Investigating secondary students’ generalization, graphing, and construction of figural patterns for making sense of quadratic functions

Journal of Mathematical Behavior 54 (2019) 100689

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Journal of Mathematical Behavior journal homepage: www.elsevier.com/locate/jmathb

Investigating secondary students’ generalization, graphing, and construction of figural patterns for making sense of quadratic functions

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Karina J. Wilkie Monash University, Melbourne, McMahons Rd, Frankston, Victoria 3199, Australia

ARTICLE INFO

ABSTRACT

Keywords: Figural growing pattern construction Figural growing pattern generalization Multiple-representational activities Quadratic functional relationships Secondary mathematics

An important aim in school mathematics is to help students experience algebra’s power to express generality. Researchers have been investigating figural growing pattern generalization as an early route to developing students’ understanding of functional relationships. More recently studies have been considering ‘open’ or ‘free’ tasks involving figural growing pattern construction. This study explored twelve Years 7–12 Australian students’ intuitions and connection of meanings for quadratic functions through growing pattern generalization, graphing, and construction activities during individual task-based interviews. The students who evidenced complementary correspondence and covariation views when generalizing appeared to handle pattern construction activities in productive ways, regardless of their prior formal study of quadratic functions. Areas of difficulty with graphing growing patterns and potential benefits of pattern construction tasks for exploring quadratic equations and for formative assessment are discussed.

1. Introduction Algebra’s power to express generality can be overlooked by students in their experience at school if they are simply taught to manipulate algebraic expressions without coming to appreciate where algebraic equations originate. Students adopt a collection of procedures involving entities that are not meaningful to them and algebra remains mysterious (Mason, 2017). Promoting students’ opportunities for algebraic generalization has been widely advocated in the literature and for some time, as important for students’ effective learning (e.g., Kaput, 2008; Kendal & Stacey, 2004; Kieran, 2007; Mason, 1996). Recent research attests to the potential for stimulating learners’ intuitive powers to specialize and to generalize (Mason, 2017). One such avenue has been the generalization of figural growing patterns, even with much younger students (e.g., Blanton, Brizuela, Gardiner, Sawrey, & Newman-Owens, 2015; Warren & Cooper, 2008). Arcavi (2003) illustrated visualization’s powerful role in such figural pattern tasks, proposing that visualization can be the analytic process of coming to a generalization. In such a visual context with concrete materials, primary students have shown that they can grapple with the concept of functional relationships, and variables being represented by alphabetic letters in rules for growing patterns, before formal introduction to algebra at secondary school (Wilkie, 2016). It has been argued that many functions concepts are effectively taught through visualization. One gains a better understanding of a given concept, outcome, or notion by seeing it in as many contexts and from as many perspectives as possible (Kleiner, 1988). Eisenberg (2002) even suggested that a non-visual approach to functions and their associated conceptions actually impedes a learner’s development in making sense of functions. Much of the research in the literature on a functional approach to learning algebra with figural growing patterns has focused on E-mail address: [email protected]. https://doi.org/10.1016/j.jmathb.2019.01.005 Received 22 February 2018; Received in revised form 8 August 2018; Accepted 19 January 2019 Available online 15 February 2019 0732-3123/ © 2019 Elsevier Inc. All rights reserved.

Journal of Mathematical Behavior 54 (2019) 100689

K.J. Wilkie

generalizing linear relationships and at primary levels of schooling (e.g., Lee & Freiman, 2004; Lannin, 2005; Markworth, 2010). Yet it is unclear if or how this approach to learning might be incorporated effectively in older secondary students’ study of non-linear functions. The research literature has documented students’ difficulties with understanding functions in general (e.g., Doorman, Drijvers, Gravemeijer, Boon, & Reed, 2012; Eisenberg, 2002; Knuth, 2000; Schwartz & Yerushalmy, 1992), and quadratic functions in particular (Ellis & Grinstead, 2008; Lobato, Hohensee, Rhodehamel, & Diamond, 2012; Metcalf, 2007; Parent, 2015). There are also pervasive findings on students’ difficulties in translating between different representations of functions. Over recent decades there have been calls internationally to prioritise algebra teaching and learning (Hodgen, Küchemann, & Brown, 2010) and significant reform efforts in the United States to promote functional approaches for teaching algebra (Kilpatrick & Izsák, 2008). In the Australian context, pervasive fundamental algebra misconceptions have been found even among tertiary mathematics students (Bardini & Pierce, 2014). Yet there remain differing views on how best to promote students’ understanding of algebra and the most appropriate time and way to introduce functions concepts to students (Kieran, 2007). Traditionally, quadratic functions are introduced after linear functions via algebraic manipulation of equations (solving and factorising) and graphical representation (translations). Ellis and Grinstead (2008) suggested beginning instead with exploring quantitative relationships that are quadratic in nature. One avenue for such exploration, that also resonates with theoretical perspectives on moving from the concrete to the abstract (Fyfe, McNeil, Son, & Goldstone, 2014) is quadratic figural growing patterns. Patterns “offer a powerful vehicle for understanding the dependent relations among quantities that underlie mathematical functions” (Moss, Beatty, Barkin, & Shillolo, 2008, p. 156) so there is the potential to consider pattern generalization tasks for exploring these relationships. There is more to learn about how best to support students’ development of connected meanings for functions, both linear and nonlinear, and in different contexts. This study focuses on a particular learning context and with different types of tasks and representations for exploring functional relationships visually: quadratic figural growing patterns. It addresses the following research questions: 1 What intuitions of functional relationships do students at different year levels evidence when generalizing and graphing quadratic figural growing patterns? 2 What intuitions do the students evidence when describing connections between/among different representations for the same quadratic functional relationship? 3 How do the students’ intuitions when generalizing figural quadratic growing patterns relate to their ability to construct their own figural growing pattern for a given quadratic equation? The following section provides an overview of theoretical perspectives on intuitions of function and empirical findings on students’ learning of quadratic functions. 2. Context and background This study drew on theoretical perspectives in the research literature about approaches to understanding functional relationships, and on empirical research findings about students’ learning of quadratic functions. These are overviewed in the following two subsections. A third section provides information on the curriculum context for the Australian students in the study. 2.1. Theoretical perspectives on approaches to understanding functional relationships The concept of function is foundational to mathematics learning (Sfard, 1991; Vinner & Dreyfus, 1989). There are dualistic conceptions of function noticeable in the literature, for example, process/object (Breidenbach, Dubinsky, Hawks, & Nichols, 1992), point-wise/global (Monk, 1992) and operational/structural (Sfard, 1992). These dichotomies are similar to each other in conceiving a function as a computational process or as a static construct, views which are nonetheless complementary to each other (Sfard, 1992). Historically among mathematicians the view of functions structurally as objects developed over time and emerged from an initial processual view. Sfard (1992) asserted that school students also need to experience the processual nature of functions for as long as possible to help them develop foundational operational understandings. This resonates with Mason’s (2017) call to help students experience where equations – as expressions of generalization – originate, and at an early stage in their learning. Pattern generalization is one activity in which students experience both the processual nature of functions and the key purpose of algebraic equations: expressing generality. Intuitions about functional relationships based on their processual nature and through everyday experience have been evidenced by very young children (Blanton et al., 2015; Leinhardt, Zaslavsky, & Stein, 1990). The two key intuitions are viewing a function as covariation between two variables (for example as time passes, a tree grows taller) and as correspondence between two sets (for example, counting objects one by one aloud to establish cardinal equivalence) (Leinhardt et al., 1990). Both of these intuitions are important for developing students’ functional thinking (Smith, 2008). There are differing definitions of covariation in the literature (Thompson & Carlson, 2017) but for the purposes of this study, one definition that is useful in the context of pattern generalization and based on quantitative reasoning defines covariation as coordinating two varying quantities while also attending to the ways which they change in relation to each other simultaneously (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002). A covariation approach to functions supports students’ learning about constant and nonconstant rates of change as they learn to reason about change itself – the direction of change, the amount of change, the average rate of change, and the instantaneous rate of change (Thompson & Carlson, 2017). Lobato et al. (2012) investigated Year 8 students’ early 2

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exploration of non-constant rates of change in the context of distance, speed, and acceleration over time. They found that a covariation approach of exploring first and second differences in tables of values supported students’ learning about the quadratic function’s characteristic that the rate of change of its rate of change is constant. A correspondence approach to functions considers the relationship between two variables described with a rule (equation) so that a particular instance of one variable can be used to calculate the matching (corresponding) instance of the other variable (for example, for the rule y = x2 – 4, when x = 3, y = 5). As with Sfard’s emphasis on the complementary nature of seeming dichotomies in mathematics (1991; 1992), researchers have emphasised the importance of students being able to connect covariation and correspondence intuitions of function rather than use them in isolation or only in specific contexts, so that they attend to how these two intuitions relate to each other in functional relationships (Küchemann, 2010; Thompson & Carlson, 2017). Figural growing pattern exploration lends itself to developing both correspondence and covariation intuitions about function. For example, for the bathroom tiles catalogue quadratic growing pattern used in this study (Appendix 2), a correspondence intuition might be evidenced by finding the operation that any item number (n) squared plus four times that item number will give you the number of grey squares (t) for that item (n2 + 4n = t). When using a covariation intuition, a student might attend to the tiles increasing in area starting from 32 (then 42, then 52) and that each time there is a new tile added (the item number increases by 1), the difference in number of grey squares between the two tiles increases by 2. Students learning about functional relationships through experiencing pattern generalization has been theorised from a semiotic perspective (Radford, 2010; Rivera, 2013). Functional thinking is seen as involving the senses and is mediated by signs such as oral linguistic terms, physical gestures, and written symbols. These signs are considered constitutive parts of thinking (Radford, 2010). Rivera similarly described figural pattern generalization in terms of multimodality – that gestural, verbal, and arithmetical expression overall show the way in which a student concretely and physically sees the objects in a pattern. Thus, in exploring students’ functional thinking, paying attention to their use of different signs together provides evidence of how they are reflecting on and conceptualising patterns. A student’s gestures alongside their verbalisations and written work can give additional insight into their use of particular intuitions. For example, with the bathroom tile catalogue pattern example in the previous paragraph, a correspondence intuition is evidenced by a student pointing first to the item number underneath a particular tile, and then to the inside square within the figure of the tile, while saying that if you multiply the item number by itself, that will give you the same amount shown in the figure. If he/she repeats the same gestures for several items, there is additional evidence of recognising the indeterminacy of the pattern (Radford, 2010). A student pointing to an aspect of a figure repeatedly along the line of figures in the pattern while explaining the changes to that aspect in relation to the change in the item number, provides evidence of attending to the covariation of the variables. In addition to the importance of developing complementary correspondence and covariation intuitions of function, connecting multiple representations of function is also emphasised in the literature as foundational, or rather the idea of building connections among areas of representational activity (Moschkovich, Schoenfeld, & Arcavi, 1993). Mathematical representations cannot be understood in isolation since the systems in mathematics and its learning are structured so that different representations within a system are richly related to one another (Goldin & Shteingold, 2001). Schwartz and Yerushalmy (1992) challenged the widespread practice of introducing students to algebraic equations in isolation (equations-based approach focused on symbolic manipulation) rather than connecting them to other representations of functions from the outset. A cognitive structure helpful for the study’s design and data analysis intersects the previously mentioned dual conceptions of process and object views of function with different types of representations (Moschkovich et al., 1993). Additionally, Romberg, Carpenter, and Fennema (1993) suggested differentiating between the actions of interpreting or constructing a representation. Interpretation refers to a students’ actions through which they make sense of or acquire meaning from a specific representation. Construction requires a student to generate a new representation or parts of a representation that is not provided (Nitsch et al., 2015). The resultant framework is presented in Fig. 1. In this study and mostly within a process view of function (older students might draw on an object view when interpreting graphs), figural growing patterns (linear and quadratic; concrete materials and drawings) were considered as a type of visual ‘realworld’ context and the students both interpreted given patterns and constructed their own patterns. They also constructed and

Fig. 1. A framework for making sense of functions, adapted from Moschkovich et al. (1993) and Romberg et al. (1993) (source: Wilkie, 2016). 3

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interpreted the other four types of representations across the variety of patterning tasks. In considering the use of activities for learning conceptually about functions, Thompson (1994) emphasised the need to find worthwhile situations that are sufficiently favourable for prompting multitudes of representational activity so that they orient students to notice connections among them in that particular situation. He argued that there is a risk of students learning about functions in inflexible and representation-specific ways if they are unable to see something remaining the same as they move among a table, graph, and equation. The study in this article investigated the potential of figural pattern generalization situations for engendering multiple-representational activities with quadratic functional relationships. 2.2. Research on students’ learning of quadratic functions In an extensive review of the literature on functions learning and teaching, Leinhardt et al. (1990) categorised students’ difficulties and misconceptions as relating to: what is and is not a function; the meaning of correspondence in the formal (set-theoretic) definition of function (one-to-one and many-to-one); a tendency to inappropriate linearity; continuous versus discrete graphs; translating between representations; and graphical interpretation. Recent research has highlighted students’ difficulties in connecting their knowledge of the simplest functions – linear functions – across different contexts and representations (Wilkie & Ayalon, 2018). Studies have found that secondary students often struggle to understand non-constant rates of change when introduced to non-linear functions (Herbert & Pierce, 2012). In terms of student difficulties, translations between representations appear to have directionality – moving from a graph to an equation involves different psychological processes than moving from an equation to a graph, since it involves more difficult actions such as pattern detection rather than simply plotting points (Janvier, 1987). Tasks in many textbooks tend to focus on translating from equations to graphs rather than the reverse (e.g., Hodgen, Küchemann, & Brown, 2010; Leinhardt et al., 1990); findings have suggested that students prefer this translation (Chang, Cromley, & Tran, 2015). Studies on quadratic functions have found that students struggled with translating from parabolic graphs to equations (Zaslavsky, 1987, as cited in Leinhardt et al., 1990) and with translations between equations and graphs in either direction for those quadratic functions containing a zero-value coefficient (they appear to students to be missing; e.g., y = x2 + 4) (Zaslavsky, 1997). In studies on students’ understanding of quadratic functions, excessive linearity adherence was also documented. For example, several secondary students who were given three labelled points on a graph from a parabola, used two of them to find the gradient of the straight line passing through them and then substituted that value for the leading coefficient in the general quadratic equation (a in y = ax2 + bx + c) (Zaslavsky, 1987, as cited in Leinhardt et al., 1990). Students also interpreted the leading coefficient as the ‘slope’ of the parabola, suggestive of a conflation of linear and quadratic properties through a lack of understanding the nature of quadratic growth (Ellis & Grinstead, 2008). Several studies demonstrated that students’ difficulties with knowing whether to represent graphical data as discrete points or a continuous line were related to issues with the subtleties underlying the nature and representation of variables in a particular context (Leinhardt et al., 1990). Markovits, Eylon, and Bruckheimer, (1986) found that students tended to ignore domain and range restrictions for a function. Students tended to focus on individual points on a graph regardless of whether or not they were connected with a line – to ‘see’ a line as serving to connect points, not as possessing meaning in itself. This suggests these students were adhering to only a point-wise view of function (not global) and misunderstood what a line represents graphically. Issues with connecting multiple representations of quadratic functions have also been documented. A doctoral study of US high school students’ approaches to quadratic tasks (traditional tasks and those involving multiple representations) found evidence that the students perceived the different characteristics of a quadratic function (such as the axis of symmetry, the turning point, orientation, y-intercept, maximum/minimum, and location of roots) as isolated pieces of information (Parent, 2015). It appeared that the students applied procedural strategies to specific parts of a task, but had difficulties connecting their understanding across task contexts or representations. They also demonstrated a preference for solving problems algebraically rather than using tables or graphs, even when doing so was less efficient, and showed a preference for the general form of the equation rather than turning point form (y = a(x – h)2 + k). In another doctoral study, Metcalf (2007) studied US undergraduate pre-calculus students and similarly found that their knowledge was context-specific and representation-specific, and that they evidenced a lack of flexibility in moving among representations. This suggests that difficulties with functions can persist beyond the years of school algebra. To address student difficulties with multiple representations for functions, Wilmot et al. (2011) proposed a learning progression in which students advance from making connections between two representations to connections among representations and also learn to compare and contrast information given in multiple representations of the same function. In the study described in this article, the students were prompted with open questions regarding what connections they noticed first between, and then among different representations for the same figural growing pattern functional relationship. 2.3. Research on students’ learning with quadratic figural growing patterns There are fewer findings in the research literature on students’ generalization of quadratic patterns as compared to linear patterns. Although there is substantial evidence of linear growing pattern generalization benefiting students’ functional thinking (e.g., Markworth, 2010; Rivera, 2013; Walkowiak, 2014) there is more to understand about if or how such a visual approach and context might support students’ learning about quadratic and other more complex non-linear relationships or complement other (equationsbased and graphical) approaches in school algebra teaching and learning (Dörfler, 2008; Orton, Orton, & Roper, 1999). As with research on students’ linear pattern generalization approaches, there is some evidence in the literature that a covariation 4

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intuition on its own does not easily support explicit generalization of a quadratic pattern. One study in England collected data on three cohorts of pre-service teachers’ attempts at quadratic pattern generalization (arrangements of dots) in the context of a written examination. They found that without exception, the participants created a table of values and used differencing in their attempts to find the 5th, 10th, 50th, and nth terms (Orton et al., 1999). Hershkowitz, Arcavi, and Bruckheimer, (2001) also found that pre-service teachers across several countries tended to employ a covariation intuition in trying to generalize the well-known quadratic matchstick square grid pattern (through creating a table of values and calculating differences) and were not able to find an explicit rule. There is also evidence that students who draw on a correspondence intuition are more likely to succeed with finding an algebraic equation for a quadratic growing pattern. Hershkowitz et al. (2001) also studied the different responses of secondary students from several different countries in generalizing the matchstick grid pattern. They found that students (13 to 14 years old), who had previously experienced linear pattern generalization and expressing them algebraically, worked together on generalizing the grid pattern using their correspondence intuition and were successful. The study highlighted students’ production of different equivalent algebraic equations for the pattern, based on their different visualizations of the given figure’s structure. Researchers surmised that such tasks have the potential to help learners’ visual and symbolic reasoning and encourage interconnections between them. Some studies have found that the structure of the figures in a particular quadratic growing pattern play a role in students’ likelihood of using visualization successfully to find an explicit generalization. Rivera (2010) distinguished between patterns that are low or high in “pattern goodness” or “Gestalt effect”. Low-goodness patterns have a complex (or unbalanced) structure that either has no easily discernible parts or consists of parts that have no natural divisions. High-goodness patterns have an orderly, balanced, and harmonious structure, which allows learners to discern quantifiable aspects and specify an algebraically useful formula (p. 304). Orton et al. (1999) found that while no pre-service teachers could find a general rule for quadratic growing pattern involving triangular arrangements of dots (lower-pattern goodness), some succeeded with dots arranged in square-based structures (higherpattern goodness). More recently, researchers have demonstrated that lower secondary students could explicitly generalize quadratic growing patterns that are not obviously arranged as square-based structures (such as a triangular ‘staircase’) by transforming them in some way, for example joining two ‘staircases’ together into a rectangular array (Rivera, 2013; Steele, 2008). There is very little in the literature about students’ learning through constructing their own figural growing patterns or the invention of their own generalities, either with linear or quadratic functions. Such activity was advocated by Dörfler (2008) for supporting students to develop an operative (processual) approach to generalization. He asserted that with typical pattern generalization tasks (finding the nth term) it is possible for students to engage in pattern detection and description without actually becoming aware of the functional concepts embedded in the task. He suggested the inclusion of different representations for the same growing pattern, including tables and Cartesian graphs. Recently, Rivera and Becker (2016) studied lower secondary students’ responses to such “semi-free” tasks. Students were provided with the figures for items #1 and #2, from which to construct and generalize their own growing patterns (including finding more than one explicit rule for the same pattern they constructed). They found that a majority opted to create known (already familiar) linear patterns. The few students who opted to construct a quadratic pattern instead of a linear pattern, used triangular figures, which needed additional teaching support for successful generalization. Rivera and Becker (2016) advocated future research on such open algebra tasks that are more engaging and challenging for students, because less predictable, more rigorous, and more supportive of deep thinking. In the study described in this article, the students were given an open task where they were provided with a quadratic algebraic equation (not item figures) and asked to construct their own growing pattern (with concrete materials and/or by drawing). 2.4. Curriculum context for Years 7–12 students in the study Students’ experience of functions tends to be orientated around the prototype examples used by teachers and textbooks, with linear and quadratic functions traditionally taught first (Schwarz & Hershkowitz, 1999). The following overview of the learning context for the students in this study is based on content from the national curriculum (Australian Curriculum Assessment & Reporting Authority, 2018), the students’ textbooks Maths World published by Macmillan Education (prescribed by the students’ school), and the author’s prior secondary mathematics teaching experience in a similar context. In the Australian curriculum, a hybrid of traditional (equations–based) and reform (functions–based) approaches has been adopted for algebra and functions (Sutherland, 2002). Content related to early learning of functions is introduced across several topics and include: finding the rule for sequences, manipulating linear expressions, solving linear equations algebraically and graphically, finding ratios and rates, and constructing Cartesian graphs. There is no explicit mention of the word ‘function’ in the Australian curriculum until the advanced Year 10 and higher–level Year 11 levels (Mathematical Methods). The Years 7 and 8 Maths World textbooks include separate chapters titled ‘Algebra’ (equations-based), ‘Solving equations’, and ‘Coordinates and graphs’. At middle secondary levels (Years 9 and 10), the Australian curriculum prescribes solving quadratic equations using algebraic techniques and exploring the connection between algebraic and graphical representations of relations, such as simple quadratics, circles, and exponentials. Transformations of parabolas and factorisation of quadratic expressions are prescribed for Advanced Year 10 levels (Australian Curriculum Assessment & Reporting Authority, 2018). The students’ Year 9 Maths World textbook chapter titled ‘Linear and non-linear graphs’ introduces students to graphing parabolas. The main focus on quadratic functions is in the Year 10 text in two chapters titled ‘Quadratic equations’ (factorisation and solving with the quadratic formula) and ‘Quadratic graphs’ (different equation forms and transformations). Students who opt to study the higher-level and more rigorous mathematics subjects at Years 11 and 12 (such as Mathematical Methods and Specialist Mathematics) are introduced to Calculus and a wider range of function types. The following section provides information on the design of the study and the tasks used in the interviews with the students. 5

Interpretation of smaller grey squares in a ‘bathroom tile’ growing pattern and extension of pattern; initial local action and then global action to find equivalent algebraic rules; defining pronumerals; filling in table of values; translating among figural, descriptive / algebraic and tabular representations Figures of square bathroom tiles; discrete familiar quadratic growth situation Contextualized; number of tile in bathroom company catalogue as independent variable and number of grey squares in a tile as dependent variable; use of covariation or correspondence and algebraic representation of variables with defined pronumerals expected

Action

6

Variables

Situation

Figural quadratic pattern generalization; relating two discrete variables with one or two equivalent descriptive/algebraic rule/s

Focus

Task 1 A – generalization of quadratic tiles pattern (high goodness) Graphing ordered pairs from a quadratic figural growing pattern; explaining connections between different representations of the same quadratic relationship Graph construction using previously completed table of values; choosing discrete or continuous graph; interpretation of graph; translating among figural, descriptive / algebraic, tabular and graphical representations; explaining connections among multiple representations for the same function Figures of square bathroom tiles; discrete realistic quadratic growth situation Contextualized; number of tile in bathroom company catalogue as independent variable and number of grey squares in a tile as dependent variable; use of covariation or correspondence concepts to explain links between variables in different representations expected

Task 1B – graph construction & interpretation – same tiles pattern

Table 1 An overview of the quadratics tasks used in the study using Leinhardt et al.’s (1990) categorisations.

Figures of triangular trees made up of square tiles; discrete familiar quadratic growth situation Contextualized; number of years of tree’s age as independent variable and number of square tiles as dependent variable; use of covariation or correspondence and algebraic representation of variables with defined pronumerals expected

Interpretation of squares in a ‘pine tree’ growing pattern and extension of pattern; initial local action and then global action to find algebraic rule/s; defining pronumerals; possible use of and translation between different representations

Interpretation of algebraic equation t = n2 + 2n and generically defined pronumerals (n and t); construction of first four items of a growing pattern using concrete materials or drawings; explanation of approach/es Abstract quadratic algebraic equation; generically defined pronumerals (n and t) Initially abstract (use of n and t in provided equation) but contextualized by students (use of counters, matchsticks, and/or drawings) and discrete; awareness of both variables in pattern expected; structural arrangement of figures to show quadratic growth of pattern expected

Figural quadratic pattern generalization; relating two discrete variables with one or two equivalent descriptive/algebraic rule/s

Task 3 – generalization of quadratic trees pattern (low goodness)

Figural quadratic pattern construction

Task 2 (open) – pattern construction for given quadratic equation

K.J. Wilkie

Journal of Mathematical Behavior 54 (2019) 100689

Journal of Mathematical Behavior 54 (2019) 100689

K.J. Wilkie

3. Research design This study investigated Years 7–12 students’ responses during one-on-one interviews to a series of linear and quadratic tasks involving generalization, graphing, and construction of figural growing patterns. This article focuses on three of the quadratics tasks and discusses evidence of students’ intuitions of function and conceptual connections across representations of function at different year levels (before, during, and after introduction to quadratic functions in their school mathematics program). The use of video-recording for capturing the students’ gestures, verbalisations and writing was informed by previously described semiotic theoretical perspectives on figural pattern generalization (Radford, 2010; Rivera, 2013). The design of the tasks was based on findings in the literature on students’ learning with figural patterns (e.g., Kaput, 2008; Markworth, 2010; Mason, 1996; Radford, 2010; Rivera, 2010, 2013), the researcher’s experience teaching quadratic functions at secondary levels, and prior research on pattern generalization with upper primary and lower secondary students (e.g., Wilkie, 2016) and on quadratic functions challenging tasks with Year 10 students (Wilkie, 2015). Table 1 provides an overview of the three quadratic tasks reported on in this article, using Leinhardt et al.’s (1990) categorisations. The interviewer (author) requested at the beginning that the student verbalise their thoughts out loud throughout the interview as much as possible. The interviewer read aloud each of the task questions in turn to the student (Appendices 1 and 2 present the questions for Tasks 1A, 1B and 3.), while making additional verbal prompts in the moment to encourage further verbal explanation or gestural demonstration from the student. In the first task (Appendix 2 bathroom tiles catalogue) the students were asked to extend a square-based (high pattern goodness) quadratic figural growing pattern by drawing additional items, describe the structure of the figures (verbally and by colouring the drawing), and then find two equivalent algebraic equations. This was to find evidence of the students’ use of visualization for generalization and ability to visualize the structure of the same figures in more than one way. They were also asked to fill in a table of values for the first six figures and to describe any connections they could see between their rule/s and the table. They were asked to explain how they would try to find an algebraic equation for the pattern if they were only given the table of values (and not the figures). This additional prompt was included to seek evidence of other covariation/correspondence strategies the students might draw on with other representations of the pattern, i.e., for translation from a table to an equation (since the reverse translation is emphasised more frequently in textbooks). Part B of the first task was designed to investigate students’ graphical construction for the same quadratic pattern, their interpretation of it, and any conceptual connections to other (previously constructed or provided) representations – the figural pattern itself, the descriptive or algebraic rule/s, and the table of values. The students were provided with a labelled set of axes (no scale): tile number (x-axis) and total number of grey squares (y-axis). In the second task, the students were asked to construct their own growing pattern for the provided equation: t = n2 + 2n (t = number of sticks/counters/things in an item; n = stage/item number). Concrete materials, such as coloured circular counters, square counters, and coloured matchsticks, were offered to the students, along with a variety of coloured pens. The intent of this open task was to learn about what intuitions the students drew on and if/how they used visualisation in reverse – to translate from an equation to their own figural growing pattern. The third task (Appendix 3 pine trees) was similar to Task 1 A in that the students were given another quadratic figural growing pattern to extend, explain, and generalize by finding an algebraic rule. The difference was that the figures were not in an obvious square-based arrangement; instead, they resembled trees with triangular structures on either side of the ‘trunk’. The intent was to see if and how the students might draw on previous experience with Task 1 (the bathroom tiles pattern) or use different intuitions with a pattern of lower pattern goodness or Gestalt effect (Rivera, 2010). 3.1. Participants and data collection Twelve high-achieving secondary students, a boy and girl at each year level (Years 7 to 12), from a large middle SES independent school in Melbourne were invited to participate. (Approximately 40% of students in Victoria attend independent (non-government) schools). The school’s Head of Mathematics deemed them to be high achieving, based on previous achievement results and the perceptions of their own mathematics teacher. A deliberate choice was made to recruit such students based on the reasoning that any misconceptions or difficulties they evidenced were surmised as likely to be more widespread among students overall and reflect genuine problematic areas (Goldenberg & Kliman, 1988). The students were also from the full range of secondary school year levels and therefore had differing or no prior experience with quadratic functions. This was intended to provide an exploratory sense of whether or not their intuitions and their meanings for and across representations might be developmental in nature. For ease of reading the study’s findings, the students have been assigned alphabetical pseudonyms, beginning with A for Year 7, and through to F for Year 12 (see Appendix 1). The students were interviewed one-on-one for approximately one hour in their own school setting during a school day (usually during a timetabled mathematics lesson). They were encouraged to ‘think aloud’ throughout the interview so as to provide insights into their self-explanations and conceptions (Booth, McGinn, Barbieri, & Young, 2017) in tackling the functions tasks. Their verbalisations were transcribed and analysed along with their videoed gestures, actions, and writing, so as to attend overall to their use of semiotic signs as evidence of their thinking (Radford, 2010; Rivera, 2013). The video camera was positioned overhead and pointed downwards so that a student’s hand movements and voice were captured, but anonymously. The intent was to help the students feel more comfortable with being videotaped and with expressing their thoughts aloud. They were informed that the transcripts and video recordings were only to be used for the study and would not be accessible by their own mathematics teachers. The students’ written 7

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work on the hard-copy task handouts (A3 in size) were also collected and used alongside the transcriptions and video footage. 3.2. Data analysis To address the first research question on the students’ intuitions of function for generalizing and graphing quadratic growing patterns, the data analysis sought evidence of: - the use of recursive (variation of the dependent variable only), covariation, and/or correspondence intuitions in the students’ choice of strategies, verbal language, gestures, and writings (Tasks 1 A and 3 in the appendices); - the students’ highest levels of generalization for each growing pattern using coding procedures established in prior studies with a research team (see Wilkie, 2016) (recursive, explicit descriptive, explicit algebraic expression/equation); - the students’ ability to find equivalent generalizations either with one intuition (or complementary intuitions; and - the students’ understandings about quadratic graphical representation through drawn features, verbal explanations, and gestures with their constructed graph of the Task 1 bathroom tiles growing pattern. To address the second research question on the students’ intuitions evidenced when describing connections across different representations of the same quadratic functional relationship, the data analysis documented specific instances in students’ spontaneous descriptions throughout the different tasks in which they compared or explained how different representations were related to each other in showing the same feature. These responses were analysed for evidence of types of translations between/among representations (pattern context, algebraic equation (rule), table of values, and graph) and of variation, covariation, and/or correspondence intuitions. More particularly, the interview also included scripted questions in the tasks asking students to describe what connections they could see, first between their rule/s and table, and then among their rule/s, table, and graph. Attention was also paid to the verbalisation of the students’ ideas, to seek insight into whether it was likely to be prior knowledge, a new application of prior knowledge, or a new idea that had emerged from the experience of the tasks in the interview. For example, a prompt and comprehensive response using formal mathematical language, such as “This is the turning point form of a quadratic equation” was considered evidence of prior knowledge. A student noticing from their table for example, “Oh, this is the same as what I already found with the pattern” was considered suggestive of knowledge application, whereas pausing and then exclaiming, “Oh, that’s crazy!” was considered suggestive of an emerging idea. To address the third research question about the students’ strategies and use of intuitions to construct their own figural growing pattern from a given quadratic equation, the data analysis included looking for evidence in the students’ verbalisations and gestures of the use of covariation and/or correspondence intuitions, and also examining the physical features and structure of the students’ constructed pattern (either with materials and/or drawn). 4. Discussion of findings The three sub-sections in this discussion address each of the research questions in turn on students’ correspondence/covariation intuitions: when generalizing and graphing given quadratic figural growing patterns; when describing connections between/among multiple representations of the same quadratic functional relationship; and when constructing their own figural growing pattern for a given quadratic equation. For additional information, Appendix 1 presents the summary of relevant task responses for each student in three tables, one for each research question (from Years 7–12 and by pseudonym). 4.1. Students’ intuitions when generalizing and graphing quadratic figural growing patterns Prior to being presented with the bathroom tiles quadratic growing pattern, the students had all demonstrated their ability to generalize two linear figural growing patterns and find a correct algebraic expression or equation (see Appendix 1). This suggested that they all were familiar with the process of generalizing a figural growing pattern prior to tackling the quadratics tasks. The older students tended to write full algebraic equations and to describe each linear relationship using covariation, whereas the younger students were more likely to write algebraic expressions and describe each relationship using correspondence. The students were initially presented with the quadratic bathroom tiles pattern (Appendix 2) and later with the pine trees pattern (Appendix 3) (high and then low pattern goodness). Table 2 presents the students’ approaches, level of generalization, and ability to find more than one equivalent rule for each of the two quadratic growing patterns: the bathroom tiles catalogue (Task 1) and the growing pine trees (Task 3). It can be seen from Table 2 that more students showed evidence of both correspondence and covariation intuitions when generalizing the bathroom tiles pattern, than with the pine trees pattern. They were also more likely to find equivalent rules with the bathroom tiles pattern based on different visualizations, for example, The middle square of grey tiles here, it’s got side length 3. So the number of squares would be the tile number which would be x squared… there’s always the same amount on each side as the tile number. So I’d do a plus 4x… (David, Yr 10) I guess you could do the whole area of the larger square minus the 4 corner ones… So I guess you could think of it as x plus 2 squared… then you can minus just 4, which is the white ones. (David, Yr 10) Additionally, Esther (Yr 11), after also finding these two rules and filling out the table of values, noticed a third rule from the 8

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Table 2 Students’ responses to two quadratic pattern generalization tasks of differing pattern goodness. Bathroom tiles (higher pattern goodness)

Pine trees (lower pattern goodness)

12

8

– 3 1 – 2 6

1 4 – 1 1 1

LEVEL OF GENERALIZATION Recursive Descriptive explicit rule Algebraic expression Algebraic equation

– – 5 7

1 – 6 1

NUMBER OF (EQUIVALENT) RULES FOUND Zero One Two Three

– 3 8 1

1 7 – –

Num. students who attempted task APPROACH/ES Recursive Correspondence Covariation Correspondence - > covariation - > correspondence Covariation - > correspondence Both correspondence & covariation

table, as shown in Fig. 2. Her exclamation suggested that finding another rule in this way had surprised her: They’re all – wait because it’s like five times one and then two times six and then three times seven and then four times eight and then five times nine… Okay, n because that’s one [pointed to the 1 in 5 × 1 written underneath the first ordered pair] and then times five – or yeah, four plus n [wrote n x (4 + n) underneath the table]. Oh that’s crazy! (Esther) It appeared that most students used a correspondence approach (similar to David’s previous examples) to find both rules, but also described the function using covariation language – how the tiles were changing from item to item as the catalogue number increased (and also not changing, as with the constant of the four corner white squares), for example: So, for tile #1 the base is 3 and then that becomes 4, that becomes 5 [points to bottom horizontal length of tiles #2 and #3]. So, for tile #4 it would be 6… And then for tile #5 there’d be 7. (Daisy, Yr 10) Fred (Yr 12) evidenced only a covariation approach for this task and found only one rule with it, then simplified it algebraically, rather than visualize a different structure with the figures, to find an equivalent expression: It’s going to increase by one vertically and horizontally, so the area will increase proportionally and similarly tile 5, it will be 7 wide and 7 units in height… It increases by one every time… so therefore the whole area of the square will be 2 plus n squared and then to find the number of grey squares you simply subtract the number of white squares and that’s going to be constant [wrote (n + 2)2 – 4]… That simplifies to n squared plus 4n [wrote n2 + 4n]. (Fred) Ethan (Yr 11) additionally demonstrated how to derive the algebraic equation for the bathroom tiles pattern by using simultaneous equations derived from the table of values, suggestive of prior knowledge. A puzzling difference between the students’ responses to the two patterns was that for the second (pine trees) pattern, most students reverted to writing an algebraic expression rather than a full equation with both variables represented. Perhaps the more challenging task meant that the students’ focus was solely on finding an explicit rule and ‘forgot’ the other variable in the moment, even though for the previous linear and quadratic tasks they had not. Most of the students initially wrote down ‘2n’ and related it using correspondence to the central column of the pine trees, and then tried to generalize the triangular structures separately (one on each side of the trunk), as shown by Cam’s (Yr 9) colouring in Fig. 3. This tree pattern took the students much more time to generalize, and they tended to persist in trying different strategies when one was unsuccessful. For example, Benita (Yr 8) and Daisy eventually constructed their own table of values and used it with a correspondence approach to find the expression x2 + x, as shown in Fig. 4. They used similar explanations of starting with the x2 term and trying to find the resultant difference in terms of the year number or ‘x’.

Fig. 2. Additional equivalent rule found by student from table of values (Yr 11 G). 9

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Fig. 3. Visualizing the pine trees pattern as a trunk with triangles on each side (Cam).

Fig. 4. Using a table of values and correspondence to generalize a pattern of low pattern goodness (Benita).

Adam (Yr 7) recognised the recursive triangular number sequence of the side structures in the pattern’s figures and attempted to use factorials (e.g., ‘4!’), albeit incorrectly (to represent 4 + 3 + 2 + 1 rather than 4 × 3 × 2 × 1), in his rule. He also experimented (unsuccessfully) with ‘n2’ in a rule using correspondence intuitions, having recognised that the pattern was non-linear. Ethan was able to draw on his prior knowledge of the formula for the triangular number sequence and doubled it (for both sides of the tree). Overall, it appeared that the students attempted first to use a correspondence view of the pine tree figures, but when they could not easily relate the structure to the year number, reverted to using a numerical sequence for the side branches to find a correspondence that way instead – a higher level of abstraction when visualization with the concrete figures proved unhelpful. None of the students transformed the triangles into a rectangular array, found in previous research (Rivera, 2013; Steele, 2008). It is possible that such a strategy requires prior experience of it and perhaps explicit teaching support. (Daisy actually asked the researcher after the interview to be shown how to visualize the figures’ structure for generalizing it and this transformation strategy was demonstrated to her.) Yet several students did know to focus on finding a multiplicative correspondence even though the triangular arrangement actually emphasised additive recursion (as previously seen with Adam). This is shown in the example in Fig. 5, where David wrote 1 × 0, 2 × 1, 3 × 2 etc. underneath the figures to represent the number of side squares. From the students’ responses, it can be seen that most could successfully use a correspondence approach with the figures in a quadratic growing pattern, even prior to formal learning about quadratic functions, if the figures were arranged in a multiplicative (square or rectangular array) structure. If the figures were lower in pattern goodness, then the students tended to maintain a correspondence approach but shifted to a more abstract representation, such as a table of values or number sequence, and used trial and

Fig. 5. Using correspondence with numerical sequence to generalize part of a pattern of low pattern goodness (the side triangular structures) (David). 10

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error to find a rule (eventually). It appeared that the older students were more efficient with this process because they had prior knowledge about quadratic relationships having a constant second difference. Once they had established this, they began with a squared term (n2) and experimented with the remaining difference. A more difficult pattern task with a non-zero leading coefficient might have produced different responses. After generalizing, the students (except Cathie (Yr 9) who ran out of time) were asked to graph the bathroom tiles growing pattern. There were provided with unscaled labelled axes. All of the Years 10–12 students plotted points and connected them to draw an accurate but continuous quadratic curve starting from the origin (0, 0), appearing to overlook the discrete nature of both variables. Both Year 8 students also drew accurate but continuous quadratic curves that started from (1, 5) – the point for the first tile in the catalogue. Adam didn’t add any scale but drew a continuous curve starting from the origin. Alyssa (Yr 7) plotted points (with an inaccurate vertical scale where zero was higher than the line of the x-axis) and then joined them in a straight line with a ruler. A few of the older students (Daisy and Ethan) did mention domain restrictions when talking about their graphs, for example that the turning point would be in another quadrant, and that the catalogue’s tile numbers could not be negative (yet they did include the origin in their graph). The students’ responses across the six year levels suggest that they had developed prior knowledge of the conventions for constructing a curved line graph by plotting points and connecting them. Only Alyssa created a linear graph, even though her correct rule for the pattern was 4x + x2 and she had also commented on the increasing first differences in her table of values: “It increases the first one by 7 and then 9 and then 11. So I guess you could do so by 7 plus 2, plus 2 every time.” It is possible that the small segment of this particular curve could be visually misconstrued as linear, particularly with an inaccurate scale and the lack of graph paper. She did comment as she ruled the line, “I know that is going to be a steady increase, that they should be in a line.” This could relate to a tendency to inappropriate linearity (Leinhardt et al., 1990) or is perhaps suggestive of some confusion in relating her covariational reasoning with a table to a graphical representation, between representing a constant increase (which she represented correctly in a previous linear graphing task) or an increasing increase (which she noticed in the table of values but still drew as a straight line). Representing discrete variables as continuous in a graphical representation seems suggestive of a lack of experience with the differences between the two types, or prior familiarity with mostly abstract graphs when learning algebra and real-life graphs when learning statistics. The concrete context in this task seems to have been overlooked by all of the students when constructing their graphs according to previous experience. Related to this, Cam drew an accurate column graph, suggesting prior experience of column graphs for ‘real’ data and only abstract ordered pairs being graphed with plotted points. Or it could be an attempt to represent discrete data as separate columns, rather than draw a continuous line. Wilkie’s (2016 ESM) study of Year 7 students found that a majority could not construct a correct graph of a growing pattern situation; several attempted to construct column graphs. And when asked to represent a real-life situation in multiple ways, no students chose a graph. These findings highlight graphing knowledge that might be isolated in specific pockets of experience according to domain (algebra and functions or statistics). The students’ lack of attention to domain restrictions because of the context and the need for discrete data representation could also suggest possible graphing misconceptions about points, and lines being made up of infinite points, which don’t appear to have been addressed, even by upper secondary levels. 4.2. Students’ descriptions of connections between/among multiple representations Table 3 presents the numbers of students who used certain conceptions of function in describing verbally a connection between or among different representations in Task 1 (bathroom tiles generalization and graphing task). Given the nature of pattern generalization, it is not surprising that the most frequent connection expressed by the students was between the figural growing pattern itself and an explicit rule for it, represented by an algebraic equation. This resonates with Thompson’s (1994) emphasis on students’ awareness of the situation of the task and also the concept of “concreteness fading” to help students connect to abstract concepts (Fyfe et al., 2014). In this activity, the task situation of the bathroom tiles catalogue is likely to be most salient with the figural representation of the growing pattern. A majority of the students described this connection in terms of a correspondence conception, for example, Your tile number, there’s that amount of tiles on the side and it’s the square in the middle with that too, so if x is the tile number then it’d be 4x for the outside, then plus x squared. (Alyssa) I just went n plus 2; the same went for all of these [pointed along the row of figures] because they all also had plus 2 then squared Table 3 Students’ use of different conceptions to connect different representations of the same function (n = 12). Conception

Pattern < > Rule

Rule < > Table

Table < > Graph

Graph < > Rule

Rule < > Table v Graph

Variation Covariation Correspondence Both correspondence & covariation Sub-total

– 1 9 1 11

– 3 2 4 9

1 7 – – 8

– – 3 – 3

– 3 2 – 5

11

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to make it square. And since each and every one of those squares have 4 corners out on the outside, it was minus 4. (Cam) Fred used covariation to describe the connection between the figural pattern and his explicit rule. As mentioned previously, he had also struggled to construct a correct growing pattern for a given quadratic rule, which seemed to relate to a lack of attention to a correspondence view and overreliance on covariation. Fiona (Yr 12) was the only student who showed no evidence of being able to express a connection between the figural pattern and its rule. She had used a table of values to find a rule rather than visualising the structure of the figures to find different equivalent rules. She had also shown difficulty with constructing her own growing pattern and had drawn clumps of circles without a discernible structure, again appearing to rely on numerical values (ordered pairs) generated from the rule. This suggests that her understandings of quadratic functions may be representation-specific and were not easily transferred to a figural structure context. Ben (Yr 8) showed evidence of drawing on both covariation and correspondence conceptions to connect the pattern and its equivalent rules: This is a cross [pointed to tile #1] and this is a bigger cross [pointed to tile #2] and then an even bigger cross [pointed to tile #3] and so just by expanding the cross we get a 4 by 4 in the middle and then 4 on the outside [his drawing of tile #4], which is basically a double version of this [pointed to tile #2] except with the corners remaining the same. (Ben) Most of the students also evidenced being able to connect the tabular representation of the growing pattern with the rule/s they had found. Several of the older (Years 10–12) students explained their prior knowledge that a constant second difference (dependent variable) in the table meant that the function had a squared number or power of two in it, for example, I always look at the differences to see what, what degree or what kind of function it is. So I’d look at the difference which is – between these two it’s 7 [pointed to totals for tiles #1 and #2], between these two it’s 9 [pointed to totals for tiles #2 and #3], between these two it’s 11, 13, 15. So that’s the first difference and then if you get the difference between the differences or a second difference, it’s constant. So, I take that to mean that it’s got a variable to the power of 2. (David) Benita evidenced complementary covariation and correspondence conceptions with the table of values and one of her rules (x2 + 4x): If I look at this rule for example [pointed to x2 + 4x], if I square numbers – 1, 4, 9 – it’ll be plus 3 plus 5 [wrote down sequence of squared numbers] just like these [pointed to similar pattern of difference of 2–7, 9, 11 – that she had written previously underneath her table of values] and then plus 4x would be consistent. (Benita) None of the students evidenced complementary covariation and correspondence conceptions to explain connections to or from their constructed graph, suggesting that although most tended to use correspondence to plot points, they viewed the finished line of their graph in terms of covariation: different views for different tasks. Interestingly, some students demonstrated a shift from correspondence while generalizing to covariation while describing connections between their graph and their table of values. They tended to relate the increasing rate of change along the curve of the graph to the increase in first differences across the table row (dependent variable) as the independent variable increased, shown by triangular gestures and arrows drawn by a student on the graph in Fig. 9. Cam explained similarly with his column graph by connecting the increasing height difference between consecutive columns to the first differences in his table of values. Even some of the younger Years 7 and 8 students evidenced making this connection between a non-constant rate of change shown on the graph and in the table, but their language was unsurprisingly less formal than the older students as shown in the difference between the following two quotes by a younger and an older student. The rise from here is small and here it’s like bigger. (Benita) So you can clearly see that obviously this isn’t linear, it’s a curve, it’s increasing, the rate of change is not constant. (David) Daisy actually drew extra points on her graph (at (1.5, 7), (2.5, 9) and (3.5, 11)) to demonstrate that the first differences from her table of values formed a straight line. Fewer students were able to express connections between their graph and their algebraic equation/s than between their graph and table of values; those who could were all older students. They tended to use a correspondence conception and focused on the turning point form of the equation (one of the equivalent rules found from the pattern t = (n + 2)2 – 4) and how you could use it to find the turning point for the graph. David additionally explained how the turning point form gave information about translation: You can also see with this second formula [(x + 2)2 – 4] about taking it away you can see that’s also true, it’s always 4 less you can see that. So, you can imagine if you just graphed this, the x plus 2 squared, you could see it following it, just 4 above [traced parallel curve just above his curve] (David) These responses by the older students suggests that previous teaching about the meanings of the specific terms in quadratic equations, and particularly turning point form, may have supported their ability to connect their graph to their equation. Across three representations, a few students used a covariation conception to describe connections. Ben additionally referred back to the figural pattern to demonstrate the increasing rate of change between the figures, with the turning point form of the equation matching the increasing side lengths of the square arrays of the figures (by two each time) and matching the first differences in the table, and also matching the rate of change on the graph: Square numbers… increase by a different amount every time and so this [taps pencil repeatedly along graph curve] increases by a different amount every time, a bit like square numbers, and again here [taps pencil along his table of values] … They both have 12

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squares in them [his two rules] because of the increasing length [taps pencil along table of values again] of the tile [makes square shape with both hands]. So, you have to put squares in there to account for the increasing difference [taps pencil along table next to handout]. (Ben) These findings suggest that the students’ conceptions of quadratic functions appeared to be translation-specific and usually between two representations, for example, correspondence between the figures and their equations (Ben was an exception), correspondence between equations and graphs when constructing the graph, and covariation between graphs and tables when interpreting the graph. Yet the students did demonstrate using a table of values for both correspondence (in trying to generalize the rule from it or using the points to construct a graph) and covariation (in examining the first and second differences). The students appeared less likely to draw on both conceptions together when the translation involved a graphical representation. 4.3. Students’ construction of their own quadratic figural growing pattern Ten of the twelve students (due to interview duration) were given a quadratic algebraic equation and asked to construct their own growing pattern to match it. (Earlier in the interview the students had also been asked to construct their own linear growing patterns, first without and then with a provided rule t = 2n + 3 (Appendix 1, third table); all of students were successful with both activities except for Cam, who could not generalize his first pattern, and Cathy, who had trouble with both tasks). As previously mentioned, they were offered a selection of concrete materials to use if they chose. Adam and Ben opted to use plastic square tiles (and Ben additionally drew his pattern). Six of the students drew squares, arranged them in a square array, and then added extra side lengths or ‘arms’ to the array. They all showed evidence of using a correspondence approach in referring to the ‘n2’ term as being represented by the square array, and the ‘2n’ term by the side lengths or arms. Benita said, “I’ll just divide these into separate parts [circled each term in the given equation] and then I’ll have a square in the middle and then just have these two arms sticking out” as shown in the example in Fig. 6. Three of these students explicitly referred to the previous bathroom tiles generalization task as having influenced their response to this pattern construction task, for example, I’m going to use boxes again… Like the middle box, kind of like the tiles one, but instead of having all of the sides you only have two sides” (Esther) Ben arranged his ‘2n’ square tiles along two adjoining sides of the square array (using the plastic tiles provided). While he was using the materials, he was explaining the additional squares around the edge using covariation language, then suddenly paused and said, “So it’s just a square number. It’s the square of – should I make up a second rule? … I’ll sketch them”. He drew his pattern and said (and wrote) “t = (n + 1)2 – 1. I just noticed that these are squares [traced finger around outside of figure] minus the top right corner, as I’ve put them.” (See Fig. 7). Even though he had not yet been introduced to non-linear functions, the experience of constructing a quadratic growing pattern in this particular arrangement drew his attention visually to the turning point form of a quadratic equation, depicted as the square array itself and a single square missing in the top corner. The three remaining students constructed noticeably different growing patterns, all of them lacking a square-based structure (low in pattern goodness), one of them incorrect. Interestingly, they all chose to draw circular shapes to construct their figures, rather than squares, unlike the other students’ patterns. Esther sought to create figures that were partially structural. She used a correspondence approach with the given equation in two parts – horizontal lines of a ‘matchstick’ tower for the ‘n2’ term and then circles on either side for the ‘2n’ term. I asked her how she decided where to draw the circles (as they appeared to be randomly placed on either side of the matchstick towers) and she replied, “It doesn’t really matter.” This suggested that although she could use her correspondence intuitions to find the correct number of objects to use in each figure, corresponding to each term in the equation, she did not evidence a clear understanding of growth and covariation in a quadratic pattern, of each figure being changed repeatedly each time to form the next item in the pattern. Fiona also used a correspondence approach, not to find amounts for each term in turn, but for each figure in total. She then drew ‘clumps’ of circles in an ad hoc manner – not arranged structurally – also suggesting a lack of understanding of growth and covariation in this type of concrete context: I’m not a super creative person… so I prefer –I find it more natural just to kind of just work out what it would be in my head, like,

Fig. 6. Using correspondence to construct a quadratic growing pattern from a given algebraic equation (Benita). 13

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Fig. 7. Noticing the difference of two squares during growing pattern construction (Ben).

mathematically so, the number. (Fiona) Interestingly, she commented on the numerical rather than structural similarities to the previous bathroom tiles task: “It’s kind of similar to the last one in the sense that the difference between them goes up by 2 each time.” After the interview, she lamented that she had worked hard to do well in mathematics, but thought she didn’t really understand it at times, like with the pattern tasks today. It appeared that she interpreted her preference for numerical representation as problematic in this context with open generalization tasks. In contrast to the other two students, Fred focused on a covariation view of the algebraic equation and grappled with how to add the correct number of dots to each preceding figure to make the next one (he used different colours to show the dots added at each stage), as shown in Fig. 8 (the fourth figure is incorrect). Both Fred and Fiona had also not been able to find equivalent rules for the previous bathroom tiles pattern, which was higher in pattern goodness, suggesting some difficulty with visualizing the figural structures using correspondence. Given that these students had been taught functions for several years and were also studying (optional) higher level calculus-based mathematics units, it seems possible that being able to visualize a quadratic functional relationship in a different way to equations and graphs is not necessarily developed by default over time. They also had surprisingly drawn inaccurate curves in the graphing task. These differing responses to the quadratic pattern construction task highlight that previous experience and success with generalizing a quadratic pattern with high ‘pattern goodness’ and finding more than one visualization leading to equivalent rules, appeared to support the subsequent complementary use of correspondence and covariation intuitions for constructing a quadratic growing pattern from a given algebraic equation. 5. Conclusion This study explored the potential of figural growing patterns as a context for learning about quadratic functional relationships with multiple representations. It involved task-based individual interviews with twelve high-achieving Years 7–12 students who

Fig. 8. Difficulty using covariation to construct a quadratic growing pattern from a given algebraic equation (t = n2 + 2n) (Fred). 14

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Fig. 9. Demonstrating the non-constant rate of change with arrows in quadratic graph (Benita).

responded to various generalization, graphing, and construction activities. Insights were sought into their correspondence/covariation intuitions of quadratic functions and connections between/among representations at different year levels through experiencing various closed and open quadratic pattern tasks. In generalizing provided quadratic figural growing patterns, the students demonstrated that patterns with higher pattern goodness (Rivera, 2010) supported them, not only to find different equivalent algebraic equations, but to use correspondence and covariation concepts in complementary ways. This suggests that the sequencing and choice of types of quadratic figural growing patterns for generalization is an important consideration for helping students connect the algebraic meaning of the x2 term in a second degree polynomial to its geometric meaning – a square-based structure (or one that can be transformed into such) – and even to the meaning of the word ‘quadratic’ (from the Latin ‘quadratum’ meaning ‘square’). This study, however, only examined patterns involving a leading coefficient of one (for the x2 term) and several of the students evidenced noticeable competency with these simpler quadratic functions. Research on different arrangements or structures for more complex quadratic growing patterns, yet still with high pattern goodness, would be worthwhile for further investigation of generalization, graphing, and construction tasks. This study found that these students all demonstrated (in the context of graphing a figural growing pattern) that they had not developed an understanding of how to represent discrete variables, despite being prescribed in the upper primary national curriculum content (Australian Curriculum Assessment & Reporting Authority, 2018). This issue was highlighted many years ago in a review of research (Leinhardt et al., 1990). It seems that despite the hybrid of traditional equations based and reform functions-based approaches in the students’ curriculum context, there could still be the need to address the understanding of graphs, such as how students actually perceive lines and points on a graph. It would be worthwhile to consider if explicitly including more graphing activities with discrete variables for linear, quadratic, and other relationships, as well as scatter plots, might help students learn to distinguish between continuous and discrete variables, and visualize lines as an arrangement of infinite points. It is possible that this approach might also support students’ making sense of the set theoretic definition of function and the need for restricted domains in different types of functions, highlighted in the literature as difficult for students (Markovits et al., 1986). This study also highlighted that even students with existing complementary correspondence and covariation intuitions in one context can evidence difficulty with connecting them in a different context when graphs are involved. This suggests the need for earlier exploration of quadratic growth with students (Ellis & Grinstead, 2008) and with covariation and correspondence intuitions (Küchemann, 2010; Smith, 2008), which emphasise the processual nature of functions, rather than progressing so quickly to transformations of graphs, which emphasises an object view of function (Breidenbach et al., 1992), as was found in the students’ Year 10 textbook (Maths World, published by Macmillan Education). This finding also resonates with Sfard’s (1992) recommendation to develop students’ operational view of function (as a computational process) more extensively in the foundational years before moving to a structural view (function as a static construct or abstract object), since developmentally they need an understanding of the underlying processes for creating functional relationships. This study contributes to the literature on ‘open’ or ‘free’ generalization tasks in which students construct their own growing patterns (Dörfler, 2008; Rivera & Becker, 2016) by offering evidence that constructing and exploring the various geometric structures for growing pattern figures have the potential to help students connect the meanings for different quadratic equation forms through visualization. This was even evidenced by a Year 8 boy (Ben) before formal study of quadratics, who ‘discovered’ the turning point form of the equation for his constructed pattern (which was also high in pattern goodness) through noticing its structure in an unexpected way. The Year 12 students’ difficulties (and younger students’ success) with finding equivalent rules for a given quadratic figural pattern and also creating their own figural growing patterns suggest that visualization in a figural growing pattern context 15

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may not be developmentally dependent on school teaching about functions, but further research in classroom contexts is warranted. This study also found that a few of the students’ difficulties in constructing a quadratic growing pattern, even for upper secondary students already studying calculus, uncovered some potential areas for their further learning, suggesting the potential of such tasks for formative assessment. These areas include learning to connect the parts of a quadratic equation to the covariation of the function (a multi-row table of values would be useful for this) and learning about the nature of a quadratic’s rate of change. These findings suggest that open growing pattern construction tasks after generalization experience might help students make algebraic, geometric, and visual connections both across multiple representations of function and between correspondence and covariation intuitions. There is more to understand about the use of visual contexts such as figural growing patterns for learning about quadratic functions, but this study contributes to the literature by providing evidence that linking the activities of generalization, graphing, and construction could provide an additional context to traditional equations-based and graphical approaches. Eisenberg (2002) actually argued that “unwillingness to stress the visual aspects of mathematics in general, and of functions in particular, is a serious impediment to students’ learning” (p. 152). Drawing on a visual context such as figural growing patterns might support students in learning conceptually about what actually makes a function quadratic in nature, and concretely about the different forms of quadratic equations. Further classroom research for designing and assessing such multi-representational activities would be valuable. This study included tasks that were sequenced to build from pattern generalization to pattern construction for a similar type of quadratic function to investigate students’ attention to their prior experience and to bi-directional connections between representations. It would be worthwhile to investigate tasks that move from the provided graph of a pattern to a table of values, constructing a figural growing pattern that matches, and then generalization – translations not usually explored by traditional tasks found in textbooks. It would also be valuable to investigate students’ growing pattern construction for different quadratic equation forms to explore the potential for this visual context to support their learning of concepts such as quadratic factorisation and transformations. Future research efforts are planned with teachers and classes to investigate types of quadratics patterning tasks that might prove useful for learners at differing levels of understanding. Appendix A. 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