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

Rethinking Norm Psychology

Norms permeate human life. Most of people's activities can be characterized by rules about what is appropriate, allowed, required, or forbidden-rules that are crucial in making people hyper-cooperative animals. In this article, I examine the current cognitive-evolutionary account of "norm psychology" and propose an alternative that is better supported by evidence and better placed to promote interdisciplinary dialogue. The incumbent theory focuses on rules and claims that humans genetically inherit cognitive and motivational mechanisms specialized for processing these rules. The cultural-evolutionary alternative defines normativity in relation to behavior-compliance, enforcement, and commentary-and suggests that it depends on implicit and explicit processes. The implicit processes are genetically inherited and domain-general; rather than being specialized for normativity, they do many jobs in many species. The explicit processes are culturally inherited and domain-specific; they are constructed from mentalizing and reasoning by social interaction in childhood. The cultural-evolutionary, or "cognitive gadget," perspective suggests that people alive today-parents, educators, elders, politicians, lawyers-have more responsibility for sustaining normativity than the nativist view implies. People's actions not only shape and transmit the rules, but they also create in each new generation mental processes that can grasp the rules and put them into action.

Resisting the urge to calculate: The relation between inhibitory control and perceptual cues in arithmetic performance

Subtle visual manipulations to the presentation of mathematical notation influence the way that students perceive and solve problems. While there is a consistent impact of perceptual cues on students' problem-solving, other cognitive skills such as inhibitory control may interact with perceptual cues to affect students' arithmetic problem-solving performance. We present an online experiment in which college students completed a version of the Stroop task followed by arithmetic problems in which the spacing between numbers and operators was either congruent (e.g., 2 + 3×4) or incongruent (e.g., 2+3 × 4) to the order of precedence. We found that students were comparably accurate between problem types but might have spent longer responding to problems with congruent than incongruent spacing. There was no main effect of inhibitory control on students' performance on these problems. However, an exploratory analysis on a combined performance measure of accuracy and response time revealed an interaction between problem type and inhibitory control. Students with higher inhibitory control performed better on congruent versus incongruent problems, whereas students with lower inhibitory control performed worse on congruent versus incongruent problems. Together, these results suggest that the relation between inhibitory control and arithmetic performance may not be straightforward. Furthermore, this work advances perceptual learning theory and contributes new findings on the contexts in which perceptual cues, such as spacing, influence arithmetic performance.

Test of Times New Roman: effects of font type on mathematical performance

Mathematics is presented in a variety of font types across materials (e.g., textbooks, online problems); however, little is known about the effects of font type on students' mathematical performance. Undergraduate students (N = 121) completed three mathematical tasks in a one-hour online session in one of three font conditions: Times New Roman (n = 45), Kalam (n = 41), or handwriting (n = 35). We examined whether font type impacted students' performance, as measured by accuracy and response time, on the Perceptual Math Equivalence Task, error identification task, and equation-solving task. Compared to students in the Kalam and handwriting conditions, students in the Times New Roman condition were less accurate on the Perceptual Math Equivalence Task in which they judged whether two expressions were equivalent or not equivalent. We did not find differences between conditions in performance on error identification and equation-solving tasks. The findings have implications for research and practice. Specifically, researchers and educators may choose font types in which they present mathematics information with consideration, as font types may impact students' mathematical processing and performance.

Space in Hand and Mind: Gesture and Spatial Frames of Reference in Bilingual Mexico

Speakers of many languages prefer allocentric frames of reference (FoRs) when talking about small-scale space, using words like "east" or "downhill." Ethnographic work has suggested that this preference is also reflected in how such speakers gesture. Here, we investigate this possibility with a field experiment in Juchitán, Mexico. In Juchitán, a preferentially allocentric language (Isthmus Zapotec) coexists with a preferentially egocentric one (Spanish). Using a novel task, we elicited spontaneous co-speech gestures about small-scale motion events (e.g., toppling blocks) in Zapotec-dominant speakers and in balanced Zapotec-Spanish bilinguals. Consistent with prior claims, speakers' spontaneous gestures reliably reflected either an egocentric or allocentric FoR. The use of the egocentric FoR was predicted-not by speakers' dominant language or the language they used in the task-but by mastery of words for "right" and "left," as well as by properties of the event they were describing. Additionally, use of the egocentric FoR in gesture predicted its use in a separate nonlinguistic memory task, suggesting a cohesive cognitive style. Our results show that the use of spatial FoRs in gesture is pervasive, systematic, and shaped by several factors. Spatial gestures, like other forms of spatial conceptualization, are thus best understood within broader ecologies of communication and cognition.

Representing Variability: The Case of Life Cycle Diagrams

Two foundational concepts in biology education are 1) offspring are not identical to their parents, and 2) organisms undergo changes throughout their lives. These concepts are included in both international and U.S. curricular standards. Research in psychology has shown that children often have difficulty understanding these concepts, as they are inconsistent with their intuitive theories of the biological world. Additionally, prior research suggests that diagrams are commonly used in instruction and that their features influence student learning. Given this prior work, we explored the characteristics of life cycle diagrams and discuss possible implications for student learning. We examined 75 life cycle diagrams from books, including five biology or general science textbooks and 25 specialized trade books focusing on biology for children. We also examined 633 life cycle diagrams from a publicly available online database of science diagrams. Most diagrams failed to show any within-species variability. Additionally, many diagrams had perceptually rich backgrounds, which prior research suggests might hinder learning. We discuss how the design characteristics of diagrams may reinforce students' intuitive theories of biology, which might make it difficult for students to understand key biological concepts in the future.

Number concepts: abstract and embodied

Numerical knowledge, including number concepts and arithmetic procedures, seems to be a clear-cut case for abstract symbol manipulation. Yet, evidence from perceptual and motor behaviour reveals that natural number knowledge and simple arithmetic also remain closely associated with modal experiences. Following a review of behavioural, animal and neuroscience studies of number processing, we propose a revised understanding of psychological number concepts as grounded in physical constraints, embodied in experience and situated through task-specific intentions. The idea that number concepts occupy a range of positions on the continuum between abstract and modal conceptual knowledge also accounts for systematic heuristics and biases in mental arithmetic, thus inviting psycho-logical approaches to the study of the mathematical mind.This article is part of the theme issue 'Varieties of abstract concepts: development, use and representation in the brain'.

Developmental Dyscalculia and Automatic Magnitudes Processing: Investigating Interference Effects between Area and Perimeter

The relationship between numbers and other magnitudes has been extensively investigated in the scientific literature. Here, the objectives were to examine whether two continuous magnitudes, area and perimeter, are automatically processed and whether adults with developmental dyscalculia (DD) are deficient in their ability to automatically process one or both of these magnitudes. Fifty-seven students (30 with DD and 27 with typical development) performed a novel Stroop-like task requiring estimation of one aspect (area or perimeter) while ignoring the other. In order to track possible changes in automaticity due to practice, we measured performance after initial and continuous exposure to stimuli. Similar to previous findings, current results show a significant group × congruency interaction, evident beyond exposure level or magnitude type. That is, the DD group systematically showed larger Stroop effects. However, analysis of each exposure period showed that during initial exposure to stimuli the DD group showed larger Stroop effects in the perimeter and not in the area task. In contrast, during continuous exposure to stimuli no triple interaction was evident. It is concluded that both magnitudes are automatically processed. Nevertheless, individuals with DD are deficient in inhibiting irrelevant magnitude information in general and, specifically, struggle to inhibit salient area information after initial exposure to a perimeter comparison task. Accordingly, the findings support the assumption that DD involves a deficiency in multiple cognitive components, which include domain-specific and domain-general cognitive functions.

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.

Embodied cognition and STEM learning: overview of a topical collection in CR:PI

Embodied learning approaches emphasize the use of action to support pedagogical goals. A specific version of embodied learning posits an action-to-abstraction transition supported by gesture, sketching, and analogical mapping. These tools seem to have special promise for bolstering learning in science, technology, engineering, and mathematics (STEM) disciplines, but existing efforts need further theoretical and empirical development. The topical collection in Cognitive Research: Principles includes articles aiming to formalize and test the effectiveness of embodied learning in STEM. The collection provides guideposts, staking out the terrain that should be surveyed before larger-scale efforts are undertaken. This introduction provides a broader context concerning mechanisms that can support embodied learning and make it especially well suited to the STEM disciplines.