Students' understanding of the mathematical equality and inequality relations : a developmental approach

Abstract

The motivation for this study was the desire to make the path to learning elementary algebra as 'generalised arithmetic' more clearly defined for both students and teachers.In the initial learning of algebra, algebraic expressions are transformed to equivalent other forms and techniques are developed for solving simple equations. Both facets require students to have a thorough understanding of arithmetic equality ' properties' if the developed procedures and techniques are to be adequately understood. The same can be claimed also with respect to arithmetic inequality and the solving of inequations.The specifics of the research described in this thesis entailed: (i) the identification of the properties of the equality and inequality relations considered to be the arithmetic roots from which algebraic procedures emanate; and (ii) consideration of what could constitute 'understanding' of the properties identified in (i).The research activity involved the design and development of an instrument referred to as the Mathematical Equality and Inequality Understanding Survey (the MEIUS). Specifically, the MEIUS has the following design features: (a) for the Equality Relation, the properties are exemplified using 'small numbers', 'larger numbers', and 'algebraic numbers; (b) for the Inequality Relation the properties are exemplified in 'small number' and algebraic numbers' only. The resulting Survey consists of three Stages for the Equality Relation and two Stages for the Inequality Relation.Through consideration of MEWS responses, levels were devised in order to determine 'understanding' of the relation properties. The levels were associated with the developed MEWS Thought Process Model. The MEWS has a tight protocol for administration designed to ascertain, in a valid and reliable manner, the 'thought processing' which a student employs when responding to an Item.The field work of the research involved the administration of the MEIUS to two hundred and fifty seven (257) Grades 7 to 10 students in ten (10) Tasmanian High Schools. Overall the sample consisted of 137 females and 120 males.The experience revealed that the MEWS components can be conveniently administered within the school context. Subsequent analyses of responses, using an elaborate but readily comprehended response 'scoring' procedure, indicate that there is a great deal of potentially useful information concerning student understanding of the relation properties which could be obtained in a specific school setting. Such knowledge could be used to indicate the need for remediation, on the one hand, or to identify 'readiness' to proceed or apply, on the other.Comprehensive analyses of the data gathered have been made with 'implications for teaching' firmly in mind. Links between the various relation properties and procedures for 'simplifying' expressions and solving simple equations are pointed out, in juxtaposition to the information of the proportion of a teaching year group that has demonstrated the various MEWS Levels of Understanding of the properties. Thus, the analyses can be of assistance to teachers and curriculum designers in anticipating the degree of need for remediation, as well as deciding on expressions' and solving simple equations or inequations.In considering aspects of 'remediation' the Study proposes cognitively sound approaches to teaching a number of 'selected' properties of equality. The properties have been 'selected' for their significance to the algebra topics identified.In summary, this Study has two tangible products:1. The Mathematical Equality and Inequality Understanding Survey (the MEWS) with its sound cognitive and content bases, tight protocol for administration and elaborate response 'scoring', leading to the MEWS Thought Process Model articulated in Levels;2. The identification and articulation of links between the analyses of responses in terms of the MEWS Thought Process Model and the application of the relation properties to aspects of elementary algebra, where algebra is considered as 'generalised arithmetic'.It is claimed that both these concrete products have the potential to make a valuable contribution to the teaching and learning of algebra.