1 Introduction

The importance of numeracy to conceptual understanding has been recognised in subjects across the curriculum. In history, conceptual understanding involves thinking about the past as having structure and direction; thus requiring comprehension of chronological conventions and “temporal concepts like ‘now’, ‘then’, ‘before’, ‘after’, ‘sequential’, ‘concurrent in time (=instantaneous)’ and ‘over time (=has duration)’” (Blow et al. 2012, p. 26). On the other hand, lack of numeracy can inhibit learning. For example, Quinnell et al. (2013) identified a lack of well-developed numeracy as a barrier to learning science, concluding that it is essential for science teachers to:

explicitly reveal to students that quantitative skills (what they deem to be “maths”) are interwoven within the sciences, and that the ability to use these skills fluidly and confidently by scientists is an essential part of practicing the discipline (p. 814).

This suggests that numeracy needs to be seen as an integral part of subjects across the curriculum, rather than merely as an educational by-product that develops as a result of studies in these subjects (e.g., Lee 2009). Therefore, within each subject there will be opportunities to foster the development of students’ numeracy capabilities alongside discipline learning. However, if all teachers are to effectively exploit numeracy learning opportunities that exist in subjects across the curriculum, then they need to be able to identify these opportunities, design appropriate tasks, and implement these tasks in their classrooms. As this capacity is likely to encompass a range of cognitive and non-cognitive characteristics, the construct of teacher identity may provide useful insights into how teachers can be supported to embed numeracy into the subjects they teach.

This article arises from a study in which it has been suggested that teacher identity could be used as an analytic lens to investigate ways of helping teachers to promote numeracy learning (Bennison 2015). The purpose of this article is to propose that a sociocultural approach, specifically an adaptation of Valsiner’s (1997) zone theory, could be used to address the following research question:

In what ways can a sociocultural approach contribute to an understanding of how teachers can be supported to embed numeracy into the subjects they teach?

The next section provides background information about what numeracy entails, how promoting numeracy learning has been approached in various international settings, and the context in which the study reported on in this article was conducted. The proposed sociocultural approach, Valsiner’s (1997) zone theory, is introduced in the third section along with a discussion of how this approach has been used to gain insights into learning in mathematics education, leading to the proposal that such an approach has potential for achieving understanding of a teacher’s situated identity in the context of promoting numeracy learning. The case studies of two teachers are presented in Sect. 5 to illustrate this approach, with details of the methodology provided in Sect. 4. The cases are compared in Sect. 6 and suggestions for further research are offered in the final section.

2 Background

There are two issues that must be dealt with before it is possible to investigate ways to support teachers to effectively to embed numeracy into the subjects they teach. Firstly, there are different ways of defining numeracy and secondly, there is a variety of approaches to promoting numeracy learning in schools. In this section, common threads in some of the definitions of numeracy, mathematical literacy, and quantitative literacy are identified and an overview of the approaches to numeracy (the term used in this article) that are taken in different international contexts is presented. The section concludes with information about numeracy in Australia; that is, the context in which the study reported on in this article was conducted.

2.1 A common understanding of numeracy

Although the term numeracy is used widely in the United Kingdom, Australia, and many other English-speaking countries, the terms mathematical literacy and quantitative literacy are used in other countries. Numeracy was first defined as the mirror image of literacy, but involving quantitative thinking (Ministry of Education 1959). Later, Steen (2001) drew on a number of definitions of numeracy and mathematical literacy to define quantitative literacy as “the capacity to deal effectively with the quantitative aspects of life” (p. 6). The importance of this capacity is seen in the inclusion of mathematical literacy in international student assessments (Programme for International Student Assessment, commonly referred to as PISA) where it is defined as:

an individual’s capacity to formulate, employ and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts and tools to describe, explain and predict phenomena. It assists individuals in recognising the role that mathematics plays in the world and to make well-founded judgments and decisions needed by constructive, concerned and reflective citizens (OECD 2014, p. 37).

There is also increasing recognition of the important role schools play with numeracy being explicitly included in curriculum documents. For example, in the recently implemented Australian Curriculum, numeracy is described as involving:

students in recognising and understanding the role of mathematics in the world and having the dispositions to use mathematical knowledge and skills purposely (ACARA 2014).

Although the numerous definitions for numeracy, mathematical literacy, and quantitative literacy vary in emphasis, it is possible to identify five common elements. These elements are encapsulated in a model of numeracy developed by Goos et al. (2014). In this model context is the central dimension in which mathematical knowledge and tools are used with initiative, risk taking, flexibility, and confidence (dispositions), with these four dimensions embedded in a critical orientation that enables a numerate individual to make judgments and support or challenge arguments. Goos et al. (2014) have defined each of these dimensions and represented the model diagrammatically (see Fig. 1); using it in empirical studies to describe classroom activities in terms of the dimensions of the model and to enable teachers to articulate their personal conceptions of numeracy.

Fig. 1
figure 1

A model of numeracy for the twenty-first century (Goos et al. 2014)

Full size image

2.2 Approaches to promoting numeracy learning

There are three approaches that have been used internationally in efforts to promote numeracy learning in schools. Firstly, in some countries, such as South Africa, mathematical literacy subjects are offered as an alternative to mathematics subjects (Venkatakrishnan and Graven 2006). A second approach is to integrate mathematics and other subjects. For example, revisions to mathematics curricula, over recent years in some European countries, have resulted in an increased emphasis on cross-curricular links (European Commission 2011).

In both of these approaches the emphasis is on mathematics; however, whereas mathematics is the discipline that underpins numeracy, the context in which mathematics is used is central to numeracy and what distinguishes it from mathematics (Steen 2001). Steen (2001) argued that school subjects other than mathematics provide a range of contexts for students’ numeracy development. This suggests that an across the curriculum approach to numeracy, one that enables students to develop their numeracy capabilities in subjects beyond the mathematics classroom, has greater potential to empower students than approaches that seek to develop numeracy capabilities solely through mathematics subjects. This third approach is the one that has been taken in Australia, Scotland and Ireland (see ACARA 2014; Department of Education and Skills 2011; Education Scotland n.d., respectively) where numeracy is seen as part of all subjects.

2.3 The Australian context

Improving the numeracy capabilities of students has been a priority for Australian Governments for over a decade (e.g., DEETYA 1997). The most recent expression of this priority, the Melbourne Declaration on Educational Goals for Young Australians (MCEETYA 2008), has guided the development of the Australian Curriculum (ACARA 2014), being progressively implemented in Australian schools from 2011. This new curriculum, by including numeracy as a general capability to be developed across the curriculum, recognises that all teachers have a responsibility to promote students’ numeracy development. Although it has been recognised for some time in Australia that numeracy is best developed across the curriculum (e.g., DEETYA 1997), such an approach has remained problematic. For example, a survey of beginning secondary teachers from all disciplines found that around half of those surveyed saw themselves as teachers of numeracy, and only one-third felt adequately prepared to teach numeracy (Milton et al. 2007).

An across the curriculum approach to numeracy is supported by the Australian Professional Standards for Teachers (AITSL 2012) which are used as the basis for the accreditation of pre-service teacher education programs and teacher registration. These standards set out what teachers need to know and be able to do in order to support students’ numeracy development.

Student performance in numeracy is measured by the National Assessment Plan–Literacy and Numeracy (NAPLAN) and reported against national minimum standards. This national numeracy testing is conducted annually with students in Grades 3, 5, 7, and 9 with school NAPLAN results published alongside demographic information on a publicly available website that enables comparison of a school’s performance with national averages and the performance of schools of similar demographics. Furthermore, schools need to demonstrate improved NAPLAN results to secure additional funding. This can place pressure on schools to prepare students for NAPLAN tests at the expense of developing the broader numeracy capabilities described previously (e.g., Thomson et al. 2013); thereby influencing school organisation, curriculum, and pedagogy (Hardy 2015). The next section outlines the theoretical approach taken in this study.

3 Theoretical framework

There are three aspects to the theoretical approach taken in this study: a common understanding of numeracy, teacher identity, and the sociocultural approach that was taken. Goos et al.’s (2014) numeracy model (see Sect. 2.1) was used to describe classroom activities and teachers’ personal conceptions of numeracy. This section sets out how identity was conceptualised to capture the factors that influence a teacher’s capacity to embed numeracy into the subjects they teach and provides an explanation of why and how the sociocultural approach taken; that is, the adaptation of Valsiner’s (1997) zone theory used by Goos (2013), can be used to understand the current and predict possible future identities of participating teachers.

3.1 Teacher identity

There are many conceptualisations of identity (e.g., Gee 2001; Sfard & Prusak 2005; Wenger 1998). However, there is general recognition that identity is complex, is situated, and changes over time. Wenger (1998) described identity development as a learning trajectory that occurs as an individual negotiates meaning through individual cognition and social interactions. These two dimensions can be seen in the framework for mathematics teacher identity developed by Van Zoest and Bohl (2005). However, the multifaceted nature of this framework raises the question of what cognitive and social characteristics should be considered to overcome the challenge of designing an empirical study that captures the complexity of teacher identity without creating onerous practical limitations (Enyedy et al. 2005).

Identities are situation specific, thus it possible to theorise about what factors might influence a teacher’s capacity promote numeracy learning. By examining characteristics that have potential to affect whether and how a teacher might utilise numeracy learning opportunities, a framework for identity as an embedder-of-numeracy, organised around five domains of influence (knowledge, affective, social, life history, and context), was developed (Bennison 2015). One of the limitations of this conceptual framework is that it provides a snapshot of a teacher’s identity at one point in time. Therefore, although the framework has potential to be useful for capturing the complexity of teacher identity, it does not reveal trajectories of identity formation over time (Wenger, 1998). Sociocultural perspectives are most suited to investigating the complexity associated with the developing identities of teachers (Lerman 2001) and one such approach that has the potential to enable the dynamic and changing nature of identity to be studied is Valsiner’s (1997) zone theory.

3.2 Valsiner’s zone theory

Drawing on the work of Vygotsky and other psychologists, Valsiner (1997) viewed development (or learning) as the result of interactions between an individual and their social context that result in new ways of behaving. These new behaviours are subsequently internalised and become part of the individual’s repertoire. In order to represent the internal processes, Valsiner redefined Vygotsky’s (1978) zone of proximal development (ZPD) as the collection of ways that an individual could develop as a result of interactions with their environment and the people in it; therefore, this zone depends on the knowledge and beliefs that an individual brings to a situation. To represent the external interactions, Valsiner defined two additional zones: the zone of free movement (ZFM) includes affordances and constraints within the individual’s environment that influence how they are able to act and the zone of promoted action (ZPA) includes factors within the environment that promote new ways of behaving. As these two zones work together, Valsiner suggested that they be considered as a ZFM/ZPA complex. He argued that development could be directed by structuring successive ZFM/ZPA complexes for the learner. However, development in a particular way cannot be guaranteed because an individual is free to accept or reject the actions that are being promoted.

3.3 A zone theory approach to understanding learning

Several researchers have used Valsiner’s (1997) zone theory as a theoretical framework in mathematics education research. In this section, some of this research is reviewed in order to illustrate how differing zone configurations can lead to suggestions about ways to assist teachers to increase their capacity to identify and exploit numeracy learning opportunities.

Blanton et al. (2005) and Bansilal (2011) both used the ZFM/ZPA complexes created by the teacher for students to investigate student learning. In their study, Blanton et al. (2005) found instances where teachers professed to promote particular actions but did not permit these actions in the classroom. For example, one teacher professed to promote conjecturing and argumentation but limited student opportunities for this by employing only short answer and leading questions. Blanton et al. argued that the teachers in their study could be assisted to structure an expanded ZFM for students; thus enabling student learning. From the learners’ perspective, there was insufficient overlap in the ZFM/ZPA complex the teacher had created for them for learning to occur. In the example above, the type of questioning employed by the teacher did not permit the actions the teacher wanted to promote (i.e., conjecturing and argumentation).

However, overlap of the zones of promoted action and free movement does not guarantee learning. Bansilal (2011) found an absence of student learning when instructional decisions made by a teacher, analysed as a series of ZFM/ZPA complexes, were outside her students' zone of proximal development; that is, there was no overlap when the students’ cognitive attributes were mapped onto the ZFM/ZPA complex they experienced. Although these two studies focused on student learning, they illustrate how the configuration of the three zones influences development and enable analysis of why learning may or may not have occurred.

If Valsiner’s (1997) zone theory is to be used to provide insights into how to direct teacher learning, then it is important to consider how the ZFM/ZPA complex experienced by the teacher interacts with the teacher’s ZPD. This approach was taken by Goos (2005), who mapped factors known to influence teachers’ use of technology onto the ZPD, ZFM, and ZPA of participating teachers. The goal of her study was to understand how these teachers developed identities as users of technology as they moved from pre-service to beginning teachers. Although Goos did not explicitly define what she meant by identity as a user of technology, her analysis involved categorising participant responses to interview questions as elements of the ZPD, ZFM, and ZPA then filling in the zones to examine how the changing relationships between the zones shaped a teacher’s identity. Recently, Goos (2013) argued that a zone theory approach also provides a way to understand teacher learning and inform research that seeks to change practice. In this later article, she defined the ZPD as “a set of possibilities for development of new knowledge, beliefs, goals and practices created by the teacher’s interaction with the environment, the people in it, and the resources it offers” (p. 523); the ZFM as the constraints and affordances provided by the teacher’s professional context; and the ZPA as activities that the teacher can be involved in that promote certain ways of teaching. Such an approach, she claimed, enables the complexity of teacher learning and development to be analysed, while still allowing for the influence of the teacher to direct their own learning by seeking out professional development or modifying their environment (i.e., by reorganising elements of their ZPA and ZFM, respectively). Therefore, the approach taken by Goos (2013), which has teacher learning as a central focus, has the potential to enable sense to be made of the interactions between cognitive, affective, social, context, and life history aspects associated with an identity as an embedder-of-numeracy (Bennison 2015); thus, offering the potential to understand and promote change to this identity.

In order to illustrate how this sociocultural approach might contribute to an understanding of the ways teachers can be supported to embed numeracy into the subjects they teach, the case studies of two teachers, Karen and Erica (pseudonyms), are presented. Details of the methodology employed in the study are given in the next section.

4 Research design and methods

The study, from which the two cases were drawn, was conducted over 2 years (2013–2014), employed collective case study methodology (Stake 2003), and involved eight teachers from two Australian secondary schools. The teachers were recruited from those teachers participating in a larger project (hereafter referred to as the Numeracy Project) and were selected because their participation in the Numeracy Project indicated an interest in developing their capacity to provide numeracy learning opportunities for students across a range of disciplines (English, history, science, and mathematics). Furthermore, the professional development component of the Numeracy Project meant that the teachers had access to a range of activities that assisted them to promote numeracy learning.

4.1 Numeracy Project

The Numeracy Project forms part of the teachers’ shared professional context (i.e., zone of free movement), contributes to their zone of promoted action, and has the potential to increase the knowledge needed to embed numeracy in subjects across the curriculum as well as change beliefs about numeracy (i.e., alter elements of the teachers’ zone of proximal development); therefore, a brief overview of this project is provided.

The Numeracy Project was conducted over 3 years (2012–2014) in six schools in two states within Australia, three in Queensland and three in Victoria. Generalist primary teachers and specialist secondary teachers from a range of disciplines participated in the project, which investigated the potential of a professional development intervention based on the numeracy model (Goos et al. 2014). Teachers took part in a series of professional development workshops that were followed by school visits where researchers observed the implementation of numeracy focused tasks and provided ongoing support for teachers. The workshops promoted engagement with the numeracy model to enable teachers to develop a rich personal conception of numeracy and provided opportunities to plan and share numeracy rich tasks in a range of disciplines.

As the case studies presented here represent the participants’ identities early in the study, the influence of the Numeracy Project on the participating teacher’s ZPD was yet to be realised. Data collected later in the study may provide evidence of the impact of the Numeracy Project on the teachers’ ZPD through changes in the teachers’ knowledge, beliefs, and practices.

4.2 Data collection

Data were collected over a 2-year period to enable testing and refinement of the framework for identity as an embedder-of-numeracy (Bennison 2015). Two methods of data collection were employed. Firstly, each teacher participated in a scoping interview to ascertain information about their background, beliefs about numeracy, school context, and the opportunities they have had to learn about how to promote numeracy learning. Interviews were guided by questions that included:

  • What opportunities have you had to develop your capacity to embed numeracy into the subjects you teach? (e.g., pre-service teacher education program, professional development)

  • What do you see as your role in developing the numeracy capabilities of your students?

  • What are the organisational structures in your school? (e.g., timetable, length of lessons, number of lessons per week)

The second source of data comprised lesson observations followed by post-lesson interviews. In the lessons, teachers implemented a numeracy focused task. Field notes recorded the timing and content of the lesson, questions that the teacher asked, and what the teacher and students wrote on the whiteboard. Lesson artefacts, such as task sheets and PowerPoint presentations, were collected to enable a complete account of each lesson to be composed. In the post-lesson interviews teachers were asked about the tasks that were used, student learning, and teacher learning. Interview questions included:

  • Why did you use this particular task?

  • What aspects of the numeracy model can you see in the task?

  • Were there any difficulties in implementing any of the tasks you used? What were they? How could you overcome these if you were to use these tasks again?

Scoping and post-lesson interviews were semi-structured with the guiding questions informed by the characteristics within the framework for identity as an embedder-of-numeracy (Bennison 2015, summarised in Table 1). Previously, each of these characteristics had been identified and defined in a nuanced way to reflect how the characteristic might impact on a teacher’s capacity to promote numeracy learning. For example, although Shulman (1987) identified seven types of knowledge needed for teaching, only three of these types of knowledge were included in the framework. The rationale for including only these types of knowledge was that a teacher would be best placed to support students’ numeracy learning if they had: curriculum knowledge (CK) to identify numeracy demands and opportunities in a subject; mathematics content knowledge (MCK) that is inherent in a subject; and pedagogical content knowledge (PCK) to design tasks that support numeracy development alongside subject learning. All interviews were audio-recorded and later transcribed.

Table 1 Conceptual framework for identity as an embedder-of-numeracy (Bennison 2015)
Full size table

4.3 Data analysis

Interview transcripts were analysed using content analysis of the text to identify aspects of each teacher’s zone of proximal development, zone of free movement, and zone of promoted action. This analysis was guided by mapping the characteristics in the framework for identity as an embedder-of-numeracy (Table 1) onto each of the zones in a way consistent with the definitions of the zones provided by Goos (2013). The mapping presented below is incomplete because of the complexity of teacher identity and how the characteristics listed in Table 1 can contribute to more than one zone. However, the mapping did enable segments of text in the transcripts to be coded as belonging to a particular zone depending on the nature of the comment. Transcripts were independently coded by a colleague and discrepancies discussed and resolved.

Goos (2013) interpreted the zone of proximal development as the set of possible ways in which a teacher might develop. In the context of embedding numeracy across the curriculum, these possibilities depend on the teacher’s level of knowledge needed for teaching numeracy and their current attitudes and beliefs about numeracy. These cognitive attributes are the result of a teacher’s experiences while at school and university, and as a teacher in their current and previous schools. It could therefore be argued that the ZPD includes characteristics from the life history, knowledge, and affective domains. For example, information that a teacher provided about their pre-service teacher education program (life history domain) was coded as belonging to the teacher’s ZPD as it gave some indication of knowledge (knowledge domain) that the teacher may have developed as a result of this program.

As the zone of free movement is related to the teacher’s professional context and “suggests which teaching actions are permitted” (Goos 2013, p. 523, emphasis in original), this zone will be constituted by characteristics that provide affordances and constraints on the teacher’s practice. In the current study, the ZFM could include interactions that a teacher has with others, both within and beyond the school setting, along with the school policy environment and resources that are available within the school. This suggests that characteristics from both the social and context domains contribute to a teacher’s ZFM. For example, information a teacher provided about interactions with students (social domain) that led to the perception that highly structured activities were required (constraint) as well as information about a school policy (context domain) that promoted an across the curriculum approach to numeracy (affordance) were both coded as belonging to the ZFM.

Finally, Goos (2013) interpreted the zone of promoted action as activities that “promote certain teaching approaches” (p. 523, emphasis in original). Therefore, in the current study, this zone can be interpreted as the opportunities that a teacher has to learn about promoting numeracy learning through the subjects they teach. Goos (2013) provided examples of how these activities could be self-initiated or come from research projects. A third possibility is that this support comes from the teacher’s professional context in the form of interactions with colleagues or as a result of school policies (e.g., Goos et al. 2007). Therefore, a teacher’s ZPA may be influenced by characteristics within the social and context domains. For example, information a teacher provided about interactions with colleagues (social domain) was coded as part of the ZPA if it related to colleagues supporting the teacher to embed numeracy into subjects they teach (e.g., mentoring). Alternatively, if the teacher reported that colleagues viewed numeracy within the realm of the mathematics department, then this information was coded as part of the ZFM.

Teachers’ personal conceptions of numeracy and the tasks used in lesson observations were analysed in terms of the numeracy model as has been done previously by Goos et al. (2014). When analysing interview transcripts, sections of text where teachers described their interpretation of numeracy were coded using the dimensions of the numeracy model; whereas for lesson observations, the dimensions of the numeracy model evident in classroom activities were identified along with potential areas where a task could be modified to more fully exploit the numeracy-learning opportunity the task provided. Judgments, based on analysis of the extent to which the teacher exploited the numeracy learning opportunities that existed in the observed lessons and comments the teacher made during interviews, were made about each teacher’s level of mathematical, pedagogical, and curriculum knowledge.

Drawing on this analysis, a narrative was constructed for each teacher. These narratives along with examples of practice were combined to form individual case studies. Drawing on data that were collected in the first year of the study, the cases of Karen and Erica are presented; thus the description of each teacher’s identity that is portrayed in this article represents their initial identity (as it relates to their teaching in science and history, respectively), which has potential to change over the course of the study. Although both teachers could be considered to be early career (within 5 years of completing their teaching qualification), they were selected because of their differing disciplinary backgrounds and teaching experiences. The examples of practice that are presented below were selected from the lessons that were observed (five and six, respectively, over the 2-year period of the study) because these lessons were representative of the way both teachers were able to identify but not fully exploit numeracy learning opportunities in the early stages of the study. Additionally, both lessons involved the use of timelines, albeit in very different ways.

5 Two case studies: Karen and Erica

Each case study presented in this section includes information about the teacher’s school, background, experiences, and an example of classroom practice. In the following section, each teacher’s identity, in the context of embedding numeracy into the subjects they teach, is analysed to illustrate how the theoretical framework explained in Sect. 3 can be used to inform interventions that aim to assist teachers to promote numeracy learning in subjects across the curriculum.

5.1 Karen: an early career science teacher

5.1.1 Karen’s professional context

The school where Karen taught had approximately 900 students and was located in a low socioeconomic area in a large metropolitan city. Recent student performance on national numeracy testing (NAPLAN) was close to schools with students from comparable backgrounds, but substantially below the national average. The principal had agreed to the school’s participation in the Numeracy Project because she saw this project as having potential to improve school performance on NAPLAN.

Classes for Grades 8 and 9 (the first 2 years of secondary school) were arranged in POD groups. This arrangement of classes was designed to assist students make the transition from primary to secondary school by reducing the number of teachers that students encountered. Under this arrangement, students had one teacher for both English and history and another teacher for both mathematics and science. Karen liked this arrangement because she could “develop a relationship [and] routines”.

According to Karen, many students were not interested in school and there was a high level of student absenteeism. She reported needing to provide students with highly structured activities because they “struggle with any activity that is out of the ordinary, or out of their routine or involves them having less guidance”.

Karen had limited access to appropriate technology for teaching. There were no computers for students in science classrooms and, although there was a laptop hire scheme, only a few students had access to personal laptops. Karen reported a few instances where lack of access to technology had impacted on the activities she was able implement (e.g., not being able to use computer simulations when teaching a lesson on radioactive decay).

5.1.2 Karen’s background and experiences

Karen completed a Bachelor of Applied Science, majoring in Biology, and a Bachelor of Education. Her studies included a first year mathematics subject and a mathematics curriculum subject designed to underpin teaching of junior secondary mathematics (Grades 8–10) and Mathematics A (a non-calculus subject that can be taken in the final 2 years of high school), but she did not complete any courses designed to promote numeracy learning in science.

In her third year of teaching at the start of the study, Karen had been at her present school since she graduated. A shortage of mathematics teachers at the school meant that Karen had few opportunities to teach science and develop her knowledge of the science curriculum. Furthermore, opportunities for Karen to participate in professional development related to numeracy had been limited and she had formed the opinion that it was “the teacher’s responsibility to develop their own skills”. To this end, she recounted how she had found mentoring from her head of department and her own reading useful and had “jumped at the opportunity” to participate in the Numeracy Project.

When asked about her understanding of numeracy, Karen described it as “mathematical concepts but in the context of real life”; seemingly a focus on only two of the five dimensions of Goos et al.’s (2014) numeracy model, even though there were other dimensions of the model present in the tasks she used during the observed lessons (e.g., the task described in Sect. 5.1.3).

Karen’s background and school experience suggest that although she may have adequate mathematics content knowledge, it is less likely that she has sufficiently developed pedagogical content knowledge for exploiting numeracy learning opportunities in science. Her pre-service program did not appear as though it had been helpful in developing the knowledge she needs to design tasks that effectively embed numeracy in science, but she had sought to begin to develop this knowledge as described above. According to Karen, “science lends itself nicely to teaching numeracy because just the nature of science uses lots of data” and in the observed lessons she demonstrated the capacity to identify numeracy learning opportunities in the science curriculum; however, as Karen had limited experience teaching science and the new science curriculum (ACARA 2014) had only recently been implemented, it is likely that it will take time for her to deepen her curriculum knowledge. Additionally, there was no strong evidence to indicate that that she saw numeracy as an integral part of learning science as advocated by Quinnell et al. (2013).

5.1.3 Promoting numeracy learning in science

The following example illustrates how Karen identified a numeracy learning opportunity but was unable to fully exploit the potential of the task she used. In this lesson, taught near the beginning of a unit on Earth Science, Karen wanted students “to understand that human history is just a sliver on the end of the timeline”. To achieve this objective, she asked students to construct a timeline using a measuring tape and a roll of paper towel. Karen presented students with a table that listed selected biological and geological events, ranging from the formation of the Earth to the appearance of the first humans (some of these events are shown in Table 2), and asked them to decide on an appropriate scale for a timeline and then locate the position of each event. Karen conducted the activity with the whole class by asking questions to elicit the correct position for each event, then getting one student to mark and label the position on the timeline. After questioning from Karen, the class decided to use 1 m to represent 1000 million years and came to a consensus that the formation of the Earth should be 4.6 m from the position of the present day (i.e., from the end of the roll of paper towel). One student was then asked to measure this distance and label the point as Formation of the Earth. The position of subsequent points were determined and labeled in the same manner.

Table 2 Events in the history of the Earth (Bennison 2014)
Full size table

Students encountered difficulties with some of the more recent events where the distances they needed to calculate and mark were less than 1 m (e.g., the location of the extinction of the dinosaurs located 65 cm from the present day, see Bennison 2014 for further information about this activity). At the end of the lesson, the timeline was placed on the classroom display board and Karen planned to add further geological events to the timeline as they were studied (e.g., major stages in the formation of the continents).

Analysis of this lesson, in terms of the numeracy model (Goos et al. 2014), indicates that the task promoted an understanding of geological time (i.e., a context within the science curriculum) by using mathematical knowledge (measurement, ratio, problem solving), and representational and physical tools (timeline and table, and measuring tape, respectively). The task enabled students’ to develop some confidence in using mathematics (dispositions) but this aspect could have been developed further if Karen had used a less teacher directed approach.

In the post-lesson interview, Karen identified the numeracy model dimensions of mathematical knowledge, tools, and context in the task but may not have seen the additional opportunities the task provided. For example, Karen’s belief that her students’ needed highly structured activities may have led to her teacher-centred approach which limited students’ opportunities for initiative and risk-taking (dispositions). Furthermore, her personal conception of numeracy and inadequate knowledge of how to design tasks that effectively bring to light all dimensions of numeracy may have prevented Karen from incorporating a critical orientation into the task; for example, by getting students to work in small groups to choose and justify a scale for the timeline (i.e., a scale that enabled the timeline to be constructed within the physical confines of the classroom).

5.2 Erica: an early career history teacher

5.2.1 Erica’s professional context

The school where Erica taught had around 1000 students and was located in a regional area where mining was the main industry. As unskilled jobs were highly paid and plentiful, Erica believed that students did not value education as much as they should. Despite this, she felt “that the vibe here is very good”. The school was considered to be in an average socioeconomic area and results from recent national numeracy testing (NAPLAN) were comparable to those of schools with students from similar backgrounds. Although there was no formal POD group arrangement for junior classes (as was the case at Karen’s school, see Sect. 5.1.1), Erica reported that this arrangement had developed naturally over the last few years and she believed that it was an “effective way of getting to know the kids and [build] relationships”. She had not encountered any negative responses from colleagues about her attempts to embed numeracy into history and thought that this was because “everyone realises … that kids need to know your basic, your basic stuff”.

5.2.2 Erica’s background and experiences

Erica completed a Bachelor of Education, with teaching areas in History and Aboriginal Studies (a subject about the experiences of Indigenous Australians), at a metropolitan university and was in her fourth year of teaching when the study began. There were no courses in her pre-service teacher education program that focused on numeracy across the curriculum, although she was required to complete a mathematics subject “because you should have a basic understanding of maths if you are going to be a teacher.” Erica had not participated in any formal professional development related to numeracy prior to her participation in the Numeracy Project. Through this project, in addition to participating in the professional development workshops and interacting with researchers and teachers from other schools, Erica was able to share ideas with another teacher at her school who was also a participant in the Numeracy Project and taught the same history subjects.

Although Erica was unsure if there was a school numeracy policy, she reported that if there was such a policy, “it’s not something that’s made obvious. Like it’s something, just one of those teacher things that you should just do and know”. One interpretation of this comment could be that while Erica saw knowing about and implementing school policies as a responsibility of each teacher, she saw no immediate imperative to do so in the case of a numeracy policy.

As Aboriginal Studies was not offered at Erica’s school, her teaching load consisted of history and English classes. She found the new history curriculum (ACARA, 2014) crowded, describing it as: “Packed, very packed. I’m finding it really hard not to be tokenistic. … The amount of things kids should know before they can make judgments and assumptions. Their assessments, it’s very hard to pack it all in”. Despite this, she believed that there was a place for numeracy in history: “History lends itself to numbers. You can’t say just ‘cos, you need to justify your reasoning and generally facts and evidence, a bit like numbers, prove your point”. However, she felt that students equated numeracy with mathematics and NAPLAN and did not necessarily see numeracy in history; mimicking students to make this point: “We don’t do numbers in history. What are timelines again? Oh yeah, that’s right, we use dates and numbers. We use dates and numbers in history all the time”.

Erica recognised the pervasive nature of numeracy: “numbers, like words, are everywhere and you use them in every part of your day”. However, her personal conception of numeracy, when expressed in this way, was focused on mathematical knowledge (especially number) with some connection to real world contexts; only two of the five dimensions of Goos et al.’s (2014) numeracy model. According to Erica, the biggest challenge she faced if she was to effectively embed numeracy into the history curriculum was to identify where she could incorporate numeracy and how to make the most of those opportunities (i.e., curriculum and pedagogical knowledge). She explained that: “just being aware of some of the opportunities … [and] having someone come into teach those things explicitly to teachers would help move that process along faster”.

5.2.3 Promoting numeracy learning in history

The need for Erica to expand her knowledge base for embedding numeracy in history is exemplified in a lesson where Erica introduced a unit on the expansion of European settlement in Australia prior to World War 1 by presenting students with a timeline that showed selected historical events from 1770 until 1918 (see Fig. 2). In the post lesson interview, when Erica was asked why she had used the timeline, she responded in the following manner:

Fig. 2
figure 2

Timeline showing selected historical events from 1770 to 1918

Full size image

kids need to see how things progressed. Also because it almost directly leads on from what they were learning previously, so they learnt all about the Industrial Revolution. At the end of the Industrial Revolution they looked at ending slavery, so it sort of let them see where it fitted, where they, where their previous knowledge fits in with the new stuff. Just so they get a visual representation, “Oh okay, I know slavery was abolished here”.

Following a whole class discussion about the timeline, Erica asked students to copy the timeline into their books; however, she did not ask students to consider a scale for the timeline.

Exposing students to the timeline placed world events in chronological order; however, the rich potential of a using a scaled timeline to develop understanding of the chronological conventions and temporal concepts that Blow et al. (2012) claimed are essential for conceptual understanding in history was missed, as the unscaled timeline provided limited opportunity for students to appreciate the time between or duration of events; that is, to fully appreciate “where their previous knowledge fits in with the new stuff”. In addition to this context within the history curriculum, the other dimensions of Goos et al.’s (2014) numeracy model evident in the task were incomplete mathematical knowledge (chronological order without consideration of scale) and use of a representational tool (timeline).

6 Discussion

When analysed in terms of the zones of proximal development, free movement, and promoted action, the case studies of Karen and Erica shed light on how these two teachers might be supported to strengthen their capacity to embed numeracy into the subjects they teach (i.e., science and history, respectively). In addition, comparing the two cases enables some tentative assertions to be made about possible starting points for interventions that seek to assist teachers to promote numeracy learning across the curriculum.

6.1 Supporting Karen to embed numeracy in science

At the beginning of the study, Karen’s ZPD seemed to include the mathematics content and (developing) curriculum knowledge necessary for embedding numeracy in science; however, she could be assisted to develop a richer personal conception of numeracy, broaden her understanding of the connection between numeracy and learning science, and expand her knowledge for designing tasks that support numeracy and discipline learning.

The ZFM/ZPA complex that Karen experienced contained contradictory elements. On one hand, she had permission to embed numeracy in science (i.e., the new curriculum and her principal-sanctioned participation in the Numeracy Project) and she had several opportunities to learn about how to do this (i.e., the Numeracy Project, her own reading, and mentoring). On the other hand, the pedagogical approach she felt able to take was influenced by factors that included pressure to improve student’s NAPLAN results, lack of access to appropriate technology, and her perceived need to take a teacher-centred approach.

This analysis suggests that there is some overlap when Karen’s ZPD is mapped onto her ZFM/ZPA complex. Her trajectory towards a teacher identity where she has greater capacity to embed numeracy in science could be aided by promoting actions that help her to restructure her ZFM so that there is increased overlap between her ZFM and ZPA. Such actions could involve helping her to find alternative ways of using the technology that is available to her or assisting her to design tasks that allow students more freedom that she was currently prepared to give them.

6.2 Supporting Erica to embed numeracy in history

When the study commenced, Erica’s ZPD included a narrow personal conception of numeracy and limited knowledge of how to utilise numeracy learning opportunities within the history curriculum. As she had studied a mathematics subject during her pre-service teacher education program, it is likely that her knowledge of the mathematics that is inherent in the history curriculum is adequate and no evidence was found to contradict this assumption.

Within Erica’s ZFM/ZPA complex, although the Numeracy Project promoted an across the curriculum approach to numeracy, Erica wanted to be told where numeracy learning opportunities existed and given appropriate tasks rather than being prepared to seek these opportunities for herself. This desire may have been a result of the implementation of the new curriculum, which gave her permission to embed numeracy in history, but also placed her under pressure to find time to cover the content, with numeracy as something extra to be fitted in; or as a result of her inadequate knowledge base, which may have led to a lack of confidence in her ability to seek out and design numeracy focused tasks.

Drawing on this analysis, suggests that currently there is limited, if any, overlap between Erica’s ZPD, ZFM, and ZPA. Professional development activities, such as the workshops that are part of the Numeracy Project, are unlikely assist Erica to develop the capacity to embed numeracy in history because her current knowledge, beliefs, and perception of her professional context do not allow her access to the ideas presented. Assisting Erica’s trajectory towards a teacher identity where she has greater capacity to embed numeracy in history could be achieved by helping her to see how numeracy can support discipline learning in history; thereby alleviating some of the time pressure she feels (i.e., helping her to restructure her ZFM). However, this would need to be done in conjunction with providing her with opportunities develop a richer personal conception of numeracy and knowledge she needs for embedding numeracy into history; in other words, assisting her to expand her ZPD.

6.3 Common themes and differences

The potential for Karen and Erica to develop their capacity to embed numeracy in science and history, respectively (i.e., their ZPD), differs as a result of their different educational backgrounds and teaching experiences. However, at this point in time Karen appears to have a greater knowledge base for embedding numeracy than Erica. Both teachers have fairly limited personal conceptions of numeracy, when analysed in terms of Goos et al.’s (2014) numeracy model, and neither seems to fully understand the potential for increased discipline learning that embedding numeracy brings (e.g., Blow et al. 2012; Quinnell et al. 2013).

For both teachers, their ZFM/ZPA complex included a national curriculum (ACARA 2014) in which numeracy is seen as part of subjects across the curriculum and participation in a research project designed to support this approach. However, Karen and Erica interpreted their ZFM/ZPA complexes in different ways. The new curriculum (ACARA 2014) and Numeracy Project gave Karen permission to embed numeracy in science, whereas the new curriculum put Erica under pressure to cover the content and the Numeracy Project did not deliver the type of professional development she needed. In terms of classroom practice, Karen’s main constraint seemed to be the limitations imposed by what she felt her students allowed her to do, whereas for Erica it appeared to be the time pressure she felt as a result of the introduction of the new curriculum and her limited knowledge for embedding numeracy in history.

Although similarities were identified in the cases of Karen and Erica, the differences in terms of the way each teacher’s ZPD interacts (or not) with her ZFM/ZPA complex, suggests that different approaches (as described in Sects. 6.1 and 6.2) are needed to help these two teachers to develop the capacity to promote numeracy learning. At the beginning of the study, there was some overlap between Karen’s ZPD and her ZFM/ZPA complex; therefore, without further intervention and in the absence of any negative influences, her capacity to embed numeracy in science is likely to strengthen over time. The suggestions made in Sect. 6.1 could be expected to speed up this development by enabling her to enact in the classroom what is being promoted through her participation in the Numeracy Project, being mentored, and her self-directed reading. However, Erica appeared to have little, if any, overlap between her ZPD, ZFM, and ZPA. Therefore, further development of her capacity to embed numeracy in history is unlikely without further intervention. Such an intervention would need to create a ZFM/ZPA complex that includes some overlap with her current ZPD.

7 Concluding remarks

Being numerate “empowers people by giving them the tools to think for themselves, to ask questions of experts and to confront authority” (Steen 2001, p. 2). Therefore, providing opportunities for students to develop their numeracy capabilities needs to be seen an essential part of school education. Moreover, not only do subjects across the curriculum provide the contexts needed for numeracy development (Steen 2001), numeracy is an integral part of learning in all subjects (e.g., Blow et al. 2012; Quinnell et al. 2013). However, if teachers are to effectively exploit numeracy learning opportunities, then they need to possess a range of attributes that will facilitate this approach. One way of investigating these attributes is through the lens of teacher identity (Bennison 2015). However, the nature of teacher identity means that, in designing any empirical study, consideration needs to be given to not only what data should be collected but also how the factors that contribute to a teacher’s identity in a particular situation interact over time; that is, capture both the complexity and dynamic nature of identity.

Utilising the characteristics included in the framework for identity as an embedder-of-numeracy (Bennison 2015) to guide the design of an empirical study enables identification of what data should be collected; thereby, enabling the complexity of teacher identity, situated in the context of promoting numeracy learning, to be captured. In this article, Valsiner’s (1997) zone theory has been proposed as one way of providing insights into how these characteristics interact and change over time; thus, capturing the dynamic nature of this situated identity.

Although the case studies of Karen and Erica that were presented in this article are necessarily brief, they do exemplify some of the ways in which the sociocultural approach advocated here contributes to an understanding of how teachers from all disciplines can be supported to embed numeracy into the subjects they teach. The case studies of Karen and Erica suggest that if a teacher is to be best placed to embed numeracy into the subjects they teach, then they need have appropriate knowledge (mathematical, pedagogical, and curriculum), a rich personal conception of numeracy, and a belief that numeracy is an integral part of the subjects they teach. In addition, there are other factors, including access to appropriate resources (e.g., digital technology), which will impact on the ways in which this situated identity might develop. Further research is needed to investigate the impact of these factors and explore how the characteristics within the framework for identity as an embedder-of-numeracy (Bennison 2015) fully map onto Valsiner’s (1997) zones. This will afford an increased understanding of this particular situated teacher identity and inform the design of programs to assist teachers so that they are better able to promote numeracy learning across the curriculum.