents may not be able to pay for that; they may not even know where to look for it. So it’s giving them that opportunity. It’s just making sure that everyone does have access to the same things. Making sure that nobody is left trailing behind.	Offering extra support, especially for children who may not receive additional support outside of school
Practical arrangements with assessments	ST6: If one pupil requires additional time for an assessment, instead of singling them out, instead of sending them to separate accommodation, I would decide to just obviously run it by managers and why not, to say, ‘look, what if I just add on an extra half an hour to everyone’s assessment time’ and they would generally say yes. Other schools that I have been in have always said, ‘yeh that’s fine, give them as long as they need. As long as they’re not totally making an absolute mockery out of it’.	Giving all pupils opportunities for extra time, instead of singling out pupils as having special needs
Digital opportunities to access content after school	ST6: We do a lot of, sort of, digital learning and stuff here so everything that we do pretty much goes online so everybody’s got access to that, nobody is now benefitting from ‘my parents are rich therefore I can buy this past book, I can buy past papers, I can afford a tutor every single week’. We put everything online so that they don’t need to go and buy resources.	Offering access to digital resources beyond school time

Early-years teachers

Practices perceived as equitable by early-years teachers originated by shared beliefs that ‘teaching should start where the pupil’s knowledge is’ (EY3) and that ‘it is very important to observe children’s progression regularly’ (EY2). Teachers shared beliefs about the importance of play during early years, scaffolding, hands-on activities and use of concrete/visual materials, talk and whole-class sharing, and outdoor learning. Table 2 illustrates the variation of quotes among early-years teachers.

Primary teachers

The use of concrete and visual materials was also addressed by primary teachers. Besides, as explained earlier, those self-identifying as early-years teachers are, in fact, registered as primary teachers by the GTCS. Nevertheless, some other practices emerged specifically by those teachers self-identifying exclusively as primary (teaching children of ages 7-12). Unlike early years, primary teachers talked about specific measurements such as standardised tests to collect information about students’ learning. Also, they talked about direct questioning by the teacher, differentiation by task (giving children to opportunity to choose the level of difficulty), whole-class sharing of different strategies, and technology-aided practices. Table 3 illustrates the variation of quotes among primary teachers.

Secondary teachers

Similar to primary teachers, providing differentiated tasks for students was also discussed by their secondary colleagues. In addition, the secondary teachers reported a number of different practices they considered equitable. Namely: provision of additional sessions for exams, practical arrangements regarding assignments and assessment, and digital opportunities for students to access content and materials online after school. Table 4 illustrates the variation of quotes among secondary teachers.

Discussion and conclusions

In this section, we focus on two key ideas that emerged from our findings: (a) the range of teachers’ self-reported equitable practices, and our interpretations as to whether these practices are indeed equitable, and (b) what this study reveals regarding transitioning from one school level to another. Both key ideas provide important contributions to the field for reasons discussed in the subsequent pages. We invite readers to ‘zoom out’ from the Scottish context of this study and consider these ideas in relation to other educational contexts as well.

Teachers’ self-reported practices – equitable or not?

Questions about how to better support pre- (Boylan, 2009; McLeman & Vomvoridi-Ivanovic, 2012) and in-service teachers (Bartell, 2013; Planas & Civil, 2009) to address issues of equity and social justice in the mathematics classroom have been raised by several colleagues. A consequent question regards the evidence-based practices that are equitable. Along these lines, our research question here is concerned with the self-reported practices teachers perceived as equitable, and the extent to which their perceptions are justified by research.

The polarisation observed in teachers’ responses concerning group composition in the mathematics classroom (ability vs mixed-ability grouping) reflects the wider debate on grouping in Scottish education (Hamilton & O’Hara, 2011). In Scotland, as described in the Education (Additional Support for Learning) Act 2004, the government has a legal obligation to provide additional support for learning to all children. However, teaching practices in Scottish mathematics classrooms often move away from ‘additional’ to an expansive interpretation of ‘different from’ in the language used in the 2004 Act (Swanson et al., 2017). As a result, even though ability grouping is regularly reported as a form of inequality and social exclusion (Gates, 2019; Taylor et al., 2019), it is often employed in many mathematics classrooms (Hamilton & O’Hara, 2011; Swanson et al., 2017). It is quite possible that teachers perceive ability grouping as a common-sense solution to a ‘problem’ of efficiency in facilitating the attainment of specific skill objectives. As we noted in the previous paper with the same participants (Xenofontos & Hizli Alkan, 2022), this can also be interpreted as teachers’ reproducing policy discourse in their practice. If teachers understand equity in terms of outcomes, then supporting skill mastery through focused ability grouping would seem to be an equity-based practice. There are many studies that find better long-term success in mixed-ability or random grouping (see for example the work of Jo Boaler, in Boaler, 2006; Boaler & Staples, 2008), and some that study mathematics classrooms guided by ‘relational equity’ of equal opportunity (Boaler, 2008), but such scholarship remains unusual (Boaler, 2006, 2008; Yolcu, 2019). Teachers’ understanding of equity might reflect their teacher education experiences and early professional enculturation, generally characterised as training in technical skills (Tatto et al., 2009), ‘political forgetting’ associated with political liberalism (Swanson, 2017), and unable to address issues of equity because of naïve assumptions about political neutrality of mathematics (Felton-Koestler and Koestler, 2017).

We believe equitable teaching practices (should) stem from an understanding of equity defined in terms of opportunity, and that the sociopolitical causes of marginalisation demand the support and empowerment of all children (especially those marginalised) as mathematics learners (Bartell et al., 2017; Berry & Wickett, 2009). However, instruction believed to be good for all learners does not always reflect the important critical awareness of marginalisation, which necessitates attention to why such generally good practices do not address inequity. At the beginning of this article, we refer to several related studies which, even though they have different foci, propose teaching strategies that explicitly link school mathematics to its sociopolitical dimensions and marginalisation (see Banes et al., 2018; Boaler, 2006, 2008; Hand, 2012; Jamar & Pitts, 2005; Moschkovich, 2013; van Es et al., 2017). Such strategies are significantly different from generally recommended practices for all mathematics learners in ways that target equity rather than assuming that equity will take care of itself if a teacher means well.

The reported practices of several participants in this study demonstrate the view that good teaching in general can and will resolve patterns of inequity. For example, the importance of language and discussion in mathematics classrooms, the provision of opportunities for children to select tasks of differentiated challenges/demands, and the use of various concrete materials for the development of deep conceptual understanding. We consider these practices important. Yet very few teachers explicitly associated their practices with how to support children that seem to be underperforming as a result of social exclusion and marginalisation. Even fewer explained any direct mechanisms by which thee practices could support the goals of equity. This finding requires further examination: How effective are ‘good’ teaching ideas/practices in addressing, endorsing, and promoting equity, when they do not explicitly bring marginalisation into teachers’ intentionality? For example, ST1 shared his use of a class webpage to extend access to mathematical content and materials beyond the school day. Even though digital opportunities to connect with mathematics after school is well intended, it assumes that his students and their families have digital devices and internet access at home. Our study indicates that almost none of the reported teaching practices explicitly brought poverty into the discussion about classroom practice, as opposed to discussions at the level of educational policy (Education Scotland, 2018; Scottish Government, 2016, 2017). Although the teachers’ discourse mirrors the policy rhetoric, it simplifies it in a reduction to blander ‘best practices’ less critical of social contexts.

As we note above, it appears that teachers ground their perceptions of equitable practices in the micro-management of skill training. In contrast with Ladson-Billings’ (1995) six habits and the nine equitable teaching practices outlined by Bartell et al. (2017), the reported conceptions of equity lack such an expanded sense of the learner as situated in a community that is in turn embedded in a society. Equity orientations require a constant shifting among micro- and macro-levels of interpretation in which a learner does more than practice skills. Teachers would ideally understand themselves as challenging the status quo, and as acting in leadership roles through mathematical thinking; they need to see themselves as political beings. Along these lines, mathematics is no longer a subject to learn but a set of ways of thinking and acting as members of a community. This is the difference that is ignored when research on social justice is seen as extraneous to the purposes of mathematics education (Xenofontos et al., 2021).

Transitioning from one school level to another

Transitioning from one school level to another can be critical for many children’s mathematical performances, self-efficacy, motivation, and engagement (Deieso & Fraser, 2019; Denner et al., 2016; Martin et al., 2015), especially for students from marginalised backgrounds (Gilbert et al., 2021; Scottish Government, 2016; West et al., 2010). Our findings reveal similarities and differences in the practices perceived as equitable and employed by teachers at early years, primary, and secondary schools. We observe some overlaps between the practices reported by early-years and primary teachers (i.e. use of concrete materials, opportunities for whole-class sharing and conversation), as well as those reported by primary and secondary teachers (i.e. differentiation of tasks, use of digital technology), implying the potential for bringing teachers together across levels to discuss equity components of learner transitions. The debate regarding ability vs mixed-ability grouping is another common issue that might be appropriate for discussions among teachers and administrators across levels; it appears to concern teachers across school levels, reflecting long-standing discussions in the context of Scotland (Hamilton & O’Hara, 2011; Swanson et al., 2017). Despite such similarities, however, our findings suggest a shift in the underpinnings of teachers’ philosophies regarding practices at each school level. On the one side of the school continuum, practices reported by early-years teachers generally stress the social aspects of teaching and learning, with emphases on play, experiential learning, communication and talk. On the other side, practices reported by secondary teachers seem to favour cognitive aspects of learning, with an emphasis on summative assessment and examinations. The reported practices of primary teachers shared similarities with those of both early-years and secondary colleagues. These findings align with the foci of previous studies about kindergarten-primary (Benner et al., 2017; Niehues et al., 2021) and primary-secondary transitions (Deieso & Fraser, 2019; Sdolias & Triandafillides, 2008). Ball et al. (2008) raise the importance of teachers developing knowledge of the mathematical horizon, which, indirectly, refers to the subject-matter transitioning of children, as schooling progresses. Here, we stress how teachers’ development of knowledge of a pedagogical horizon (pedagogical experiences children are exposed to at different school levels) is equally important. To address possibly negative cognitive, affective, and social impacts on children as they transition from one school level to another, we recommend that teacher education programs (both pre- and in-service) bring teachers from different school levels together, instead of providing them with fragmented views of schooling. The commonly mentioned equity-related practices would be a platform for discussions of equity in the context of transitioning.

Closing thoughts

In closing this paper, we would like to reflect upon our theoretical/methodological decision to approach our work from a VT perspective. As Cheng (2016) notes, VT has mostly been used to understand and interpret the teaching of specific content, while there is limited research on its use for teasing out confusing aspects of management concepts. If we view classroom orchestration as a form of ‘knowledge management’ (Cheng, 2016, p. 283), then we can imagine extensions of this research into facilitating teachers’ comprehension of the distinctions between practices generally recommended as good for all pupils, methods of instruction that specifically address marginalisation, and the nuances among them. Our work exemplifies the application of VT to examine teachers’ perspectives on their own classroom practices. In this respect, VT has shown to be a useful tool in enabling us document and understand the variations of perspectives at different school levels, as well as cross-level overlaps.

Every study carries a set of limitations. In this article we focus on teachers’ perceptions of their practices based on interview data. Nevertheless, we are aware that there are often discrepancies between what teachers report in interviews and questionnaires, and what they actually do in practice. Therefore, future research should focus on the extent to which self-reported practices described as related to equity are enacted in the classroom. Do the described practices actually look like what the teachers report? Are they as frequently employed as the teachers imply? Also, despite the interesting findings on transitioning across school levels indicated by our research, future studies with a greater number of participants is needed for mapping emerging level-specific characteristics and dynamics, as well as to gain deeper insights into transitioning processes themselves. To address such fragmented views of schooling, more collaborative and coherent approaches beyond the context of Scotland are required.

In this paper, we do not examine the specific definitions that teachers hold regarding sociopolitical concepts such as equity, social justice, diversity, and inclusion. However, this is an important topic for future research. We have gathered related data through our card activity, presented in Table 1. This will allow us to explore the explicit relationship between teachers’ perceptions, contextual sociopolitical influences, and working definitions as outlined in academic literature. This is a direction we plan to pursue in the near future. An additional direction for future research would compare the practices of teachers who articulate their pedagogical choices in terms of equity with those who frame their decisions in terms of social justice. While these two perspectives are often indistinguishable in interviews with teachers, they originate in different discourses, and have the potential to indicate substantially different sources of support for the selection of instructional practices among available options. Equity is commonly associated with issues of fairness in terms of access or outcomes, whereas social justice has the potential to emphasise the pursuit of dignity and recognition (Appelbaum, 2019; Stathopoulou & Appelbaum, 2016). A discourse of fairness leads to a focus on policies and equivalencies, distribution of services and comparative performance. A discourse of dignity and recognition guides reflection on the experiences of individuals in a cultural and political context. Teachers who explain their decisions in terms of dignity and recognition might emphasise different perspectives on the student experience or the broader impact on the community than those teachers who focus on the fairness of their teaching practices. Here, too, broadening the context beyond Scotland would help to tease out those features of discourse that are specific to a geopolitical location and those that could be useful in supporting the professional growth of teachers trans-nationally, in different regions of the world.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are not publicly available for confidentiality reasons, as they include information that may disclose participants’ identity. The parts of the data that do not include personal information are available from the corresponding author upon request.

Additional information

Funding

This work was supported by The Carnegie Trust for the Universities of Scotland (grant reference: RIG008710).

Notes on contributors

Constantinos Xenofontos

Constantinos Xenofontos is Professor of Mathematics Education at Oslo Metropolitan University, Norway. He previously worked as a primary school teacher in Cyprus and as a mathematics teacher educator in Cyprus and Scotland. His research focuses on how mathematics teachers’ knowledge, beliefs, and practices are shaped by social, cultural, and political factors. He is particularly interested in supporting teachers to develop sociocultural and sociopolitical awareness, specifically on the challenges experienced by pupils from marginalised groups when learning mathematics.

Sinem Hizli Alkan

Sinem Hizli Alkan graduated as a primary mathematics teacher and worked in a primary school in Finland before pursuing her Master’s in Curriculum and Instruction. She holds a PhD in education from the University of Stirling, where she also worked as a lecturer in the Initial Teacher Education programme, before joining Anglia Ruskin University as a senior lecturer in 2023. Her research interests include teachers’ curriculum-making practices, socio-cultural aspects of teaching mathematics—especially in relation to language diversity—and teachers’ networks and reflexivity.

Peter Appelbaum

Peter Appelbaum is Professor of Education and Director of Education Studies and Curriculum Studies Programs at Arcadia University in suburban Philadelphia, USA. His research focus is mathematics and social justice in and out of school. Appelbaum is the Founding Director of the Youth Mathematician Laureate Project (ymlp.org), and the author of Embracing Mathematics: On Becoming a Teacher and Changing with Mathematics (Routledge) and The Creative Math Teacher’s Book of Lists (Brill).

Notes

1 Deprivation - Scottish Index of Multiple Deprivation (webarchive.org.uk).