Teaching and Learning Fraction and Rational Numbers: The Origins and Implications of Whole Number Bias

2005 ◽  
Vol 40 (1) ◽  
pp. 27-52 ◽  
Author(s):  
Yujing Ni ◽  
Yong-Di Zhou
Keyword(s):  
Teaching And Learning ◽  
Rational Numbers ◽  
Whole Number

Special Characteristics of Rational Numbers

1984 ◽  
Vol 31 (6) ◽  
pp. 10-12
Author(s):  
Dora Helen B. Skypek
Keyword(s):  
Rational Number ◽  
Teaching And Learning ◽  
Rational Numbers ◽  
Unlimited Number ◽  
Coding Schemes ◽  
Whole Number

The characteristics of rational numbers must be considered in a variety of interpretations and coding schemes. It is this variety that, if not sorted and carefully developed in appropriate contexts, results in confusion in teaching and learning about rational numbers. Although a discussion of interpretations necessarily involves the use of coding conventions, the two will be treated separately as special characteristics of the rational numbers. Another important charac-teristic. which is difficult to separate from interpretations and coding conventions. is this: unlike a whole number (or an integer), a rational number has an unlimited number of “behavioral clones.” These clones have different names, but they behave in exactly the same way. Still other Still other characteristics to be considered are the density and order of the rational number.

Semantic alignment across whole-number arithmetic and rational numbers: evidence from a Russian perspective

2017 ◽  
Vol 24 (2) ◽  
pp. 198-220 ◽  
Author(s):  
Yulia A. Tyumeneva ◽  
Galina Larina ◽  
Ekaterina Alexandrova ◽  
Melissa DeWolf ◽  
Miriam Bassok ◽  
...  
Keyword(s):  
Rational Numbers ◽  
Whole Number ◽  
Semantic Alignment ◽  
Whole Number Arithmetic

scholarly journals The relational nature of rational numbers

Pythagoras ◽  
2015 ◽  
Vol 36 (1) ◽  
Author(s):  
Bruce Brown
Keyword(s):  
Conceptual Change ◽  
Number Field ◽  
Rational Number ◽  
Teaching And Learning ◽  
Field Research ◽  
Mathematical Structure ◽  
Rational Numbers ◽  
Number Line ◽  
Complex Nature ◽  
Research Projects

It is commonly accepted that the knowledge and learning of rational numbers is more complex than that of the whole number field. This complexity includes the broader range of application of rational numbers, the increased level of technical complexity in the mathematical structure and symbol systems of this field and the more complex nature of many conceptual properties of the rational number field. Research on rational number learning is divided as to whether children’s difficulties in learning rational numbers arise only from the increased complexity or also include elements of conceptual change. This article argues for a fundamental conceptual difference between whole and rational numbers. It develops the position that rational numbers are fundamentally relational in nature and that the move from absolute counts to relative comparisons leads to a further level of abstraction in our understanding of number and quantity. The argument is based on a number of qualitative, in-depth research projects with children and adults. These research projects indicated the importance of such a relational understanding in both the learning and teaching of rational numbers, as well as in adult representations of rational numbers on the number line. Acknowledgement of such a conceptual change could have important consequences for the teaching and learning of rational numbers.

Characteristics of teaching and learning single-digit whole number multiplication in china: the case of the nine-times table

ZDM ◽  
2019 ◽  
Vol 51 (1) ◽  
pp. 81-94
Author(s):  
Shu Zhang ◽  
Yiming Cao ◽  
Lidong Wang ◽  
Xinlian Li
Keyword(s):  
Teaching And Learning ◽  
Single Digit ◽  
Whole Number

scholarly journals Textbook: an Analysis in the Light of the Ontosemiotic Focus on Rational Numbers

2020 ◽  
Vol 22 (5) ◽  
pp. 122-142
Author(s):  
Patricia Pujol Goulart Carpes ◽  
Eleni Bisognin
Keyword(s):  
Elementary School ◽  
Mathematical Knowledge ◽  
Teaching And Learning ◽  
Rational Numbers ◽  
Instructional Process ◽  
Mathematical Instruction ◽  
Problem Situations ◽  
Curriculum Guidelines ◽  
Textbook Design ◽  
Current Curriculum

Background: Many teachers consider the textbook the primary guide for the curriculum materialisation. This work analyses the content of the rational numbers in a textbook used in the 7th-grade of elementary school. For this analysis, the theoretical and methodological tools of the Ontosemiotic Approach to Knowledge and Mathematical Instruction (OSA) are used. Objective: Our goal was to understand the level of didactic suitability of the instruction process in the textbook. Design and setting: Thus, we analysed a section of the textbook about rational numbers using the categories described by OSA. Concepts, procedures, problem situations, definitions and arguments used by the authors were also analysed. Results: From the results obtained, we could infer that there are examples of the concepts worked in class, but without the proper definitions and arguments, hindering generalisation. The analysis also allowed us to highlight didactic-mathematical knowledge that can guide the teacher concerning the textbooks' possibilities and limitations, to achieve more didactic suitability in the process of teaching and learning rational numbers. Conclusions: the book under study should not be considered as planning for an instructional process to meet the current curriculum guidelines.

scholarly journals Do we have a sense for irrational numbers?

2017 ◽  
Vol 2 (3) ◽  
pp. 170-189 ◽  
Author(s):  
Andreas Obersteiner ◽  
Veronika Hofreiter
Keyword(s):  
Correct Response ◽  
Response Times ◽  
Number Sense ◽  
Irrational Number ◽  
Rational Numbers ◽  
Cube Root ◽  
Irrational Numbers ◽  
Whole Numbers ◽  
Symbolic Notation ◽  
Whole Number

Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers’ whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers.

Students' Strategies for Fair Shares

2004 ◽  
Vol 10 (3) ◽  
pp. 132-135
Author(s):  
Shannon O. S. Driskell
Keyword(s):  
Everyday Life ◽  
Prior Knowledge ◽  
Conceptual Knowledge ◽  
Middle Grades ◽  
Rational Numbers ◽  
Computational Fluency ◽  
Informal Knowledge ◽  
Whole Number ◽  
Fraction Concepts ◽  
Fair Shares

Children often begin to construct an informal understanding of fractions before entering school as they learn to share their crayons or snacks fairly with friends. NCTM (2000) recommends that teachers recognize and build on each child's informal knowledge of fractions during grades K–2. In grades 3–5, children should be actively engaged in constructing conceptual knowledge about fraction concepts, with an emphasis on computational fluency as they progress into grades 6–8. The NCTM (2000) further suggests that “The study of rational numbers in the middle grades should build on students' prior knowledge of whole-number concepts and skills and their encounter with fractions, decimals, and percents in lower grades and in everyday life” (p. 215).

On natural numbers, integers, and rationals

1949 ◽  
Vol 14 (2) ◽  
pp. 81-84
Author(s):  
Frederic B. Fitch
Keyword(s):  
Natural Number ◽  
Rational Numbers ◽  
Simple Theory ◽  
Natural Numbers ◽  
Guiding Principle ◽  
Relative Product ◽  
Whole Number

A theory of natural numbers will be outlined in what follows. This theory will also be extended to give an account of positive and negative integers and positive and negative rational numbers. The system of logic used will be that of Whitehead and Russell's Principia mathematica with the simple theory of types. It will be assumed that the reader is familiar with the more elementary properties of relations and with such notions as the relative product of two relations, the square of a relation, the cube of a relation, and the various other whole-number powers of relations.The guiding principle of this theory is that the natural number zero is to be regarded as the relation of the zevoth power of a relation А to А itself, and the natural number 1 is to be regarded as the relation of the first power of a relation А to А itself, and the natural number 2 is to be regarded as the relation of the square of a relation А to А itself, and so on.

scholarly journals Probing the neural basis rational numbers: the role of inhibitory control and magnitude representations

2021 ◽  
Author(s):  
Miriam Rosenberg-Lee
Keyword(s):  
Inhibitory Control ◽  
Rational Number ◽  
Mathematics Curriculum ◽  
Parietal Lobe ◽  
Rational Numbers ◽  
Number Processing ◽  
Neural Basis ◽  
Good Correspondence ◽  
Whole Number

Rational numbers, such as fractions, decimals and percentages, are a persistent challenge in the mathematics curriculum. An underappreciated source of rational number difficulties are whole number properties that apply to some, but not all, rational numbers. I contend that mastery of rational numbers involves refining and expanding whole number representations. Behavioral evidence for the role inhibitory control and magnitude-based processing of rational numbers support this hypothesis, although more attention is needed to task and stimuli selection, especially among fractions. In the brain, there is scant evidence on the role of inhibitory control in rational number processing, but surprisingly good correspondence, in the parietal lobe, between the handful of neuroimaging studies of rational numbers and the accumulated whole number literature. Decimals and discrete nonsymbolic representations are fruitful domains for probing the neural basis role of whole number interference in rational number processing.

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