Teachers’ understanding and use of mathematical structure

Mathematical structure is central to the interconnectedness of numerical, spatial or logical relationships, but it is not known how well teachers understand this concept or implement it in pedagogical practice. In this paper, we examine three junior secondary mathematics teachers’ understanding of mathematical structure and observe whether they apply it in their teaching. Case study data comprised teacher interviews and mathematics lesson observations. Analysis of interviews and teacher-directed communication (utterances) in lessons utilised an emergent framework for categorising mathematical structure: connections to other learning (C), recognising patterns (R), identifying similarities and differences (I) and generalising and reasoning (G). Findings from the interviews indicated that teachers supported the development of mathematical structure, but the interview responses were not generally reinforced by their utterances in mathematics lesson observations. Analysis of teacher utterances revealed superficial understanding and use of mathematical structure although there was some evidence of CRIG components in the teacher-directed communication for individual cases.


Introduction
Australia's declining performance on the international assessments, Trends in Mathematics and Science Study (TIMSS) (Mullis et al. 2015) and the OECD Programme for International Student Assessment (PISA) (Thomson et al. 2019) raises concerns about the effectiveness of mathematics teaching and students' engagement and learning of mathematics (Dinham 2013). A decreasing number of students are studying higher levels of mathematics in secondary school, a trend that continues beyond school, with fewer students studying mathematics at university and even fewer entering and completing mathematics teacher education programs (Smith et al. 2018). Engagement in learning mathematics has been found to decrease over the years of schooling (Attard 2011), and disengagement in mathematics learning related to procedural pedagogical practices (Mason et al. 2009). The reliance on procedural practices may also place a burden on the learner's memory, and consequently, the learner fails to develop conceptual understanding of the subject (Cragg et al. 2017).
A procedural approach is represented by mathematics teachers' overuse of textbook exercises and worksheets (Lokan et al. 2003;Vincent and Stacey 2008). Increasing pressure to perform well in examinations and implement curriculum may explain teachers' use of procedurally-focused pedagogies, even when they believe that conceptual understanding is important. Teachers are often persuaded to adopt procedural tasks due to time constraints resulting in less time teaching conceptual knowledge (Saeki et al. 2015). Boaler (2015) found that students lacked confidence in their mathematical abilities when they did not understand mathematical concepts learned through memorised and repeated procedures. Learners do need to know what procedures to use when solving problems, but procedural learning needs to be connected to the problem through conceptual understanding.
Building on the seminal work of Skemp (1976) on relational and instrumental understanding, Hiebert and Lefevre (1986) made the distinction between procedural and conceptual knowledge. Procedural knowledge is simply a sequence of actions that can be learned with or without meaning. Conceptual knowledge is rich in relationships, implying a link to existing cognitive structures, making it a higher-order thinking skill, which, in terms of mathematical thinking, refers to the development of generalisation and abstraction. Mason et al. (2009) described structure as the bridge that connects procedural and conceptual knowledge. Understanding a concept is attained by connecting the mathematical procedures when solving a problem to achieve deeper thinking about mathematics. Teachers' awareness of mathematical structure could support linking procedural and conceptual knowledge, but to use this awareness, teachers need to develop a deep understanding of mathematical structure through their mathematical content and pedagogical knowledge.
Conceptual knowledge of mathematics is also related to productive disposition. Woodward et al. (2017) identified that when a teacher's productive disposition supported students' conceptual knowledge of mathematics, students were more likely to have a positive attitude towards their learning. Mason et al. (2009) argued that a productive disposition is one where the learner develops an appreciation of mathematics through an understanding of mathematical structures and structural relationships. Students who are not encouraged to observe mathematical structure in their learning are obstructed from thinking deeply about mathematics. When teachers' pedagogical practices are developing structural thinking by relating and connecting the mathematical concepts and relationships in meaningful ways, then students can begin to think mathematically and become more engaged and motivated to learn the subject.
The notion of mathematical structure is implied in the Australian Curriculum: Mathematics (AC:M) (ACARA 2012) through the proficiency strands of understanding, fluency, problem-solving and reasoning. However, it is not known how teachers attend to and connect these processes in their teaching of mathematical concepts and relationships. Mason (2018) noted that teachers make pedagogical choices that have the power to promote structural thinking, and yet teachers themselves may be unsure about how these choices support the learning of complex mathematical relationships.
An exploratory descriptive study was designed to investigate secondary mathematics teachers' existing understandings and use of mathematical structure in their practice through pedagogical utterances. As an initial phase, the research aims to develop an observational framework to identify features of teacher practice that reflect attention to mathematical structure (Gronow 2016, Unpublished MRes thesis). Based on "structures of attention" (Mason 2004), the study develops and trials a connections, recognising patterns, identifying similarities and differences and generalising (CRIG) framework. These findings can then inform a second phase design study that implements a training program for pre-service teachers/professionals on how to support mathematical structure in teaching and learning (Gronow 2020, Unpublished PhD thesis).
Three research questions were examined: 1. How effective are teachers in demonstrating their understanding and use of mathematical structure, and can mathematical structure be categorised using an observational framework (CRIG)? 2. Do teachers promote structural thinking in their students? 3. Is there a discrepancy between what teachers say about mathematical structure and their classroom practice? If so, what is the nature of the discrepancy?

Mathematical structure
The concept of mathematical structure has a rich history in mathematics education, but not one that is clearly understood by many teachers of mathematics (Richland et al. 2012). Historically, Taylor and Wade (1965) proposed a theoretical definition of structure as the formation and arrangement of a system of mathematical properties. This definition points to the notion of mathematical relationships that is taken up by other theorists in later years. A more simplistic view of structure was suggested by Jones and Bush (1996) who used a "building blocks" metaphor to describe mathematical structure, stating that mathematical structure is like the foundation of a building, on which the content is built. They identified structural thinking in mathematics as a vehicle for helping students understand and answer the "why" questions in mathematics. Barnard (1996) also described it in terms of cognitive blocks of information rich in structure. However, these views reduce the notion of structure to the combination of individual parts that do not seem to be connected or conceptually different. A more integrated view was advanced by Schmidt et al. (2002) who recognised that a deeper knowledge of mathematical structure enables one to make connections between mathematical concepts. Based on the Mason's seminal work on mathematical thinking (Mason et al. 1982), Mason et al. (2009) describe mathematical structure as connecting mathematical relationships, recognising patterns, identifying similarities and differences and generalising results. They defined mathematical structure as "the identification of general properties which are instantiated in particular situations as relationships between elements or subsets of elements of a set" (p. 10). Other researchers, Fischbein and Muzicant (2002), Stephens (2008) and Stephens and Armanto (2010), described "structure" as synonymous with relational thinking (Skemp 1976). Vale (2013) identified students' relational thinking when they solved problems by using known relationships, such as equivalence. Similarly, Mulligan and Mitchelmore (2009) associate structure with young children's ability to recognise patterns and relationships. An awareness of mathematical structure is demonstrated through representing relationships between the properties of mathematical concepts albeit simple or complex. Mason (2018) recognised relationships between properties as being instantiated into the heart of school mathematics, and in arithmetic, it is the properties of numbers that are most useful; calculations are the byproduct of and not the focus of attention.
Teacher knowledge and "noticing" of mathematical structure Studies that examine teachers' understanding of mathematical structure have assessed key components of mathematical structure and relationships: making connections within mathematics learning (Albert 2012;Jones and Bush 1996;Richland et al. 2012), recognising patterns (Mulligan and Mitchelmore 2009;Stephens 2008), identifying similarity and difference (Barnard 1996) and generalising and reasoning (Mason 2018;Vale et al. 2011;Watson and Mason 2005).
Other studies have attended to mathematical structure through teachers' understanding of what they notice in students' mathematical thinking. Early descriptions of teacher noticing were proposed by Mason (2002) and later described by Sherin et al. (2011). Van Es and Sherin (2008) described teacher noticing as attending to events and making sense of those events as they occur in mathematical thinking. A framework for noticing mathematical thinking has emerged (Jacobs et al. 2010;Kaiser et al. 2017; van Es 2011) focused on identifying students' relational thinking. Scheiner (2016a) advanced the construct of teacher noticing by arguing for a convergent model of how students abstract and connect mathematical structures. Scheiner pointed out that not all we attend to is consciously perceived. Noticing can include implicitly attending to an action or behaviour without awareness of it. Equally, we can explicitly attend to an action or behaviour with complete awareness. Scheiner proposes that teacher noticing is not just about what one sees through the teacher's eyes but also includes what teachers are noticing with their own "mind's eye." Similarly, Choy and Dindyal (2017) discuss that developing teachers' "eyes" and developing their minds to make sense of mathematical relationships are crucial for developing learning experiences. Teacher noticing is essential for teachers' awareness of what they attend to and the sense they make of what they notice.
The notion of teacher understanding of mathematical structure can be seen to cross pedagogical and content boundaries; it bridges what to teach with how to teach. Vale et al. (2011) linked pedagogical content knowledge (PCK) (Shulman 1987) and teacher mathematical knowledge (TMK) (Hill et al. 2005) to mathematical structure and structural thinking. Ball et al. (2008) argued that the work of Shulman was key to research on teaching and built on this through specialised mathematical content knowledge (MCK) to understand teachers' professional knowledge. Blömeke et al. (2011) used MCK and teachers' mathematical content and pedagogical knowledge (MCPK) to determine the cultural differences of primary teacher education programs across 15 countries. Teachers' MCPK was identified as having a significant impact on students' mathematical performance. Further studies (Buchholtz 2017;Ivars et al. 2018) have linked teacher noticing to MCK.
Teaching for mathematical understanding, through deep MCK, also enhances student engagement in mathematics. In a study on mathematics in the middle school years, Attard (2010) found that teachers who give clear explanations of mathematical procedures and concepts know their students' learning needs and provide the students with the foundation for prolonged engagement in mathematics.
A fundamental understanding of mathematical structure can support teachers' professional learning and can positively influence confidence and self-efficacy. Vale et al. (2011) educated practising out-of-field mathematics teachers to appreciate mathematical structure. An appreciation and awareness of structure developed through the teachers' recognition of mathematical relationships and properties, resulting in a deepening of their structural understanding of mathematics. The teachers' confidence improved as they were able to make the connections between mathematical relationships and properties to promote students' structural thinking. Vale et al. (2019) recommended teachers use prompts that include making connections, looking for patterns, knowing what stays the same or differs and shifting reasoning towards generality so students develop a structural understanding of mathematics.
Teachers' awareness and use of mathematical structure impact on their interpretation and implementation of the curriculum and the way they promote mathematical thinking. The New South Wales (NSW) Mathematics Syllabus for the Australian Curriculum (Board of Studies NSW 2012) identifies mathematical structure in the Working Mathematically processes, which reflect the proficiency strands of the AC:M. The key components of structure are embedded in the processes of communicating, problemsolving, reasoning, understanding and fluency. However, there is limited research on how teachers utilise their knowledge of structure in relation to the explicit teaching of the AC:M proficiencies and the Working Mathematically processes. Cavanagh's (2006) research on teachers' understanding of Working Mathematically processes was conducted when Working Mathematically was quite new in New South Wales. He found that there was a limited understanding of Working Mathematically processes held by teachers. Thirty-nine teachers were interviewed to understand their interpretation and implementation of Working Mathematically when teaching mathematics. Cavanagh's findings showed that teachers did use some of the processes to develop students' structural thinking. Further research is needed to investigate the AC:M proficiency strands and the NSW Working Mathematically processes in practice and how mathematical structure is utilised in relation to these strands.
Theoretical framework: structural thinking Scheiner (2020) identified the diversity of theories in mathematical education, and this is particularly relevant to research concerning structure. A number of theoretical approaches have been developed on mathematical structure in teaching and learning mathematics. Notably, Dubinsky's (1991) Action Process, Organisation and Schemas (APOS) theory describes the construction of concepts through knowing the procedures used. Following the work of Hiebert and Lefevre (1986), Gray and Tall (1994) combined procedural and conceptual knowledge describing "proceptual" thinking as the ability to think flexibly across mathematical processes and concepts. Schwarz et al. (2009) proposed a model of mathematical thinking of recognising, building, constructing and consolidating. This new construct required establishing relationships and connections between known constructs. More recently, models have been developed that provide more dynamic and holistic perspectives of mathematical structure. Bass (2017) focused on theory building which required students to recognise, articulate and name the mathematical concepts so as to identify the common structures of different problems. Scheiner and Pinto (2019) identified several models that identify cognitive processes that have connections with mathematical structures, building on Scheiner's (2016b) convergent model of abstraction of mathematical structures.
The theoretical framework developed in this study can be traced to Skemp's (1976) seminal work on learning mathematics. Skemp (1976) made a distinction between instrumental and relational understanding of mathematics. He explained that instrumental understanding was like having a fixed plan that consisted of a starting point and finishing point, with explicit instructions and directions of how to complete the plan. Relational understanding involved building up a conceptual structure or a schema that offered an unlimited number of starting points towards any finishing point, with multiple paths to get there. In this process, learning involves understanding and exploring interrelationships between mathematical concepts. His ideas about instrumental and relational understanding in mathematics learning remain central to new theories relating to procedural and conceptual knowledge (Hiebert and Lefevre 1986). Skemp's contribution is reflected in the work of Mason and colleagues who developed the notion of relational or structural thinking as mathematical thinking and noticing. Mason (2004) considered thinking processes as a demonstration of structural thinking when referring to structures of attention to dissect schemas when solving a problem. He identified cognitive processes when learning mathematics as "what learners are attending to" (p. 17). He promoted structures of attention as what learners need to be aware of, what they notice and how they attend to it. The same focus of attention can be applied to teachers considering their structural thinking processes. More recently, Ivars et al. (2018) integrated Mason's (2004) structures of attention into a framework for teacher noticing of students' mathematical thinking. They believed that teacher noticing of mathematical thinking would improve if there was a framework for them to follow.

Development of a framework: Connections, Recognising Patterns, Identifying Similarities and Differences and Generalising and Reasoning (CRIG)
This study builds upon the five modes of what people attend to or notice, developed by Mason (2004) as "structures of attention" when learning mathematics. The extra demand for teachers is to know how to communicate their mathematical thinking so that others make sense of it. That is, the teacher has to know how others think about the mathematics and what is happening when they are doing mathematics. Mason Mason identified these modes as what teachers attend to and designed problems that embedded these features of mathematical structure. He utilised diagrams or expressions to draw the learner's attention towards structural thinking skills to solve the problems. He did this by asking probing questions, such as What do you see when you look at this picture? What do you look at? What questions come to mind? (Mason 2004, p. 9.) Mason invites the learner to look for relationships between numbers, symbols and shapes. The learner is directed to reflect on the structure of the problem by considering the key components of connecting relationships, recognising the patterns, identifying the similarities and differences and generalising and reasoning about a result. Mason viewed these key components from a thinking mathematically perspective and not from mathematics as procedural learning. He argued that these key components of structure need to be identified and promoted by the teacher, as mathematical thinking skills that can support students' structural thinking. In noticing his own mathematical thinking, Mason raises the key question-What do mathematics teachers think when communicating with learners? (Mason 2018). The CRIG framework was developed from Mason's (2004) five forms. The first form of holding wholes (gazing) is synonymous with connecting because the reference here is to viewing or gazing by making connections to what is known, what can be seen or interpreted. The second form, recognising relationships is associated with patterning. Mathematical relationships are often recognised through patterns and identifying similarities and differences. The third form, discerning details, also relates to pattern identification. The two forms of perceiving properties and reasoning are aligned with processes of generalising and reasoning.
An overview of the four components of mathematical structure, referred to as the CRIG framework, is provided as follows: & Connections (C) between contexts or concepts to allow for an informal process of mathematical understanding. Making connections with prior learning of mathematics supports students' reasoning. & Recognising patterns (R) identifies the importance of patterning, awareness of patterns and reproducing patterns as essential for mathematical development. The association with structure generates mathematical knowledge and understanding. & Identifying similarities and differences (I) is primarily built on sorting and classifying objects into like or unlike categories. Equivalence, as a different notion of sameness, can also develop through experience and extends into more subtle differences in mathematical representations. & Generalising and reasoning (G) as Mason (2008) described, generalising is an innately human activity that should be built upon to develop a more in-depth and exciting experience of mathematics. Generalising and reasoning encompasses the other three components as a higher-order thinking skill. Through identifying similarities and differences, one can recognise patterns connected to a general rule.
Using Stephen's (2008) example, □ +17 = 15 + 24, the CRIG components may be utilised by a student to find the missing number. Instead of adding or subtracting, the focus could be on structural relationships of equivalence; that is, since 17 is 2 more than 15, to maintain equivalence, the missing number is 2 less than 24. Connections (C) to prior learning and awareness of mathematical structure require that learners notice particular connections, patterns, similarities, differences, logical (and analogical) connections and generalisations. Identifying similarities and differences (I) between the numbers and using the notion of equivalence could support a student's generalising and reasoning (G) that there is a general rule for all equations like this. Looking for and recognising the underlying structure of number and arithmetical processes is an essential aspect of mathematical thinking. Bishop et al. (2016) referred to features of mathematical structure that are reflective of the CRIG components, such as: recognising connections between structures; seeing the fundamental properties of commutativity, associativity and distributivity as generalised patterns and looking for similarities and differences between known problems and making generalisations about whole numbers.
The CRIG components can be identified in a teacher's communication with learners and may be integrated into a teacher's pedagogical practice. Vale et al. (2011) reported on a professional development program to improve mathematical pedagogy and content knowledge of out-of-field teachers of mathematics. The authors found evidence of awareness and knowledge of mathematical structure when teachers recognised properties and understood that these properties can form a generalisation. Further to this, they described teachers making connections by observing and using similarities and differences and by developing relationships and properties. Vale et al. found evidence of mathematical structure that aligns with the CRIG components and demonstrates their interrelated nature. They do not exist in isolation and can appear as combinations of each. For example, a pattern of numbers such as 2, 4, 6, … is immediately recognised as the even numbers (connections), the pattern (recognising patterns) is the difference between each consecutive number (identifying similarities and differences) and knowing that the pattern progresses to 8, 10, 12, …, n, n + 2, … leads to a general rule to explain the pattern (generalising and reasoning).

Method
An exploratory study of three secondary mathematics teachers' understanding and use of mathematical structure comprising of a series of interviews and classroom lesson observations was conducted over a 1-month period. The study adopted an interpretive research paradigm (Cohen et al. 2013) that aimed to describe, through qualitative analyses, teachers' perceptions and views about their own understanding of mathematical structure. Integral to the design of the study was the development of the CRIG framework to identify teachers' awareness of mathematical structure and whether structural thinking is promoted when teaching mathematics.

Context and participants
The study took place in one comprehensive secondary boys' school in metropolitan Sydney. The school comprised approximately 600 students from backgrounds of high socioeconomic status with approximately 15% of students with language backgrounds other than English. The student population was representative of an average academic standard on national assessment (ACARA 2018). All eight teachers on the mathematics staff were invited to participate, and of this group, three volunteered as case studies. Background information on the three case study teachers (identified as Teacher A, B and C) is given in Table 1.
Teacher A was a trained science teacher but only taught mathematics at this school. Teacher B was the least experienced teacher of 3 years with a university mathematics teaching degree. Teacher C was the most experienced of 17 years with mathematics and physical education teaching qualifications and held the position of head of department.

Data sources, instruments and procedures
Two types of data were collected from the three participants: a structured interview and classroom observations of three sequential mathematics lessons. The instruments and data collection procedures were piloted with three mathematics teachers from a comparable school, prior to the main study. After the piloting of the instruments, a discussion with the piloting teachers resulted in the instruments and data collection procedures being modified and refined in preparation for the main study. After observing the pilot teachers in the classroom, the researcher reformatted the lesson observation template to simplify the recording of teacher utterances.
The data collection occurred during a 4-week period of teaching during term 2 (April-June). Interviews were conducted 1 week prior to the classroom observations.

Interviews
The three teachers were interviewed individually using a structured interview comprising six questions (see Table 2). For consistency and clarity in interpreting the interview questions, it was considered necessary to provide teachers with a definition of Some authors describe mathematical structure as the building blocks of mathematical learning. Mathematical structure can be found in connecting mathematical concepts, recognising and reproducing patterns, identifying similarities and differences, and generalising and reasoning results. Students who perform structural thinking use these skills without always considering them when solving problems. Many students need to be taught these skills when introduced to concepts as a reminder of how to think mathematically.
Questions 1 and 2 focused on teachers' understanding of mathematical structure, Question 3 was concerned with identifying the teachers' knowledge of where mathematical structure appears in the NSW syllabus and Questions 4, 5 and 6 focused on teacher noticing of their students' structural thinking. Each interview lasted approximately 10 min and was audio recorded and then transcribed and entered into NVivo 10 (QSR International 2018) qualitative data analysis software for coding and analysis.

Lesson observations
Following the interviews, three consecutive 1-hour mathematics lessons were observed over a 1-week period for each teacher. The researcher was positioned at the back of the classroom and did not communicate with the students or teacher during the lesson. The students were advised that the visiting researcher was observing the teacher's pedagogical practice and that no student data would be collected. Using the noticing as an expert (Miller 2011) technique, the researcher identified and recorded teacher-directed communications (utterances) during the lesson that referred to the mathematics of the lesson.
As ethical consent for digital (audio and video) recording could not be obtained for classroom observations, the teachers' utterances, such as phrases and notable examples In what way does the NSW K-10 mathematics syllabus for the Australian Curriculum-Mathematics identify mathematical structure?
4 How do you encourage your students to use structural thinking skills in mathematics?

5
How would you recognise structural thinking in any of your students? 6 What are the benefits of students using structural thinking in mathematics?
that explicitly or implicitly referred to any CRIG component, were entered into a lesson observation template for recording purposes. Later, these records were transferred to an Excel spreadsheet for analysis. Table 3 presents background information regarding the classes and lessons observed for each teacher. For each lesson, teachers revised content from the NSW Mathematics K-10 syllabus (Board of Studies NSW 2012) in preparation for the end-of-term examination. Teachers were not expected to use the CRIG processes in their teaching and were instructed to teach the lesson according to their regular practice. Although the lesson content varied, there were ample opportunities for the teachers to use instructional language and strategies that could reflect the CRIG components. All three teachers used a teachercentred approach with an initial modelling and explanation phase, followed by discussion between the teacher and students. Teachers modelled responses to worked examples on the blackboard and provided similar examples for the students to complete. The teachers then reviewed the responses from individual students at the front of the class and shared these with the whole class. At times, students were engaged with similar examples either from their designated mathematics textbook or an online learning program using their laptop computers. The lessons comprised tasks to assist students' preparation for examinations. While the tasks across the nine lesson topics required conceptual understanding and problem-solving, most teacher instruction prompted procedural responses. However, teachers drew upon CRIG components inadvertently to revise and explain concepts and processes.

Data analysis
The three teachers' interview transcripts, providing a total of 18 responses from the six questions, were coded for words and phrases associated with the CRIG components, as NVivo primary nodes. In total, 29 CRIG references were identified. For Questions 4, 5 and 6, further analysis using NVivo enabled coding of the nine interview responses for teacher noticing of their students' structural thinking. Three broad themes emerged: recognising students' structural thinking, student engagement when thinking structurally and how structural thinking benefitted students.
Analysis of the lesson observation data focused explicitly on 227 recorded teacher utterances relating to one of the four CRIG components of mathematical structure. Initial coding of these utterances was subject to independent coding for 20% of responses where discrepancies were found for 3% of these responses. These discrepancies were resolved after the researcher and an independent coder reached agreement on the categorisation of the CRIG examples.

Results
The presentation of results and discussion focuses primarily on the interviews and lesson observations to identify common patterns and individual differences in the three cases. The CRIG components provide a framework to interpret and describe the teachers' use and understanding of mathematical structure.

Teacher interviews
The next section summarises the interview responses according to teachers' interpretation and understanding of mathematical structure, teachers' references to the CRIG components and teacher noticing of students' structural thinking.

Teachers' interpretation and understanding of mathematical structure
The teachers' responses to the first interview question provide insights into their understanding and use of mathematical structure. Responses to the first question reflected three different interpretations of the meaning of mathematical structure such as building blocks of knowledge, curriculum and programming structures and structure of mathematical thinking processes. Comments based on a "building block" metaphor (Jones and Bush 1996)  The teachers' use of the "building block" metaphor reflects the idea that new knowledge is connected to and built upon previously learning. This aligned with the connections component of the CRIG framework, but the teachers' views reflected the idea of connecting rather than Barnard's perspective as cognitive blocks, or small separate structures of a greater whole.
Mathematical structure was also referred to by Teachers A and B in terms of the organisation of the mathematical content in programming and the curriculum. Teacher A compared the difference between the way the curriculum is designed with how teachers present the content in the classroom, and Teacher B referred to the structure of mathematical knowledge and programming content: So, I think the structure that teachers use is different to the structure of the curriculum, as it must be more generic. [Teacher A] I think putting, like putting earlier work at the beginning of it … things you need to do it as a review before that. [Teacher B] Teachers A and C referred to mathematical structure related to students' ability to justify and reason in their written solutions to mathematical problems, representing the CRIG component of generalising and reasoning: If they were able to structure the [written] solutions. [Teacher A] My big thing with structure is to get them to show their process, their thinking.

[Teacher C]
The teachers also indicated their familiarity with mathematical structure as it appears in the curriculum in their response to Question 3. Teacher B expressed a concern of not knowing where mathematical structure appeared in the curriculum:

I don't know how to answer it to be honest [Teacher B]
Teacher A suggested that mathematical structure was about the sequential order of the syllabus.
In the way it's set out in the stages, I think you are picking the strand and I think the structure is how you are developing through the strand. [Teacher A] Teacher C associated mathematical structure to the Working Mathematically processes of the NSW K-10 Mathematics syllabus (Board of Studies NSW 2012).
In our syllabus, mathematical structure comes in the Working Mathematically.

Teachers' references to the CRIG components
The interview transcripts were analysed for references to the CRIG components. The frequency of these references is given in Table 4.  Table 4 shows that 29 references to a CRIG component were made by the teachers during the interviews. There were marked differences between each teacher's responses and the frequencies between the CRIG components. Teacher A made the least number of references to the CRIG components, one for connections and one for recognising patterns. Teacher B made 10 references: five for connections, four for identifying similarities and differences and one for generalising and reasoning. Teacher C provided the most responses with five for connections, two for recognising patterns, five for identifying similarities and differences and five for generalising and reasoning.
Recognising patterns and generalising and reasoning had the lowest frequency of the all the CRIG components for Teachers A and B. Teacher B did make references to connections and identifying similarities and differences, but the overall references to the CRIG components by Teachers A and B do not indicate their overall awareness of mathematical structure.
Teacher C's responses reflect an awareness of mathematical structure. Teacher C's references were insightful; for example, the generalising and reasoning reference showed the relationship of generalising to the development of a rule, "I realise they were looking for the generalisation and coming up with the formula".
Teacher C's statements reflect an understanding of mathematical structure through rich mathematical relationships. Teacher C, the most experienced mathematics teacher with a teaching qualification in mathematics and as the head of department, demonstrated a greater awareness of mathematical structure than either Teacher A, who was not trained as a mathematics teacher, or Teacher B, who was a novice teacher.
The participants' responses are consistent with the results of Miller (2011) who showed that expert teachers, like Teacher C, notice what to react to in a classroom more than novice and out-of-field mathematics teachers. Novice teachers, like Teacher B, focused on different aspects of teaching (Borko and Livingston 1989), and out-of-field teachers, like Teacher A, need to cultivate an appreciation of mathematical structure (Vale et al. 2011).

Teacher noticing of students' structural thinking
Analysis of the responses to interview Questions 4, 5 and 6 identified that teachers believed structural thinking was a reflection of students' mathematical ability and that students would not achieve effectively without it. The following are exemplars of teachers' statements that reflect this opinion: But the lower ability kids don't have that ability to structure. [Teacher A] If they don't have that structure in the beginning, then they'll become completely lost when they have challenging questions. [Teacher B] (With structural thinking) they will be able to access the higher order thinking questions, if they do not have structural thinking, they really get stuck. [Teacher C] These responses reflect the teachers' view that structural thinking in mathematics is necessary for achievement and engagement. The analysis of transcripts indicated that this interpretation of mathematical structure was fixed or reflected innate ability in their students. The responses from all three teachers, particularly Teacher A, highlighted the teachers' problem that "lower ability kids don't have that ability to structure". Teacher B stated that students need structural thinking to attempt challenging tasks, and Teacher C related structural thinking to higher-order problems. These views may explain why the teachers focussed largely on procedural instructional practices in their lessons. Table 5 presents frequencies of teachers' interview responses that are indicative of noticing students' structural thinking through the three themes of student engagement, teacher noticing of students' structural thinking and benefits of structural thinking. These three themes were identified in the NVivo analysis process of the interview responses. The statements (n = 48) reflected teachers' understanding of mathematical structure when they were talking about their students' behaviours.
Exemplars of teacher statements categorised by the three themes in Table 5 are given below.
The teachers associated the first theme, of student engagement, with structural thinking through mathematical achievement, challenging mathematical tasks and confidence in mathematics.
So, we need to identify the structure that they need to learn and then once they can see their achievements they are going to be engaged. [Teacher A] If they don't have that structure in the beginning, then they'll become completely lost when they have challenging questions. [Teacher B] The students who don't have the structural thinking lack confidence in their ability. [Teacher C] The teachers made more references to how they noticed students' structural thinking, including how the students presented their mathematical thinking through their written work, explaining their thinking and students' responses to the teacher's questions. When asked about benefits of structural thinking for students, the teachers said it reinforced their students' knowledge, prepared them for deeper understanding and gave students the ability to complete harder questions.

Lesson observations
All three teachers were involved in teaching content from the Number and Algebra and Measurement and Geometry strands of NSW Mathematics K-10 syllabus (Board of Studies NSW 2012). The outcomes for the strands of Number and Algebra and Measurement and Geometry include: connect mathematical ideas; recognise and explain mathematical relationships with reasoning; compare, order, calculate and generalise number properties and create and display number patterns. In total, 227 utterances made by teachers that referenced the CRIG framework were documented across nine lessons. Table 6 presents frequencies of the utterances coded as one of the four CRIG components. Given that the lessons often reflected teachercentred practices, the 227 documented utterances represent only the components of the CRIG framework, and other aspects of the teachers' talk were not recorded.
Table 6 data confirm that the teachers made utterances that could be categorised into one of four CRIG components, but it is not clear from these data if they were encouraging students' structural thinking. Most of the generalising and reasoning utterances were not explicit, without depth in identifying mathematical relationships. The inconsistency between the frequency of generalising and reasoning utterances in Table 6 and in the teacher interview statements revealed large discrepancies between the teachers' understanding and use of mathematical structure in the language of instruction. Table 7 gives exemplars of utterances made by each of the teachers, coded for CRIG components which ranged from superficial or procedural-type language to support students' attention to structural thinking (see Table 8 following).
These exemplars, gleaned from the lesson observations, raise questions about the depth of teachers' attention to mathematical structure in their questioning and language of instruction. The utterances ranged from superficial statements or closed questions to a more in-depth, analytical approach aimed at relational understanding. Hence, all utterances were subject to secondary coding, as either analytical or superficial. Table 8 presents the secondary coding of utterances as superficial or analytical, showing differences for each teacher. The analytical and superficial utterances were coded in reference to the context of the teacher's commentary at that time in the lesson. Analytical responses were categorised when the teacher asked a question or gave an explanation which required students' structural thinking and probed for conceptual and relational understanding. These utterances were beyond remembering a fact, procedure or a yes/no response typical of superficial understanding. For example, in the recognising patterns exemplars, the analytical statement refers to the teacher demonstrating an understanding of reciprocal relationships using appropriate language "inverse". The superficial statement involves remembering the two times' table or counting by twos. Exemplars of teachers' analytical and superficial utterances are provided below.    Remember, you do multiplication before you do addition or subtraction [Teacher A].

Recognising patterns
Analytical The secondary coding of utterances in Table 8 shows that the number of utterances coded as analytical is almost half that of superficial. It is hard to know from these results whether the particular CRIG components highlighted as analytical were supporting a structural understanding of the lesson topic or if the students were developing structural understanding, as this study did not collect data from the students.
The number of analytical connections and generalising and reasoning utterances was higher than the superficial utterances, demonstrating that the teachers did use structural awareness at times. There were few analytical references to recognising patterns and identifying similarities and differences which indicates that teachers did not articulate these in their pedagogy but they may still have been aware of them. The high frequency of teachers' superficial utterances appears to reflect a lack of structural awareness.

Discussion
The focus of this study was to describe secondary mathematics teachers' existing understandings and use of mathematical structure in their practice through an analysis of their teacher-directed communication (utterances). As an initial phase, the research aimed to develop and trial an observational framework (CRIG) to identify features of teacher practice that reflected attention to mathematical structure.
The first research question asked how effective teachers are in demonstrating their understanding and use of mathematical structure. It appears from the interview data that there is evidence, although inconsistent, that the teachers' do have an understanding of mathematical structure and believe it is important in students' mathematical thinking. In the interview responses, the teachers' elicited varying interpretations of mathematical structure from simplistic notions of conceptual "building blocks" through to understanding mathematical relationships and generalisations. One teacher articulated that some students cannot think structurally on the basis that students lack the ability or the requisite background knowledge for structural thinking. It was also revealed that the teachers believed that students needed to be taught how to think structurally because structural thinking does not always occur naturally, reinforcing Stephens and Armanto's (2010) assertion that structural thinking needs to be taught.
In their pedagogical practice, the teachers' utterances often referred to features of mathematical structure in accordance with Mason's (2002) views; however, these references were mostly superficial and inconsistent across lesson topics and teachers. In line with Scheiner (2016a), teachers may have used mathematical structure but did not necessarily notice that they were doing so. It was not possible to provide evidence of this as the teachers were not interviewed post-lesson.
The opportunity for the teachers to demonstrate their approach to mathematical structure in their pedagogy may have been limited by the conditions of the study. At the time, all classes were involved in revision lessons for end-of-term examinations, and these lessons were predominantly teacher centred. Identifying mathematical structure in the teachers' communication was possibly hampered because the lesson purpose was not conducive to practices that supported structural thinking. This could be likened to regular teaching practice that relies on procedural methods and completion of tasks or exercises, not necessarily exam preparation. Teacher B, in the interview, acknowledged that there were no opportunities to develop students' structural understandings, as the focus was on students' performance in examinations. However, the teacher's classroom utterances reflected thinking about structure at times.
The CRIG components developed as an observational framework proved feasible in categorising teacher utterances with a high level of inter-coder reliability obtained. At times, utterances could be coded for more than one CRIG category so this complexity needed to be addressed. Further refinement and validation of the CRIG framework as a tool for measuring teacher attention to structure is essential if it is to prove a reliable tool for observation and professional learning.
The second question asked if teachers promote structural thinking in their students. It was found that the teachers associated student engagement with structural thinking. They thought that this directly impacted on student achievement, setting challenging mathematical tasks and developing confidence in mathematics for their students. Cavanagh's (2006) results are reinforced in this study as the teachers recognised the benefits of structural thinking but lacked the ability to explain structural thinking or direct students' thinking in the classroom towards structural thinking. Mason et al. (2009) raise this issue that teachers may notice students' structural thinking but are not able to articulate or build upon it. Teachers may need to adopt a broader view of structuring as a generic ability or as something that happens when a learner's attention is prompted towards important mathematical ideas and relationships. This is an important distinction since teachers not only need to be aware of the importance of structural thinking in their teaching but also need to notice whether students are developing structural thinking.
The third research question asked if there was a discrepancy between what teachers say about mathematical structure and what they do in practice. The data certainly revealed this discrepancy, but it remains unclear why such a disconnect exists. Teachers' interview responses, supporting mathematical structure, were not reflected in 65% of the classroom utterances when the CRIG components were applied. Further, the analysis of teachers' utterances from the lesson observations did not indicate they were attending to mathematical structure in a deep way. The superficial/analytical distinction between the utterances may reflect to some extent the discrepancy with the teachers' interview responses; that is, the teachers claimed that structure was important without understanding the nature of mathematical structure or how to prompt attention to the features and relationships that comprise those structures. However, these teachers may have been attending to mathematical structure in other forms such as feedback on student learning or in other lessons that were not the subject of these observations. Cavanagh's (2006) findings that teachers have limited understanding of Working Mathematically further support the idea that the teachers may have a "hidden" understanding of mathematical structure, but their ability to articulate it clearly and to attend to it explicitly in their teaching requires the teacher to pay closer attention to their practice.
By adopting Mason's (2002) approach to noticing, the CRIG framework can be utilised in professional learning as a form of directing teachers' attention to mathematical structure. Mason's (2004) structures of attention highlights how one approaches and deals with what is happening when doing mathematics. Mason (2002) asserted that "every act depends on noticing" (p. 7), and he introduced the term "awareness" to characterise the ability to notice. For mathematics teachers, this awareness comes from the ability to identify mathematical relationships in their thinking and to include these in their instruction, so they are aligned with students' thinking and understandings (Krupa et al. 2017).

Limitations
A key limitation is the small number of participants and the scope of the data collection, and as such, these findings are limited to these three cases and do not permit generalisation. This study provides a descriptive account of the findings based essentially on Mason et al.'s (2009) theoretical approach to mathematical structure. The analysis might have been more explanatory in terms of supporting broader theoretical perspectives on mathematical structure, but given the pilot nature of the study, it could then inform the design and implementation of a larger study on pre-service mathematics teacher noticing of structural thinking (Gronow 2020, Unpublished PhD thesis).
The research method could have been improved by videoing lessons and monitoring teacher planning and practice over a much longer timeframe. There was limited opportunity to access a range of teaching practices given school schedules and mathematics class timetables. An additional limitation was that there was no post-interview or follow-up to monitor any change in the teachers' pedagogy or their awareness of mathematical structure following this study. The study's focus may have also been improved by including an investigation into the teachers' awareness of and attention to Working Mathematically processes identified in the mandated NSW Mathematics K-10 syllabus (Board of Studies NSW 2012).
In the development of the CRIG framework, three problems were raised regarding categorising teacher utterances according to the CRIG components. First, an assumption was made that if an utterance was allocated to a CRIG component, then it provided evidence of the teacher's structural awareness. However, it became obvious that not all utterances allocated to a CRIG component were sufficiently deep or analytical enough to warrant awareness of mathematical structure. The second problem was that not all utterances could be assigned to a single CRIG component. To address this, utterances could be allocated to a new combined CRIG component. A third issue was that the notion of a discrete category may have been limited in that there was no attempt to indicate qualitatively different levels or progressive features of structural awareness. This could be best achieved through longitudinal observations of teaching practice.

Conclusions and implications
This study revealed that the teachers believed structural thinking was beneficial to students' learning of mathematics but these views were not reflected to a large extent in their responses from interviews or from mathematics lesson observations. Analysis of teacher utterances revealed superficial understanding and use of mathematical structure, although there was some evidence of the use of CRIG components in 35% of the teacher-directed communication for individual teachers. They did not directly articulate students' structural thinking when teaching, and their consideration of mathematical structure was not obvious in their communications and pedagogical practice. This aligns with Cavanagh's (2006) findings that teachers believed that mathematical content was best delivered through a teacher-centred classroom with students working quietly on their own. Mason's (2002) work on noticing, followed by the development of structures of attention (Mason 2004) and the evolution of mathematical structure (Mason et al. 2009) form the basis of how this study has developed an emergent framework for mathematics teachers to notice structural thinking in their teaching.  recently describes "structures of attention" as a micro view of teachers' links to perception, thinking and reasoning. Explicitly communicating the CRIG framework is one way that teachers' awareness of mathematical structure can develop while providing a focus of what to attend to when teaching. Mason (2011) pointed to the necessity for what teachers attend to must be on the same path as the students or else communication is lost.
Following further development in the future, the CRIG framework may offer teachers an opportunity to attend to mathematical structure in their pedagogy and to model structural thinking for their students. Consideration should be given to lesson design and how teachers' understanding and use of mathematical structure could be developed through these means. Students' structural thinking may develop through the teacher explicitly modelling components of the CRIG framework with suitable problems or tasks. Capturing this development would require closer consideration of the interaction and communication between teacher and student.
Further studies may uncover how the CRIG framework could improve students' engagement in mathematics learning (Attard 2011) and improve productive dispositions (Mason et al. 2009;Woodward et al. 2017). There remains a need to investigate how both primary and secondary teachers can utilise and apply the CRIG framework in their pedagogical practice for students across a wide age range. To understand mathematical structure, teachers may need tailored professional learning that addresses teachers' varying views and MCPK. Mason et al. (2009) pointed out that mathematical structure cannot be directly taught as a topic or as a process skill. However, by attending to the CRIG framework, teachers can develop an awareness of mathematical structure. Engaging in the CRIG framework supports the AC:M proficiency strands (ACARA 2012) and Working Mathematically processes within the New South Wales K-10 mathematics syllabus (Board of Studies NSW 2012). Effective teacher professional learning and pre-service programs can introduce the CRIG framework in alignment with the AC:M proficiency strands and Working Mathematically processes. Further research on how the CRIG framework could be developed in pre-service teachers' pedagogical practices was conducted as a second phase to this study.
Funding information This study was supported by the NSW Catholic Education Commission through the Br. John Taylor Education Research Fellowship awarded to the first author in November 2014. Parts of this analysis were reported in the paper, Teachers' understanding and use of mathematical structure (Gronow, Mulligan, & Cavanagh, 2017), 40 years on: We are still learning!: Proceedings of the 40th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 286-292). Melbourne, MERGA.