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Teaching Classroom Mathematics: Linking Two Pedagogical Models for Promoting Student Engagement and Conceptual Connections

This paper explains how an original conceptual framework model for mathematics pedagogy, the Australian Curriculum Conceptual Rubric (ACCR), has continued to be used successfully by the author in pre-service and in-service teacher education programs over the past ten years or more. Now further enhanced by a deeper reflection upon Peter Sullivan’s Six Principles (2011) for the effective teaching of classroom mathematics, the ACCR is based on four preparatory “big questions” that the teachers may ask of themselves and their students. The model is also a sequenced system of conceptual “rubrics” whose aim is to encourage, in new teachers especially, a beginning sense of hierarchical mathematics concept building and connectedness. Using Sullivan’s Principles for corroboration, the ACCR presents some useful ideas for helping teachers to keep track of the important elements of practical, effective teaching, and to use engaging and meaningful language in their classrooms.


Introduction
Mathematics as a discipline is usually agreed to be some kind of negotiated, organised construct of knowledgeand the exact nature of this construct has been a source of debate and fascination for a very long time. Yet whether the structure of mathematics is primarily envisaged as hierarchically fixed, or as organically evolving, or as something in between, less experienced mathematics teachers often have difficulty in accessing any kind of clear visual representation of a connected mathematics. This paper continues the author's interrogation of a more practical model for looking at mathematical connections for teaching that is consistent with the Australian Curriculum for mathematics (AC, ACARA, 2013).
The framework described in this paper, the Australian Curriculum Conceptual Rubric (ACCR), was first introduced five years ago in a paper that explored the usefulness of the mapping resources provided to some pre-service mathematics teachers (Ormond, 2016). Four "key" questions were developed into a suite of tabular rubrics and maps, which consisted of visual organisers at both macro-and micro-levels, and were based upon four "conceptual areas". In the ACCR documents provided to students these four questions are consistently colour-coded thus in order to facilitate teachers' understanding of the associated framework: • What meaningful patterns are there in this maths content? • What sort of fluency do students need in this maths? • What relationships and connections can be seen in this maths? • What practical relevance does this maths have to students' everyday lives?
circle diagram (see Appendix A). In the last few years "project based learning" in collaborative cross-disciplinary STEM (Science, Technology, Engineering and Mathematics) classrooms has elevated student inquiry into even more practical, practice-based approaches, which in turn require real expertise in mathematics teachers as they strive in more complex scenarios to draw out key mathematical ideas and formalise their students' understanding (Edmunds et al., 2017). These strategies seem far removed from adherence to Jean Piaget's original step-like, sequential stages of learning (Piaget, 1952)yet the history of mathematical knowledge building has always focused, and still does, on some sort of construction. Social constructivism itself has consistently relied upon "scaffolding" up through a hierarchy of "negotiated" learning (Bauersfeld, 1995;Vygotsky,1962).
Mathematics is of course about more than its building blocks: it is just as much about the processes of joining these together. Warren (2003) defined "knowledge of mathematical structure" as "knowledge of mathematical objects and the relationship between the objects and the properties of those objects" (italics added). Commentators have long advised that mathematical learning is best carried out in a social context that encourages observation of the connections and relationships between mathematical ideas (Kilpatrick, 2010;Sfard, 1991Sfard, , 2008Skemp, 1986). Whether such understanding systemically evolves out of itself, or travels in a more predictably linear fashion, the process of establishing such links (operational or relational learning) must still precede (structural) understanding. Askew et al. (1997) reminded us of the importance of connectionist teaching and learning, and of classroom teachers finding a sort of balance between transmission and discovery approaches. Askew et al. found in their 1997 study that teachers who appeared to establish stronger mathematical connections for their students were also generally happier to employ strategies of open-ended problem solving and risk-taking. They found that "the challenge of an application", even when students have not "learnt a skill in advance of being able to apply it" could in fact result in very meaningful learning. Anna Sfard also describes "operational" and "structural" ways of learning and understanding as interlinking with each other in a "backwards-forwards" fashion, gradually moving students towards a solid comprehension of an idea, a "new, and then permanent, awareness of that object" (Ormond, 2000).
Sfard named this final moment of realization as "reification" (1991), and later as "commognition" (2008); and argued that getting right that balance of student inquiry and conceptual accomplishment in a mathematics lesson is the most important pedagogical skill for teachers. As Siemon et al. (2019) explain, students' deep mathematical understanding in a classroom is the result an amalgam of teaching strategies that are dependent on "fine-grained research that is focused on individual student learning trajectories and intimate analyses of classroom pedagogical practices, as well as large-scale research that explores how student populations typically engage with the big ideas of mathematics over time" (p.1) The ACCR model here discussed is likewise based upon the belief that the recognition of mathematical ideas and relationships provides the most lasting kind of reification. And just as helping students to reach such moments in the mathematical journey should be a chief aim of the connectionist teacher, much more overt information about mathematical connections needs to be offered to teachers, especially inexperienced ones. This paper argues that the "big ideas" of mathematics need to be broken into some clear conceptual markers that guide new teachers through the pedagogical waters.
The Australian Curriculum (AC) is currently presented in near-equivalent forms in Australian classrooms across the states, and offers teachers a hierarchical framework for teaching mathematics. The three "strands", or content areas, are supported by clear and consistent "working mathematically" themes running across all strands and years: the four proficiency strands of understanding, fluency, problem solving and reasoning. The sequential presentation of Australian Journal of Teacher Education Vol 46, 4, April 2021 39 the AC, with the Achievement Standards, and the mathematics content for each year laid out with the strands placed side-by-side, is a necessary tool for classroom teachers. The Curriculum outlines a consistency of content inclusion and a multi-strand approach for a particular year group. The AC does indeed help teachers to make some mathematical connections. Yet a basic hurdle is still faced by many practising teachers, the everyday challenge of translating it into engaging and meaningful mathematics lessons that also create conceptual links for their students. This is because the links between the mathematical ideas still need first to be made by the teachers themselves, before they can work successfully with their students. A national curriculum might offer an organized planner for mathematics teaching, but the AC is still often not quite enough for the fundamental, early support of some teachers. Thomas (2015) criticised the AC for its relative failure to "focus on the development of conceptual skills to enhance capacity to analyse, evaluate, and create", something needed "to ensure that learning is contextualised" (p. 96). Mulligan et al. (2012) claimed that while the AC might offer "on the surface … the opportunity for greater conceptual and connected knowledge and the development of teaching practices that focus on relational thinking", it does relatively little to help new teachers to understand the nature of that "relational thinking" (p. 65). More than a curriculum content sequence, even populated with associated "proficiencies", is needed for that.
Many new teachers, especially primary teachers, often also need assistance to acquire their own deeper understanding of mathematics (Hohensee, 2017;Tanisli & Kose, 2013), and sometimes even with their own actual "reification" of fundamental ideasin other words, help with soundly constructing their own Pedagogical Content Knowledge (PCK) (Shulman, 1986). Teachers need continuing support in developing their professional expertise. In an analysis of the offerings of mathematics educators over the past fifty years, Schoenfeld stated that, "We have made great progress in understanding teacher thinking and learning, but there is much to be done in supporting teacher growth." (Schoenfeld, 2016, p. 518). Brent Davis's 2014 research analysed some teachers' understanding of the mathematics they bring to students, and focused on "the tacit dimensions of teachers' mathematics knowledge" and how these may surface and coalesce to form their idiosyncratic understandings. Davis (2012Davis ( , 2015 talks of his observations of teachers' "emergent mathematical knowledge" and defines their "realisations"and those of their students -as the collection of "all manner of associations that a learner might draw on and connect in efforts to make sense of a mathematical construct grouping" (Davis, 2012, p. 3). Davis's work has focused on "the complexity of teacher knowledge" and upon on "strategies to uncover it, analyse it, and resynthesise it in ways that will make it more available for teaching" (Davis & Renert, 2014).
The ACCR also attempts to bring teachers' half-forgotten understandings and knowledges to the surface. Such realisations as described by Davis may be promoted in the form of complementary verbal-symbolic and visual-spatial-kinaesthetic presentations of the concepts (Skemp, 1986), which a teacher can facilitate by covering the multiple representations that connect with both the left-and right-brain aspects of complex mathematical ideas. Visual concept mapping of such multi-representational strategies via the ACCR model has been used extensively in the author's mathematics education teaching of ITE and, more recently, in-service teacher students; and its success as an approach was explored in research concerning its specific usefulness for presenting Number and Algebra concepts (Ormond, 2016). It was stated then: Just as a sense of balance is needed in the strategic use of those well-researched classroom teaching orientations, the same is no doubt true for mathematics education programs for pre-service teachers. Teaching how to teach needs to be modelled at every point. As structured as the discipline of mathematics may be described to be, it has been argued in this paper that even more structure and guidance is needed to support improved teacher PCK. …The guidance needed should be one that attempts to enable or unearth teachers' sense of mathematical connections… Perhaps there is something about a minute analysis of the scoping and sequencing of a mathematics strand that can liberate novice teachers to experiment with inquiry processesknowing that a sound understanding of some sort of master plan will mean that they can diverge a little without getting lost. (p. 28-29) It needs to be said also that the author's notion of four "key pedagogical ideas" is something more (and less) than the consideration of the ontological meaning and structure of mathematical thought discussed so often in the literature. The four ACCR conceptual areas are in fact pedagogical prompts for practical classroom teaching (how to talk about the mathematics), with two of the areas concerned with the "big picture" nature of all mathematics patterns and relationshipsand two more grounded in procedures to establish numerical skills -fluency with facts and everyday use of mathematics. The teaching students in the earlier study "consistently demonstrated a marked improvement in both their pedagogical and their content understandings", and each "made at least one reference to such experiences" as … • a sudden awareness or realisation; • a brand new big picture overview of the content and the pedagogy; • a new sense of the deep connections in the mathematics; and/or • a sense of continuity and scope in the mathematics. (p.17) In more recent years since the study reported on in 2016, a higher proportion of students had been teaching mathematics, either in-field or out-of-field, for several yearsyet similar student affirmations as to a new sense of "connectedness" in their PCK have been frequently offered. (Some of these can be seen in Appendix B.) Additionally, some new observations and reflection have now value-added to the author's teaching approach. It was felt that, at the same time as more clearly "seeing" the mathematics, many new teachers also need more practical access to a language about it, in which they ask important mathematical questions of themselves and their students, questions that relate to its very nature as a discipline. In doing this, they are also adhering to the first two of Sullivan's 2011 Principles, those of "articulating goals" and "making connections". This is clearly seen in his references to "communicating to students … the key ideas that underpin the concepts, … the goals of the teaching," and "explaining to them how [the teacher] hopes they will learn" (Principle 1); and "creating and connecting students with stories that both contextualise and establish a rationale for the learning" (Principle 2).
In short, as well as finding the links in the mathematics, many mathematics teachers need assistance with developing ways to talk about them in clear language to their students (O'Halloran, 2005). The ACCR has continued to provide one more strategy for ensuring a more comprehensive presentation and "articulation" of mathematical ideas in a classroom, encouraging teachers to define as clearly as possible the desirable range and coverage of these ideas. To use the colour coding again: each time they prepare to teach, they should be considering the necessary fluencies and everyday uses of the mathematics, and balancing these with an examination of its meaningful patterns and deeper mathematical relationships. In the next sections the various tiers of the ACCR tabular rubrics are explained, and, by providing some examples of their use in the author's teacher education classes, they are linked to this improved strategy for facilitating more effective classroom language.

The Tiers of the Australian Curriculum Conceptual Rubric (ACCR)
Two "dispositions" that the ACCR attempts to encourage in mathematics teachers were mentioned earlier -a willingness to develop an understanding both of the importance of locating and using scope and sequence content information; and of developing a sense of the relationships in mathematics within and across the three key strands. The ACCR was originally written to offer practical support for pre-service and early career mathematics teachers in relation to "big picture thinking". This is especially important for primary classrooms, where many new teachers fall into the habit of presenting mathematics in isolated snapshots that do not connect together sensibly for their students. As explained, the four "big questions" of the ACCR were operationalised as four "conceptual areas" for all school mathematics teaching. These are seen on the left-hand-side in Figure 1, which associates the conceptual areas with the questions a teacher could be thinking about when preparing classwork, and also what he or she could pose as "good questions" (Sullivan & Clarke, 1991) in the classroom. Figure 2 illustrates the 12 overarching, cross-strand components of the ACCR, across all years of schooling. David Clarke used the analogy of "grain-size" in teachers' pedagogical decision-making about classroom assessment types (Clarke, 1997). In a useful analogy for assessment types he contrasted fine-grained micro-level "sand" with the macro-level of the "boulder". If the actual Curriculum itself is seen as the sand, Figure 2 is essentially the boulder of the ACCR, as each of the 12 components of this rubric attempts to capture a highly complex idea with as few words as possible. This figure therefore explains the first "tier" of the model, showing how the conceptual areas can be seen to be strand-specific, while at the same time inviting more explicit unpacking of depth and detail.

Big questions for the teacher's classroom preparation
Big questions for the students in the classroom

RECOGNISING PATTERNS in MATHS
What meaningful patterns are there in this maths content?
"What patterns can we see here in this maths?"

MATHS FACTS and FLUENCIES
What sort of fluency do students need in this maths?
"What maths do we need to really learn here, and be able to remember?"

EQUIVALENCE and RELATIONSHIPS in MATHS
What relationships and connections can be seen in this maths?
"How does this maths fit together …? Say the same thing as …?"

NUMERACY for EVERYDAY LIFE
What practical relevance does this maths have to students' everyday lives?
"How do we use this maths in our lives?"   It could also be noted here that Figure 2 appears to be conceptually consistent with the general NNLP conceptual framework (ACARA, 2020, Version 3, p. 5). The NNLP refers to student "outcomes" of understanding that a student "says, does, or produces" and their "developmental levels" are envisaged as partial or full segments of a circle, the "elements". It is interesting that all of the full or almost full segments emphasise Recognising Patterns and Equivalence and Relationships, and most of them fit comfortably with Facts and Fluencies and Numeracy for Everyday Life. (See Appendix A for a more detailed comparison.) As with any attempt to summarise a complex system in a few words, these first rubrics are of course deceptively simple, and each actually implies a much deeper meaning. Yet it is argued here that in their apparent simplicity lies their strength, in terms of clear visual organisers with which less experienced teachers may at least start. In the next section, the objective of helping teachers to "locate and use scope and sequence content information" will be considered more closely, further explaining the machinery of the ACCR resource.

The ACCR and Understanding Mathematical Scope and Sequence
It was explained earlier that the ACCR resource aims to encourage in teachers two habits of mind. The first of these, that willingness to seek out and use the relevant scope and sequence of the mathematics content, is now discussed and exemplified further.
The encouragement of these "dispositions" has been incorporated into the author's mathematics education programs for pre-service and in-service teachers with some success, according to their feedback (Ormond, 2016). In asking the pre-service teachers and retraining teachers to step outside an often more limited view of mathematics and to look at it both holistically and structurally, many found a sudden metacognitive clarity as to what they were actually trying to do in a mathematics classroom.
Student 7 (2014) The two "visual strategies" for pedagogical understanding were defined in this earlier paper as "the vertical perspective (scoping and sequencing the Australian Curriculum up through the years)" and "the horizontal perspective (introducing the … conceptual areas across content for connecting the mathematics)" (Ormond, 2016, p. 16). These perspectives are exemplified in unit resources, a PowerPoint teaching slide example of which can be seen in Figure 3. The various rubrics offered in the ACCR may also be seen, as in Figure 3, in terms of "reading down" and "reading across". To achieve a more holistic view of scope and sequence than presented by the AC, one may choose a strand and examine in its rubric the 12 conceptual framework components across three schooling phases. Figure 4, a more detailed, second-tier rubric than Figure 2 (second-tier in that it focuses only on one strand) provides an example using Measurement and Geometry. (The generic "all-phase" conceptual areas are seen highlighted grey in the last column, and can be compared with the first-tier Figure 2.) The MATHS FACTS and FLUENCIES components are seen to move from fluencies around drawing, describing and manipulating (Years Foundation to 2); through to  an understanding of measurements of all attributes, and of simple geometric categories (Years 3 to 6); to the skills of carrying out indirect measurement, of transformations and of the generalisations of geometric properties in two-and three-dimensions (Years 7 to 10). The NUMERACY for EVERYDAY LIFE components involve estimation skills in all the attribute areas, and practical uses of measurement and scale. The RECOGNISING PATTERNS components again move students from early experimentation with the attributes through to more and more complex generalisations of properties (Years Foundation to 6); and then to geometric generalities based on algebra and geometric proof (Years 7 to 10). Pattern recognition can be used to develop and consolidate the fluencies and to highlight commonalities, differences and congruencies (Smith et al., 2007).  The EQUIVALENCE and RELATIONSHIPS component is always the most predominant in terms of interrogating underlying mathematical connections, both within and between the strands. More will be said about this in later sections of the paper which examine that "sense of the connectedness and correspondences". Yet as far as scoping and sequencing in this strand is concerned, the ever-present mathematical theme of "what is the same and what is not the same"invariance and variance, as described by Goos et al. (2017, p. 228)can clearly be seen. The conceptual area focused on equality and inequality is seen to move from an evolving system of comparisons and categorisations of attributes (Years Foundation to 6); to such notions as congruence and similarity, or Pythagorean triad theory, or transformations on trigonometric functions (Years 7 to 10). A final explanatory point to be made here concerns the source of these 12 "components" and these are of course the AC Content Descriptions themselves (supported by the associated proficiencies and Achievement Standards). The most detailed rubric offered in the ACCR suite is seen in Fig. 5, where for the strand in question a list of relevant content descriptions by code is provided for each conceptual area and student learning phase. With content codes included, this is a third-tier rubric. Again, brevity is a limitation of such a skeleton framework, and further detail is required by teachers for planning tasks and lessons but this is amply provided in the AC itself, and the links to the Content Descriptions have been provided to start this process. If a teacher wishes to emphasise fluency or everyday numeracy, he or she can use this rubric to quickly identify some relevant AC descriptions. Interestingly, because of the complexity of the mathematics and the richness of the Descriptions themselves (intended by Peter Sullivan and the other ACARA mathematics writers to provide breadth, with implied but not always highly elaborated depth), some of these codes are seen repeatedly appearing in more than one of the conceptual area categories.

Australian Curriculum Conceptual Rubric, MEASUREMENT and GEOMETRY
(For an example of this see the third slide in Fig. 6, where codes are repeated in several conceptual areas in both the Year 3 and Year 4 lists.) If a Content Description code is repeated, it is hoped that the teacher will reflect on it through the lens of the different "big questions" so as to establish the focus of the lesson. A more specific example of how these ideas have been used in a pre-service mathematics education classroom is offered here, this time in relation to primary level Number and Algebra. Seen below in the first two slide shots in Figure 6 are examples from a selection of PowerPoint slides which tasked the teaching students with planning a lesson based on the number fact 3  7 = 21. They were asked to think about content and teaching approaches for Years 3, 4, and 5, also considering the scope and sequence of the ideas. They were then required to organise their "brain-stormed" ideas into the four conceptual areas and to match them with appropriate AC content descriptions, using the AC website and the ACCR document together. The next part of the lecture, exemplified in the last two slide shots in Figure 6, involved listening to a description of how, with some deeper thought, the four conceptual areas could be mapped across Years 3 to 5 so as to gain a sense of the concept-building involved. The pre-service teachers could then use this fairly simplistic example to model and develop other lessons based around times tables or number facts. The four pedagogical questions were a constant theme in the explanation. The ACCR document allows teachers to isolate quickly the AC content descriptions and conceptual areas for one year level, or a small number of consecutive years … for example, Year 3.

Theoretical framework
Again, to summarise the last three slides … It also allows them to consider In depth a particular "conceptual area" (or "big question") across several consecutive years (reading down and across) … for example, Year 3 Patterning: what is before and after?

The ACCR and Understanding Mathematical Connectedness: Within the Strands
The next two sections of the paper now focus more upon the second of these teacher competencies, that is, developing a "sense of the connectedness and correspondences in mathematics", both within and across the strands. To return again to the visual organisers, the first-tier, most large-scale ACCR rubric is used, first seen in Figure 2. Figure 7 below is essentially a repetition of Figure 2, but with the notion of conceptual "direction" highlighted. So just what else can such "rubrics" offer teachers for their preparation of classroom experiences, and the language of those classrooms? This is now considered.
Firstly, as seen in Figure 7, teachers should be able to see at a glance the consistent nature of mathematics content, and how all of the conceptual areas apply "down" all of the topic concepts covered. This also corresponds neatly with the notion of the four AC proficiency strands 2 applying equally across all strands and year levels. That is, in all three strands it is necessary to problem-solve and to reason (mostly RECOGNISING PATTERNS and reasoning about EQUIVALENCE and RELATIONSHIPS); to achieve personal fluency (mostly MATHS FACTS and FLUENCIES and NUMERACY for EVERYDAY LIFE); and to gain a connected understanding (a synthesis of all four conceptual areas). In Figure 8 is seen a more detailed conceptual rubric for Years 7 to 10/10A, across all three strands. As it also includes AC Content Description codes it is working at the third-tier level of detail. (It should be contrasted with Figure 4, which was also third-tier but showed a scope and sequence across years for just one strand.) Two more specific examples, using PowerPoint slides from postgraduate pedagogy enrichment units concerning transition (Years 6 and 7) mathematics, in two different strands, consider similar ideas to those in Figure 6. In Figure 9 a geometric task had been chosen and teachers were asked to explain the use or relevance of three of the conceptual areas to the creation of this task. In Figure 10, teachers were asked to consider the links within a topic in prealgebra/algebra, where patterning concepts are used to develop the notion of function.

Features of circles: angle
Naming'the'features'of'circles! Categories,+features+and+measurements+of+shapes!

Year 6
Measuring*objects*with*appropriate*units! Patterns: in realising that there is an even number of congruent triangles each time, whose number is the same as the number of polygon sides.

Equivalence and relationships:
understanding that there is a general rule here and that there exists a logical relationship between circles and regular polygons inscribed inside them. In both cases above, the "connections" being examined are those between the conceptual areas within particular strands, and by topic. However, deeper relationshipsexploring that "deep ontological gap between operational and structural conceptions" (Sfard, 1991, p.4)can also be considered using the ACCR. These are seen in the links between the strands that illustrate the coherence of mathematics as a whole. It could also be argued that in this kind of reflective practice lies the richest source of engaging ideas for conversation in a mathematics classroom, especially in regards to "good questioning" (Sullivan & Clarke, 1991). This is discussed next.

The ACCR and Understanding Mathematical Connectedness: Across or Between the Strands
Looking at the different ("across") perspective in Figure 11, the cohesive nature of mathematics and just what an attempted disclosure of this cohesion may do for the learner is suggested. It can be observed that teaching in all three strands of mathematics needs attention to the same four conceptual areas: • RECOGNISING PATTERNS applies to topics involving numbers, algebra, graphs, measurement, space, shape, time, chance and data. • MATHS FACTS and FLUENCIES involves learning and recalling important facts about numbers, algebra, graphs, and measurement of space, shape, time, chance and data; • NUMERACY for EVERYDAY LIFE involves choosing tools, instruments and mental strategies on a regular basis to solve daily problems occurring with numbers, algebra, graphs, and the measurement of space, shape, time, chance and data; and, perhaps most importantly of all, • all mathematical reasoning depends on a deeper understanding of the notion of mathematical EQUIVALENCE and RELATIONSHIPS: what is the same, and not the same, in representations of numbers, algebra, graphs, measurement, space, shape, time, chance and data.
In other words, mathematics should not be "taught in isolated pockets" (Ormond, 2016) according to topics set for the day or the week: it is both a science and an art in that it links logically and cohesively across topics (eventually, as we know, merging conceptually more and more between the strands themselves, as calculus does for statistical analysis, for example). Such a rubric contributes to that common language that can be used to talk about all topics, one that helps students to associate mathematical ideas from the different strands. The four questions themselves, spoken aloud as they are or paraphrased for students, can steer these conversations (also see Fig. 1): • "What patterns can we see here in this maths?" • "What maths do we need to really learn and be able to remember?" • "How does this maths fit together …? Say the same thing as …?" • "How do we use this maths in our lives?" Goos et al. (2017) remind us that "invariancethat is, a relationship or property that stays the same and does not change when some change is permittedis a key aspect of geometric thinking (Johnston-Wilder & Mason, 2005)" (p. 228). The conceptual area of EQUIVALENCE and RELATIONSHIPS in MATHS is based here on a very similar sentiment concerning all mathematical reasoning; and it is therefore this conceptual area of the ACCR that has the strongest connection with that second disposition for cultivating a "sense of the connectedness and correspondences" in the discipline.
It is argued here that the discipline of mathematics is largely built on noticing, using and proving when entities are the "same" or "not the same". Equivalence can also be thought of, semantically, as direct "correspondence". From solving simple linear equations in one unknown variable to integrating a definite integral to find a quantity equal to the area under a curve, the process of preserving operational and functional equivalences (balancing sides of the linear equation, using a trigonometric identity to re-cast an integrand in a form that can be more easily integrated) is maintained. The constraints in a linear inequality problem which can be understood by graphing lines to show the region corresponding to the solutions, as opposed to that region outside not corresponding to the solutions, is another example. Factorising a fifth-degree polynomial expression in x is about breaking it down into an equivalent product of algebraic expressions that are in the end as close as possible to all being first-degree onesthis in turn corresponds to finding the possible roots of the polynomial when expressed as a function, which in turn correspond to the x-intercept values of that curve. Hypothesis testing in statistics problems is based on finding, within set limits of confidence, the likelihood of two population means or proportions not being equal to each otherand this can be represented graphically with variously overlapping curves ("how much the same are they?") that illustrate the population distributions.
Sometimes the "equivalence" is in the form of exact equality of quantities, as in solving equations or evaluating definite integrals or factorising a polynomial expression to achieve an equivalent expressionsometimes the equivalence is by nature more of a correspondence, one that links between a verbal-symbolic representation of a mathematical idea with its visual-spatial-graphical one. An example is in the sketching of a parabola y = f(x) which, through its corresponding non-factorised, factorised, or "completed square" forms shows any or all x-intercepts, a y-intercept, and maximum or minimum points. Sometimes the equivalence is a kind of "shorthand", as in the expression of a consistent set of linear equations in matrix form, one which is more easily "reduced" to a set of solutions for the unknowns. Table 1 provides some other examples of this, across all three strands. Example 1 shows "correspondences" across the three strands, in terms of a common representation (a divided circle). Example 1 therefore also displays a shared relationship between the strands: proportional thinking can be used in fractions, degree relationships in a circle, and chance concepts. Example 2 shows how the notion of equality and inequality is important, albeit in different ways, in all three strands. And in both examples, the inherent relationships within are also described, seen in the second row of the examples, each time.
Whichever way one looks at it, equivalence is everywhere in mathematics, and truths about equivalence and non-equivalence invariably indicate important conceptual relationships. This idea is intrinsic to mathematical sense-making of all kinds, and is therefore worthy of a "conceptual area" that is far deeper than just noting patterns or obeying operational rules; thus, EQUIVALENCE and RELATIONSHIPS is the most complex of the four pedagogical areas in the ACCR, and also the most embedded in the relational learning that can result in Sfard's "reification". Such a complex idea needs to be carefully scaffolded for many new teachers, who in turn must do the same, if in a simpler way, for their students in classrooms. And this can be looked at from two limiting perspectives, depending on the expertise of the mathematics teachers. Those with well-developed mathematics understanding may still need help with, as Freudenthal expressed it in 1983, "unclogging" their "automatisms" or "original sources of insight". (Davis would call this "substructing.") When one "sees" the function simultaneously as an algebraic equation and as a sketched curve without even thinking about it, one has to work very hard to appreciate the very different perspective of a lower secondary classroom student. On the other hand, pre-service teachers with less mathematical confidence may have never actually had to contemplate mathematics from an ontological or structural view at all: they may still need to build a sense of its connectedness before sharing it with their students.

Reasoning about space, shape and time Reasoning about chance and data
Example 1 EQUIVALENCE A fraction of a whole can be shown with a shaded sector of a circle.
The area of a shaded sector in a circle can be found using the angle its connecting rays makes at the centre of the circle.
A shaded sector in a circle can be used to describe the probability of an event occurring.
 RELATIONSHIPS  This depends on the area of the circle being the "whole".  This depends on there being 360 degrees in one full turn of the circle, and the area of the sector being the angle/360 times the area of the circle.


This depends on the area of the circle being equal to the probability of 1 (certainty).
→ RELATIONSHIPS → Proportional thinking is correspondingly used across strands, in fractions concepts, degree relationships in a circle, and chance concepts Example 2 EQUIVALENCE Solving an equation involves correctly balancing and maintaining equality across both sides of the working.
Categorising types of quadrilaterals is about recognising equal or unequal sides or angles.
Calculating that an equal chance of something occurring is equal to 0.5, and impossibility is 0.  RELATIONSHIPS  This is based on an understanding of inverse operational relationships in arithmetic.
 This is based on recognising correspondences in geometric properties that are able to be generalised.
 This is based on an understanding that probabilities are related to fractional numbers between and including 0 and 1.

→ RELATIONSHIPS →
Within all three strands, the concept of exact equality is a key mathematical idea. Table 1. Using common representations of concepts, and mathematical "correspondences", across the three strands. Figure 13. Selected slide from a Postgraduate Certificate mathematics education lecture, prompting the notion of Equivalence and Relationships applying across mathematics, across the three strands.
It could be argued that ideas of equivalence -and nonequivalence -occur right throughout mathematics, and are indeed essential to the very nature of all mathematics.
Also briefly consider the next slide, which offers a simple "conceptual framework" for primary mathematics and its teaching, aligned to the Australian Curriculum.
32 > 28 28 < 15 × 2 Understanding mathematical connectedness both within and between the strands is a very challenging task for novice teachers, needing self-discipline and reflection. The concept has been introduced by the author to pre-service and in-service teachers in several ways in mathematics education scenarios (see Figs. 13 and 14). While the representations in Figure  13 comment on the notion of the blue conceptual area being evident across the mathematical strands, Figure 14 emphasises that within each strand, "teaching inputs" that involve pattern exploration, fluency and practice and application in real life contexts all lead to the "learning outcomes" that are dependent upon these processes, and that can formalise or reify for students the "key ideas that underpin the concepts" (Sullivan, 2011). That is what is meant by "relational learning".
It was explained earlier that Sullivan's six fundamental Principles for effective teaching provided the impetus for this re-invigorated thinking around the ACCR model. Of its four "big" pedagogical questions, two involve a great deal more thought and effort from teachers. While MATHS FACTS and FLUENCIES and NUMERACY for EVERYDAY LIFE correspond to important features of Principle 2: Making connections and Principle 6. Promoting fluency and transfer, the other two conceptual areas, RECOGNISING PATTERNS and EQUIVALENCE and RELATIONSHIPS, each relate to at least five of the six Principles 3 . Table 2 represents the author's exploration of the links between the Sullivan Principles and the conceptual areas. The first impression upon looking at it is its multi-coloured natureand this has confirmed the author's sense that the conceptual areas weave through all of the Principles. It also appears to indicate that the "four big questions" of 3 It needs to be mentioned here also that Principle 4: Differentiating challenges, with its sensitivity to student differences, underpins all successful mathematics teaching and learning, and so is assumed to be an all-encompassing aim. It is in fact inherent in all four of the pedagogical questions, and also relates strongly to the AITSL Graduate Teacher Standard 1, Knowing how students learn, and Standard 4, Create and maintain supportive and safe learning environments (AITSL, 2011). This Principle relies on careful preparation of lessons that consider the likely prior knowledge of students and employ differentiation of tasks for both less and more mathematically capable students. The other five Principles are all embedded by this aim, so mapping of this Principle has not been attempted in Table 2.
Good mathematics teaching involves a cyclical development of teaching inputs working towards learning outcomes. Remember: the four ACCR "conceptual areas" for teaching are provided as support for using the Australian Curriculum (AC) for mathematics. The ideas are not intended in any way to replace or re-interpret the AC in a different way, but to enrich it.
the ACCR are indeed a "distillation" of the most important features of effective classroom teaching.
The MATHS FACTS and FLUENCIES and NUMERACY for EVERYDAY LIFE conceptual areas are of course important aspects of a mathematics classroom (something testified to by the prominence many less confident mathematics teachers place on pure drill and memorisation). The first row of Table 2 illustrates the significance of Principles 2 and 6 to these conceptual areas. Interestingly, Principles 2 and 6 still correspond to the other two conceptual areas also, as seen in the blue and green highlights. This is not surprising, as "stories" and "contexts" are so important to the exploration of mathematical patterns, as much as they are to a sense that the mathematics is relevant to everyday life and needs to be learned and internalised. Similarly, building on prior knowledge to create new understandings and connections is unavoidable for effective work on fluency and numeracy. Table 2 supports the argument that the conceptual areas of RECOGNISING PATTERNS and EQUIVALENCE and RELATIONSHIPS much more closely encapsulate the essence of Sullivan's Principles. (See Tab. 2 and Fig. 15.) Earlier in the paper it was suggested (and illustrated in Figs. 3 and 4) that the four conceptual areas can be used to develop a sense of the scope and sequence of a topic's concepts. It was also suggested earlier that recognising mathematical relationships and speaking a coherent mathematical language link best with both Principle 1: Articulating goals and Principle 2: Making connections. These Principles relate to RECOGNISING PATTERNS in the sense that "key ideas that Table 2. The four "conceptual areas" of mathematics teaching, represented also as pedagogical questions, related using the ACCR colour codes to five of Sullivan's six Principles for effective mathematics teaching.
Other Sullivan Principles are also here briefly reflected upon. Principle 3: Fostering engagement encourages teachers to "use a variety of forms of representation", "challenging" students with "rich tasks" that depend on them having "time and opportunities to make decisions". Such tasks are at the heart of exercises that employ looking for patterns and finding links and relationships. The effective practice of Principle 5: Structuring lessons depends on attention to the scope and sequence of concepts; to the prior knowledge of students; to the desirable range of difficulty; to cooperative group work on noticing and recording mathematical trends; and to the teacher's subsequent formalisation of the key relationships between the concepts. Principle 6: Promoting fluency and transfer contributes both to procedural fluency and to students' own formalisation (reification) of the key concepts.

Conclusion
One final piece of analysis is seen in Table 3, which represents the culmination of this work and is now being shared with the author's current mathematics education students. It summarises in a practical way the links between the two pedagogical models discussed in this paper. Using this table allows teachers, pre-service and in-service, to frame their understanding of the four conceptual areas in terms of the foundational principles for all classroom mathematics teaching that have been devised by Sullivan.

Big questions for the teacher's classroom preparation/ "Big questions for the students"
The key points of five of Sullivan's Principles, linked specifically to the four ACCR "big questions" (distilled from Figure 1 and the analysis in Table 2) What sort of fluency do students need in this maths?
"What maths do we need to really learn here, and be able to remember?" • Build on what students know, mathematically and experientially (2) • Use students' reports to the class as learning opportunities (5) • Fluency is important, and it can be developed …. by short everyday practice of mental processes What practical relevance does this maths have to students' everyday lives?
"How do we use this maths in our lives?" • Build on what students know, mathematically and experientially (2) • Create, and connect students with, stories that both contextualise and establish a rationale for the learning.
(2) • Fluency is important, and it can be developed … by practice and reinforcement (6) What meaningful patterns are there in this maths content?
"What patterns can we see here in this maths?" • Identify key ideas that underpin the concepts (1) • Create, and connect students with, stories that … contextualise (2) • Engage students by utilising a variety of rich and challenging tasks that allow students time and opportunities to make decisions (3) • Foster communication and both individual and group responsibilities What relationships and connections can be seen in this maths?
"How does this maths fit together …? Say the same thing as …?" • Identify key ideas that underpin the concepts (1) • Communicate to students that these are the goals of the teaching (1) • Explain to them how you hope they will learn (1) • Build on what students know, mathematically and experientially (2) • Create, and connect students with, stories that both contextualise and establish a rationale for the learning (2) • Engage students by utilising a variety of rich and challenging tasks which use a variety of forms of representation (3) • Use teacher summaries of key mathematical ideas (5) • Fluency is important, and it can be developed … by prompting transfer of learnt skills (6) Table 3. The four ACCR pedagogical questions and the key corresponding Sullivan descriptors, related using the ACCR colour codes. (Also see Fig.1.) The relationship between the two frameworks is also seen in Figure 15, where Figure  14 has been enhanced with extra links to Sullivan's Principles. The relative predominance of those two conceptual areas, green and blue, can be seen, now also representing RECOGNISING PATTERNS as the primary source of teaching inputs and EQUIVALENCE and RELATIONSHIPS as that of student learning outcomes related to these inputs. Indeed, Sullivan's "key ideas that underpin the concepts" (Principle 1, 2011) provide the central goal of effective mathematics teaching. This paper has interpreted the body of knowledge that is mathematics from the author's perspective and understanding of the needs of the new classroom teacher, who must convey that knowledge, in a logical way, to students. It has suggested the value of initially using a fairly spare "structure" of conceptual keystones and themesone that nevertheless implies a complex internal organisation of ideasin a first venture into the deeper ontological questions that accompany any study of mathematics.
The author further believes that any organized and sequenced Primary/Secondary mathematics curriculum can be interpreted in this way, and has argued that such a visual and thematic framework may help teachers to keep track of some powerful tools for practical, effective teaching. By keeping the important elements of big picture thinking for PCK in the forefront of their minds as they prepare tasks and plan potential classroom conversations, the focus may remain on student-centred, connected mathematics teaching. The paper has also argued that Sullivan's six Principles for the effective classroom delivery of mathematics underpin the very fundamental questions of the ACCR framework, questions that teachers may use to enhance both their own, and their students', understanding of the mathematics.

APPENDIX A
Conceptual framework for the National Numeracy Learning Progression (ACARA, 2020, p. 5). Appendix A Figure 1. The first tier of the four ACCR "conceptual areas" of mathematics teaching, across all three AC strands, mapped against the NNLP elements of its conceptual framework.

APPENDIX B
Voluntary course-end comments from students concerning support of conceptual mapping materials.
Student 1, email October 2017, teaching secondary mathematics Thank you again for your wonderful workshop materials which I will definitely look over again even after the Course is over.
Student 2, email November 2017, teaching secondary mathematics I really want to thank you for the interesting and extremely valuable units. I know that I will keep going back to the presentations, documents and all the additional materials you have so kindly provided.
Student 3, email October 2018, teaching secondary mathematics Thank you so much for the course content. This unit has really put an understanding to what I have been doing in the classroom. It has given me content knowledge that was missing in my CV.
Student 4, email November 2018, teaching secondary mathematics I just wanted to say thank you for the learning opportunity this course has allowed me and for all the work you have put into creating this course. I have gained so much from undertaking this course and am becoming more confident every day that I will be able to effectively support future maths students in their journey through this amazing subject. I am excited about how much there is to learn from this course and although the assessments are completed and the course is coming to an end for 2018, I am looking forward to further breaking down the 2018 content over the summer holidays in my journey towards mastery.
Student 5, email June 2019, teaching secondary mathematics Thanks so much for the wonderful delivery of your course … The course outline, reading material, schedule, lecture notes etc were very clear and I feel that the course has prepared me well for teaching maths in my school. It has opened a new world to me and given possible new directions.
Student 6, email June 2020, teaching secondary mathematics I would also like to thank you because I have learnt so much doing this course and your slides and audio presentation are very clear, well-sequenced and very informative. I now know the importance of the conceptual areas in mathematics, i can navigate confidently with the ACARA and appreciate the importance of giving students as much opportunities to do problem solving task. I now understand the importance of developing conceptual areas using different mode of presentation to children -especially in the primary years, as it will give them a strong foundation for high school. My favourite topic was learning about Numbers and Algebra. I had so many misconceptions that I never knew I had, but you clarified the many misconceptions I had. Thank you once again for being an excellent lecturer and your passion is evident in your work.