scholarly journals Creating a context for learning: Activating children’s whole number knowledge prepares them to understand fraction division

2017 ◽  
Vol 3 (1) ◽  
pp. 31-57 ◽  
Author(s):  
Pooja Gupta Sidney ◽  
Martha Wagner Alibali
Keyword(s):  
Prior Knowledge ◽  
Mathematical Thinking ◽  
Conceptual Structure ◽  
Analogical Transfer ◽  
Arithmetic Operations ◽  
Fraction Division ◽  
Whole Numbers ◽  
Whole Number ◽  
Fraction Concepts ◽  
Division Problems

When children learn about fractions, their prior knowledge of whole numbers often interferes, resulting in a whole number bias. However, many fraction concepts are generalizations of analogous whole number concepts; for example, fraction division and whole number division share a similar conceptual structure. Drawing on past studies of analogical transfer, we hypothesize that children’s whole number division knowledge will support their understanding of fraction division when their relevant prior knowledge is activated immediately before engaging with fraction division. Children in 5th and 6th grade modeled fraction division with physical objects after modeling a series of addition, subtraction, multiplication, and division problems with whole number operands and fraction operands. In one condition, problems were blocked by operation, such that children modeled fraction problems immediately after analogous whole number problems (e.g., fraction division problems followed whole number division problems). In another condition, problems were blocked by number type, such that children modeled all four arithmetic operations with whole numbers in the first block, and then operations with fractions in the second block. Children who solved whole number division problems immediately before fraction division problems were significantly better at modeling the conceptual structure of fraction division than those who solved all of the fraction problems together. Thus, implicit analogies across shared concepts can affect children’s mathematical thinking. Moreover, specific analogies between whole number and fraction concepts can yield a positive, rather than a negative, whole number bias.

Confounding Whole-Number and Fraction Concepts When Building on Informal Knowledge

1995 ◽  
Vol 26 (5) ◽  
pp. 422-441 ◽  
Author(s):  
Nancy K. Mack
Keyword(s):  
Prior Knowledge ◽  
Individualized Instruction ◽  
Whole Numbers ◽  
Symbolic Representations ◽  
Informal Knowledge ◽  
Fourth Grade Students ◽  
Whole Number ◽  
Fraction Concepts ◽  
Addition And Subtraction ◽  
Development Of Students

This study examined the development of students' understanding of fractions during instruction with respect to the ways students' prior knowledge of whole numbers influenced the meanings and representations students constructed for fractions as they built on their informal knowledge of fractions. Four third-grade and three fourth-grade students received individualized instruction on addition and subtraction of fractions in a one-to-one setting for 3 weeks. As students attempted to construct meaning for symbolic representations of fractions, they overgeneralized the meanings of symbolic representations for whole numbers to fractions, and they overgeneralized the meanings of symbolic representations for fractions to whole 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).

scholarly journals PARTITIVE FRACTION DIVISION: REVEALING AND PROMOTING PRIMARY STUDENTS’ UNDERSTANDING

2020 ◽  
Vol 11 (2) ◽  
pp. 237-258
Author(s):  
Kamirsyah Wahyu ◽  
Taha Ertugrul Kuzu ◽  
Sri Subarinah ◽  
Dwi Ratnasari ◽  
Sofyan Mahfudy
Keyword(s):  
Design Research ◽  
Primary Students ◽  
Graphical Representations ◽  
Fraction Division ◽  
Symbolic Representations ◽  
Whole Number ◽  
Division Problems ◽  
Field Notes ◽  
Support Students ◽  
Teaching Experiments

Students show deficient understanding on fraction division and supporting that understanding remains a challenge for mathematics educators. This article aims to describe primary students’ understanding of partitive fraction division (PFD) and explore ways to support their understanding through the use of sequenced fractions and context-related graphical representations. In a design-research study, forty-four primary students were involved in three cycles of teaching experiments. Students’ works, transcript of recorded classroom discussion, and field notes were retrospectively analyzed to examine the hypothetical learning trajectories. There are three main findings drawn from the teaching experiments. Firstly, context of the tasks, the context-related graphical representations, and the sequence of fractions used do support students’ understanding of PFD. Secondly, the understanding of non-unit rate problems did not support the students’ understanding of unit rate problems. Lastly, the students were incapable of determining symbolic representations from unit rate problems and linking the problems to fraction division problems. The last two results imply to rethink unit rate as part of a partitive division with fractions. Drawing upon the findings, four alternative ways are offered to support students’ understanding of PFD, i.e., the lesson could be starting from partitive whole number division to develop the notion of fair-sharing, strengthening the concept of unit in fraction and partitioning, choosing specific contexts with more relation to the graphical representations, and sequencing the fractions used, from a simple to advanced form.

scholarly journals Grade 9 learners’ understanding of fraction concepts: Equality of fractions, numerator and denominator

Pythagoras ◽  
2021 ◽  
Vol 42 (1) ◽  
Author(s):  
Methuseli Moyo ◽  
France M. Machaba
Keyword(s):  
Qualitative Case Study ◽  
Equivalent Fractions ◽  
Written Responses ◽  
Whole Numbers ◽  
Study Approach ◽  
Complex Concept ◽  
Physical Manipulatives ◽  
Fraction Concepts ◽  
Developmental Intervention

Our research with Grade 9 learners at a school in Soweto was conducted to explore learners’ understanding of fundamental fraction concepts used in applications required at that level of schooling. The study was based on the theory of constructivism in a bid to understand whether learners’ transition from whole numbers to rational numbers enabled them to deal with the more complex concept of fractions. A qualitative case study approach was followed. A test was administered to 40 learners. Based on their written responses, eight learners were purposefully selected for an interview. The findings revealed that learners’ definitions of fraction were neither complete nor precise. Particularly pertinent were challenges related to the concept of equivalent fractions that include fraction elements, namely the numerator and denominator in the phase of rational number. These gaps in understanding may have originated in the early stages of schooling when learners first conceptualised fractions during the late concrete learning phase. For this reason, we suggest a developmental intervention using physical manipulatives to promote understanding of fractions before inductively guiding learners to construct algorithms and transition to the more abstract applications of fractions required in Grade 9.

scholarly journals Discussion on the structure of atomic nuclei

1929 ◽  
Vol 123 (792) ◽  
pp. 373-390 ◽  
Keyword(s):  
Accurate Determination ◽  
Essential Point ◽  
Whole Numbers ◽  
Atomic Nuclei ◽  
Whole Number ◽  
Nuclear Atom ◽  
The Individual ◽  
The Masses ◽  
Made In

Sir Ernest Rutherford: It was on March 19, 1914, that the Royal Society held its last discussion on the constitution of the atom—just fifteen years ago. I had the honour to open the discussion on that occasion, and the other speakers were Mr. Moseley, Profs. Soddy, Nicholson, Hicks, Stanley Allen, S. P. Thomp­son. In my opening remarks I put forward the theory of the nuclear atom and the evidence in support of it, while Mr. Moseley gave an account of his X-ray investigations, which defined the atomic numbers of the elements, and showed how many gaps were present between hydrogen number 1 and uranium number 92. Prof. Soddy drew attention to the existence of isotopes in the radioactive series, and also to a remarkable observation by Sir Joseph Thomson and Dr. Aston, who had obtained two parabolas in the positive ray spectrograph of neon, and he suggested that possibly the ordinary elements might also consist of mixture of isotopes. I think you will find that the remarks and suggestions made in this discussion fifteen years ago have a certain pertinence to-day. In particular Hicks and Stanley Allen drew attention to the importance of taking into account the magnetic fields in the nucleus, although at that time we had very little evidence on that point, and even to-day our information is very scanty. What has been accomplished in the intervening period ? On looking back we see that three new methods of attack on this problem have been developed. The first, and in some respects the most important, has been the proof of the isotopic constitution of the ordinary elements, and the accurate determination of the masses or weights of the individual isotopes, mainly due to the work of Dr. Aston. This has led in a sense to an extension of the original ideas of Moseley. The experiments of the latter fixed the number of possible nuclear charges, while Aston has shown that there are a large number of species of atoms each defined by its nuclear charge, although their masses and their nuclear constitution may be different. The essential point brought out in the earlier work of Dr. Aston was that the masses of the elements are approxi­mately expressed by whole numbers, where oxygen is taken as 16—with the exception of hydrogen itself. But the real interest, as we now see it, is not the whole number rule itself, but rather the departures from it.

Bridging the Gap: Using Interactive Computer Tools to Build Fraction Schemes

2002 ◽  
Vol 8 (6) ◽  
pp. 356-361
Author(s):  
John Olive
Keyword(s):  
Mathematics Instruction ◽  
Elementary Grades ◽  
Mathematical Development ◽  
Computer Tools ◽  
Whole Number ◽  
Fraction Concepts ◽  
Teaching Fractions ◽  
Number Knowledge

Teaching fractions has been a complex and largely unsuccessful aspect of mathematics instruction in the elementary grades for many years. Students' understanding of fraction concepts is a big stumbling block in their mathematical development. Some researchers have pointed to children's whole-number knowledge as interfering with, or creating a barrier to, their understanding of fractions (Behr et al. 1984; Streefland 1993; Lamon 1999). This article illustrates an approach to constructing fraction concepts that builds on children's whole-number knowledge using specially designed computer tools. This approach can help children make connections between whole-number multiplication and their notion of a fraction as a part of a whole, thus bridging the gap between whole-number and fraction knowledge.

Connecting Research and Practice – The Case of Multiplicative Thinking

2019 ◽  
pp. 535-540
Author(s):  
Dianne Siemon
Keyword(s):  
Formative Assessment ◽  
School Mathematics ◽  
Assessment Tools ◽  
Evidence Based ◽  
Middle Years ◽  
Research And Practice ◽  
Whole Numbers ◽  
Middle Years Of Schooling ◽  
Multiplicative Thinking ◽  
Division Problems

There is very little of any substance that can be achieved in school mathematics, and beyond without the capacity to recognise, represent and reason about relationships between quantities, that is, to think multiplicatively. However, research has consistently found that while most students in the middle years of schooling (i.e., Years 5 to 9) are able to solve simple multiplication and division problems involving small whole numbers, they rely on additive strategies to solve more complex problems involving larger numbers, fractions, decimals, and/or proportion. This paper describes how this situation can be addressed through the use of evidence-based formative assessment tools and teaching advice specifically designed to support the development of multiplicative thinking.

Divisibility and the base-ten numeration system

1964 ◽  
Vol 11 (8) ◽  
pp. 563-568
Author(s):  
George Spooner
Keyword(s):  
The Other ◽  
Numeration System ◽  
Whole Numbers ◽  
Whole Number ◽  
The One

There exists in the arithmetic of whole numbers several tests of divisibility which permit us to determine whether or not a given whole number is divisible by another given whole number without our having to divide the one given number by the other. The purpose of this paper is to examine the mathematical rationale or justification behind some of these tests. In our discussion we shall say that one whole number is divisible by another if and only if the remainder is zero.

Verbal Multiplication and Division Problems: Some Difficulties and Some Solutions

1986 ◽  
Vol 33 (8) ◽  
pp. 26-33
Author(s):  
A. Dean Hendrickson
Keyword(s):  
Arithmetic Operations ◽  
Specific Instruction ◽  
Division Problems

Verbal problems that involve multiplication and division are difficult for children to solve. Many of these difficulties arise because of their limited under tanding of these arithmetic operations. Their experience with the different kinds of situation that call for these operations is also limited. At the same time, these problem cannot be categorized easily because the situation that require these operations are varied. Nonetheless. multiplication is often taught only as “repeated addition” and division only as “repeated subtraction.” Children must have specific instruction in all the situation that require multiplication and division as arithmetic operations if they are to apply them successfully to verbal problems.

Sense Making and the Solution of Division Problems Involving Remainders: An Examination of Middle School Students' Solution Processes and Their Interpretations of Solutions

1993 ◽  
Vol 24 (2) ◽  
pp. 117-135 ◽  
Author(s):  
Edward A. Silver ◽  
Lora J. Shapiro ◽  
Adam Deutsch
Keyword(s):  
Middle School ◽  
Middle School Students ◽  
Student Performance ◽  
Semantic Processing ◽  
Mathematical Thinking ◽  
Sense Making ◽  
School Students ◽  
Solution Processes ◽  
Division Problems ◽  
Division With Remainder

In this study, about 200 middle school students solved an augmented-quotient division-with-remainders problem, and their solution processes and interpretations were examined. Based on earlier research, semantic-processing models were proposed to explain students' success or failure in solving division-with-remainder story problems on the basis of the presence or absence of an adequate interpretation provided by the solver after obtaining a numerical solution. In this study, students' solutions and their attempts and failures to “make sense” of their answers were analyzed for evidence that supported or refuted the hypothesized semantic-processing models. The results confirmed that the models provide a solid explanation of students' failure to solve division-with-remainder problems in school settings. More generally, the results indicated that student performance was adversely affected by their dissociation of sense making from the solution of school mathematics problems and their difficulty in providing written accounts of their mathematical thinking and reasoning.

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