scholarly journals Fraction magnitude: Mapping between symbolic and spatial representations of proportion

2020 ◽  
Vol 6 (2) ◽  
pp. 204-230
Author(s):  
Michelle Ann Hurst ◽  
Marisa Massaro ◽  
Sara Cordes
Keyword(s):  
Rational Numbers ◽  
Number Line ◽  
Future Research ◽  
Spatial Representations ◽  
Improve Performance ◽  
Comparison Task ◽  
Magnitude Comparison ◽  
Number Lines ◽  
Research And Education ◽  
Partitioning Strategy

Fraction notation conveys both part-whole (3/4 is 3 out of 4) and magnitude (3/4 = 0.75) information, yet evidence suggests that both children and adults find accessing magnitude information from fractions particularly difficult. Recent research suggests that using number lines to teach children about fractions can help emphasize fraction magnitude. In three experiments with adults and 9-12-year-old children, we compare the benefits of number lines and pie charts for thinking about rational numbers. In Experiment 1, we first investigate how adults spontaneously visualize symbolic fractions. Then, in two further experiments, we explore whether priming children to use pie charts vs. number lines impacts performance on a subsequent symbolic magnitude task and whether children differentially rely on a partitioning strategy to map rational numbers to number lines vs. pie charts. Our data reveal that adults very infrequently spontaneously visualize fractions along a number line and, contrary to other findings, that practice mapping rational numbers to number lines did not improve performance on a subsequent symbolic magnitude comparison task relative to practice mapping the same magnitudes to pie charts. However, children were more likely to use overt partitioning strategies when working with pie charts compared to number lines, suggesting these representations did lend themselves to different working strategies. We discuss the interpretations and implications of these findings for future research and education. All materials and data are provided as Supplementary Materials.

The fractions SNARC revisited: Processing fractions on a consistent mental number line

2018 ◽  
Vol 71 (8) ◽  
pp. 1761-1770 ◽  
Author(s):  
Elizabeth Y Toomarian ◽  
Edward M Hubbard
Keyword(s):  
Number Line ◽  
Mental Number Line ◽  
Snarc Effect ◽  
Spatial Representations ◽  
Comparison Task ◽  
Single Digit ◽  
Magnitude Processing ◽  
Number Comparison ◽  
Within Subjects ◽  
Solid Foundation

The ability to understand fractions is key to establishing a solid foundation in mathematics, yet children and adults struggle to comprehend them. Previous studies have suggested that these struggles emerge because people fail to process fraction magnitude holistically on the mental number line (MNL), focusing instead on fraction components. Subsequent studies have produced evidence for default holistic processing but examined only magnitude processing, not spatial representations. We explored the spatial representations of fractions on the MNL in a series of three experiments. Experiment 1 replicated Bonato et al.; 30 naïve undergraduates compared unit fractions (1/1-1/9) to 1/5, resulting in a reverse SNARC (Spatial-Numerical Association of Response Codes) effect. Experiment 2 countered potential strategic biases induced by the limited set of fractions used by Bonato et al. by expanding the stimulus set to include all irreducible, single-digit proper fractions and asked participants to compare them against 1/2. We observed a classic SNARC effect, completely reversing the pattern from Experiment 1. Together, Experiments 1 and 2 demonstrate that stimulus properties dramatically impact spatial representations of fractions. In Experiment 3, we demonstrated within-subjects reliability of the SNARC effect across both a fractions and whole number comparison task. Our results suggest that adults can indeed process fraction magnitudes holistically, and that their spatial representations occur on a consistent MNL for both whole numbers and fractions.

scholarly journals How the Abstract Becomes Concrete: Irrational Numbers are Understood Relative to Natural Numbers and Perfect Squares

2018 ◽  
Author(s):  
Purav Patel
Keyword(s):  
Teaching And Learning ◽  
Number System ◽  
Number Line ◽  
Natural Numbers ◽  
Comparison Task ◽  
Irrational Numbers ◽  
Magnitude Comparison ◽  
Number Systems ◽  
Number Line Estimation ◽  
Perfect Squares

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like the square root of 2, is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers – specifically, the radicands of radical expressions – as natural numbers. Strategy self-reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task-specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.

Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task

2009 ◽  
Vol 62 (8) ◽  
pp. 1598-1616 ◽  
Author(s):  
Gaëlle Meert ◽  
Jacques Grégoire ◽  
Marie-Pascale Noël
Keyword(s):  
Mental Representation ◽  
Priming Effect ◽  
Rational Numbers ◽  
Numerical Comparison ◽  
Natural Numbers ◽  
Comparison Task ◽  
Magnitude Comparison ◽  
Distance Effects ◽  
Holistic Representation ◽  
Common Components

This study investigated whether the mental representation of the fraction magnitude was componential and/or holistic in a numerical comparison task performed by adults. In Experiment 1, the comparison of fractions with common numerators (x/a_x/b) and of fractions with common denominators (a/x_b/x) primed the comparison of natural numbers. In Experiment 2, fillers (i.e., fractions without common components) were added to reduce the regularity of the stimuli. In both experiments, distance effects indicated that participants compared the numerators for a/x_b/x fractions, but that the magnitudes of the whole fractions were accessed and compared for x/a_x/b fractions. The priming effect of x/a_x/b fractions on natural numbers suggested that the interference of the denominator magnitude was controlled during the comparison of these fractions. These results suggested a hybrid representation of their magnitude (i.e., componential and holistic). In conclusion, the magnitude of the whole fraction can be accessed, probably by estimating the ratio between the magnitude of the denominator and the magnitude of the numerator. However, adults might prefer to rely on the magnitudes of the components and compare the magnitudes of the whole fractions only when the use of a componential strategy is made difficult.

scholarly journals Magnitude integration in the Archerfish

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Tali Leibovich-Raveh ◽  
Ashael Raveh ◽  
Dana Vilker ◽  
Shai Gabay
Keyword(s):  
Animal Model ◽  
Grocery Store ◽  
The Body ◽  
Water Tank ◽  
Comparison Task ◽  
Magnitude Comparison ◽  
Mathematical Abilities ◽  
In The Wild ◽  
Numerical Magnitudes ◽  
Shortest Queue

AbstractWe make magnitude-related decisions every day, for example, to choose the shortest queue at the grocery store. When making such decisions, which magnitudes do we consider? The dominant theory suggests that our focus is on numerical quantity, i.e., the number of items in a set. This theory leads to quantity-focused research suggesting that discriminating quantities is automatic, innate, and is the basis for mathematical abilities in humans. Another theory suggests, instead, that non-numerical magnitudes, such as the total area of the compared items, are usually what humans rely on, and numerical quantity is used only when required. Since wild animals must make quick magnitude-related decisions to eat, seek shelter, survive, and procreate, studying which magnitudes animals spontaneously use in magnitude-related decisions is a good way to study the relative primacy of numerical quantity versus non-numerical magnitudes. We asked whether, in an animal model, the influence of non-numerical magnitudes on performance in a spontaneous magnitude comparison task is modulated by the number of non-numerical magnitudes that positively correlate with numerical quantity. Our animal model was the Archerfish, a fish that, in the wild, hunts insects by shooting a jet of water at them. These fish were trained to shoot water at artificial targets presented on a computer screen above the water tank. We tested the Archerfish's performance in spontaneous, untrained two-choice magnitude decisions. We found that the fish tended to select the group containing larger non-numerical magnitudes and smaller quantities of dots. The fish selected the group containing more dots mostly when the quantity of the dots was positively correlated with all five different non-numerical magnitudes. The current study adds to the body of studies providing direct evidence that in some cases animals’ magnitude-related decisions are more affected by non-numerical magnitudes than by numerical quantity, putting doubt on the claims that numerical quantity perception is the most basic building block of mathematical abilities.

scholarly journals The legacy of Dr Earl Russell

2018 ◽  
Vol 87 (1) ◽  
pp. 14-15
Author(s):  
Sandra Botros
Keyword(s):  
Pain Management ◽  
Residency Program ◽  
Korean War ◽  
Medical Officer ◽  
Future Research ◽  
Pain Medicine ◽  
Pain Specialist ◽  
The Korean War ◽  
Research And Education ◽  
First Pain

Dr Earl Russell (1920-2008) was a Canadian anesthetist and pain specialist who spent the majority of his career as a Western University faculty member and a pain physician in Southwestern Ontario. Dr Russell obtained his medical degree at Western, graduating in the class of 1950, and went on to serve in the Korean War as a medical officer. It was in Korea that he began developing a keen interest in pain medicine, using self-taught anesthesia skills to help soldiers suffering from frostbite. He returned to Canada and focused the rest of his career on the practice and advancement of pain medicine, and endowed the Earl Russell Chair in Pain Management in order to fund future research and education in the field. This article highlights the importance of his contributions to the field, in particular through his creation of the Earl Russell Chair, and how this led to the first Pain Medicine residency program in Canada at Western University.

scholarly journals From Which Direction Does the Empire Strike (Back)?

2021 ◽  
Vol 12 ◽  
Author(s):  
Katharina Theresa Halicki ◽  
Moritz Ingendahl ◽  
Maren Mayer ◽  
Melvin John ◽  
Marcel Raphael Schreiner ◽  
...  
Keyword(s):  
Opposite Direction ◽  
Spatial Representation ◽  
Future Research ◽  
Spatial Representations ◽  
The Right ◽  
Attack And Defense

In cultures with left-right-script, agentic behavior is mentally represented as following a left-to-right trajectory, an effect referred to as the Spatial Agency Bias (SAB, Suitner and Maass, 2016). In this research, we investigated whether spatial representations of activities are universal across activities by analyzing the opposite concepts of “attack” and “defense”. Both behaviors involve similar actions (e.g., fighting) but may differ in perceived agency. Moreover “defense” is necessarily always a response to an attack and may therefore be represented by a trajectory in the opposite direction. Two studies found the classic SAB for activities representing attacking but a reduction (Study 1) and reversal (Study 2) for activities involving defense. Although the spatial representation of defense on the right was much weaker and less unequivocal than that of attack on the left, the results suggest that the spatial representations of defense and attack are located in different positions. Apparently not all actors and all activities are spatially represented on the left with a left-to-right trajectory but position and direction depend on the perceived agency. Directions for future research and applications of our findings are discussed.

scholarly journals Physiological Demands of Extreme Obstacle Course Racing: A Case Study

2019 ◽  
Vol 16 (16) ◽  
pp. 2879
Author(s):  
Timothy Baghurst ◽  
Steven L. Prewitt ◽  
Tyler Tapps
Keyword(s):  
Heart Rate ◽  
The United States ◽  
Future Research ◽  
Improve Performance ◽  
Cognitive Demands ◽  
Unique Nature ◽  
Physiological Demands ◽  
Road Races ◽  
Training Opportunities

Obstacle course races are a popular source of recreation in the United States, providing additional challenges over traditional endurance events. Despite their popularity, very little is known about the physiological or cognitive demands of obstacle course races compared to traditional road races. The purpose of this study was to investigate the physiological effects of participation in an extreme obstacle course race. The participant was a 38-year-old Caucasian male, who completed an extreme obstacle course race over a 24-h period. Exercise intensity, steps taken, energy expenditure, and heart rate were recorded over the event’s duration using an Actigraph Link GT9X-BT accelerometer and a Polar heart rate monitor. Results reflected the unique nature of obstacle course racing when compared to traditional endurance events, with ups-and-downs recorded in each variable due to the encountering of obstacles. This case study provides a glimpse into the physiological demands of obstacle course racing, and suggests that the cognitive demands placed on competitors may differ to traditional endurance events, due to the challenges of obstacles interspersed throughout the event. With the popularity of obstacle course racing, and to enhance training opportunities, improve performance, and decrease the incidence of injuries, future research should further investigate the physiological and cognitive demands of obstacle course races of various distances and among diverse populations.

Fixated in unfamiliar territory: Mapping estimates across typical and atypical number lines

2019 ◽  
Vol 73 (2) ◽  
pp. 279-294
Author(s):  
Sabrina Michelle Di Lonardo ◽  
Matthew G Huebner ◽  
Katherine Newman ◽  
Jo-Anne LeFevre
Keyword(s):  
Numerical Range ◽  
Number Line ◽  
Strategy Use ◽  
Reverse Direction ◽  
Tracking Data ◽  
Spatial Processes ◽  
Number Lines ◽  
Number Line Estimation ◽  
The Right ◽  
Typical Range

Adults ( N = 72) estimated the location of target numbers on number lines that varied in numerical range (i.e., typical range 0–10,000 or atypical range 0–7,000) and spatial orientation (i.e., the 0 endpoint on the left [traditional] or on the right [reversed]). Eye-tracking data were used to assess strategy use. Participants made meaningful first fixations on the line, with fixations occurring around the origin for low target numbers and around the midpoint and endpoint for high target numbers. On traditional direction number lines, participants used left-to-right scanning and showed a leftward bias; these effects were reduced for the reverse direction number lines. Participants made fixations around the midpoint for both ranges but were less accurate when estimating target numbers around the midpoint on the 7,000-range number line. Thus, participants are using the internal benchmark (i.e., midpoint) to guide estimates on atypical range number lines, but they have difficulty calculating the midpoint, leading to less accurate estimates. In summary, both range and direction influenced strategy use and accuracy, suggesting that both numerical and spatial processes influence number line estimation.

Performance Appraisal, Performance Management and Improving Individual Performance: A Motivational Framework

2006 ◽  
Vol 2 (2) ◽  
pp. 253-277 ◽  
Author(s):  
Angelo S. DeNisi ◽  
Robert D. Pritchard
Keyword(s):  
Performance Management ◽  
Performance Appraisal ◽  
Performance Improvement ◽  
Individual Performance ◽  
Future Research ◽  
Improve Performance ◽  
Performance Management Systems ◽  
Recent Trends ◽  
And Performance ◽  
The Right

Performance appraisal has been the focus of considerable research for almost a century. Yet, this research has resulted in very few specific recommendations about designing and implementing appraisal and performance management systems whose goal is performance improvement. We believe that a reason for this is that appraisal research became too interested in measurement issues and not interested enough in ways to improve performance, although some recent trends in the area have begun moving the field in the right direction. We review these trends and their genesis, and propose a motivational framework as a means of integrating what we have learned and generating proposals for future research that focus on employee's performance improvement.

Dissociation between magnitude comparison and relation identification across different formats for rational numbers

2017 ◽  
Vol 24 (2) ◽  
pp. 179-197 ◽  
Author(s):  
Maureen E. Gray ◽  
Melissa DeWolf ◽  
Miriam Bassok ◽  
Keith J. Holyoak
Keyword(s):  
Rational Numbers ◽  
Magnitude Comparison
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