Perceptual learning modules in mathematics: enhancing students' pattern recognition, structure extraction, and fluency

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.

Connecting Adaptive Perceptual Learning and Signal Detection Theory in Skin Cancer Screening

Combining perceptual learning techniques with adaptive learning algorithms has been shown to accelerate the development of expertise in medical and STEM learning domains (Kellman & Massey, 2013; Kellman, Jacoby, Massey & Krasne, 2022). Virtually all adaptive learning systems have relied on simple accuracy data that does not take into account response bias, a problem that may be especially consequential in multi-category perceptual classifications. We investigated whether adaptive perceptual learning in skin cancer screening can be enhanced by incorporating signal detection theory (SDT) methods that separate sensitivity from criterion. SDT-style concepts were used to alter sequencing, and separately to define mastery (category retirement). SDT retirement used a running d' estimate calculated from a recent window of trials based on hit and false alarm rates. Undergraduate participants used a Skin Cancer PALM (perceptual adaptive learning module) to learn classification of 10 cancerous and readily-confused non-cancerous skin lesion types. Four adaptive conditions varied either the type of adaptive sequencing (standard vs. SDT) or retirement criteria (standard vs. SDT). A non-adaptive control condition presented didactic instruction on dermatologic screening in video form, including images, classification schemes, and detailed explanations. All adaptive conditions robustly outperformed the non-adaptive control in both learning efficiency and fluency (large effect sizes). Between adaptive conditions, SDT retirement criteria produced greater learning efficiency than standard, accuracy-based mastery criteria at both immediate and delayed posttests (medium effect sizes). SDT sequencing and standard adaptive sequencing did not differ. SDT enhancements to adaptive perceptual learning procedures have potential to enhance learning efficiency.

From Image to Diagnosis: Characterizing Sources of Error in Histopathologic Interpretation

An accurate histopathologic diagnosis on surgical biopsy material is necessary for the clinical management of patients and has important implications for research, clinical trial design/enrollment, and public health education. This study used a mixed methods approach to isolate sources of diagnostic error while residents and attending pathologists interpreted digitized breast biopsy slides. Ninety participants including pathology residents and attendings at major United States medical centers reviewed a set of 14 digitized whole slide images of breast biopsies. Each case had a consensus-defined diagnosis and critical region of interest (cROI) representing the most significant pathology on the slide. Participants were asked to view unmarked digitized slides, draw their own participant region of interest (pROI), describe its features, and render a diagnosis. Participants' review behavior was tracked using case viewer software and an eye-tracking device. Diagnostic accuracy was calculated in comparison to the consensus diagnosis. We measured the frequency of errors emerging during four interpretive phases: 1) detecting the cROI, 2) recognizing its relevance, 3) using the correct terminology to describe findings in the pROI, and 4) making a diagnostic decision. According to eye tracking data, both trainees and attending pathologists were very likely (about 94% of the time) to find the cROI when inspecting a slide. However, trainees were less likely to consider the cROI relevant to their diagnosis. Pathology trainees were more likely (41% of cases) to use incorrect terminology to describe pROI features than attending pathologists (21% of cases). Failure to accurately describe features was the only factor strongly associated with an incorrect diagnosis. Identifying where errors emerge in the interpretive and/or descriptive process and working on building organ-specific feature recognition and verbal fluency in describing those features are critical steps for achieving competency in diagnostic decision making.

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.

The Impacts of Three Educational Technologies on Algebraic Understanding in the Context of COVID-19

The current study investigated the effectiveness of three distinct educational technologies-two game-based applications (From Here to There and DragonBox 12+) and two modes of online problem sets in ASSISTments (an Immediate Feedback condition and an Active Control condition with no immediate feedback) on Grade 7 students' algebraic knowledge. More than 3,600 Grade 7 students across nine in-person and one virtual schools within the same district were randomly assigned to one of the four conditions. Students received nine 30-minute intervention sessions from September 2020 to March 2021. Hierarchical linear modeling analyses of the final analytic sample (N = 1,850) showed significantly higher posttest scores for students who used From Here to There and DragonBox 12+ compared to the Active Control condition. No significant difference was found for the Immediate Feedback condition. The findings have implications for understanding how game-based applications can affect algebraic understanding, even within pandemic pressures on learning.

What influences students’ abilities to critically evaluate scientific investigations?

Critical thinking is the process by which people make decisions about what to trust and what to do. Many undergraduate courses, such as those in biology and physics, include critical thinking as an important learning goal. Assessing critical thinking, however, is non-trivial, with mixed recommendations for how to assess critical thinking as part of instruction. Here we evaluate the efficacy of assessment questions to probe students’ critical thinking skills in the context of biology and physics. We use two research-based standardized critical thinking instruments known as the Biology Lab Inventory of Critical Thinking in Ecology (Eco-BLIC) and Physics Lab Inventory of Critical Thinking (PLIC). These instruments provide experimental scenarios and pose questions asking students to evaluate what to trust and what to do regarding the quality of experimental designs and data. Using more than 3000 student responses from over 20 institutions, we sought to understand what features of the assessment questions elicit student critical thinking. Specifically, we investigated (a) how students critically evaluate aspects of research studies in biology and physics when they are individually evaluating one study at a time versus comparing and contrasting two and (b) whether individual evaluation questions are needed to encourage students to engage in critical thinking when comparing and contrasting. We found that students are more critical when making comparisons between two studies than when evaluating each study individually. Also, compare-and-contrast questions are sufficient for eliciting critical thinking, with students providing similar answers regardless of if the individual evaluation questions are included. This research offers new insight on the types of assessment questions that elicit critical thinking at the introductory undergraduate level; specifically, we recommend instructors incorporate more compare-and-contrast questions related to experimental design in their courses and assessments.

Promoting scientific literacy in evolution through citizen science

Evolutionary understanding is central to biology. It is also an essential prerequisite to understanding and making informed decisions about societal issues such as climate change. Yet, evolution is generally poorly understood by civil society and many misconceptions exist. Citizen science, which has been increasing in popularity as a means to gather new data and promote scientific literacy, is one strategy through which people could learn about evolution. However, despite the potential for citizen science to promote evolution learning opportunities, very few projects implement them. In this paper, we make the case for incorporating evolution education into citizen science, define key learning goals, and suggest opportunities for designing and evaluating projects in order to promote scientific literacy in evolution.

Spatial Alignment Facilitates Visual Comparison in Children

Visual comparison is a key process in everyday learning and reasoning. Recent research has discovered the spatial alignment principle, based on the broader framework of structure-mapping theory in comparison. According to the spatial alignment principle, visual comparison is more efficient when the figures being compared are arranged in direct placement-that is, juxtaposed with parallel structural axes. In this placement, (1) the intended relational correspondences are readily apparent, and (2) the influence of potential competing correspondences is minimized. There is evidence for the spatial alignment principle in adults' visual comparison (Matlen et al., 2020). Here, we test whether it holds for children. Six- and eight-year-old children performed a same-different task over visual pairs. The results indicated that direct placement led to faster and more accurate comparison, both for concrete same-different matches (matches of both objects and relations) and for purely relational matches-evidence that the same structural alignment process holds for visual comparison in 6- and 8-year-olds as in adults. These findings have implications for learning and education.

Optimal Learning Under Time Constraints: Empirical and Simulated Trade-offs Between Depth and Breadth of Study

Learners are often constrained by their available study time, typically having to make a trade-off between depth and breadth of learning. Classic experimental paradigms in memory research treat all items as equally important, but this is unlikely the case in reality. Rather, information varies in importance, and people vary in their ability to distinguish what is more or less important. We test the impact of this trade-off in the study of Graduate Record Examination (GRE)-synonym word pairs. In our empirical Study 1, we split our stimuli set, with some items (focal) being afforded more rounds of retrieval practice than other items (non-focal). All conditions had the same total number of trials (i.e., constant study time), but differed in the number of focal words (breadth) and repetitions (depth). The conditions differed significantly in both mean performance and variance on the day-delayed test. Using this empirical data as a base, we then conducted a simulation (Study 2) modeling depth-breadth trade-offs under various conditions of learner forecasting accuracy and test coverage. In Study 2, we found that a medium-depth medium-breadth strategy was appropriate for most of the learning situations covered by our simulation, but that learners with a well-calibrated understanding of importance may benefit from a more targeted high-depth, low-breadth approach. Our results highlight the complexity of navigating the depth-breadth trade-off. Models of learning strategy optimization will need to account for learner forecasting sensitivity, which itself is likely an interaction between relatively stable individual differences and shifting contextual factors.

Novel Education Modules Addressing the Underrepresentation of Skin of Color in Dermatology Training

BACKGROUND

Representative images of pathology in patients with skin of color are lacking in most medical education resources. This particularly affects training in dermatology, which relies heavily on the use of images to teach pattern recognition. The presentation of skin pathology can vary greatly among different skin tones, and this lack of representation of dark skin phototypes challenges providers' abilities to provide quality care to patients of color.In Botswana and other countries in sub-Saharan Africa, this challenge is further compounded by limited resources and access to dermatologists. There is a need for improved and accessible educational resources to train medical students and local medical providers in basic skin lesion description and diagnosis.

OBJECTIVES

We examined whether online Perceptual and Adaptive Learning Modules (PALMs) composed of representative dark skin images could efficiently train University of Botswana medical students to more accurately describe and diagnose common skin conditions in their community.

METHODS

Year 4 and 5 medical students voluntarily completed PALMs that teach skin morphology, configuration, and distribution terminology and diagnosis of the most common dermatologic conditions in their community. Pre-tests, post-tests and delayed-tests assessed knowledge acquisition and retention.

RESULTS

PALMs training produced statistically significant (P < .0001) improvements in accuracy and fluency with large effect sizes (1.5, 3.7) and good retention after a 12.5-21-week median delay. Limitations were a self-selected group of students, a single institution, slow internet connections, and high drop-out rates.

CONCLUSIONS

Overall, population-specific PALMs are a useful tool for efficient development of pattern recognition in skin disease description and diagnosis.

Mastering Electrocardiogram Interpretation Skills Through a Perceptual and Adaptive Learning Module

Although accurate interpretation of the standard 12-lead electrocardiogram (ECG) is fundamental to diagnosing heart disease, several prior studies report low accuracy rates among medical students, residents, and practicing physicians. The objective of this study was to determine if an online ECG Perceptual and Adaptive Learning Module (ECG PALM) is an efficient instrument to teach ECG interpretation. The ECG PALM consists of 415 unique ECG tracings with associated pretest, posttest, and delayed tests, each using 30 additional ECGs to gauge the effectiveness and durability of training. Between 2013 and 2015, a total of 113 third-year and 156 fourth-year medical students and 34 first-year, 41 second-year, and 37 third-year emergency medicine residents completed the PALM and associated tests. We measured two mastery criteria: accuracy, the percentage of correct interpretations, and fluency, the percentage of images interpreted accurately within 15 seconds. The ECG PALM produced statistically significant improvements (0.0001 < p < 0.0045) in student and resident performance for both accuracy (effect size = 0.9 to 3.2) and fluency (effect size = 2.5 to 3.1) following training ranging from 46 ± 24 minutes (R3s) to 88 ± 32 minutes (third-year medical students). Medical students and residents performed significantly better on a test the year following training (delayed test) than those without prior ECG PALM training (pretest). The fluency of R3 residents in classifying the 15 diagnostic categories was less than 60% for nine of the 15 diagnoses and greater than 80% for only one. Following PALM training, fluency was higher than 80% for seven of the 15 categories and less than 60% for only two categories. Accuracy in recognizing ST-elevation myocardial infarctions (STEMIs) was high both before and after PALM training for R3s, but fluency was only 64% for anterior STEMIs on the pretest, increasing to 93% following PALM training. These observations suggest that the ECG PALM is an effective and durable supplemental tool for developing mastery in interpreting common ECG abnormalities.

One Instructional Sequence Fits all? A Conceptual Analysis of the Applicability of Concreteness Fading in Mathematics, Physics, Chemistry, and Biology Education

To help students acquire mathematics and science knowledge and competencies, educators typically use multiple external representations (MERs). There has been considerable interest in examining ways to present, sequence, and combine MERs. One prominent approach is the concreteness fading sequence, which posits that instruction should start with concrete representations and progress stepwise to representations that are more idealized. Various researchers have suggested that concreteness fading is a broadly applicable instructional approach. In this theoretical paper, we conceptually analyze examples of concreteness fading in the domains of mathematics, physics, chemistry, and biology and discuss its generalizability. We frame the analysis by defining and describing MERs and their use in educational settings. Then, we draw from theories of analogical and relational reasoning to scrutinize the possible cognitive processes related to learning with MERs. Our analysis suggests that concreteness fading may not be as generalizable as has been suggested. Two main reasons for this are discussed: (1) the types of representations and the relations between them differ across different domains, and (2) the instructional goals between domains and subsequent roles of the representations vary.

What do radiologists look for? Advances and limitations of perceptual learning in radiologic search

Supported by guidance from training during residency programs, radiologists learn clinically relevant visual features by viewing thousands of medical images. Yet the precise visual features that expert radiologists use in their clinical practice remain unknown. Identifying such features would allow the development of perceptual learning training methods targeted to the optimization of radiology training and the reduction of medical error. Here we review attempts to bridge current gaps in understanding with a focus on computational saliency models that characterize and predict gaze behavior in radiologists. There have been great strides toward the accurate prediction of relevant medical information within images, thereby facilitating the development of novel computer-aided detection and diagnostic tools. In some cases, computational models have achieved equivalent sensitivity to that of radiologists, suggesting that we may be close to identifying the underlying visual representations that radiologists use. However, because the relevant bottom-up features vary across task context and imaging modalities, it will also be necessary to identify relevant top-down factors before perceptual expertise in radiology can be fully understood. Progress along these dimensions will improve the tools available for educating new generations of radiologists, and aid in the detection of medically relevant information, ultimately improving patient health.

Practicing Connections: A Framework to Guide Instructional Design for Developing Understanding in Complex Domains

Research suggests that expert understanding is characterized by coherent mental representations featuring a high level of connectedness. This paper advances the idea that educators can facilitate this level of understanding in students through the practicing connections framework: a practical framework to guide instructional design for developing deep understanding and transferable knowledge in complex academic domains. We start by reviewing what we know from learning sciences about the nature and development of transferable knowledge, arguing that connectedness is key to the coherent mental schemas that underlie deep understanding and transferable skills. We then propose features of instruction that might uniquely facilitate deep understanding and suggest that the connections between a domain’s core concepts, key representations, and contexts and practices of the world must be made explicit and practiced, over time, in order for students to develop coherent understanding. We illustrate the practicing connections approach to instructional design in the context of a new online interactive introductory statistics textbook developed by the authors.

Literature review of perceptual learning modules in medical education: What can we conclude regarding dermatology?

INTRODUCTION

Visual pattern recognition is important in many different medical fields, particularly in dermatology. Perceptual learning modules (PLM) are software programs developed to enhance visual pattern recognition through the sequential presentation of images that trainee must quickly diagnose. The aim of this literature review was to determine the scope and effectiveness of PLM in medical education.

METHODS

We carried out a search of MEDLINE, EMBASE, Web of Science and ERIC from its inception through to July 1, 2017. All articles describing an educational intervention based on perceptual learning in a medical field were included. Two investigators worked independently on study selection and data extraction.

RESULTS

Of 191 references selected, 5 studies were included in the final analysis: 3 before-after studies and 2 randomized studies comparing 12 to 236 trainees taking PLM with 12 to 316 trainees not taking PLM. Four studies reported a statistically significant increase in diagnostic accuracy (lower error rate) and fluency (shorter response time) following PLM interventions (dermatology, pathology, echocardiography), with long-term persistence of the effect in three studies.

CONCLUSION

PLM is a promising educational tool to teach pattern recognition that may be used in dermatology and other medical fields to improve diagnostic accuracy and rapidity in daily practice.

Visual chunking as a strategy for spatial thinking in STEM

Working memory capacity is known to predict the performance of novices and experts on a variety of tasks found in STEM (Science, Technology, Engineering, and Mathematics). A common feature of STEM tasks is that they require the problem solver to encode and transform complex spatial information depicted in disciplinary representations that seemingly exceed the known capacity limits of visuospatial working memory. Understanding these limits and how visuospatial information is encoded and transformed differently by STEM learners presents new avenues for addressing the challenges students face while navigating STEM classes and degree programs. Here, we describe two studies that explore student accuracy at detecting color changes in visual stimuli from the discipline of chemistry. We demonstrate that both naive and novice chemistry students' encoding of visuospatial information is affected by how information is visually structured in "chunks" prevalent across chemistry representations. In both studies we show that students are more accurate at detecting color changes within chemistry-relevant chunks compared to changes that occur outside of them, but performance was not affected by the dimensionality of the structure (2D vs 3D) or the presence of redundancies in the visual representation. These studies support the hypothesis that strategies for chunking the spatial structure of information may be critical tools for transcending otherwise severely limited visuospatial capacity in the absence of expertise.

Deliberate practice as an educational method for learning to interpret the prepubescent female genital examination

BACKGROUND

Correct interpretation of the prepubescent female genital examination is a critical skill; however, physician skill in this area is limited.

OBJECTIVE

To complement the bedside learning of this examination, we developed a learning platform for the visual diagnosis of the prepubescent female genital examination and examined the amount and rate of skill acquisition.

PARTICIPANTS AND SETTING

Medical students, residents, and fellows and attendings participated in an on-line learning platform.

METHODS

This was a multicenter prospective cross-sectional study. Study participants deliberately practiced 158 prepubescent female genital examination cases hosted on a computer-based learning and assessment platform. Participants assigned the case normal or abnormal; if abnormal, they identified the location of the abnormality and the specific diagnosis. Participants received feedback after every case.

RESULTS

We enrolled 107 participants (26 students, 31 residents, 24 fellows and 26 attendings). Accuracy (95 % CI) increased by 10.3 % (7.8, 12.8), Cohen's d-effect size of 1.17 (1.14, 1.19). The change in specificity was +16.8 (14.1, 19.5) and sensitivity +2.4 (-0.9, 5.6). It took a mean (SD) 46.3 (32.2) minutes to complete cases. There was no difference between learner types with respect to initial (p = 0.2) or final accuracy (p = 0.4) scores.

CONCLUSIONS

This study's learning intervention led to effective and feasible skill improvement. However, while participants improved significantly with normal cases, which has relevance in reducing unnecessary referrals to child protection teams, learning gains were not as evident in abnormal cases. All levels of learners demonstrated a similar performance, emphasizing the need for this education even among experienced clinicians.

Simultaneous training on overlapping grapheme phoneme correspondences augments learning and retention

An important component of learning to read is the acquisition of letter-to-sound mappings. The sheer quantity of mappings and many exceptions in opaque languages such as English suggests that children may use a form of statistical learning to acquire them. However, whereas statistical models of reading are item-based, reading instruction typically focuses on rule-based approaches involving small sets of regularities. This discrepancy poses the question of how different groupings of regularities, an unexamined factor of most reading curricula, may affect learning. Exploring the interplay between item statistics and rules, this study investigated how consonant variability, an item-level factor, and the degree of overlap among the to-be-trained vowel strings, a group-level factor, influence learning. English-speaking first graders (N = 361) were randomly assigned to be trained on vowel sets with high overlap (e.g., EA, AI) or low overlap (e.g., EE, AI); this was crossed with a manipulation of consonant frame variability. Whereas high vowel overlap led to poorer initial performance, it resulted in more learning when tested immediately and after a 2-week-delay. There was little beneficial effect of consonant variability. These findings indicate that online letter/sound processing affects how new knowledge is integrated into existing information. Moreover, they suggest that vowel overlap should be considered when designing reading curricula.

Cognitive Task Analysis for Implicit Knowledge About Visual Representations With Similarity Learning Methods

Visual representations are prevalent in STEM instruction. To benefit from visuals, students need representational competencies that enable them to see meaningful information. Most research has focused on explicit conceptual representational competencies, but implicit perceptual competencies might also allow students to efficiently see meaningful information in visuals. Most common methods to assess students' representational competencies rely on verbal explanations or assume explicit attention. However, because perceptual competencies are implicit and not necessarily verbally accessible, these methods are ill-equipped to assess them. We address these shortcomings with a method that draws on similarity learning, a machine learning technique that detects visual features that account for participants' responses to triplet comparisons of visuals. In Experiment 1, 614 chemistry students judged the similarity of Lewis structures and in Experiment 2, 489 students judged the similarity of ball-and-stick models. Our results showed that our method can detect visual features that drive students' perception and suggested that students' conceptual knowledge about molecules informed perceptual competencies through top-down processes. Furthermore, Experiment 2 tested whether we can improve the efficiency of the method with active sampling. Results showed that random sampling yielded higher accuracy than active sampling for small sample sizes. Together, the experiments provide the first method to assess students' perceptual competencies implicitly, without requiring verbalization or assuming explicit visual attention. These findings have implications for the design of instructional interventions that help students acquire perceptual representational competencies.

Analysis of Perceptual Expertise in Radiology - Current Knowledge and a New Perspective

Radiologists rely principally on visual inspection to detect, describe, and classify findings in medical images. As most interpretive errors in radiology are perceptual in nature, understanding the path to radiologic expertise during image analysis is essential to educate future generations of radiologists. We review the perceptual tasks and challenges in radiologic diagnosis, discuss models of radiologic image perception, consider the application of perceptual learning methods in medical training, and suggest a new approach to understanding perceptional expertise. Specific principled enhancements to educational practices in radiology promise to deepen perceptual expertise among radiologists with the goal of improving training and reducing medical error.

Preschoolers and multi-digit numbers: A path to mathematics through the symbols themselves

Numerous studies from developmental psychology have suggested that human symbolic representation of numbers is built upon the evolutionally old capacity for representing quantities that is shared with other species. Substantial research from mathematics education also supports the idea that mathematical concepts are best learned through their corresponding physical representations. We argue for an independent pathway to learning "big" multi-digit symbolic numbers that focuses on the symbol system itself. Across five experiments using both between- and within-subject designs, we asked preschoolers to identify written multi-digit numbers with their spoken names in a two-alternative-choice-test or to indicate the larger quantity between two written numbers. Results showed that preschoolers could reliably map spoken number names to written forms and compare the magnitudes of two written multi-digit numbers. Importantly, these abilities were not related to their non-symbolic representation of quantities. These findings have important implications for numerical cognition, symbolic development, teaching, and education.

Natural Alternatives to Natural Number: The Case of Ratio

The overwhelming majority of efforts to cultivate early mathematical thinking rely primarily on counting and associated natural number concepts. Unfortunately, natural numbers and discretized thinking do not align well with a large swath of the mathematical concepts we wish for children to learn. This misalignment presents an important impediment to teaching and learning. We suggest that one way to circumvent these pitfalls is to leverage students' non-numerical experiences that can provide intuitive access to foundational mathematical concepts. Specifically, we advocate for explicitly leveraging a) students' perceptually based intuitions about quantity and b) students' reasoning about change and variation, and we address the affordances offered by this approach. We argue that it can support ways of thinking that may at times align better with to-be-learned mathematical ideas, and thus may serve as a productive alternative for particular mathematical concepts when compared to number. We illustrate this argument using the domain of ratio, and we do so from the distinct disciplinary lenses we employ respectively as a cognitive psychologist and as a mathematics education researcher. Finally, we discuss the potential for productive synthesis given the substantial differences in our preferred methods and general epistemologies.

Orienting to see what's important: Learn to ignore the irrelevant

The current study used a triad judgment task to assess whether blocking by comparison type in a triad judgment task could lead people to pay less attention to surface-level (irrelevant) features and pay more attention to deep (structural) features of information. A sample of 313 participants recruited through Mechanical Turk participated in this study. On each triad, participants were asked to evaluate which of two source scenarios went best with the target scenario. Three types of triads were constructed with materials related to the ability to perceive ethical issues within the practice of psychology. One type of triad contrasted a scenario that was similar to the target in terms of surface-level features and a scenario that was similar in terms of deep features (similar surface-similar deep, SS/SD). A second triad type contrasted a scenario that was similar in terms of deep features with an unrelated scenario (similar deep-unrelated, SD/U). The third contrasted a scenario that was similar in terms of surface features with an unrelated scenario (similar surface-unrelated, SS/U). There were 10 triads of each type. We found that with all the blocking orders except two, participants reliably chose the SS scenario over the SD scenario for the SS/SD triads. However, participants who started out with the SD/U triads did not. The results provide evidence that people often have the ability to perceive the deep, but they are distracted by the surface-level features. The results also show that people can be oriented away from being distracted by the surface-level features so that they can see the deep.

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.

Perceptual support promotes strategy generation: Evidence from equation solving

Over time, children shift from using less optimal strategies for solving mathematics problems to using better ones. But why do children generate new strategies? We argue that they do so when they begin to encode problems more accurately; therefore, we hypothesized that perceptual support for correct encoding would foster strategy generation. Fourth-grade students solved mathematical equivalence problems (e.g., 3 + 4 + 5 = 3 + __) in a pre-test. They were then randomly assigned to one of three perceptual support conditions or to a Control condition. Participants in all conditions completed three mathematical equivalence problems with feedback about correctness. Participants in the experimental conditions received perceptual support (i.e., highlighting in red ink) for accurately encoding the equal sign, the right side of the equation, or the numbers that could be added to obtain the correct solution. Following this intervention, participants completed a problem-solving post-test. Among participants who solved the problems incorrectly at pre-test, those who received perceptual support for correctly encoding the equal sign were more likely to generate new, correct strategies for solving the problems than were those who received feedback only. Thus, perceptual support for accurate encoding of a key problem feature promoted generation of new, correct strategies. Statement of Contribution What is already known on this subject? With age and experience, children shift to using more effective strategies for solving math problems. Problem encoding also improves with age and experience. What the present study adds? Support for encoding the equal sign led children to generate correct strategies for solving equations. Improvements in problem encoding are one source of new strategies.

Grounded and embodied mathematical cognition: Promoting mathematical insight and proof using action and language

We develop a theory of grounded and embodied mathematical cognition (GEMC) that draws on action-cognition transduction for advancing understanding of how the body can support mathematical reasoning. GEMC proposes that participants' actions serve as inputs capable of driving the cognition-action system toward associated cognitive states. This occurs through a process of transduction that promotes valuable mathematical insights by eliciting dynamic depictive gestures that enact spatio-temporal properties of mathematical entities. Our focus here is on pre-college geometry proof production. GEMC suggests that action alone can foster insight but is insufficient for valid proof production if action is not coordinated with language systems for propositionalizing general properties of objects and space. GEMC guides the design of a video game-based learning environment intended to promote students' mathematical insights and informal proofs by eliciting dynamic gestures through in-game directed actions. GEMC generates several hypotheses that contribute to theories of embodied cognition and to the design of science, technology, engineering, and mathematics (STEM) education interventions. Pilot study results with a prototype video game tentatively support theory-based predictions regarding the role of dynamic gestures for fostering insight and proof-with-insight, and for the role of action coupled with language to promote proof-with-insight. But the pilot yields mixed results for deriving in-game interventions intended to elicit dynamic gesture production. Although our central purpose is an explication of GEMC theory and the role of action-cognition transduction, the theory-based video game design reveals the potential of GEMC to improve STEM education, and highlights the complex challenges of connecting embodiment research to education practices and learning environment design.

Changing the precision of preschoolers' approximate number system representations changes their symbolic math performance

From early in life, humans have access to an approximate number system (ANS) that supports an intuitive sense of numerical quantity. Previous work in both children and adults suggests that individual differences in the precision of ANS representations correlate with symbolic math performance. However, this work has been almost entirely correlational in nature. Here we tested for a causal link between ANS precision and symbolic math performance by asking whether a temporary modulation of ANS precision changes symbolic math performance. First, we replicated a recent finding that 5-year-old children make more precise ANS discriminations when starting with easier trials and gradually progressing to harder ones, compared with the reverse. Next, we show that this brief modulation of ANS precision influenced children's performance on a subsequent symbolic math task but not a vocabulary task. In a supplemental experiment, we present evidence that children who performed ANS discriminations in a random trial order showed intermediate performance on both the ANS task and the symbolic math task, compared with children who made ordered discriminations. Thus, our results point to a specific causal link from the ANS to symbolic math performance.

Non-formal mechanisms in mathematical cognitive development: The case of arithmetic

The idea that cognitive development involves a shift towards abstraction has a long history in psychology. One incarnation of this idea holds that development in the domain of mathematics involves a shift from non-formal mechanisms to formal rules and axioms. Contrary to this view, the present study provides evidence that reliance on non-formal mechanisms may actually increase with age. Participants - Dutch primary school children - evaluated three-term arithmetic expressions in which violation of formally correct order of evaluation led to errors, termed foil errors. Participants solved the problems as part of their regular mathematics practice through an online study platform, and data were collected from over 50,000 children representing approximately 10% of all primary schools in the Netherlands, suggesting that the results have high external validity. Foil errors were more common for problems in which formally lower-priority sub-expressions were spaced close together, and also for problems in which such sub-expressions were relatively easy to calculate. We interpret these effects as resulting from reliance on two non-formal mechanisms, perceptual grouping and opportunistic selection, to determine order of evaluation. Critically, these effects reliably increased with participants' grade level, suggesting that these mechanisms are not phased out but actually become more important over development, even when they cause systematic violations of formal rules. This conclusion presents a challenge for the shift towards abstraction view as a description of cognitive development in arithmetic. Implications of this result for educational practice are discussed.

Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance

What basic capacities lay the foundation for advanced numerical cognition? Are there basic nonsymbolic abilities that support the understanding of advanced numerical concepts, such as fractions? To date, most theories have posited that previously identified core numerical systems, such as the approximate number system (ANS), are ill-suited for learning fraction concepts. However, recent research in developmental psychology and neuroscience has revealed a ratio-processing system (RPS) that is sensitive to magnitudes of nonsymbolic ratios and may be ideally suited for supporting fraction concepts. We provide evidence for this hypothesis by showing that individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university entrance exam in algebra. We suggest that the nonsymbolic RPS may support symbolic fraction understanding much as the ANS supports whole-number concepts. Thus, even abstract mathematical concepts, such as fractions, may be grounded not only in higher-order logic and language, but also in basic nonsymbolic processing abilities.

The simple advantage in perceptual and categorical generalization

Recent research in relational learning has suggested that simple training instances may lead to better generalization than complex training instances. We examined the perceptual-encoding mechanisms that might undergird this "simple advantage" by testing category and perceptual learning in adults with simplified and traditional (more complex) Chinese scripts. In Experiment 1, participants learned Chinese characters and their English translations, performed a memorization test, and generalized their learning to the corresponding characters written in the other script. In Experiment 2, we removed the training phase and modified the tests to examine transfer based purely on the perceptual similarities between simplified and traditional characters. We found the simple advantage in both experiments. Training with simplified characters produced better generalization than training with traditional characters when generalization relied on either recognition memory or pure perceptual similarities. On the basis of the results of these two experiments, we propose a simple process model to explain the perceptual mechanism that might drive this simple advantage, and in Experiment 3 we tested novel predictions of this model by examining the effect of exposure duration on the simple advantage. We found support for our model that the simple advantage is driven primarily by differences in the perceptual encoding of the information available from simple and complex instances. These findings advance our understanding of how the perceptual features of a learning opportunity interact with domain-general mechanisms to prepare learners for transfer.

Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes

This study presents evidence that humans have intuitive, perceptually based access to the abstract fraction magnitudes instantiated by nonsymbolic ratio stimuli. Moreover, it shows these perceptually accessed magnitudes can be easily compared with symbolically represented fractions. In cross-format comparisons, participants picked the larger of two ratios. Ratios were presented either symbolically as fractions or nonsymbolically as paired dot arrays or as paired circles. Response patterns were consistent with participants comparing specific analog fractional magnitudes independently of the particular formats in which they were presented. These results pose a challenge to accounts that argue human cognitive architecture is ill-suited for processing fractions. Instead, it seems that humans can process nonsymbolic ratio magnitudes via perceptual routes and without recourse to conscious symbolic algorithms, analogous to the processing of whole number magnitudes. These findings have important implications for theories regarding the nature of human number sense - they imply that fractions may in some sense be natural numbers, too.

Patterns of response times and response choices to science questions: the influence of relative processing time

We report on five experiments investigating response choices and response times to simple science questions that evoke student "misconceptions," and we construct a simple model to explain the patterns of response choices. Physics students were asked to compare a physical quantity represented by the slope, such as speed, on simple physics graphs. We found that response times of incorrect answers, resulting from comparing heights, were faster than response times of correct answers comparing slopes. This result alone might be explained by the fact that height was typically processed faster than slope for this kind of task, which we confirmed in a separate experiment. However, we hypothesize that the difference in response time is an indicator of the cause (rather than the result) of the response choice. To support this, we found that imposing a 3-s delay in responding increased the number of students comparing slopes (answering correctly) on the task. Additionally a significant proportion of students recognized the correct written rule (compare slope), but on the graph task they incorrectly compared heights. Finally, training either with repetitive examples or providing a general rule both improved scores, but only repetitive examples had a large effect on response times, thus providing evidence of dual paths or processes to a solution. Considering models of heuristics, information accumulation models, and models relevant to the Stroop effect, we construct a simple relative processing time model that could be viewed as a kind of fluency heuristic. The results suggest that misconception-like patterns of answers to some science questions commonly found on tests may be explained in part by automatic processes that involve the relative processing time of considered dimensions and a priority to answer quickly.

Spatial Factors Influence Arithmetic Performance: The Case of the Minus Sign

Spatial features of mathematical equations may influence how people solve and interpret those equations. This study examined whether manipulations of spatial features affected how participants solved and interpreted equations involving the minus sign. Undergraduate participants ( N = 91) solved multioperation arithmetic equations involving addition, subtraction, and multiplication (e.g. 25 − 3 + 2 × 5 = __). We varied the spacing of the final three operands and the position of the first operator relative to the adjacent operands. Participants also generated a story problem to correspond with a given equation. We evaluated the procedures that participants used in solving the equations by analysing both their solutions and their written work. Both close spacing of the final three operands and position of the first operation sign influenced the procedures that participants used. Both of the spatial manipulations also influenced participants’ interpretations of the conceptual structure of the equations, as revealed in the story problems that they generated. These results have implications for understanding how people process mathematical symbols and for mathematics education.

The Psychophysics of Algebra Expertise: Mathematics Perceptual Learning Interventions Produce Durable Encoding Changes

Mathematics requires thinking but also pattern recognition. Recent research indicates that perceptual learning (PL) interventions facilitate discovery of structure and recognition of patterns in mathematical domains, as assessed by tests of mathematical competence. Here we sought direct evidence that a brief perceptual learning module (PLM) produces changes in basic information extraction. Accuracy and speed of undergraduate participants' encoding of equations was assessed in a psychophysical task at pretest and delayed posttest. In between, the experimental group completed an Algebraic Transformations PLM, which involved identifying valid transformations of equations. Relative to controls, PLM participants showed reliable changes in encoding equations, detectable psychophysically 24 hours later. Encoding improvements were shown robustly by participants who were initially less proficient at algebra and were negligible for participants who were initially proficient. These results provide direct evidence for durable changes in information encoding produced by a PL intervention targeting a complex mathematical skill.

The Role of Search Speed in the Contextual Cueing of Children's Attention

The contextual cueing effect is a robust phenomenon in which repeated exposure to the same arrangement of random elements guides attention to relevant information by constraining search. The effect is measured using an object search task in which a target (e.g., the letter T) is located within repeated or nonrepeated visual contexts (e.g., configurations of the letter L). Decreasing response times for the repeated configurations indicates that contextual information has facilitated search. Although the effect is robust among adult participants, recent attempts to document the effect in children have yielded mixed results. We examined the effect of search speed on contextual cueing with school-aged children, comparing three types of stimuli that promote different search times in order to observe how speed modulates this effect. Reliable effects of search time were found, suggesting that visual search speed uniquely constrains the role of attention toward contextually cued information.

Applying perceptual and adaptive learning techniques for teaching introductory histopathology

Background:

Medical students are expected to master the ability to interpret histopathologic images, a difficult and time-consuming process. A major problem is the issue of transferring information learned from one example of a particular pathology to a new example. Recent advances in cognitive science have identified new approaches to address this problem.

Methods:

We adapted a new approach for enhancing pattern recognition of basic pathologic processes in skin histopathology images that utilizes perceptual learning techniques, allowing learners to see relevant structure in novel cases along with adaptive learning algorithms that space and sequence different categories (e.g. diagnoses) that appear during a learning session based on each learner's accuracy and response time (RT). We developed a perceptual and adaptive learning module (PALM) that utilized 261 unique images of cell injury, inflammation, neoplasia, or normal histology at low and high magnification. Accuracy and RT were tracked and integrated into a “Score” that reflected students rapid recognition of the pathologies and pre- and post-tests were given to assess the effectiveness.

Results:

Accuracy, RT and Scores significantly improved from the pre- to post-test with Scores showing much greater improvement than accuracy alone. Delayed post-tests with previously unseen cases, given after 6-7 weeks, showed a decline in accuracy relative to the post-test for 1^st-year students, but not significantly so for 2^nd-year students. However, the delayed post-test scores maintained a significant and large improvement relative to those of the pre-test for both 1^st and 2^nd year students suggesting good retention of pattern recognition. Student evaluations were very favorable.

Conclusion:

A web-based learning module based on the principles of cognitive science showed an evidence for improved recognition of histopathology patterns by medical students.

Adaptive response-time-based category sequencing in perceptual learning

Although much recent work in perceptual learning (PL) has focused on basic sensory discriminations, recent analyses suggest that PL in a variety of tasks depends on processes that discover and select information relevant to classifications being learned (Kellman & Garrigan, 2009; Petrov, Dosher, & Lu, 2005). In complex, real-world tasks, discovery involves finding structural invariants amidst task-irrelevant variation (Gibson, 1969), allowing learners to correctly classify new stimuli. The applicability of PL methods to such tasks offers important opportunities to improve learning. It also raises questions about how learning might be optimized in complex tasks and whether variables that influence other forms of learning also apply to PL. We investigated whether an adaptive, response-time-based, category sequencing algorithm implementing laws of spacing derived from memory research would also enhance perceptual category learning and transfer to novel cases. Participants learned to classify images of 12 different butterfly genera under conditions of: (1) random presentation, (2) adaptive category sequencing, and (3) adaptive category sequencing with 'mini-blocks' (grouping 3 successive category exemplars). We found significant effects on efficiency of learning for adaptive category sequencing, reliably better than for random presentation and mini-blocking (Experiment 1). Effects persisted across a 1-week delay and were enhanced for novel items. Experiment 2 showed even greater effects of adaptive learning for perceptual categories containing lower variability. These results suggest that adaptive category sequencing increases the efficiency of PL and enhances generalization of PL to novel stimuli, key components of high-level PL and fundamental requirements of learning in many domains.

Middle cerebral artery bifurcation aneurysms: an anatomic classification scheme for planning optimal surgical strategies

BACKGROUND

Changing landscapes in neurosurgical training and increasing use of endovascular therapy have led to decreasing exposure in open cerebrovascular neurosurgery. To ensure the effective transition of medical students into competent practitioners, new training paradigms must be developed.

OBJECTIVE

Using principles of pattern recognition, we created a classification scheme for middle cerebral artery (MCA) bifurcation aneurysms that allows their categorization into a small number of shape pattern groups.

METHODS

Angiographic data from patients with MCA aneurysms between 1995 and 2012 were used to construct 3-dimensional models. Models were then analyzed and compared objectively by assessing the relationship between the aneurysm sac, parent vessel, and branch vessels. Aneurysms were then grouped on the basis of the similarity of their shape patterns in such a way that the in-class similarities were maximized while the total number of categories was minimized. For each category, a proposed clip strategy was developed.

RESULTS

From the analysis of 61 MCA bifurcation aneurysms, 4 shape pattern categories were created that allowed the classification of 56 aneurysms (91.8%). The number of aneurysms allotted to each shape cluster was 10 (16.4%) in category 1, 24 (39.3%) in category 2, 7 (11.5%) in category 3, and 15 (24.6%) in category 4.

CONCLUSION

This study demonstrates that through the use of anatomic visual cues, MCA bifurcation aneurysms can be grouped into a small number of shape patterns with an associated clip solution. Implementing these principles within current neurosurgery training paradigms can provide a tool that allows more efficient transition from novice to cerebrovascular expert.

Metaphorical motion in mathematical reasoning: further evidence for pre-motor implementation of structure mapping in abstract domains

The theory of computation and category theory both employ arrow-based notations that suggest that the basic metaphor "state changes are like motions" plays a fundamental role in all mathematical reasoning involving formal manipulations. If this is correct, structure-mapping inferences implemented by the pre-motor action planning system can be expected to be involved in solving any mathematics problems not solvable by table lookups and number line manipulations alone. Available functional imaging studies of multi-digit arithmetic, algebra, geometry and calculus problem solving are consistent with this expectation.