Mental Representations in Fraction Comparison

2011 ◽  
Vol 58 (6) ◽  
pp. 480-489 ◽  
Author(s):  
Thomas J. Faulkenberry ◽  
Benton H. Pierce
Keyword(s):  
Mental Representations ◽  
Reaction Times ◽  
Cross Product ◽  
Numerical Distance ◽  
Regression Analyses ◽  
Problem Size ◽  
Comparison Task ◽  
Strategy Types ◽  
Problem Size Effect ◽  
Fraction Comparison

In this study, we investigated the mental representations used in a fraction comparison task. Adults were asked to quickly and accurately pick the larger of two fractions presented on a computer screen and provide trial-by-trial reports of the types of strategies they used. We found that adults used a variety of strategies to compare fractions, ranging among just knowing the answer, using holistic knowledge of fractions to determine the answer, and using component-based procedures such as cross multiplication. Across all strategy types, regression analyses identified that reaction times were significantly predicted by numerical distance between fractions, indicating that the participants used a magnitude-based representation to compare the fraction magnitudes. In addition, a variant of the problem-size effect (e.g., Ashcraft, 1992) appeared, whereby reaction times were significantly predicted by the average cross product of the two fractions. This effect was primarily found for component-based strategies, indicating a role for strategy choice in the formation of mental representations of fractions.

scholarly journals A study on congruency effects and numerical distance in fraction comparison by expert undergraduate students

2019 ◽  
Author(s):  
Nicolas Morales ◽  
Pablo Dartnell ◽  
David Maximiliano Gomez
Keyword(s):  
Undergraduate Students ◽  
Cognitive Complexity ◽  
Procedural Knowledge ◽  
Response Times ◽  
Numerical Distance ◽  
Comparison Task ◽  
Common Component ◽  
Individual Level ◽  
Level Data ◽  
Fraction Comparison

School mathematics comprises a diversity of concepts whose cognitive complexity is still poorly understood, a chief example being fractions. These are typically taught in middle school but many students fail to master them, and misconceptions frequently persist into adulthood. In this study we investigate fraction comparison, a task that taps into both conceptual and procedural knowledge of fractions, by looking at performance of highly mathematically skilled young adults. Fifty-seven Chilean engineering undergraduate students answered a computerized fraction comparison task, while their answers and response times were recorded. Task items were selected according to a number of mathematically and/or cognitively relevant characteristics: (a) whether the fractions to be compared shared a common component, (b) the numerical distance between fractions, and (c) the applicability of two strategies to answer successfully: a congruency strategy (a fraction is larger if it has larger natural number components than another) and gap thinking (a fraction is larger if it is missing fewer pieces than another to complete the whole). In line with previous research, our data indicated that the congruency strategy is inadequate to describe participants’ performance, as congruent items turned out to be more difficult than incongruent ones when fractions had no common component. Although we hypothesized that this lower performance for congruent items would be explained by the use of gap thinking, this turned out not to be the case: evidence was insufficient to show that the applicability of the gap thinking strategy modulated either participants’ accuracy rates or response times (although individual-level data suggests that there is an effect for response times). When fractions shared a common component, instead, our data displays a more complex pattern that expected: an advantage for congruent items is present in the first experimental block but fades as the experiment progresses. Numerical distance had an effect in fraction comparison that was statistically significant for items without common components only. Altogether, our results from experts’ reasoning reveal nuances in the fraction comparison task with respect to previous studies and contribute to future models of reasoning in this task.

scholarly journals Symbolic number comparison is not processed by the analogue number system: different symbolic and nonsymbolic numerical distance and size effects

2017 ◽  
Author(s):  
Attila Krajcsi ◽  
Gabor Lengyel ◽  
Petia Kojouharova
Keyword(s):  
Size Effects ◽  
Goodness Of Fit ◽  
Numerical Cognition ◽  
Reaction Times ◽  
Number System ◽  
Error Rates ◽  
Numerical Distance ◽  
Comparison Task ◽  
Number Comparison ◽  
Symbolic Number

Dominant numerical cognition models suppose that both symbolic and nonsymbolic numbers are processed by the Analogue Number System (ANS) working according to Weber’s law. It was proposed that in a number comparison task the numerical distance and size effects reflect a ratio-based performance which is the sign of the ANS activation. However, increasing number of findings and alternative models propose that symbolic and nonsymbolic numbers might be processed by different representations. Importantly, alternative explanations may offer similar predictions to the ANS prediction, therefore, former evidence usually utilizing only the goodness of fit of the ANS prediction is not sufficient to support the ANS account. To test the ANS model more rigorously, a more extensive test is offered here. Several properties of the ANS predictions for the error rates, reaction times and diffusion model drift rates were systematically analyzed in both nonsymbolic dot comparison and symbolic Indo-Arabic comparison tasks. It was consistently found that while the ANS model’s prediction is relatively good for the nonsymbolic dot comparison, its prediction is poorer and systematically biased for the symbolic Indo-Arabic comparison. We conclude that only nonsymbolic comparison is supported by the ANS, and symbolic number comparisons are processed by other representation.

scholarly journals Brain Mechanisms of Arithmetic: A Crucial Role for Ventral Temporal Cortex

2018 ◽  
Vol 30 (12) ◽  
pp. 1757-1772 ◽  
Author(s):  
Pedro Pinheiro-Chagas ◽  
Amy Daitch ◽  
Josef Parvizi ◽  
Stanislas Dehaene
Keyword(s):  
Numerical Cognition ◽  
Parietal Lobe ◽  
Temporal Cortex ◽  
Classical Problem ◽  
Brain Regions ◽  
Inferior Temporal Cortex ◽  
Problem Size ◽  
Ventral Temporal Cortex ◽  
Intracranial Electroencephalography ◽  
Problem Size Effect

Elementary arithmetic requires a complex interplay between several brain regions. The classical view, arising from fMRI, is that the intraparietal sulcus (IPS) and the superior parietal lobe (SPL) are the main hubs for arithmetic calculations. However, recent studies using intracranial electroencephalography have discovered a specific site, within the posterior inferior temporal cortex (pITG), that activates during visual perception of numerals, with widespread adjacent responses when numerals are used in calculation. Here, we reexamined the contribution of the IPS, SPL, and pITG to arithmetic by recording intracranial electroencephalography signals while participants solved addition problems. Behavioral results showed a classical problem size effect: RTs increased with the size of the operands. We then examined how high-frequency broadband (HFB) activity is modulated by problem size. As expected from previous fMRI findings, we showed that the total HFB activity in IPS and SPL sites increased with problem size. More surprisingly, pITG sites showed an initial burst of HFB activity that decreased as the operands got larger, yet with a constant integral over the whole trial, thus making these signals invisible to slow fMRI. Although parietal sites appear to have a more sustained function in arithmetic computations, the pITG may have a role of early identification of the problem difficulty, beyond merely digit recognition. Our results ask for a reevaluation of the current models of numerical cognition and reveal that the ventral temporal cortex contains regions specifically engaged in mathematical processing.

scholarly journals Numerical cognition in action: Reaching behavior reveals numerical distance effects in 5- to 6-year-olds

2018 ◽  
Vol 4 (2) ◽  
pp. 286-296 ◽  
Author(s):  
Christopher D. Erb ◽  
Jeff Moher ◽  
Joo-Hyun Song ◽  
David M. Sobel
Keyword(s):  
Spatial Representation ◽  
Numerical Cognition ◽  
Spatial Information ◽  
Numerical Comparison ◽  
Digital Display ◽  
Numerical Distance ◽  
Tight Coupling ◽  
Comparison Task ◽  
The Right ◽  
Trial Participants

This study investigates how children’s numerical cognition is reflected in their unfolding actions. Five- and 6-year-olds (N = 34) completed a numerical comparison task by reaching to touch one of three rectangles arranged horizontally on a digital display. A number from 1 to 9 appeared in the center rectangle on each trial. Participants were instructed to touch the left rectangle for numbers 1-4, the center rectangle for 5, and the right rectangle for 6-9. Reach trajectories were more curved toward the center rectangle for numbers closer to 5 (e.g., 4) than numbers further from 5 (e.g., 1). This finding indicates that a tight coupling exists between numerical and spatial information in children’s cognition and action as early as the preschool years. In addition to shedding new light on the spatial representation of numbers during childhood, our results highlight the promise of incorporating measures of manual dynamics into developmental research.

scholarly journals Symbolic distance: Unfamiliar versus familiar space

Psihologija ◽  
2007 ◽  
Vol 40 (1) ◽  
pp. 93-110 ◽  
Author(s):  
Radmila Stojanovic ◽  
Suncica Zdravkovic
Keyword(s):  
Reaction Time ◽  
Linear Relation ◽  
Mental Representations ◽  
Reaction Times ◽  
Distance Effect ◽  
Professional Orientation ◽  
Symbolic Distance Effect ◽  
Symbolic Distance ◽  
Male Subjects

The symbolic distance effect was investigated using both realistic distances and distances represented on the map. The influence of professional orientation and sex on mental visualization was measured. The results showed that an increase of distance leads to an increase in reaction time. The slope for realistic distances was steeper. Male subjects always had longer reaction times, although the effect differs for the two types of distances. Professional orientation did not play a role. The obtained relation between reaction time and distance is a confirmation of theories proposing that mental representations encompass structure and metric characteristics. The confirmed role of the effect of symbolic distance additionally supports Kosslyn?s theory: there is a linear relation between the time and distance.

scholarly journals Implicit Priming Reveals Decomposed Processing in Fraction Comparison

2019 ◽  
Author(s):  
Jessica A. Nejman ◽  
Thomas J. Faulkenberry
Keyword(s):  
Mental Representations ◽  
Numerical Magnitude ◽  
Unique Challenge ◽  
Number Comparison ◽  
Priming Paradigm ◽  
Whole Number ◽  
Comparison Trial ◽  
The Individual ◽  
Experimental Trials ◽  
Fraction Comparison

Fractions present a unique challenge in early mathematics instruction, as they require focusing not on the individual symbols that make up the fraction, but rather a mental combination of the two into a single numerical magnitude. Previous studies have given conflicting accounts of how adults form these complex mental representations. Whereas some studies indicate that mental representations of fractions are holistic and are based upon the fraction’s numerical magnitude, others have indicated support for decomposed processing, where separate representations of the numerator and denominator are formed. In the present study, we tested this decomposed processing account using an implicit priming paradigm. In a series of experimental trials, the comparison of two fraction magnitudes (“which is larger?”) primed a subsequent comparison trial with whole numbers. Using Bayesian analyses, we found that when people compared two fractions with common denominators, they were faster in the subsequent whole number comparison. However, when two fractions with common numerators were compared, the subsequent whole number comparison was slower. This indicates that representations of the fraction components were activated in the fraction comparison, and these residual activations primed the subsequent whole number comparison. These data give further support to the notion of decomposed processing in fraction comparison.

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