Why Learning Common Fractions Is Uncommonly Difficult: Unique Challenges Faced by Students With Mathematical Disabilities

2016 ◽  
Vol 50 (6) ◽  
pp. 651-654 ◽  
Author(s):  
Daniel B. Berch
Keyword(s):  
Learning Disability ◽  
Rational Number ◽  
Rational Numbers ◽  
Significant Delay ◽  
Comprehensive Understanding ◽  
Mathematical Learning ◽  
Formal Instruction ◽  
Different Types ◽  
Common Fractions ◽  
Decimal Fractions

In this commentary, I examine some of the distinctive, foundational difficulties in learning fractions and other types of rational numbers encountered by students with a mathematical learning disability and how these differ from the struggles experienced by students classified as low achieving in math. I discuss evidence indicating that students with math disabilities exhibit a significant delay or deficit in the numerical transcoding of decimal fractions, and I further maintain that they may face unique challenges in developing the ability to effectively translate between different types of fractions and other rational number notational formats—what I call conceptual transcoding. I also argue that characterizing this level of comprehensive understanding of rational numbers as rational number sense is irrational, as it misrepresents this flexible and adaptive collection of skills as a biologically based percept rather than a convergence of higher-order competencies that require intensive, formal instruction.

EXPRESS: People’s Preferences for Different Types of Rational Numbers in Linguistic Contexts

2022 ◽  
pp. 174702182210763
Author(s):  
Xiaoming Yang ◽  
Yunqi Wang
Keyword(s):  
Rational Number ◽  
General Pattern ◽  
Rational Numbers ◽  
The Other ◽  
Multiple Factors ◽  
Corpus Studies ◽  
Different Types ◽  
The One ◽  
General Linguistic ◽  
Preference Experiment

Rational numbers, like fractions, decimals, and percentages, differ in the concepts they prefer to express and the entities they prefer to describe as previously reported in display-rational number notation matching tasks and in math word problem compiling contexts. On the one hand, fractions and percentages are preferentially used to express a relation between two magnitudes, while decimals are preferentially used to represent a magnitude. On the other hand, fractions and decimals tend to be used to describe discrete and continuous entities, respectively. However, it remains unclear whether these reported distinctions can extend to more general linguistic contexts. It also remains unclear which factor, the concept to be expressed (magnitudes vs. relations between magnitudes) or the entity to be described (countable vs. continuous), is more predictive of people’s preferences for rational number notations. To explore these issues, two corpus studies and a number notation preference experiment were administered. The news and conversation corpus studies detected the general pattern of conceptual distinctions across rational number notations as observed in previous studies; the number notation preference experiment found that the concept to be expressed was more predictive of people’s preferences for number notations than the entity to be described. These findings indicate that people’s biased uses of rational numbers are constrained by multiple factors, especially by the type of concepts to be expressed, and more importantly, these biases are not specific to mathematical settings but are generalizable to broader linguistic contexts.

Editorial comment: As we read

1969 ◽  
Vol 16 (2) ◽  
pp. 82-83
Keyword(s):  
Editorial Comment ◽  
Rational Number ◽  
Number System ◽  
Rational Numbers ◽  
The Other ◽  
Grade Levels ◽  
Common Fractions ◽  
The Subject ◽  
Percentage Ratio

To many, child and teacher alike, rational numbers are not very rational. They resist our intuitive powers. Computation with these mystic numbers borders on the occult. Too often the computational skills needed are developed with an intricate system of memory and tidy little tricks. Il is not difficult to understand why fractional notation is of relatively recent vintage. Applications of the rational numbers are equally mystifying and constitute a formidable problem for teachers at all grade levels. Not infreq uently students remember with distaste their introduction to the applications of these numbers. Even teachers shy away from the subject. Do you have difficulty teaching common fractions, percentage, ratio, and the other assorted topics associated with the rational-number system? This issue of The Arithmetic Teacher will be of particu lar value to you.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

scholarly journals Morphological and Behavioral Effects in Zebrafish Embryos after Exposure to Smoke Dyes

Toxics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 9
Author(s):  
Kimberly T. To ◽  
Lindsey St. Mary ◽  
Allyson H. Wooley ◽  
Mitchell S. Wilbanks ◽  
Anthony J. Bednar ◽  
...  
Keyword(s):  
Dose Dependence ◽  
Behavioral Effects ◽  
Zebrafish Embryos ◽  
Comprehensive Understanding ◽  
Anthraquinone Dyes ◽  
Locomotor Movement ◽  
Different Types ◽  
Dose Dependent Increase ◽  
Dose Dependent ◽  
Significant Mortality

Solvent Violet 47 (SV47) and Disperse Blue 14 (DB14) are two anthraquinone dyes that were previously used in different formulations for the production of violet-colored smoke. Both dyes have shown potential for toxicity; however, there is no comprehensive understanding of their effects. Zebrafish embryos were exposed to SV47 or DB14 from 6 to 120 h post fertilization (hpf) to assess the dyes’ potential adverse effects on developing embryos. The potential ability of both dyes to cross the blood–brain barrier was also assessed. At concentrations between 0.55 and 5.23 mg/L, SV47 showed a dose-dependent increase in mortality, jaw malformation, axis curvature, and edemas. At concentrations between 0.15 and 7.54 mg/L, DB14 did not have this same dose-dependence but had similar morphological outcomes at the highest doses. Nevertheless, while SV47 showed significant mortality from 4.20 mg/L, there was no significant mortality on embryos exposed to DB14. Regardless, decreased locomotor movement was observed at all concentrations of DB14, suggesting an adverse neurodevelopmental effect. Overall, our results showed that at similar concentrations, SV47 and DB14 caused different types of phenotypic effects in zebrafish embryos.

Call for Manuscripts for the 2014 Focus Issue: Rational Number Sense – October 2012

2012 ◽  
Vol 18 (3) ◽  
pp. 189
Keyword(s):  
Rational Number ◽  
Number Sense ◽  
Rational Numbers

This call for manuscripts is requesting articles that address how to make sense of rational numbers in their myriad forms, including as fractions, ratios, rates, percentages, and decimals.

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

scholarly journals ELDER MISTREATMENT AND DEPRESSIVE SYMPTOMS AMONG OLDER CHINESE AMERICANS

2019 ◽  
Vol 3 (Supplement_1) ◽  
pp. S761-S761
Author(s):  
Ying-Yu Chao ◽  
Yu-Ping Chang ◽  
XinQi Dong
Keyword(s):  
Older Adults ◽  
Depressive Symptoms ◽  
Chinese Americans ◽  
Comprehensive Understanding ◽  
Elder Mistreatment ◽  
Chinese Older Adults ◽  
Financial Exploitation ◽  
Different Types ◽  
Older Chinese ◽  
Older Chinese Americans

Abstract This study aimed to examine the association between different types of elder mistreatment and depressive symptoms among U.S. Chinese older adults. Data were from the Population Study of Chinese Elderly in Chicago (PINE). Participants were 3,157 Chinese older adults who were 60 years and over (mean age = 72.8). Logistic regression analyses were performed. The results showed that participants with overall mistreatment (OR, 2.11; 95% CI, 1.83-2.43), psychological mistreatment (OR, 2.12; 95% CI, 1.78-2.51), physical mistreatment (OR, 1.82; 95% CI, 1.10-2.99), and financial exploitation (OR, 1.33; 95% CI, 1.11 – 1.60) were more likely to report more depressive symptoms. There was no significant association between sexual mistreatment and depressive symptoms (p = 0.07). Longitudinal studies are needed to obtain a more comprehensive understanding of the pathways between elder mistreatment and depressive symptoms.

scholarly journals Preschool deficits in cardinal knowledge and executive function contribute to longer-term mathematical learning disability

2019 ◽  
Vol 188 ◽  
pp. 104668 ◽  
Author(s):  
Felicia W. Chu ◽  
Kristy vanMarle ◽  
Mary K. Hoard ◽  
Lara Nugent ◽  
John E. Scofield ◽  
...  
Keyword(s):  
Executive Function ◽  
Learning Disability ◽  
Mathematical Learning

scholarly journals Approximation of rational numbers by Dedekind sums

2014 ◽  
Vol 10 (05) ◽  
pp. 1241-1244 ◽  
Author(s):  
Kurt Girstmair
Keyword(s):  
Rational Number ◽  
Rational Numbers ◽  
Dedekind Sums ◽  
Dedekind Sum

Given a rational number x and a bound ε, we exhibit m, n such that |x - s(m, n)| < ε. Here s(m, n) is the classical Dedekind sum and the parameters m and n are completely explicit in terms of x and ε.

Have You Read …?

1969 ◽  
Vol 62 (3) ◽  
pp. 220-221
Author(s):  
Philip Peak
Keyword(s):  
Real Number ◽  
Rational Number ◽  
Geometric Approach ◽  
Rational Numbers ◽  
Polynomial Equations ◽  
Real Numbers ◽  
Basic Principles ◽  
New Ideas

One of the basic principles we follow in our teaching is to relate new ideas with old ideas. Dr. Forbes has done just this in his article about extending the concept of rational numbers to real numbers. He points out how this extension cannot follow the same pattern as that of extensions positive to negative integers or from integers to rationals. If we look to a definition for motivating the extension we at best can only say, “Some polynomial equations have no rational number solutions and do have some real number solutions.” We might use least-upperbound idea, or we might try motivating through nonperiodic infinite decimals. However, Dr. Forbes rejects all of these and makes the tie-in through a geometric approach.

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