David Landy

In 4 experiments, the authors explored the role of visual layout in rule-based syntactic judgments. Participants judged the validity of a set of algebraic equations that tested their ability to apply the order of operations. In each experiment, a nonmathematical grouping pressure was manipulated to support or interfere with the mathematical convention. Despite the formal irrelevance of these grouping manipulations, accuracy in all experiments was highest when the nonmathematical pressure supported the mathematical grouping. The increase was significantly greater when the correct judgment depended on the order of operator precedence. The result that visual perception impacts rule application in mathematics has broad implications for relational reasoning in general. The authors conclude that formally symbolic reasoning is more visual than is usually proposed.(PsycINFO Database Record (c) 2016 APA, all rights …

Although the field of perceptual learning has mostly been concerned with low‐ to middle‐level changes to perceptual systems due to experience, we consider high‐level perceptual changes that accompany learning in science and mathematics. In science, we explore the transfer of a scientific principle (competitive specialization) across superficially dissimilar pedagogical simulations. We argue that transfer occurs when students develop perceptual interpretations of an initial simulation and simply continue to use the same interpretational bias when interacting with a second simulation. In arithmetic and algebraic reasoning, we find that proficiency in mathematics involves executing spatially explicit transformations to notational elements. People learn to attend mathematical operations in the order in which they should be executed, and the extent to which students employ their perceptual attention in this manner is …

Although a general sense of the magnitude, quantity, or numerosity of objects is common in both untrained people and animals, the abilities to deal exactly with large quantities and to reason precisely in complex but well-specified situations—to behave formally, that is—are skills unique to people trained in symbolic notations. These symbolic notations typically employ complex, hierarchically embedded structures, which all extant analyses assume are constructed by concatenative, rule-based processes. The primary goal of this article is to establish, using behavioral measures on naturalistic tasks, that some of the same cognitive resources involved in representing spatial relations and proximities are also involved in representing symbolic notations—in short, that formal notations are a kind of diagram. We examined self-generated productions in the domains of handwritten arithmetic expressions and …

How does the physical structure of an arithmetic expression affect the computational processes engaged in by reasoners? In handwritten arithmetic expressions containing both multiplications and additions, terms that are multiplied are often placed physically closer together than terms that are added. Three experiments evaluate the role such physical factors play in how reasoners construct solutions to simple compound arithmetic expressions (such as “2 + 3 × 4”). Two kinds of influence are found: First, reasoners incorporate the physical size of the expression into numerical responses, tending to give larger responses to more widely spaced problems. Second, reasoners use spatial information as a cue to hierarchical expression structure: More narrowly spaced subproblems within an expression tend to be solved first and tend to be multiplied. Although spatial relationships besides order are entirely formally …

One of the most important applications of grounded cognition theories is to science and mathematics education where the primary goal is to foster knowledge and skills that are widely transportable to new situations. This presents a challenge to those grounded cognition theories that tightly tie knowledge to the specifics of a single situation. In this chapter, we develop a theory learning that is grounded in perception and interaction, yet also supports transferable knowledge. A first series of studies explores the transfer of complex systems principles across two superficially dissimilar scenarios. The results indicate that students most effectively show transfer by applying previously learned perceptual and interpretational processes to new situations. A second series shows that even when students are solving formal algebra problems, they are greatly influenced by non-symbolic, perceptual grouping factors. We interpret both results as showing that high-level cognition that might seem to involve purely symbolic reasoning is actually driven by perceptual processes. The educational implication is that instruction in science and mathematics should involve not only teaching abstract rules and equations but also training students to perceive and interact with their world.

People can be taught to manipulate symbols according to formal mathematical and logical rules. Cognitive scientists have traditionally viewed this capacity—the capacity for symbolic reasoning—as grounded in the ability to internally represent numbers, logical relationships, and mathematical rules in an abstract, amodal fashion. We present an alternative view, portraying symbolic reasoning as a special kind of embodied reasoning in which arithmetic and logical formulae, externally represented as notations, serve as targets for powerful perceptual and sensorimotor systems. Although symbolic reasoning often conforms to abstract mathematical principles, it is typically implemented by perceptual and sensorimotor engagement with concrete environmental structures.

Despite their importance in public discourse, numbers in the range of 1 million to 1 trillion are notoriously difficult to understand. We examine magnitude estimation by adult Americans when placing large numbers on a number line and when qualitatively evaluating descriptions of imaginary geopolitical scenarios. Prior theoretical conceptions predict a log‐to‐linear shift: People will either place numbers linearly or will place numbers according to a compressive logarithmic or power‐shaped function (Barth & Paladino, ; Siegler & Opfer, ). While about half of people did estimate numbers linearly over this range, nearly all the remaining participants placed 1 million approximately halfway between 1 thousand and 1 billion, but placed numbers linearly across each half, as though they believed that the number words “thousand, million, billion, trillion” constitute a uniformly spaced count list. Participants in this group also …

Perceptual modules adapt at evolutionary, lifelong, and moment-to-moment temporal scales to better serve the informational needs of cognizers. Perceptual learning is a powerful way for an individual to become tuned to frequently recurring patterns in its specific local environment that are pertinent to its goals without requiring costly executive control resources to be deployed. Mechanisms like predictive coding, categorical perception, and action-informed vision allow our perceptual systems to interface well with cognition by generating perceptual outputs that are systematically guided by how they will be used. In classic conceptions of perceptual modules, people have access to the modules’ outputs but no ability to adjust their internal workings. However, humans routinely and strategically alter their perceptual systems via training regimes that have predictable and specific outcomes. In fact, employing a …

We dispute Carey's assumption that distinct core cognitive processes employ domain-specific input analyzers to construct proprietary representations. We give reasons to believe that conceptual systems co-opt core components for new domains. Domain boundaries, as well as boundaries between perceptual–motor and conceptual cognitive resources may be useful abstractions, but do not appear to reflect constraints respected by brains and cognitive systems.

This paper explores the hypothesis that schematic abstraction—rule following—is partially implemented through processes and knowledge used to understand motion. Two experiments explore the mechanisms used by reasoners solving simple linear equations with one variable. Participants solved problems displayed against a background that moved rightward or leftward. Solving was facilitated when the background motion moved in the direction of the numeric transposition required to solve for the unknown variable. Previous theorizing has usually assumed that such formal problems are solved through the repeated application of abstract transformation patterns (rules) to equations, replicating the steps produced in typical worked solutions. However, the current results suggest that in addition to such strategies, advanced reasoners often employ a mental motion strategy when manipulating algebraic forms: elements of the problem are “picked up” and “moved” across the equation line. This demonstration supports the suggestion that genuinely schematic reasoning could be implemented in perceptual-motor systems through the simulated transformation of referential (but physical) symbol systems.

We describe an intervention being developed by our research team, Pushing Symbols (PS). This intervention is designed to encourage learners to treat symbol systems as physical objects that move and change over time according to dynamic principles. We provide students with the opportunities to explore algebraic structure by physically manipulating and interacting with concrete and virtual symbolic systems that enforce rules through constraints on physical transformations. Here we present an instantiation of this approach aimed at helping students learn the structure of algebraic notation in general, and in particular learn to simplify like terms. This instantiation combines colored symbol tiles with a new touchscreen software technology adapted from the commercial Algebra Touch software. We present preliminary findings from a study with 70 middle-school students who participated in the PS intervention over a three-hour period.

Two experiments test predictions of a visual process-driven model of multi-term arithmetic computation. The visual process model predicts that attention should be drawn toward multiplication signs more readily than toward plus signs, and that narrow spaces should draw gaze comparably to multiplication signs. Although both of these predictions are verified by behavioral response measures and eye-tracking, the visual process model cannot account for patterns of early looking. The results suggest that people strategically deploy visual computation strategies.

Human concept learning depends upon perception. Our concept of “car” is built out of perceptual features such as “engine,”“tire,” and “bumper.” However, recent research indicates that the dependency works both ways. We see bumpers and engines in part because we have acquired “car” concepts and detected examples of them. Perception both influences and is influenced by the concepts that we learn. We have been exploring the psychological mechanisms by which concepts and perception mutually influence one another, and building computational models to show that the circle of influences is benign rather than vicious.

The computation-as-cognition metaphor requires that all cognitive objects are constructed from a fixed set of basic primitives; prominent models of cognition and perception try to provide that fixed set. Despite this effort, however, there are no extant computational models that can actually generate complex concepts and processes from simple and generic basic sets, and there are good reasons to wonder whether such models may be forthcoming. We suggest that one can have the benefits of computationalism without a commitment to fixed feature sets, by postulating processes that slowly develop special-purpose feature languages, from which knowledge is constructed. This provides an alternative to the fixed-model conception without radical anti-representationlism. Substantial evidence suggests that such feature development adaptation actually occurs in the perceptual learning that accompanies category …