Experts' structuring processes of equations that can be solved with varying efficiency: An analysis of gaze paths with eye-tracking

In: Bolondi,G. & Gaidoschik, M. (Eds.) (2025). Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education CERME14 (pp. 598–605). Free University of Bozen and ERME. Experts’ structuring processes of equations that can be solved with varying efficiency: An analysis of gaze paths with eye-tracking Christof Weber1, Georg Bruckmaier2 and Markus Vogel3 1Lucerne University of Teacher Education, Switzerland; [email protected] 2Northwestern Switzerland University of Applied Sciences, Windisch, Switzerland 3University of Education, Heidelberg, Germany Solving equations is fundamental to algebra yet remains a persistent challenge in mathematics teaching and learning. Drawing on a semiotic-pragmatic framework, we argue that algebraic equations lack inherent structure; instead, solvers must actively impose their own personal structure, a process known as ‘structuring’. Using eye-tracking, we analysed gaze patterns of student teachers solving equations with varying efficiency. Adaptive experts showed global relating, characterised by rapid transitions between subexpressions of both sides of the equation, and hierarchical structuring, enabling faster solutions. In contrast, routine experts exhibited local relating, marked by sequential processing, longer fixations and increased cognitive load, leading to slower and less accurate performance. These findings provide insights into the cognitive strategies of equation solving and offer implications for instructional approaches that promote more effective equation solving. Keywords: Equations, structuring processes, solving strategies, adaptive expertise, eye-tracking. Introduction This paper presents an ongoing study focusing on how experts’ reading of algebraic equations is reflected in their gaze patterns. Adopting a semiotic-pragmatic approach, we contend that equations do not inherently possess an (objective) structure that be directly “read off.” Instead, equations are strings of symbols devoid of meaning. To make meaning of an equation, we have actively to “read” a structure “into” it—essentially, impose a structure upon it. This process, known as structuring, involves identifying subexpressions and relate them to one other (e.g., Rüede, 2013). Consider the equation x – 3(x + 1) = 7 – 3(x + 1) (based on Hämmerle et al., 2018). It can be solved using two distinct solution strategies: (A) expanding 3(1 + x) on both sides and then simplifying the equation, or (B) adding 3(1 + x) to both sides. These strategies differ in two significant ways: First, strategy (B) is more efficient, requiring fewer computational steps. Second, the process of cognitive structuring—the mental framing before beginning to solve the equation—varies between the two strategies: Using the standard strategy (A) requires structuring the equation as {x} – {3}{(1 + x)} = {7} – {3}{(1 + x)}, relating the number {3} to the subexpression {(1 + x)} twice. In contrast, applying the more efficient strategy (B) involves structuring the equation as {x} – {3(1 + x)} = {7} – {3(1 + x)}, relating the left and right subexpressions {3(1 + x)} to each other. Depending on which structuring the solver imposes, the solution will be correct and achieved with varying levels of efficiency. In this context, the question arises of how structuring processes differ between experts who solve equations with varying efficiency, namely routine experts and adaptive experts (Baroody, 2003). Routine experts tend to rely on standard solution strategies, whereas adaptive experts apply more – 599 – innovative and efficient approaches. This classification is fluid, as individuals may switch categories depending on task-specific factors that influence strategy selection. The present study addresses this question by capturing and analysing the gaze paths of both expert types using eye-tracking methods. Our objective is to reconstruct ideal-typical gaze patterns of routine and adaptive experts during equation solving and explore their implications for instruction (e.g., Bikner-Ahsbahs, 2015). Theoretical Framework Structuring an equation: A necessary condition to solve the equation Algebraic expressions and equations are symbol strings requiring interpretation according to established rules. For instance, one rule permits omission of the multiplication sign, while another mandates that multiplication takes precedence over addition. Consider the expression x – 3(1 + x) which, from a mathematical perspective, must be interpreted as x – {3·(1 + x)}. However, in algebra education, it is widely acknowledged that algebraic expressions and equations are less selfexplanatory than they may initially seem. Much like interpreting mathematical visualisations, reading algebraic equations involves an active interpretative process. Hefendehl-Hebeker and Schwank (2023) describe this as the “dynamic interplay of reading off and reading into” (p. 109). This dynamic implies that a student might interpret the expression x – 3(1 + x) from an entirely different—and incorrect—structural perspective, such as (x – 3)·(1 + x). In this interpretation, the subtraction between x and 3 is prioritised before relating 3 to the subexpression (1 + x), resulting in an alternative structure. Each way a person relates parts of an expression represents an interpretative achievement of the person. To highlight this process, we use the term ‘structuring’ to describe how an individual creates an idiosyncratic structure for an expression (Rüede, 2013). This cognitive act shapes how the person subsequently manipulates and transforms the expression (Radford, 2010). For instance, a student who perceives the structure {x – 3}·{1 + x} is more likely to multiply the term. Mastering algebra, therefore, requires students to address two critical questions (Malle, 1993): What possible idiosyncratic structuring does an algebraic expression allow, and which align with mathematical rules? And: Which of these structures is most advantageous for further manipulation? Eye-tracking analyses, particularly in the domain of equation solving, offer a promising method for uncovering these cognitive processes (see Strohmaier et al., 2020, for an overview). Gaze patterns of experts when solving an equation: The eye-tracking approach The study of eye movements in mathematical activities has gained prominence in recent years through eye-tracking technology (e.g., Lilienthal & Schindler, 2019). Research spans topics like arithmetic, algebra, and geometry, focusing on activities such as proof and problem-solving. Eye-tracking excels at correlating gaze location with overt attention, though the reliability of this correlation with cognitive processing remains debated (Strohmaier et al., 2020). Few studies examine eye movements during the reading of algebraic expressions and equations. In one expert-novice study, university students rearranged simple equations like x · a = b or x/a = b to solve for x (Susac et al., 2014). Results showed a positive correlation between the number of fixations and the ratio of reaction time to response accuracy. Experts demonstrated fewer fixations—consistent with earlier studies—meaning that experts “knew where to look” (Susac et al., 2014, p. 567). Another – 600 – study (Kohlhase & Fürsich, 2016) found that experts located the equals sign in simple linear equations before shifting to operational symbols. Novices, however, often processed equations from the center or, with longer, more complex equations (such as integral equations), from left to right, mirroring text reading. A follow-up study (Kohlhase et al., 2017) showed that experts navigated complex equations by following their expression trees in depth first traversal of the hierarchical structure. Moreover, eye-tracking has identified ideal-typical procedures in pattern completion tasks (Schwank, 2001) and revealed common erroneous strategies in, e.g., reasoning with tree diagrams and 2×2 tables (Bruckmaier et al., 2019). Eye-tracking highlights task areas that capture attention and maps decisionmaking and problem-solving processes. Depending on the research focus, data can be distilled into ideal types through typifying content analysis or used to validate and refine existing ideal types. The current study Our study explores how eye-tracking can help to reconstruct solving strategies, an approach employed in previous research. We aim to understanding the mental actions taken when routine and adaptive experts encounter an equation that can be solved with varying efficiency. To observe their structuring process, we focus on their gaze paths, and address the following research questions: Where is the initial gaze directed? Which elements—such as subexpressions, operations, brackets, or the equals sign—are scrutinised more closely (indicating perceived importance)? Between which subexpressions does the gaze shift (suggesting the establishment of relations)? Which areas does the gaze focus on, and where is it absent? Hypotheses Aligned with our theoretical framework and the overarching research questions, our hypotheses of the current study explore how participants’ eye movements—indicative of varying levels of expertise—differ when solving equations of varying levels of efficiency: · Adaptive experts, who solve equations efficiently after an initial scan, will structure the equation following a top-down approach in the corresponding expression tree (see above). They will focus selectively on areas deemed relevant by the expression tree, demonstrating targeted and efficient “initial scanning” within these critical areas. · Routine experts, following a standard strategy, will distribute their attention more broadly across the task areas. Their gaze will follow a more linear, left-to-right pattern, with a particular emphasis on areas that support the standard solution approach. While eye-tracking studies typically include both qualitative and quantitative measures—such as fixation count or saccade length—our analysis in this report prioritises qualitative aspects. However, in line with the information reduction hypothesis, we anticipate that adaptive experts will exhibit fewer fixations compared to routine experts, as they reduce attention to redundant areas and concentrate on task-relevant information. Additionally, we expect adaptive experts to solve the equation more quickly than routine experts, who may take longer to complete the standard procedure. – 601 – Method Sample Seven student teachers in their early years of teacher education participated in the study. All had completed upper-secondary education and had achieved very good grades in mathematics in their Matura certificate. Their educational background indicates a solid foundation in algebra, making them well-suited for tasks involving equation solving. We could therefore reasonably expect that all participants possessed, at a minimum, the requisite knowledge to solve basic equations correctly and within a reasonable time frame, reflecting a baseline level of mathematical expertise. Materials The task set consisted of five equations—both linear and quadratic—selected based on prior studies. Each equation could be solved using either a standard method (e.g., expansion or quadratic formula) or a more efficient strategy. Examples include the linear equation x – 3(1 + x) = 7 – 3(1 + x) (see above), as well as the quadratic equations (x – 3)(x + 5) = 0 and (x – 1)(x – 2) = x – 1. This task design allowed for variation in strategy use and corresponding gaze behaviour across participants. Procedure Eye movements were recorded using an SR Research EyeLink 1000 system (500 Hz sampling rate) with a tripod-mounted camera, offering a spatial resolution of 0.01°. The system tracked a 32° horizontal and 25° vertical range on a screen with a 1920 × 1080 resolution. Participants were seated approximately 55 cm from the screen to ensure consistent tracking. After receiving instructions and an illustrative example, participants were asked to solve the equations mentally, without external tools or time constraints. As soon as they had found a solution, they were instructed to say “finish” and then state it aloud. At this point, the display and eye-tracking were paused, and participants explained their approach, which was audio-recorded. The full session— including calibration, task performance and interviews—lasted approximately 30 minutes. Initial Observations We focused on participants’ solution strategies and their gaze patterns (i.e., scan paths). For example, in solving x – 3(1 + x) = 7 – 3(1 + x), adaptive experts showed gaze transitions between x (on the very left side of the equation) and 7 (after the equals sign), indicating efforts to identify structural similarities and apply an innovative strategy. Routine experts, by contrast, exhibited more linear, leftto-right gaze paths, focusing on expanding the expression –3(1 + x) on both sides, consistent with a standard strategy approach. Preliminary findings suggest routine and adaptive experts display divergent gaze patterns, both qualitatively and quantitatively, such as their time progression and focus intensity. These distinctions informed the hypotheses underlying our current study. Analysis Traditional heatmaps and gaze plots struggle to capture both fixation locations and order, particularly for linearly structured stimuli like equations, which are read bidirectionally. To address this limitation, we employ fixation sequence diagrams (also known as sequence charts), mapping fixation order on the x-axis and fixation position on the y-axis. – 602 – We began by combining eye-tracking data and interviews to identify participants with consistent strategies for solving x – 3(x + 1) = 7 – 3(x + 1). For instance, Claudia consistently employed a shortcut strategy, recognising the identical terms 3(x + 1) on both sides of the equation. This adaptive use of her knowledge led us to classify her as an adaptive expert. In contrast, Indira adhered to the standard strategy of expanding and simplifying the expressions on both sides. Her consistent use of the standard method across all equations made her gaze patterns a representative for a routine expert. To explore how solution strategies manifest in gaze macrostructures, we analysed the gaze paths of both participants. By incorporating interview data and existing literature, we reconstructed two idealtypical gaze patterns: one representing adaptive experts and another for routine experts. However, the extent to which these ideal-typical gaze patterns can be generalised beyond the specific characteristics of Claudia and Indira remains an open question. Findings In the following section, we continue to focus on Claudia and Indira, illustrating and contrasting their gaze paths as the solve the previously introduced equation x – 3(x + 1) = 7 – 3(x + 1). This analysis serves to represent broader patterns across equations involving different solution strategies and to address our two hypotheses. Ideal-typical gaze paths of adaptive experts: The case of Claudia Claudia solved the equation correctly in just over six seconds, explaining her strategy during the subsequent interview: “I just saw, on the left and right, there is exactly the same term being subtracted. So there’s 7 left over.” In other words, she structures the equation as A – C = B – C, identifying and matching the term C. Claudia not only recognised this peculiarity but also used it purposefully, employing the innovative method by adding C (or 3(x + 1), respectively) to both sides, instead of following the standard algebraic strategy. Claudia’s gaze path comprised 39 fixations, each lasting between 50 and 350 milliseconds, amounting to approximately 6 seconds in total (see Figure 1 for her fixation sequence diagram). Her visual attention initially focused on the left side of the equation before shifting to the right. Further analysis revealed approximately a consistent gaze transition between the two sides—roughly one per second— resulting in seven transitions during her solution process. Early in her gaze path, she concentrated on the left-hand side, attending specially to the operators ‘–’, ‘⋅’ (or ‘(‘, respectively), and ‘+’, in this exact order. Notably, her gaze showed no saccades between equivalent symbols on either side of the equals sign, challenging common assumptions about the gaze strategies of adaptive experts. Figure 1: Fixation Sequence Diagram of Claudia (x-axis: fixation order) We interpret these observations as follows: Adaptive experts construct relationships that extend beyond the equals sign, relate subexpressions on the left side of the equation to those on the right in )x+1(3–7=)x+1(3–x 10 20 30 40 – 603 – a process we term global relating. For these experts, the equals sign does not merely separate the equation into two isolated components; instead, they perceive the equation as a cohesive whole. This integrative perspective allows them to relate partial terms effectively, allowing for remarkably rapid equation-solving. The sequence in which they examine the operators aligns not only with their leftto-right arrangement but also reflects a systematic, top-down traversal of the hierarchical structure within the expression tree of x – 3(x + 1) (Kohlhase et al., 2017). When this expression is written in functional notation—such as – (x, ⋅ (3, + (x, 1)))— it becomes evident that these experts approach the problem “outside in” (Rüede 2013, p. 402). This way of processing expressions, aligning with our first hypothesis, appears to be a defining characteristic of adaptive experts. Ideal-typical gaze paths of routine experts: The case of Indira Indira took 63 seconds to solve the equation, ultimately arriving at the (incorrect) solution x = 4/5. During the interview, she explained her approach: “I applied the typical math rule. First, I looked at the 3, then at the bracket, and resolved it. After that, I worked it out by hand. Because I didn’t have a notepad, it was a bit difficult to remember the numbers. [...] So first the left part and then the same on the right side, and yes, afterwards I looked at both together.” Her strategy revolved around identifying expressions in the form A(B + C), which led her to follow the standard strategy: distributing, simplifying expressions, and isolating variables sequentially on each side of the equation sequentially before comparing them. Indira’s gaze path (see Figure 2) comprised 150 fixations, ranging from 50 milliseconds to over 2 seconds, totaling approximately 63 seconds. Her visual attention unfolded in four phases: initial focus on the left (~15 s), followed by the right (~14 s), returning to the left (~12 s), and a final revisit to the right (~22 s). Throughout this process, she exhibited minimal gaze alternation between the two sides, with only three transitions across the entire 63-second period. Indira consistently fixated on both operands and operators for extended durations, a pattern observed across all four phases. Figure 2: Fixation Sequence Diagram of Indira (x-axis: fixation order) These observations align with our second hypothesis: routine experts primarily engage in local relating, linking subexpressions mainly within the same side of the equation. They tend to process each side of the equation in isolation, conceptualising the equals sign as a cognitive boundary. This )x+1(3–7=)x+1(3–x 10 20 30 40 50 60 70 80 90 100 110 130 120 150 140 – 604 – approach imposes a substantial cognitive load, particularly when mental manipulations are required. Consequently, routine experts must sustain a high level of concentration, investing significant time and cognitive resources to solve the problem. While their strategical approach can handle equations, it also heightens the risk of errors due to prolonged cognitive strain and accumulating inaccuracies. Discussion and Outlook The contrasting gaze patterns between Claudia and Indira illustrate distinct cognitive strategies in equation solving. Adaptive experts like Claudia demonstrate global relating, rapidly integrating information across the equation through hierarchical processing. In contrast, routine experts such as Indira engage in local relating, sequentially focusing on isolated segments with greater cognitive effort, resulting in longer solution times and more fixations. This segmented approach, combined with increased working memory demands, may elevate the risk of errors under cognitive strain. These findings highlight the importance of structuring in equation solving—an often overlooked aspect of algebra instruction. As Malle (1993) observes, “Many teachers are not even aware that the manipulation of algebraic expressions relies on the recognition of structure [i.e., structuring, C.W.]. As a result, they often focus on ‘endless’ manipulation practice without addressing this foundational step” (p. 254). Without targeted attention to structuring processes, instruction risks defaulting to procedural practice, with limited development of conceptual understanding. Furthermore, fixation sequence diagrams offer a promising analytic lens. By visualizing how students structure equations, these diagrams can reveal whether learners adopt globally integrative or locally fragmented strategies. This distinction not only echoes recent work by Schreiter and Vogel (2024) but also aligns with broader expertise literature: adaptive experts draw on flexible pattern recognition, while routine experts tend to follow procedural, step-by-step routines. Our preliminary findings suggest that gaze patterns may serve as predictive indicators of students’ solution strategies. As Solstad et al. 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