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Relational understanding triumphs over instrumental understanding : the case of Singapore primary four children’s understandings of odd and even numbers
Author
Teo, Kwee Huang
Supervisor
Ng, Swee Fong
Abstract
Research (e.g. Britt & Irwin (2008), Kaput (1999), and Warren (2004)) increasingly emphasises the importance of cultivating in children the habits of mind that attend to the deeper underlying structure of mathematics. Developing such awareness of the underlying structure of the arithmetic facilitates children’s transition to the learning of formal algebra. Exploring operations with odd and even numbers can be one effective means to gearing children towards structures. Attending to structures underpinning odd and even numbers forms the basis for modular arithmetic. However, children often focus on using definitions and rules to explain why some numbers are odd and others even. They make little attempt to explain why and how such definitions and rules used to differentiate odd from even numbers work. This makes it difficult for them to make meaningful algebraic generalisations and justifications when operating with odd and even numbers.
To generalise and justify operations with odd and even numbers meaningfully, children need to make reference to the structural properties underpinning odd and even numbers. For example, children use the structural understanding of odd numbers as numbers when divided in twos or divided into two equal groups would leave one remainder, reason that the one remainder in one odd number would pair up with the other one remainder in the other odd number, which explain for the even result when adding two odd numbers. Children need to reason structurally about the generalisation of sum of two odd numbers with the generalisation of odd and even numbers. This study looked at Singapore primary four children’s understandings of odd and even numbers through their justifications about sums of pairs of odd numbers, pairs of even numbers and sum of one odd and one even numbers in a variation task, and the different types of justifications these children used to support these relationships.
This study involved pen-and-paper task-based interviews with eighteen children, six from each ability group, High-Progressing (HP) group, Middle-Progressing (MP) group and Low-Progressing (LP) group. They first completed a variation task sheet which included three different addition tasks (a) sum of two even numbers, (b) sum of two odd numbers, and (c) sum of one odd number and one even number independently. Thereafter they were asked to justify the outcomes for these three tasks.
The findings suggested that children of all abilities had the capability to explain odd and even numbers relationally and to provide analytical justifications about sums of odd and even numbers. Children with relational understanding of odd and even numbers offered better and more convincing analytical justifications about sums of odd and even numbers than children with instrumental understanding or with vague understanding of odd and even numbers who could only provide empirical justifications about sums of odd and even numbers. Children with incorrect understanding of odd and even numbers could not abstract, generalise and justify about sums of odd and even numbers.
These findings offer evidence that all teachers need to teach for relational understanding of odd and even numbers. Teachers can shift children’s focus on the structures underlying the generalisations by first building up the conceptual structure of odd and even numbers with the use of small numbers, and then moving on to justifying with the use of big numbers. Teachers need to ask children questions consistently at the appropriate time and keep probing and pressing for meanings in their generalisations and justifications. Having regular continuous in-building of mathematical reasoning and encouragement to explain why and how the occurrence of the generalisations probably on a daily basis can create the many opportunities and experiences to seeing generalisations at a structural perspective. The study concludes with recommendations of possible future research studies on operations with odd and even numbers.
To generalise and justify operations with odd and even numbers meaningfully, children need to make reference to the structural properties underpinning odd and even numbers. For example, children use the structural understanding of odd numbers as numbers when divided in twos or divided into two equal groups would leave one remainder, reason that the one remainder in one odd number would pair up with the other one remainder in the other odd number, which explain for the even result when adding two odd numbers. Children need to reason structurally about the generalisation of sum of two odd numbers with the generalisation of odd and even numbers. This study looked at Singapore primary four children’s understandings of odd and even numbers through their justifications about sums of pairs of odd numbers, pairs of even numbers and sum of one odd and one even numbers in a variation task, and the different types of justifications these children used to support these relationships.
This study involved pen-and-paper task-based interviews with eighteen children, six from each ability group, High-Progressing (HP) group, Middle-Progressing (MP) group and Low-Progressing (LP) group. They first completed a variation task sheet which included three different addition tasks (a) sum of two even numbers, (b) sum of two odd numbers, and (c) sum of one odd number and one even number independently. Thereafter they were asked to justify the outcomes for these three tasks.
The findings suggested that children of all abilities had the capability to explain odd and even numbers relationally and to provide analytical justifications about sums of odd and even numbers. Children with relational understanding of odd and even numbers offered better and more convincing analytical justifications about sums of odd and even numbers than children with instrumental understanding or with vague understanding of odd and even numbers who could only provide empirical justifications about sums of odd and even numbers. Children with incorrect understanding of odd and even numbers could not abstract, generalise and justify about sums of odd and even numbers.
These findings offer evidence that all teachers need to teach for relational understanding of odd and even numbers. Teachers can shift children’s focus on the structures underlying the generalisations by first building up the conceptual structure of odd and even numbers with the use of small numbers, and then moving on to justifying with the use of big numbers. Teachers need to ask children questions consistently at the appropriate time and keep probing and pressing for meanings in their generalisations and justifications. Having regular continuous in-building of mathematical reasoning and encouragement to explain why and how the occurrence of the generalisations probably on a daily basis can create the many opportunities and experiences to seeing generalisations at a structural perspective. The study concludes with recommendations of possible future research studies on operations with odd and even numbers.
Date Issued
2017
Call Number
QA141 Teo
Date Submitted
2017