Coordinating visual and algebraic reasoning with quadratic functions

Abstract

Quadratics provide a foundational context for making sense of many important algebraic concepts, such as variables and parameters, nonlinear rates of change, and views of function. Yet researchers have highlighted students’ difficulties in connecting such concepts. This in-depth qualitative study with two pairs of Year 10 (15 or 16-year-old) students investigated the potential of figural pattern generalisation—a context not traditionally used for teaching quadratics—to stimulate students’ coordination of visual and algebraic reasoning and attention to quadratic function concepts. Theorisations of embodied visualisation, algebraic thinking, and student noticing were drawn on to analyse the pairs responding to 19 quadratic figural pattern generalisation tasks interspersed throughout their class topic on quadratic equations. It was found that students became adept at connecting the generality of different types of structural aspects of figures (square, rectangular, linear, constant/invariant) to their symbolic expression in quadratic equations. Students’ construction of numeric instantiations of figural aspects was found to support pairs in moving towards symbolic generalisation. Task prompts to find different (but equivalent) algebraic equations for the same pattern evidenced pairs beginning to distinguish among general, factorised and standard forms of quadratic equations. One pair’s attention to first and second differences (between total quantities of figures in a sequence) highlighted both the difficulty of and potential for connecting quadratic rate-of-change concepts and parameters visually. Implications for including figural pattern generalisation when teaching quadratics and suggestions for further research are shared.

Introduction

Although there are differing views on how best to learn and teach function concepts, there is widespread consensus that functions are crucial, even foundational to all mathematics (Schliemann et al., 2021; Thompson & Carlson, 2017). They require meaningful development rather than procedure memorisation (Mason, 2017). Traditional school algebra approaches have been criticised for overreliance on rote learning in abstract equations-based contexts (Rakes et al., 2010). Researchers over recent decades have highlighted the ongoing difficulties students experience in trying to make sense of algebraic ideas, such as where equations even originate (Mason, 2017).

In efforts to reform algebra teaching and learning, the inclusion of functional approaches for conceptual learning in curricula around the world is evident (Kieran, 2007; Sutherland, 2002; Yoon & Thompson, 2020). One functional approach, typically used with younger students and with linear relationships (Steele, 2008), is figural pattern generalisation: a concrete context intended to elicit visual reasoning, different views of function such as correspondence and co-variation (Smith, 2008), and conceptual connections between functional relationships and their symbolisation (e.g. Mason, 2017; Radford, 2003; Rivera, 2010). Students attend to relationships between variables—spatial aspects of figures and their increasing item numbers—and learn how to represent them with algebraic equations and pronumerals (alphabetic letters) (Smith, 2008). Yet little is known about if or how such visual approaches might support the conceptual learning of more complex types of functions, such as quadratics.

Quadratic functions are foundational to key algebraic concepts that are necessary for calculus learning and modelling in both pure and applied contexts (Suominen, 2018). In an Australian curriculum context, relevant to the students in this study, quadratics are introduced in Year 9 and initially via quadratic equations (factorising, completing the square etc.) and later with graphs and their transformations (Australian Curriculum & Reporting Authority [ACARA], 2017). The literature has highlighted significant student difficulties with understanding and connecting quadratic concepts and representations (equations, tables, and graphs) (Lobato et al., 2012). Studies have found that students experience particular difficulties with grasping the nature of quadratic growth (Ellis & Grinstead, 2008) and with different quadratic equation forms (McCallum, 2018). Unlike algebraic equations, tables and graphs, quadratic figural patterns are not a typical representation used for teaching quadratic functions (Graf et al., 2018), so little is known about if or how they might help address such student difficulties.

This study sought insights into how quadratic figural tasks that elicit visual and algebraic reasoning with a representation not typically used for teaching quadratics might support older students with making sense of quadratic growth and equation forms. This article reports on a fine-grained case study of two pairs of Year 10 (15 or 16-year-old) students for whom quadratics are a core focus prior to further study of other types of functions. Several tasks involved the symbolic generalisation (Radford, 2003) of different types of quadratic figural patterns. There is the need to understand how students’ algebraic reasoning processes for such generalisation might interact with their more traditional equations-based learning of quadratics. Does pattern generalisation seem an additional, disparate type of activity to students, or can it facilitate their making connections among quadratic concepts?

Researchers have highlighted the cognitive demand of tasks that require visual and algebraic reasoning, and flexible movement among different views and representations of function (e.g. Mason, 2017; Montenegro et al., 2018; Rivera, 2010). It was therefore considered worthwhile to research students working in pairs on the tasks, and their collaborative visual and algebraic reasoning. The research questions for the study were:

• 1. In the pairs’ interactions on quadratic figural pattern tasks, which of their foci evidenced movement towards or away from symbolic generalisation?

• 2. In what ways did the pairs’ coordination of visual and algebraic reasoning draw their attention to particular quadratic function concepts?

Background and context

The following two subsections overview theoretical perspectives on visualisation and learning quadratic functions, and the Student Noticing framework used in the study for analysing visual and algebraic reasoning processes.

Theoretical perspectives on visualisation and learning quadratic functions

Spatial reasoning and visualisation

Historically, there has been considerable interest on ‘spatial ability’ in the psychology literature and on types of ‘spatial reasoning’ in mathematics education (Mulligan, 2015) yet an ongoing lack of consensus on how to define these terms (Ramful et al., 2017). In mathematics education Battista (2007) proposed the notion of ‘spatial ability’ as being able “to see, inspect, and reflect on spatial objects, images, relationships, and transformations” (p. 843). Studies have drawn attention to the role of ‘spatial reasoning’ in mathematics learning, particularly in the context of patterning (e.g. Mulligan, 2015) and algebraic generalisation (e.g. Radford et al., 2007; Rivera, 2010). Ramful and colleagues (2017) proposed three categories of spatial reasoning: ‘mental rotation’, ‘spatial orientation’, and ‘spatial visualisation’, cautioning that ‘spatial visualisation’ as it relates to school mathematics is not yet clearly characterised.

In the mathematics education literature visualisation is conceptualised in different ways. One view is that it is a process. Zazkis et al. (1996) theorised visualisation as involving the mental construction of objects/processes that a student associates with objects/events external to them or objects/processes in their mind. Another mentalist view theorises visualisation both as a process and a product. Arcavi (2003) considered a produced visual image as ‘a visualisation’ (noun), students’ communication of what they ‘see’ via their written representations—circling or shading of figural items, calculations, symbolic equations, tables of values etc. An individual’s visualisation can be inferred by such constructions in an external medium and does not need to be mathematically ‘correct’ as such to be considered ‘visualisation’ (Arcavi, 2003). Radford (2014) argued for an embodied multimodal perspective of visualisation—perceptions, physical gestures, kinaesthetic actions, written signs, and artefacts are considered not merely to mediate thinking but part of thinking itself. Research from an embodied visualisation perspective attends to such semiotic data for insights into student reasoning.

Visualisation and figural pattern generalisation

In the context of figural pattern generalisation, Arcavi (2003) argued that not only does visualisation guide the process of developing a generalisation, it is a key component of the necessary algebraic reasoning; it is beneficial not only for illustration of a generalisation but for its actual conception. Visual representations, such as figures in a growing pattern, provide immediacy because they embed quantitative functional relationships in a concrete (rather than abstract) context. They also highlight different views of function (Smith, 2008). For example, in Fig. 1 (the ‘squares’ growing pattern used in the study), a recursive view focuses on the repeated change in quantity of grey tiles in the figure sequence (the ‘first difference’), + 5, + 7, + 9, + 11, etc. A co-variation view coordinates the change in both variables, for example, in Fig. 1, as the item (position or stage) number increases by one, the figure quantity increases constantly by two (a constant ‘second difference’). A correspondence view of function seeks a rule (equation) that can be used to find a particular variable value when given the corresponding value of the other variable. For example, in Fig. 1, adding one to the item number and then squaring will give the total number of grey tiles in the matching figure (symbolically t = (n + 1)²).

Fig. 1

‘Squares’ task* For the following growing patterns, if n = figure number and t = total number of grey tiles in that figure, try to find a rule (equation) to calculate the number of grey tiles when given any figure number in the pattern. This pattern is the same as in b) but has been shaded to highlight one way to visualise it. Try to find a different (but equivalent) equation based on the various shaded parts.

Radford (2003) highlighted the central role of embodied visualisation in his theorisation of three different levels of student generalisation with figural patterns: factual, contextual, and symbolic. He viewed the process of generalisation as a type of objectification, “related to those actions aimed at bringing or throwing something in front of somebody or at making something visible to the view” (p. 40). The means of objectification is semiotic in that students focus their attention, become aware of a generality in a pattern aspect (factual generalisation), and coordinate bodily actions, such as utterances (refined linguistic, generic, locative terms), rhythmic pointing (gesturing) across several figures, and written representations, to communicate what they visualise (contextual generalisation). They then express the generality they see in the conventional symbolic notation of algebra (symbolic generalisation).

Radford (2014) also argued that reasoning is transformed by social interaction, emphasising the importance of research that attends to students reasoning together. The value of student interaction on cognitively demanding algebra tasks was highlighted in Montenegro and colleagues’ (2018) study of middle school students’ individual (linear) pattern generalisation. They found that students did not automatically intuit algebraically useful spatial characteristics of figures individually but required interaction to help them notice and then reason with the spatial characteristics productively.

Students learning about quadratic relationships through pattern generalisation

It has been argued that a foundational understanding of functions relates to the nature of the relationship between variables (Thompson & Carlson, 2017). Correspondence and co-variation views of function are important for making sense of what a function is (Smith, 2008). With figural growing patterns, student can learn visually how variables are related to each other, with item numbers and corresponding quantifiable aspects of figures. They can learn to attend to the nature of a particular type of growth (linear, quadratic, etc.) as quantifiable aspects change from figure to figure. A key meaning for quadratic growth is that its rate of change is changing constantly (Lobato et al., 2012). With a growing pattern, this meaning can be seen visually with the constant second difference (‘constant rate of change of the rate of change’) from figure to figure.

In the literature specifically on quadratics, it is considered important for students to make sense of the reasons for representing quadratic relationships with different equation forms—‘general form’ (y = ax² + bx + c), ‘factored form’ (y = a(x – r)(x – s)), and ‘standard (or turning point) form’ (y = a(x – h)² + k). Otherwise, it is argued that students think of quadratics as a lot of disparate procedures or unrelated and “complicated tricks” (McCallum, 2018, p. 95) like factorising, completing the square, and using the quadratic formula. Quadratic figural patterns can be configured with different spatial structures that visually match the different equation forms. For example, square-based figures (such as in Fig. 1) connect the ‘x squared’ term in an (abstract) quadratic equation to a (visual) geometric meaning for ‘square’. Rectangular pattern figures (such as in Fig. 3) can highlight that a quadratic expression is factorisable if it can be represented visually as an exact rectangle, i.e. connecting binomial (factored) expressions to the area formula for rectangles (Kajander, 2018).

Studies in the literature on students’ figural pattern generalisation have tended to report on younger students learning about linear relationships, but some researchers have investigated generalisation of quadratic relationships. Orton et al. (1999) studied pre-service mathematics teachers generalising a variety of quadratic ‘dot’ growing patterns and found they experienced difficulties with triangular arrangements (for example, see the bridges in Fig. 5, each of which has two triangular sides). A majority of participants constructed a table of values for the pattern and tried unsuccessfully to use differencing to find an equation. Similarly, Hershkowitz et al. (2001) found that most of the pre-service teachers were unsuccessful in generalising the well-known quadratic matchstick grid pattern, even though it was a square arrangement. They focussed on the recursive change from item to item and not on the correspondence between the square structure and the side length (which matched the item number). Middle-school students, who had previously experienced linear generalisation, were successful with the same grid pattern, evidencing spatial reasoning and a correspondence view of function.

In a study with high-achieving pre-algebra lower secondary students, Steele (2008) found that quadratic figural patterns helped students make sense of variables, notice quadratic growth, and make connections between different representations, such as a sequence of figures and a table of values. Rivera (2010) provided examples of students using their spatial reasoning to rearrange quadratic pattern figures into shapes that could be generalised more easily, for example, joining triangles to form a rectangle. He categorised patterns as having higher ‘pattern goodness’ if their structure had easily discernible aspects that can be generalised. Figures are lower in pattern goodness if they do not embed visual aspects that can be easily intuited. Rivera’s categorisation of figural patterns (presented in Table 3) was used in the current study to design and sequence the tasks. Chua and Hoyles (2014) similarly found that lower secondary students could learn to generalise quadratic figures using visual strategies, including rearrangement into rectangular shapes.

Wilkie (2019) explored 12 high-achieving secondary students’ (Years 7 to 12) linear and quadratic figural pattern generalisation during task-based individual interviews. All the students were found to generalise symbolically the higher-goodness quadratic pattern, and some with multiple equivalent algebraic rules, which suggested the potential for figural patterns to support students’ conceptual understanding of different quadratic equation forms. Yet the lower-goodness quadratic pattern was found to elicit numeric reasoning with a table of values rather than visual reasoning. Wilkie (2021) investigated 67 secondary mathematics pre-service teachers’ creation of their own figural patterns for a given quadratic equation, to consider the role of previous school study of quadratics. Several teachers evidenced new connections between their prior knowledge of quadratic equation forms and areas of rectangles by visualising different spatial arrangements for the same generalisation, including factored and turning point forms. This finding suggested that quadratic figural patterns might be beneficial for students with some prior knowledge of quadratic equations.

The Student Noticing Framework for analysing student visual and algebraic reasoning

In a mathematical situation, information, features, visual cues, and possible patterns—which cannot be processed all at once—compete for one’s attention. Different individuals attend to different things, which has consequences for their subsequent reasoning (Lobato et al., 2013). Researchers have studied ‘teacher noticing’ for investigating the processes of professional learning (e.g. Jacobs et al., 2010; Mason, 2002), but there is an emerging body of research on ‘student noticing’. Student noticing has been defined as selecting, interpreting, and working with features salient to them from multiple sources of information (Lobato et al., 2013). Lobato et al. (2012) connected student noticing with students reasoning about unfamiliar (cognitively demanding) mathematics problems. They built on actor-oriented transfer (AOT) theorisations to conceptualise a framework that coordinates both individual and social aspects of student noticing. An AOT perspective—taking an actor’s point of view—resonates with studies that investigate students’ “interpretive engagement with the experiential world, through an interaction of prior learning experiences, task and artefactual affordances, discursive interplay with others, and personal goals” (Lobato, 2012, p. 234). Lobato and colleagues (2012) developed their Student Noticing Framework comprised of four components: ‘Centres of focus’ (CoF), ‘Features of mathematical tasks’, ‘Focusing interactions’, and ‘Nature of the mathematical activity’. The CoF emerge through student noticing, and the other three components assist in accounting for the emergence of particular CoF, as outlined in the following subsections.

Centres of focus

Centres of focus are properties, features, regularities or conceptual objects, which a student notices. They are not directly observable as such, but are inferred from semiotic data, for example, verbal utterances, physical gestures, and written inscriptions. During interaction between/among students (and/or teacher), multiple CoF are likely to be at play (Lobato et al., 2013). In the quadratic figural pattern context for this study, the Centres of focus to which pairs of students attend give insights into the resources they are using, for example, the task itself and the representations provided, their prior mathematics knowledge, and their interactions with each other. Sequences of CoF provide evidence of whether the pair’s CoF were productive for their algebraic reasoning.

Features of mathematical tasks

Instructional tasks provide the context for discourse practices—discussions between/among students (and/or teacher). The features of a task can influence what students notice because they constitute a collection of features that are available to be noticed (Hohensee, 2016). Some features may be easily discernible, or discovered later through transformation, and others may remain hidden without intervention (Montenegro et al., 2018; Rivera, 2010). The design of a task has ramifications for the CoF that are likely to emerge. For example, a task involving proportional rectangles in a growing pattern afford opportunity for students to notice and describe a quadratic relationship geometrically in terms of changes to measurement attributes (height, length, and area) of the rectangles. A triangular or ‘step’ structure may help students notice the linearly increasing rate of change but not how to find a symbolic generalisation (algebraic equation). Tables of values for a quadratic relationship afford opportunity for students to notice and describe number sequences using numeric reasoning but do not elicit spatial reasoning.

Focusing interactions

Focusing interactions refer to the discourse practices themselves—verbal utterances, physical gestures, and written inscriptions—that give rise or contribute to particular CoF (Hohensee, 2016). Lobato and colleagues (2013) adopted two of Goodwin’s (1994) discourse practices—‘Highlighting’ and ‘Coding’ (now ‘Re-naming’)—for mathematics learning and added a third termed ‘Quantitative dialogue’. Hohensee (2016) additionally included Goodwin’s (1994) category for ‘Producing and articulating material representations.’ These four types are defined and illustrated in Table 1. In the current study, the description for ‘Quantitative dialogue’ was extended to encompass verbalised symbolic generalisations (not just numerical) since these were found in the students’ interactions.

Table 1 Four types of focusing interactions in the Student Noticing Framework (Hohensee, 2016; Lobato et al., 2013) and illustrated examples with pattern generalisation

Nature of the mathematical activity

Nature of the mathematical activity (or ‘Norms of noticing’; Hohensee, 2016) describes the social ‘global’ norms that establish or regulate who is expected/allowed to speak, and what type of contribution is permitted, such as creating new strategies, explaining reasoning, critiquing others’ ideas. Such norms are seen as influencing the CoF that emerge. In this study, the students worked in self-selected pairs and were given task handouts to share between them to encourage joint written work. The tasks encouraged collaboration in pairs and students were also found at times to consult other pairs about their task responses and discuss answers, although they were encouraged to start with tackling a task themselves.

Prior research on student noticing with quadratics

There are a few studies that have used the Student noticing framework in the context of either figural growing patterns or quadratic functions. Lobato et al. (2013) demonstrated empirically the emergence of different CoF in two middle-grade classes with the same content goals on linear functions, but with dissimilar tasks, activities, and class discourse. One class participated in figural growing pattern generalisation and three CoF emerged from their data analysis, presented in Table 2.

Table 2 Three Centres of focus in linear figural pattern generalisation (Lobato et al., 2013)

The specific illustrations of this CoF imply that although students might procedurally use the constant rate of change for their generalisation of linear relationships, they may not yet notice why the coefficient of the step number is the rate of change (i.e. move beyond recursive reasoning to a covariation view of function with an equation). In another study with 7 middle-grade students, Hohensee (2016) used the Student Noticing framework to investigate their collective reasoning first with quadratic functions and then again with linear functions (learnt previously). He found that the students developed quadratic function concepts that could be leveraged productively for comparative reasoning with linear functions, in particular a co-variation view of function through noticing how quantities co-vary firstly in quadratic relationships, and then in linear relationships.

In this study, the Student Noticing framework was used as a fine-grained analytic tool for investigating pairs of students responding to different types of figural quadratic pattern generalisation tasks. It was considered an appropriate methodological means of embodied visualisation research on students reasoning together (Radford, 2014) with cognitively demanding algebra tasks. The intent was to investigate how different Centres of focus that the students evidenced related to their ability to reach symbolic generalisation with quadratic figural patterns and how the coordination of their visual and algebraic reasoning in pairs might draw their attention to particular quadratic function concepts.

Research design

A qualitative collective case study of two pairs of students working together on different types of (unfamiliar) quadratic figural pattern tasks was undertaken to analyse in detail their Centres of focus, embodied visualisation, and algebraic reasoning. Theoretical insights into useful teaching strategies, for supporting conceptual understanding of quadratic functions and addressing documented student difficulties, were sought through fine-grained in-depth analysis of students’ actions (Creswell, 2013). The unit of analysis was each pair of students (Creswell, 2013) working together in their usual classroom setting (Cobb & Bowers, 1999).

Development of the quadratic pattern generalisation tasks

For this case study, data were drawn from within-class videoing, work samples, observations, and post-task interviews of two pairs of Year 10 (15 or 16-year-old) students. They (along with their whole class) experienced 19 figural quadratic pattern generalisation tasks of increasing difficulty in a task sequence interspersed throughout their usual school topic on quadratic equations (taught by their class teacher). An overview of the pattern task sequence is presented in the Appendix.

Six illustrative generalisation tasks (highlighted in bold in the Appendix) and the students’ responses are presented in detail in the Findings section (see Figs 1, 2, 3, 4 5 and 7 for the relevant task prompts and patterns) to provide descriptive transcripts of students’ (multimodal) interactions and insights into the analysis process used for all 19 tasks. These six tasks illustrate the variety of different Centres of focus evidenced by each pair, the shifts in focus highlighting their coordination of visual and algebraic reasoning, and attention to some key quadratic function concepts.

In terms of ‘Features of mathematical tasks’ in the Student Noticing framework (Lobato et al., 2012), the task sequence was developed through categorising a variety of figural patterns using Rivera’s (2010) definitions of lower- and higher-goodness patterns, and numerical quantities of the figures, interpreted by the author (Wilkie, 2019, 2021) as ‘simple’ or ‘complex’ (see Table 3). Complex patterns were categorised as such because they involve larger quantities in their figures and therefore require greater cognitive load (they also discourage local tactics or ‘guess-and-check’—see Mason, 1996).

Table 3 Four categories of quadratic figural patterns

The six illustrative tasks discussed in this paper include four simple higher-goodness and two simple lower-goodness patterns. Prior to tackling them, both pairs had correctly symbolically generalised some higher-goodness linear and simple square-based quadratic patterns. So the process of collaborating to find an algebraic equation relating item number (n) and total number of tiles (t) had become familiar to them. The task prompts, such as to extend the pattern with a drawn figure (after and/or before the given figures), generalise, and find other visualisations and equivalent equations, were developed from the literature on learning with figural patterns (e.g. Kaput, 2008; Markworth, 2010). Written prompts in the tasks for students to highlight figures with circling/colouring were drawn from prior research with younger students on visualisation with linear patterns (Wilkie & Clarke, 2016).

Participants

After consultation with the Head of Mathematics at a large suburban non-government school in Melbourne, the Year 10 Mathematics Coordinator agreed to participate in the teaching intervention with his own class. (In Victoria, nearly 40% of students attend non-government schools.) His students were intending to study the more rigorous units in Years 11 and 12 (Mathematical Methods), which include Calculus, making functions an important focus for them. The intended data collection for the current study involved one class, but the teachers of three other Year 10 classes were also interested in experimenting with some of the tasks and so were provided with task handouts and solutions.

Year 10 had been targeted for the teaching intervention because the students’ current prescribed national curriculum places introductory study of binomial expressions and simple parabolas the year before (Year 9) (ACARA, 2017). The intent was for participants to have some prior knowledge of quadratics, as this was found to be productive with pre-service teachers (Wilkie, 2021). In consultation with the class teacher, the figural pattern tasks were scheduled throughout the usual Year 10 topic on quadratic equations (the null factor law, factorisation, real solutions, the quadratic formula, problem solving; quadratic graphs are taught as a separate topic later in the school year). It emerged after the data collection had taken place that the school had chosen to exclude the topic of quadratics from their Year 9 program, and therefore the students had not studied quadratics before and exhibited much less prior knowledge than expected by the researcher (yet no experience of quadratic figural patterns as intended).

During nine (60-min) lessons over a period of four weeks (that were dedicated to figural pattern tasks) the students in the Year 10 class all worked in self-selected pairs. Given the fine-grained intensive nature of the analysis, two pairs were considered appropriate for the study. Beforehand, the teacher was asked to suggest two pairs of students that would be likely to be at differing levels of understanding about functions and comfortable talking aloud with each other on the tasks; the four students and their parents gave their consent for video data to be collected.

Data collection

Table 4 presents an overview of the data collected at each stage of the study.

Table 4 Overview of data collection

Pair #1 (Tess and Cath) and Pair #2 (John and Matt) (gender-preserving pseudonyms) were videoed during the nine lessons. A camera for each pair was directed downwards towards their workspace to capture their verbal utterances, physical gestures, and written markings on the hard-copy task handouts. Their task handouts and other writings were collected. The pairs were actively encouraged to attempt each task first before consulting the teacher, researcher (author) or other peers. Student and teacher interviews were audio-recorded and transcribed. Each lesson concluded with a brief class discussion facilitated by the researcher in which students shared what they had noticed. The class teacher interacted with the student pairs during the task sequence lessons but with minimal teaching to encourage learning through problem solving (Van de Walle, et al., 2017) and taught his usual lessons on quadratic equations.

Data analysis

To address the first research question on students’ foci of attention during the process of generalising quadratic figural patterns, a similar analysis process to Lobato et al. (2013) was employed. The Centres of focus (CoF) from Lobato and colleagues’ research on linear pattern generalisation (Table 2) were included as initial provisional categories derived from the literature (Miles & Huberman, 1994). Further categories were derived from the author’s prior research with linear patterns (e.g. Wilkie & Clarke, 2016) and with quadratic patterns (involving pre-service teachers; e.g. Wilkie, 2021). These CoF were then refined and added to through constant comparative analysis of each pair’s video data with multiple passes (Corbin & Strauss, 2008) across 19 different quadratic pattern generalisation tasks. A colleague and previous co-researcher on linear figural pattern generalisation reviewed the data analysis for a third of the tasks using the revised CoF framework (presented in the Findings in Table 5). Attention was firstly paid to documenting moment-by-moment each pair’s Focusing interactions—verbalisations, gestures, writing—using video data (and work samples). These actions were then interpreted, for example, as attending to the relationship between variables with a particular view of function, or as noticing different generalisable aspects of a figure’s structure (such as linearly increasing side lengths of squares or added constants or missing corners). Sequences of CoF were then analysed for evidence of algebraic reasoning moving towards or away from symbolic generalisation. It is acknowledged that the coding was researcher-driven and CoF that emerged are “data talking through the ‘eyes’ of the researcher” (Corbin & Strauss, 2008, p. 33). Within the confines of the paper’s word limit, analysis tables (Tables 7, 8, 9 and 10) have been included in the Findings section to provide further transparency about the analysis process and for the reader to draw their own conclusions about the researcher’s interpretations of the pair’s documented interactions (Silverman, 2001).

Table 5 Centres of focus evidenced by pairs across 19 quadratic figural pattern generalisation tasks (

To address the second research question on ways the pairs’ coordination of visual and algebraic reasoning evidenced drawing their attention to particular quadratic function concepts, each pair’s interactions were analysed to identify sequences of CoF that particularly involved pairs in connecting any visually noticed generality in the figures to their symbolic representation in algebraic equations. These were then synthesised across the 19 tasks and four pattern types to categorise, compare, and contrast interactions within each pair and how they related to pattern type and level of generalisation reached. These themes are discussed in the following section using six illustrative tasks as previously outlined.

Findings

In the following section, six figural pattern tasks provide an illustrative context to present findings on each pairs’ Centres of focus (CoF) that evidenced movement towards or away from symbolic generalisation. Ways in which the students’ coordination of visual and algebraic reasoning in pairs were found to draw their attention to quadratic function concepts are also described.

To help the reader with deciphering the CoF in the various tables, Table 5 presents definitions for the 16 Centres of focus (CoF) that emerged from analysis of the two pairs’ quadratic pattern generalisation, along with semiotic examples. The CoF were categorised as relating to: Views of function (such as recursive or correspondence); Process of generalisation (such as capitalising on pattern goodness, or transforming pattern goodness from lower to higher through rearranging structures, or verifying proposed equations); and Features of quadratic function representations (such as pattern figures, tables of value, equations, graphs).

(Please note that some tasks involved students constructing tables of values for, and graphing, their figural patterns, so although some CoF do not appear in the six tasks shared in this paper, they are nevertheless included in Table 5 as they emerged from analysis of other tasks in the sequence.)

To address the first research question, Table 6 summarises the CoF that were found to move pairs in their interactions towards or away from symbolic generalisation across the four pattern types.

Table 6 CoF within each pair across pattern types (19 tasks) that evidenced movement towards or away from symbolic generalisation

It can be seen from Table 6 that a correspondence view of function (CoF5 – first row) and verification of generalisation (CoF6 – third row) were used by both pairs across all pattern tasks, which was encouraging, as these are considered important algebraic reasoning approaches in the literature on figural patterns (e.g. Mason, 2017; Smith, 2008; Radford et al., 2007). A correspondence approach evidences the students’ attention to how the variables (n and t) are related to each other, rather than focusing on one variable (usually the dependent variable t, as with a recursive approach) (Smith, 2008). Verifying their generalisations, and repeatedly across several tasks, is suggestive of the students attending to generality—of wanting to ensure that their equations ‘work’ for all possible values of n (Radford et al., 2007). A key CoF was CoF7 Instantiation, which was found in both pairs’ interactions, since it appeared to act as a scaffold for coordinating their visual and numeric reasoning initially, and then their visual and algebraic reasoning. These findings about specific CoF and evidence of students making conceptual connections are described in the following six subsections with the illustrative tasks arranged chronologically.

Numeric instantiations as a scaffold for moving from the visual to the symbolic: ‘Squares’ task

Figure 1 presents the task prompts and figures provided to the students in the ‘Squares’ task.

Table 7 presents each pair’s evidenced CoF and focusing interactions for the ‘squares’ pattern in Fig. 1 above.

Table 7 Each pair’s response to the ‘Squares’ task

Both pairs initially found the (correct) factored equation t = (n + 1)² using the ‘length $\times$ width’ visually discerned dimensions of each square. (This response occurred with the uniformly shaded figures.) Both pairs also found the (equivalent) general equation t = n² + 2n + 1 by responding to the three shades of grey in the second set of figures in the task. Each pair noticeably developed numeric instantiations (CoF7, e.g. ‘1² + 2.1 + 1’) for a particular figure—that represented numerically what they were seeing spatially—and then translated them into a symbolic generalisation (equation). This process seemed an essential correspondence ‘bridge’ between their initial visualisation and their later symbolic generalisation in providing both multiple numeric ‘instances’ of the functional relationship and the means for verifying they had found a correct generality.

In this task, both pairs also evidenced repeatedly checking each other’s ideas and verifying the rules they found (CoF13): a focus that supported the process of generalisation by ensuring that their equation would give the correct tile totals for different item numbers. Interestingly, at this stage neither pair evidenced explicitly expanding their first rule to verify if it was algebraically equivalent to their second rule, suggesting that they were not yet drawing on newly learnt equations-based approaches from their other lessons. (Task prompts later in the sequence did, however, elicit their attention to further verification of equivalent rules algebraically.)

Numeric totals shifting focus away from symbolic generalisation: ‘Overlapping squares’ task

Figure 2 presents the task prompts and figures provided to the students in the ‘Overlapping squares’ task.

Fig. 2

‘Overlapping squares’ task Draw figure #5 in the following pattern and then find an equation to calculate the total number of grey tiles in each figure

The squares in this task had been shaded deliberately, as with some prior tasks, to highlight for students the structural aspects of the figures (note that all three shades of grey are included for generalising the total number of grey tiles). Table 8 presents each pair’s evidenced CoF and focusing interactions for the ‘Overlapping squares’ pattern in Fig. 2.

Table 8 Each pair’s response to the ‘Overlapping squares’ task

Each pair evidenced one person building on their partner’s expressed idea to move to a higher level of generalisation. Pair #1 did evidence one person (Cath) temporarily shifting their partner’s attention to a lower level of generalisation (CoF 6—finding numeric totals—which is different to CoF7—numeric instantiations that spatially match visualisations of the structure). Tess, however, after responding by scribing Cath’s ideas, nevertheless returned to her focus on trying to symbolise her previous instantiations. Cath then joined her by verbalising Tess’ instantiations spatially (add larger and smaller squares, subtract the overlap). They collaborated on symbolising it and then made some verbal utterances in unison. Pair #2 also evidenced constructing visual instantiations (CoF7), with John focusing initially on adding the two lighter grey squares, and Matt contributing by subtracting the darker grey overlap. As with the previous ‘Squares’ task, it seemed that the various shades of grey in the provided figures supported the students’ spatial reasoning by drawing their attention to usefully generalisable parts.

Puzzling over first and second differences for finding parameters: ‘Rectangles’ task

Figure 3 presents the task prompts and figures provided to the students in the ‘Rectangles’ task.

Fig. 3

‘Rectangles’ task For the following collection* try to find three equivalent rules and shade each picture to show how each rule relates to the shaded sections

Table 9 presents each pair’s evidenced CoF and focusing interactions for the ‘rectangles’ pattern in Fig. 3.

Table 9 Each pair’s response to the ‘Rectangles’ task

It can be seen in Table 9 that Pair #1 began with a correspondence view of function, but then moved their focus to finding the first and second differences (CoF3 & 4). They found that the second difference was always ‘2’ and puzzled over how they could use the ‘2’ in their generalisation. Tess queried, “So isn’t the rate of rate of change the a or the b?” This is perhaps suggestive of seeking a conceptual connection between what they were learning about the general quadratic equation parameters (in their usual quadratic equations lessons) and their meaning in symbolic generalisation (in the pattern tasks). Or they could be trying to apply their knowledge of rate of change with linear functions—that the coefficient of the x is the rate of change (m in the general equation y = mx + c)—to quadratics. The pair eventually returned to a focus on visual reasoning with the side lengths of the rectangle to find the matching factored form of the generalisation. They did not move on to other visualisations with the figures initially but expanded their rule algebraically to give the general equation form (n² + 5n + 6). Although the figures can structurally elicit the general form (a simple n² square with the other linear and constant parts), Pair #1 did not find it this way. Their actions, however, do suggest an increased awareness at this stage of the sequence, of being able to find equivalent expressions algebraically (using procedures from their other lessons). With a complex higher goodness pattern (later in the sequence; not reported here), Pair #1 evidenced puzzling over second differences again, suggestive of at least some awareness that rate-of-change concepts do relate to quadratic parameters (halving the second difference gives the coefficient of the x² term—the parameter ‘a’ in y = ax² + bx + c).

As intended by the task prompts, Pair #2 did find three different visualisations and generalisations for the same pattern—factorised form and then two other equations based on seeing a smaller square, firstly (n + 2)², and then (n + 1)² with the remaining linear and constant parts. But surprisingly, they didn’t visualise the general form. This response highlights that students may visualise figures in ways other than intended by a task’s design, and that perhaps some of these tasks would best be open-ended, i.e. to find as many visualisations as they can find. The pair’s interactions nonetheless showed multiple sequences of coordinated visual and algebraic reasoning: creating instantiations using spatial reasoning, translating to a symbolic generalisation, and then seeking to verify using the figural structures for multiple figures. In that way both John and Matt self-corrected when the figures didn’t match what they initially proposed. This suggests further evidence of instantiations as a valuable CoF for supporting coordinated visual and algebraic reasoning, and attention to generality (Radford et al., 2007).

Seeing general equation form terms visually: ‘Flowers on stems’ task

Figure 4 presents the task prompts and figures provided to the students in the ‘Flowers on stems’ task.

Table 10 presents each pair’s evidenced CoF and focusing interactions for the ‘Flowers on stems’ pattern in Fig. 3.

Fig. 4

‘Flowers on stems’ task* Draw figures #1 and #5 and then generalise each of the following patterns (find the rule for any figure #), giving equations (in general form). (Circle/colour parts of the figures to show how they relate to each part of your equation.)

Table 10 Each pair’s response to the ‘Flowers on stems’ task

With this task, neither pair appeared to need to develop numeric instantiations to scaffold their visualisations but moved quickly to symbolic generalisation. This pattern had been designed to elicit a generalisation in general form (y = ax² + bx + c). Pair #1 evidenced (as intended) focussing on spatially matching the different parts of the flower to an appropriate generalised term to create an overall equation in general form. This response suggests that quadratic figural patterns can help students make sense of different types of functional growth through noticing each ‘type’ of structural increase in relation to its corresponding term in general form—quadratic (CoF1: ax² term), linear (CoF9: bx term), and constant or invariant (CoF8: c term). Yet Pair #2 visualised the flowers in an unanticipated way—with turning point form for the flower by itself and then an additional constant + 3 for the stem). They did not subsequently convert their rule to general form (despite the task prompt), using algebraic rearrangement (which they were learning in their other lessons) or a different structural visualisation. It is possible that with these students, the prompt ‘general equation form’ was not yet clearly understood, perhaps because of no prior quadratics study in Year 9.

Reverting to local tactics with lower-goodness patterns: ‘Bridge’ task

Figure 5 presents the task prompts and figures provided to the students in the ‘Bridge’ task.

Fig. 5

‘Bridge’ task Try to extend and then generalise the following bridge growing pattern.

This task was the 17^th figural pattern generalisation task in the sequence and the first lower-goodness pattern (in not being made up of square- or rectangle-based structures). The intent was to see if the students might focus spatially on the possibility of rearranging the figures (for example, to make squares with a missing corner). Pair #1 evidenced similar prior interactions: one person suggesting a symbolic rule and the other testing it. Tess (looking at the first figure) said “n + 2n?”, then “Oh, no, it has to be square, [pause] n² + 2n?”. She tested it herself by counting the squares in figure #2 and (incorrectly) said “No”. Cath then counted with figure #2 and disagreed, saying “Yeh, n² + 2n”. She also checked the rule with figure #3. The gestures used by both in their counting of tiles suggested that they were seeking a total numerically rather than connecting the terms to the bridge structure spatially. In this case, the pair seemed to collaborate on what Mason (1996) termed local tactics—a form of ‘guess and check’. They didn’t make any attempt to re-arrange the figures, yet successfully found a rule that worked. This suggests that a simple lower-goodness pattern on its own may not be sufficient to assess students’ algebraic reasoning, because of the possibility of a correct symbolic rule via local tactics, rather than noticing and representing generality. Pair #1’s checking of several figures (and their correct drawing of figure #4), however, does suggest some attention to generality.

Matt in Pair #2 watched John’s responses to the bridge task but did not intervene or offer anything verbally, with gestures, or in writing at all. John began by counting the number of tiles in the first two figures. After a long pause, John traced in the air (just above the paper) a horizontal then vertical line on the right side of the third figure (see Fig. 6), suggestive of trying out rearranging to create a rectangle (CoF11 Rearranging).

Fig. 6

John’s tracing gesture (arrows) on figure #3 of the bridge pattern

This was further evidenced when he wrote the instantiation “3 $\times$ 4 + 3” (CoF7 Instantiation) – which would match the rectangle he seemed to visualise, plus the vertical line of 3 tiles to the left of it. He wrote a similar instantiation (“2 $\times$ 3 + 2”) under the second figure, but unlike previous tasks, Matt didn’t respond to John’s idea. It was not clear if he understood John’s gestures, and John didn’t elaborate. John then re-counted the total number of tiles in the second figure. He then wrote “2 $\times$ 4” and “3 $\times$ 5” but there was no further evidence from any gestures that he had rearranged the bridge to make full rectangle with those dimensions; it seemed rather that he had reverted to numeric reasoning with the instantiations he had made. He then wrote “n $\times$ (n + 2)”. For this pair, it seemed that a task that was more difficult than the previous higher-goodness patterns led to each person working separately. Matt did not elicit explanations from John, and John did not offer them, nor ask for help. He seemed like he was close to ‘discovering’ the potential rearrangement of the bridge on his own but then defaulted to numeric reasoning.

Rearranging figures is the key with lower-goodness patterns: ‘Pine tree’ task

Figure 7 presents the ‘Pine tree’ pattern, given immediately after the ‘Bridge’ pattern.

Fig. 7

‘Pine tree’ task Repeat for the following pine trees growing pattern (n = number of years, t = total number of squares)

As with other tasks, Pair #1 evidenced working together by focusing initially on additive patterns: that the central ‘trunk’ of the pine trees involved constantly increasing pairs of tiles, and that a new row of branches was being added each year. As with other patterns, Cath also focused on finding the totals for each figure (CoF6 Correspondence to total) saying, “Let’s do the numbers, so there’s 2, 6”, while Tess scribed for her. Tess built on Cath’s idea and calculated first differences (CoF3 Rate of change). She initially wrote them down incorrectly (‘4, 8’) but Cath intervened (see Fig. 8).

Fig. 8

Pair #1’s written totals and differences

Tess continued to puzzle over the total additive change from figure #1 to #2 (+ 4) and also that figure #3 had an extra horizontal row of 4. She suggested “add 4 each time”, suggestive of focusing on the second difference with this pattern, but Cath disagreed, pointing out that it didn’t work consistently. After a silence of several seconds, Tess offered “It’s like n² plus that” (pointing to the item number labels under the trees: 1 year, 2 years, 3 years). Cath built on Tess’s idea and exclaimed, “n² + n!” They then verified by re-counting aloud in unison the tiles in figures #2 and #3 (not using their written totals). Cath commented, “That was easier than we thought.” It appeared that the pair had again used local tactics to guess and check a rule—made possible by the small quantities involved. With the complex lower-goodness patterns (given later in the sequence), this pair found that they were not successful with local tactics and had to return to spatial reasoning. Explicit prompting was needed to focus their attention on creating usefully generalisable aspects through rearrangement of the figures.

As with the earlier lower-goodness pattern, John began on his own, writing and verbalising ideas related to the amounts of squares in the side branches while Matt watched. John seemed to use numerical (rather than spatial) reasoning, writing ‘0, 2, 6, 12’ for the successive subtotals of the branches and then numeric instantiations to match: “2² + 2 = 6” and “3² + 3 = 12”. This was corroborated when he also said to the researcher (when queried) that he was “just looking at the numbers”. He then wrote (incorrectly) “(n – 1)² + (n – 1)”. Matt joined in then but did not build on John’s idea. It seemed that he had been reasoning separately, which once explained, evidenced visual reasoning with rearrangement (CoF11). He wrote “[1 + (n – 1)](n – 1) + 2n” (he initially divided the first term by 2 but self-corrected). After simplifying his rule to “n² + n” while John watched, Matt drew an arch across the fourth figure and explained to John verbally and with gestures his visualisation: how he had rearranged the branch ‘triangles’ to form a rectangle from the branches with side lengths “n – 1” and one more than “n – 1” with “2n” for the central ‘trunk’. He repeated his explanation for the other figures. John checked Matt’s rule numerically and agreed, “ok”. Although John had seemed to succeed with the previous low-goodness pattern with numeric reasoning, it was Matt who employed spatial reasoning with the pine trees and explained his rearrangement strategy to John. Later in the sequence, John showed that he understood Matt’s strategy by successfully rearranging complex lower-goodness patterns. This finding suggests that a key milestone is visualising a way to transform figures from lower to higher goodness so that the pattern becomes spatially ‘quadratic’.

Discussion

This fine-grained collective case study investigated two Year 10 pairs’ interactions in class on 19 quadratic figural pattern tasks. The Student Noticing framework (Goodwin, 1994; Hohensee, 2016; Lobato et al., 2013) was used to analyse each pairs’ Centres of focus (CoF) that evidenced movement towards or away from symbolic generalisation (Radford, 2003). The students’ coordination of visual and algebraic reasoning in pairs was also investigated for evidence of their attention to quadratic function concepts. As discussed below, the study found students evidencing connections between generality expressed spatially, and generality expressed symbolically, and with different quadratic equation forms. It provides evidence of the potential for visualisation to be harnessed for making sense of quadratic equations typically taught abstractly.

Overall, it was found that eight CoF evidenced students’ movement towards symbolic generalisation in their reasoning (see Table 6). Three of these foci have also been documented in the literature related to linear pattern generalisation: seeking a correspondence between item number and a quantity in figures (e.g. Lobato et al., 2013; Markworth, 2010; Smith, 2008), creating numeric instantiations to translate into symbolic equations (Wilkie & Clarke, 2016), and verifying a generalisation (Radford et al., 2007). It has been argued that students need to provide evidence of attending to generality in their reasoning for it to assessed as algebraic reasoning and not merely numeric or procedural knowledge (Dörfler, 2008; Radford, 2003). In this study with older students tackling quadratic patterns, their verifying was often spontaneous (not a task prompt) and of each other’s proposed equations. This provided evidence of their attention to generality in being aware that their equation needed to ‘work’ for any figure (i.e. any item number in the pattern). All four students frequently used the strategy of testing proposed generalisations, the absence of which other studies have highlighted as problematic because students find a rule that works only for one figure and go no further (e.g. Lannin, 2005). It is possible in this study that collaboration in pairs and the older ages of the students may have played a role in students recognising the importance of such a strategy.

Instantiations and verification were also found to be valuable actions for peer learning, with each student trying to make sense of the other’s visual and algebraic reasoning. Even when both students in the pair were struggling with a particular task, their interactions seemed to promote persistence. A possible teaching implication is that paired work on such cognitively demanding visualisation tasks may better support algebraic reasoning than individual work. The proviso is that pairs are well-matched in their interest in collaborating, which is not always the case; ineffective partnering can hinder the learning of the other (Sfard, 2001).

In this study five CoF that evidenced movement towards symbolic generalisation seemed to be elicited specifically by the quadratic nature of the patterns. Over time, the students were found to seek out different parts of a figure that could be generalised individually—the dimensions of squares or rectangles (e.g. the inside centre of the flowers in Fig. 4), linearly increasing sections (e.g. the four ‘petals’), and constants (e.g. the flower stems). For example, both pairs evidenced increasing fluency over time in being able to generalise (linearly increasing) side lengths of different types of rectangles: n + constant, mn + constant, and mn – constant; m > 1. This highlights the benefit algebraically for students in connecting the squared terms in a quadratic equation to ‘squares’, both quantitatively (x²) and geometrically (square shapes), and in experiencing factorised equation form as the multiplication of two dimensions of a rectangle. Students also evidenced becoming more adept at noticing spatial strategies such as seeing blank spaces as removing tiles (subtracting action in a visualisation) and rearranging shapes into more easily generalised squares and rectangles. The CoF that highlighted reaching a key milestone for tackling lower-goodness patterns (unreached initially by one pair) was visualising a way to transform lower-goodness figures into a usefully ‘quadratic’ structure (Montenegro et al., 2018)—to make the quadratic features more easily discernible. This finding suggests it is important to distinguish between students merely finding a correct symbolic rule and actual symbolic generalising. Evidence for algebraic reasoning with quadratics might require patterns of differing complexity and goodness.

These CoF provide evidence of students developing the important algebraic reasoning necessary for functions, through pattern generalisation: in this case ‘seeing’ algebraic terms in a quadratic equation as generalities through experiencing generalising squared (ax²), linear (bx), and constant (c) spatial structures. One pair additionally evidenced generalising side lengths of rectangular figures to find quadratic generalisations in factorised and turning point equation forms. These were hints that the students were making sense of the difference between variables and parameters in quadratic equations. Dörfler (2008) argued that such evidence, of making connections between students’ usual algebra study in school and pattern generalisation, is necessary for justifying the inclusion of pattern tasks in school mathematics. The students in this study were learning equations-based approaches, such as factorising and completing the square (with their class teacher), alongside the visual tasks throughout the intervention. Some of their interactions highlighted students seeking to connect ideas from both approaches. Some task prompts seemed to elicit this coordination, such as to find more than one possible visualisation (and its equation) for the same figural pattern and showing algebraically that generalisations were equivalent (although Pair #1 did attempt at times to bypass the visualisation route and find equivalent equations algebraically first!) Wilkie’s (2019) study with secondary pre-service teachers highlighted the potential for visual tasks to elicit new conceptual connections after prior quadratics learning. This current study with Year 10 school students beginning to learn quadratics provides emergent evidence that these connections might be made by incorporating pattern generalisation with the more traditional equations-based quadratics study as a complementary visual functions-based approach.

Two CoF were found to move students’ reasoning away from symbolic generalisation. One of these—additive change—has been documented in linear patterns research (e.g. Lannin, 2005; Lobato et al., 2013; Markworth, 2010) as relating to a recursive view of function: important for developing an understanding of co-variation with functions but not easily built on for deriving an algebraic equation (possible but mathematically demanding.) In this study of pairs, it was interesting to find one pair’s repeated oscillations between correspondence and recursive views of function, particularly when encountering lower-goodness patterns. Pre-service teachers have also been found to struggle with generalising such types of pattern (Hershkowitz et al., 2001; Orton et al., 1999) Nevertheless, grappling with both views of function together may still be productive for algebraic reasoning, because students are making sense of how each view of function ‘works’ in different representations—visual, algebraic, and numeric. This same pair additionally focused on trying to work out how to connect the second difference somehow to their equation for a quadratic pattern, which was intriguing. Pair #1 evidenced writing number sequences underneath the figures (almost like a table of values) and puzzling over the meaning of the first and second differences—particularly how they could use the information directly in their equation (generalisation). Although the pair did not ‘discover’ the connection (that halving the second difference gives the coefficient of the x² term), the fact that they were exploring it suggests the potential to teach it to students studying quadratics, perhaps by linking differences in figural patterns to tables of value and graphs for the same pattern.

Quadratic growth—as involving a rate of change of the rate of change that is constant—is a foundational idea for understanding quadratic functions (Lobato et al., 2012). It is also important for exploring understanding gradient functions before Calculus (Suominen, 2018). Secondary and tertiary students have been found to experience difficulty in distinguishing between quadratic and exponential growth (Afamasaga-Fuata’i, 2005; Wilkie, 2019). Quadratic functional relationships represented visually in a concrete context might provoke students to grapple with such a key idea as quadratic growth and distinguish it from exponential growth.

Conclusion

In providing insights into the incorporation of quadratic figural generalisation tasks in a traditionally-taught topic on quadratic equations, this study makes a contribution to the literature on harnessing visualisation for algebra learning, to address well-documented student difficulties with abstract procedural teaching. Rather than figural pattern generalisation being considered as “nice” but “independent” of more formal school algebra study (Dörfler, 2008, p. 146), it has potential as evidenced in this study, as a complementary approach that can support students in making sense of function concepts visually, numerically, and symbolically. A key implication is that drawing on visualisation for generalisation while teaching quadratic equations can support students in noticing and connecting function concepts, including views of function, quadratic growth and the meanings of different quadratic equation forms. The study also raised some questions about the timing and sequencing of such visualisation tasks and the need for further research, particularly on helping students relate quadratic rate of change to parameters in algebraic equations.

This study’s findings highlighted that students working in pairs (rather than individually) on cognitively demanding figural tasks can attend to a rich range of foci, which may evidence students moving towards or away from symbolic generalisation in their reasoning. Some of their foci were found to signal key milestones, such as knowing how to connect linearly increasing lengths and widths of rectangles to generalisations in factored equation form, or how to rearrange lower-goodness figures to make them visually ‘quadratic’. Some foci signalled a move to numeric reasoning with local tactics, particularly found with complex lower-goodness patterns. A possible teaching implication is that these foci may guide teachers in what to look for when observing and assessing their students’ understanding of quadratics, and how to respond in the moment to particular struggles. Experience with a range of higher- and lower-goodness patterns as well as more complex tasks seem to be needed, to help students move beyond savvy numeric ‘guessing’ to distinctly algebraic reasoning (Dörfler, 2008; Radford, 2003); this study provides evidence that quadratic rather than only linear patterns provide the opportunity to develop this algebraic reasoning. A limitation of this study was that the students were considerably less experienced with quadratic equations than expected and so some connections that pre-service teachers were found to make in another study (Wilkie, 2019) seemed only to emerge slightly with these students, even after several tasks. Future research with students who have already been introduced to quadratic equations and graphs will be helpful for investigating the timing for incorporating visualisation tasks.

The study provided evidence of visualisation tasks having the potential to address another documented student difficulty in school algebra—making sense of the different uses of alphabetic letters (pronumerals) in algebra, for example as unknowns, variables, arguments or parameters (e.g. Bloedy-Vinner, 2001). The students all evidenced connecting the algebraic terms from the general quadratic equation form (y, ax², bx, c) to different geometric aspects of figures and types of growth (quadratic, linear, and zero). An intriguing finding was also that a pair repeatedly tried to find out how to connect the (constant) second difference of a visually represented quadratic relationship to its algebraic equation (to achieve symbolic generalisation). Even though these students’ lack of prior experience with quadratics was considered a limitation of the study, it was heartening to find students, even at an early stage of understanding, grappling with such quadratic rate-of-change ideas. With more prior knowledge or refined task design, the connection might have been made, which would benefit such pre-Calculus students’ knowledge of functions. A constantly changing rate of change is an important meaning for quadratic growth (Lobato et al., 2012) that can support students’ later study of gradient functions (Suominen, 2018). Future research on harnessing figural patterns for introductory Calculus concepts would be worthwhile, for example on how students might build on their seeing quadratic growth concretely in figural patterns to ‘discover’ visually why halving the second difference gives the coefficient of the x² term (the parameter a) and connect it to other representations (e.g. to co-variation in tables of value and to graphical transformations). Tasks utilising a ‘non-abstract’ context for exploring parameters, alongside traditional graphical transformation approaches, could also be designed and researched for scaffolding Calculus students who struggle with understanding differentiation concepts.

Although the study highlighted the potential of incorporating quadratic figural generalisation into equations-based approaches to help students connect important quadratic concepts, it also raised the need to consider at what stage and in what ways such generalising can best support students’ learning, especially those intending to study Calculus. For students in similar curriculum contexts to the students in this study, quadratic equations and graphs are taught as separate topics. Findings from prior research with secondary pre-service teachers (Wilkie, 2021) and this study with inexperienced secondary students suggest that future research with more experienced students may provide further insights into the sequencing of quadratic tasks and experiences to elicit a richer and more connected understanding of concepts.

Appendix

Overview of the task sequence

Task focus	Functional activities
Pre-assessment	Knowledge about linear and quadratic functions, equations, graphs -Figural pattern generalisation -Multiple visualisations -Multiple representations -Figural pattern creation
Linear figural pattern generalisation	Sea star pattern (t = 6n + 1) Matchstick houses (t = 4n + 1) with three different visualisations
Square-based figural pattern generalisation & creation	Generalisation, multiple visualisations and equivalent rules, multiple representations: t = n2 t = n2 + 2 t = (n + 1)2; t = n2 + 2n + 1 t = (n + 2)2; t = n2 + 4n + 4 t = (n + 1)2 – 1; t = n2 + 2n Checkerboard pattern (t = (2n)2/2) Overlapping squares pattern* (t = n2 + 8n + 4) Create and generalise own square-based patterns
More square-based figural patterns	Simple bathroom tiles pattern: two visualisations (t = n2 + 4n and t = (n + 2)2 – 4) -Rate of change with table -Rate of change of the rate of change -Plot graph Large bathroom tiles pattern: two visualisations (t = 4n2 + (2n + 1)2 and t = (4n + 1)2 – 4n(2n + 1)) -Compare rate of change and rate of change of the rate of change with simpler version
Rectangular figural patterns	Generalisation, multiple visualisations and equivalent rules, multiple representations: t = n2 + n; t = n(n + 1) t = (n + 2)2 + (n + 2); t = n2 + 5n + 6; t = (n + 2)(n + 3) Create and generalise own rectangular-based patterns (with two equivalent rules each)
Spot the similarities (2n2 patterns)	Butterfly A pattern (t = 2[n(n + 1)] + 3) -Rate of change -Rate of change of the rate of change Butterfly B pattern (t = 2n2 + 2n) Key pattern (t = 2(n2 – 1) + n) -Similarities across three patterns in task Create another similar pattern (co-eff not 1)
Comparing and connecting (larger coefficient for n and rate of change)	Extension (before and after), generalisation, multiple representations: t = n(n + 2) – 1 + 4 Flowers on stem pattern (t = n2 + 4n + 3) -Compare tables and graphs
Time to get tricky (lower pattern goodness)	Bridge pattern (t = (n + 1)2 – 1) Pine trees pattern (t = n(n – 1) + 2n) Tilted rhombus pattern using different visualisations: t = n2 + (n – 1)2; t = n(2n – 1) – (n – 1); t = (2n – 1)2 – 2n(n – 1) -Show equivalence Create and generalise own ‘difficult’ pattern (class competition)
Creative construction	Create own figural growing patterns for: t = n2 + 8 t = n2 + 3n t = n(n + 4) t = n2 + 3n + 2 t = (n + 1)(n + 3) t = 3n2 – 2 Create growing pattern from given graph (and generalise)
Post-assessment & survey	Learning about quadratic functions, equations, graphs -Figural pattern generalisation -Multiple visualisations -Multiple representations -Figural pattern creation: for t = n2 + 6n + 8; open creation Feedback on experiences

1. * The six tasks in bold are those discussed in detail in this article