We investigated the hypothesis that the visual system is retrained to perceive the hierarchical structure of algebraic expressions, reducing high-level algebraic reasoning to basic perceptual processes. As predicted, participants who had mastered the hierarchical structure of algebra exhibited object-based attention for algebraic sub-expressions (i.e., variables around a higher-precedence operation). In addition, the extent of their object-based attention for algebraic sub-expressions predicted their performance on a purely mathematical task, with performance improving as object-based attention increased. This was not the case for participants who had not yet mastered the hierarchical structure of algebra; they did not exhibit object-based attention for algebraic structure, and their algebraic performance was unrelated to their perceptual processing. Taken together, these results are consistent with the hypothesis that mathematical expertise involves, at least in part, recycling processes in the visual system to create structured groups that honor the hierarchical structure of algebra.

Why, for some participants, was perceptual discriminability actually better between algebraic sub-expressions than within? Most of these participants were Syntax Non-Knowers. Some of these individuals may have the order of precedence exactly wrong, solving addition first—perhaps because it is easier—and only afterwards moving to multiplication. Past studies have found that a full third of college students struggle to apply the correct order of operations (Pappanastos, Hall, & Honan, 2002; see also Glidden, 2008). Perhaps more likely, the addition symbol may attract attention for purely visual reasons (e.g., it consists of lines that are vertical and horizontal, rather than slanted) or because it is more familiar, comfortable, and comprehensible, particularly for lower-performing individuals. Indeed, extensive early experience with addition may train the visual system to perceive sums as wholes, an early bias that must be overridden by later algebraic training.

A between-object advantage, however, was found even among some Syntax Knowers—including a few who performed quite well on the Algebraic Equivalence task. Some of this is presumably just noise; no behavioral index of object-based attention is going to be a perfect measure of perceptual processing. But this is also a good reminder that there are multiple routes to mathematical success. It is unlikely that every competent reasoner is going to rely on the same visuospatial perceptual strategy; some may even rely entirely on rote, explicit, linguistically encoded knowledge of the order of operations (e.g., recalling the abbreviation PEDMAS: Parentheses, then Exponents, then Division and Multiplication, then Addition and Subtraction). Object-based attention for algebraic structure, therefore, may take time to develop, emerging only after mastering algebraic syntax. For some, perceptual processes may always be overshadowed by complementary strategies.

Previous work has demonstrated object-based attention for concrete objects inferred from sensory cues (Duncan, 1984) and expectations that reflect past perceptual experience (Zemel et al., 2002). The current study extended this phenomenon to objects established on the basis of abstract relations and conceptual knowledge. In some ways, this is reminiscent of the holistic perception of written words (Ehri, 2005). Skillful readers retrain their visual system so they see written words as wholes, not collections of individual letters. Holistic word perception, however, still depends primarily on a sensory cue—the space between words—or past exposure to that particular word-form. This is sometimes true for algebraic notation, too, where algebraic precedence is associated with spatial proximity. Often, however, the hierarchical structure of an expression is not readily apparent from visual inspection alone. In the current study, for instance, addition and multiplication were spaced equally, minimizing any sensory cues indicating which variables belong together. Furthermore, during reading, only specific combinations of letters form legitimate words. In algebra, by contrast, new variables can be combined productively to create novel sub-expressions; indeed, in the current study, letters were chosen randomly from the alphabet, generating combinations that participants may have never before encountered. Despite this productive novelty, algebraic sub-expressions were perceived as unified visual objects. These visual objects could only have been constructed on the basis of the formal rules governing algebraic syntax. Basic perception was reshaped by high-level conceptual knowledge.

### The nature of mathematical expertise

The current results suggest that relying on visual processing might be a boon, not a barrier, to mathematical reasoning. This might come as a surprise. Confronted with evidence of students’ reliance on misleading, superficial visual strategies in algebra, some have argued that mathematical training should avoid and even suppress perceptual strategies (e.g., Kirshner, 1989; Kirshner & Awtry, 2004). For example, when asked to solve 4 + 4/2 + 2, some students might be led to answer “2,” incorrectly, because of the superficially tempting, perceptually strong 4 + 4 and 2 + 2 groups. Indeed, we sometimes found evidence for perceptual grouping around addition, rather than multiplication, particularly among participants who had yet to master the hierarchical syntax of algebra. But the fact that novices use perceptual strategies to arrive at incorrect answers does not imply that experts abandon such strategies entirely. Instead, experts may refine those perceptual strategies so that they become reliable, robust, and rapid routes to *correct* solutions (Goldstone et al., 2010; cf., Hutchins, 1995, and Rumelhart et al., 1986). In line with this, participants who had mastered the hierarchical syntax of algebra also exhibited object-based attention for algebraic sub-expressions. Mathematical expertise, therefore, might be better thought of as the *skillful* deployment of perception.

Thus, the mathematical expert is made more expert, on the one hand, by mastering clever notations in which conceptual relations are presented perceptually and, on the other, by retraining their visual system to perform some aspects of algebraic reasoning. Both this perspective on mathematical practice and its resistance have a long heritage. To quote Whitehead (1911, p. 61) yet again: “It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.” The resistance continues to this day. *New Mathematics* was a relatively recent, and particularly controversial, movement in education that attempted to foreground the “important operations” of mathematics, at the expense of procedural mastery (Adler, 1972). But one implication of our perspective is that mathematical training might be better spent encouraging students to adapt—not abandon—their perceptual grouping processes. Instead of minimizing students’ reliance on perceptual strategies (Kirshner, 1989; Kirshner & Awtry, 2004), education should aim to refine students’ use of perception and action, so that they rig up their perception and action systems like mathematical experts. This could take the form of explicit instruction on how the visuospatial layout of algebraic equations contains hints to the hierarchical relations that they represent. Additionally, future curricula or tools could intervene in targeted ways on the embodied routines that contribute to mathematical expertise, taking advantage of decades of research on perceptual and motor learning (Ottmar & Landy, in press).

Regardless of what we do as teachers, children pick up on the perceptual regularities of their environments, implicitly developing perceptual associations and routines. These can become obstacles, such as when children interpret the visual form of the equals sign as a cue to calculate, hindering learning in early algebra (McNeil, 2008). But they can also offer long-term benefits, such as the perceptual strategy documented in the current study. We imagine a future where computer-based tools will systematically manipulate the visual and interactive features of mathematical representations so that children pick up on the perceptual regularities that help, rather than hurt (e.g., Weitnauer, Landy, & Ottmar, 2016).

Of course, perception alone is insufficient to account for all of mathematical reasoning. However, we suspect it is a critical part of the larger, distributed system that accomplishes mathematics, a system in which resources within the skull are brought into coordination with resources outside (e.g., gestures, inscriptions), skillfully soft-assembled to respond to the situated demands of the task (Clark, 2008). These sundry resources are often “embodied,” from neural circuits that evolved for perceiving and acting, to the fleshy hands that do the literal “manual labor” of mathematics (Marghetis, Edwards, & Núñez, 2014). For example, brain circuits that evolved for perceiving motion or shifting attention are redeployed to support mathematical skills like symbolic arithmetic, where attention is shifted along a simulated number-line (Knops, Thirion, Hubbard, Michel, & Dehaene, 2009; Marghetis, Núñez, & Bergen, 2014; McCrink, Dehaene, & Dehaene-Lambertz, 2007), or solving equations, where terms are imagined to move across the equals sign (Goldstone et al., 2010). Our bodies, too, are disciplined by mathematical training. When a mathematical expression is examined, eye movements respect the expression’s hierarchical structure, starting with the highest-precedence operation and moving sequentially to gradually lower-precedence operations (Landy, Jones, & Goldstone, 2008; Schneider, Maruyama, Dehaene, & Sigman, 2012). And while gestures can shape children’s early mathematical knowledge (Goldin-Meadow, Cook, & Mitchell, 2009), even experts gesture spontaneously to express their mathematical understanding (Marghetis & Núñez, 2013). A complete understanding of mathematical cognition requires that we study mathematics as it is actually accomplished, as an embodied practice: eyes darting across the blackboard, hands scribbling away.

### The widespread role of regimented perception

While the current study has focused on retraining our perceptual apparatus to perform *algebraic* reasoning, mathematics is full of other practices that also likely depend on the regimentation of perception. Visual proofs in Euclidean geometry are unreliable when treated naively as exact depictions, but the expert geometer learns to ignore those diagrammatic features that could lead to invalid conclusions (e.g., exact length) while perceiving those features that can make valid contributions to a proof (e.g., containment; Manders, 2008). And this is not restricted to high school mathematics. Category Theory, a branch of modern mathematics, relies on a proof technique known as “diagram chasing” that relies entirely on the creation and interpretation of diagrams in which spatial locations indicate mathematical relations. Indeed, visuospatial ability is significantly greater among professional mathematicians compared to non-mathematicians, and it completely mediates the relation between basic numerical abilities and the attainment of advanced mathematical expertise (Sella, Sader, Lolliot, & Cohen Kadosh, 2016). Thus, while algebra has been our case study, we propose that mathematics more generally depends for its accomplishment on the cultural regimentation of our perceptual apparatus.

And this may be an even more general phenomenon, with regimented perception playing a role in the reproduction of many, if not most, sociocultural systems. Biases in face perception, for instance, may contribute to the reproduction of structural racism: implicit racial biases, which shape the perception of facial emotions (Hugenberg & Bodenhausen, 2003), can influence split-second decisions by law enforcement about whether or not to shoot a suspect (Correll, Park, Judd, & Wittenbrink, 2007), thus reproducing structural inequalities in safety and policing. Marx even argued that a similar process of regimented perception contributes to the reproduction of capitalist society as a whole, such that we learn to *see* the world in terms of objects to be owned (Marx, 2012). Thus, the cultivation of highly disciplined ways of seeing and acting may be a critical mechanism by which we reproduce immense sociocultural systems (Bourdieu, 1977), from structural inequality to the inferential structure of mathematics.