Tyler Marghetis

Doing arithmetic by hand: Hand movements during exact arithmetic reveal systematic, dynamic spatial processing

Mathematics requires precise inferences about abstract objects inaccessible to perception. How is this possible? One proposal is that mathematical reasoning, while concerned with entirely abstract objects, nevertheless relies on neural resources specialized for interacting with the world—in other words, mathematics may be grounded in spatial or sensorimotor systems. Mental arithmetic, for instance, could involve shifts in spatial attention along a mental “number-line”, the product of cultural artefacts and practices that systematically spatialize number and arithmetic. Here, we investigate this hypothesized spatial processing during exact, symbolic arithmetic (e.g., 4 + 3 = 7). Participants added and subtracted single-digit numbers and selected the exact solution from responses in the top corners of a computer monitor. While they made their selections using a computer mouse, we recorded the movement of their hand as indexed by the streaming x, y coordinates of the computer mouse cursor. As predicted, hand movements during addition and subtraction were systematically deflected toward the right and the left, respectively, as if calculation involved simultaneously simulating motion along a left-to-right mental number-line. This spatial–arithmetical bias, moreover, was distinct from—but correlated with—individuals’ spatial–numerical biases (i.e., spatial–numerical association of response codes, SNARC, effect). These results are the first evidence that exact, symbolic arithmetic prompts systematic spatial processing associated with mental calculation. We discuss the possibility that mathematical calculation relies, in part, on an integrated system of spatial processes.

Of magnitudes and metaphors: explaining cognitive interactions between space, time, and number.

Space, time, and number are fundamental to how we act within and reason about the world. These three experiential domains are systematically intertwined in behavior, language, and the brain. Two main theories have attempted to account for cross-domain interactions. A Theory of Magnitude (ATOM) posits a domain-general magnitude system. Conceptual Metaphor Theory (CMT) maintains that cross-domain interactions are manifestations of asymmetric mappings that use representations of space to structure the domains of number and time. These theories are often viewed as competing accounts. We propose instead that ATOM and CMT are complementary, each illuminating different aspects of cross-domain interactions. We argue that simple representations of magnitude cannot, on their own, account for the rich, complex interactions between space, time and number described by CMT. On the other hand, ATOM is better at accounting for low-level and language-independent associations that arise early in ontogeny. We conclude by discussing how magnitudes and metaphors are both needed to understand our neural and cognitive web of space, time and number.

Literal and Metaphorical Senses in Compositional Distributional Semantic Models

Metaphorical expressions are pervasive in natural language and pose a substantial challenge for computational semantics. The inherent compositionality of metaphor makes it an important test case for compositional distributional semantic models (CDSMs). This paper is the first to investigate whether metaphorical composition warrants a distinct treatment in the CDSM framework. We propose a method to learn metaphors as linear transformations in a vector space and find that, across a variety of semantic domains, explicitly modeling metaphor improves the resulting semantic representations. We then use these representations in a metaphor identification task, achieving a high performance of 0.82 in terms of F-score.

Mastering algebra retrains the visual system to perceive hierarchical structure in equations

Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual system—in particular, object-based attention—is retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressions—but only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception.

Where Does the Ordered Line Come From? Evidence From a Culture of Papua New Guinea

Number lines, calendars, and measuring sticks all represent order along some dimension (e.g., magnitude) as position on a line. In high-literacy, industrialized societies, this principle of spatial organization—linear order—is a fixture of visual culture and everyday cognition. But what are the principle’s origins, and how did it become such a fixture? Three studies investigated intuitions about linear order in the Yupno, members of a culture of Papua New Guinea that lacks conventional representations involving ordered lines, and in U.S. undergraduates. Presented with cards representing differing sizes and numerosities, both groups arranged them using linear order or sometimes spatial grouping, a competing principle. But whereas the U.S. participants produced ordered lines in all tasks, strongly favoring a left-to-right format, the Yupno produced them less consistently, and with variable orientations. Conventional linear representations are thus not necessary to spark the intuition of linear order—which may have other experiential sources—but they nonetheless regiment when and how the principle is used.

Adapting Perception, Action, and Technology for Mathematical Reasoning

Formal mathematical reasoning provides an illuminating test case for understanding how humans can think about things that they did not evolve to comprehend. People engage in algebraic reasoning by (1) creating new assemblies of perception and action routines that evolved originally for other purposes (reuse), (2) adapting those routines to better fit the formal requirements of mathematics (adaptation), and (3) designing cultural tools that mesh well with our perception-action routines to create cognitive systems capable of mathematical reasoning (invention). We describe evidence that a major component of proficiency at algebraic reasoning is Rigged Up Perception-Action Systems (RUPAS), via which originally demanding, strategically controlled cognitive tasks are converted into learned, automatically executed perception and action routines. Informed by RUPAS, we have designed, implemented, and partially assessed a computer-based algebra tutoring system called Graspable Math with an aim toward training learners to develop perception-action routines that are intuitive, efficient, and mathematically valid.

Bias and ignorance in demographic perception

When it comes to knowledge of demographic facts, misinformation appears to be the norm. Americans massively overestimate the proportions of their fellow citizens who are immigrants, Muslim, LGBTQ, and Latino, but underestimate those who are White or Christian. Previous explanations of these estimation errors have invoked topic-specific mechanisms such as xenophobia or media bias. We reconsidered this pattern of errors in the light of more than 30 years of research on the psychological processes involved in proportion estimation and decision-making under uncertainty. In two publicly available datasets featuring demographic estimates from 14 countries, we found that proportion estimates of national demographics correspond closely to what is found in laboratory studies of quantitative estimates more generally. Biases in demographic estimation, therefore, are part of a very general pattern of human psychology—independent of the particular topic or demographic under consideration—that explains most of the error in estimates of the size of politically salient populations. By situating demographic estimates within a broader understanding of general quantity estimation, these results demand reevaluation of both topic-specific misinformation about demographic facts and topic-specific explanations of demographic ignorance, such as media bias and xenophobia.