Algebra Archives - Page 2 of 3

Historical studies on the development of mathematical concepts will help mathematics teachers to relate their students’ difficulties in understanding to conceptual problems in the history of mathematics. Albrecht Heefer in his paper, argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. His arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians such as Arnauld, Leibniz, Wallis, Euler and d’Alembert. Not only does division by negative numbers pose problems for the number line, but even the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics, he argue for the introduction of negative numbers in education within the context of symbolic operations.

The number line is currently one of the important tools for teaching basic arithmetical concepts such as natural and real numbers in primary and secondary education. Hans Freudenthal (1983, 101) calls this mental object a ‘‘device beyond praise’’ and considers it a preferred vehicle to teach negative numbers (ibid, 437). In many countries the ordering of negative numbers by means of the number line is taught by the fifth grade (Howson et al. 1999).
Not everyone is convinced of the benefits of using the number line for teaching negative numbers in primary education. In fact, the very teaching of operations on negative numbers is no longer allowed in education below the age of 12 in Belgium. Negative numbers can only be used in ‘‘concrete situations’’. The examples provided are the floors of a building and the temperature scale.
The concept of an isolated negative number is an intrinsically difficult concept. Negative numbers emerged in history within the context of symbolic algebra (Heeffer 2010). This concept is best taught in secondary education and more specifically within an algebraic context.

Author: Albrecht Heefer

Published online: 10 March 2011
Springer Science+Business Media B.V. 2011

Several researchers, including Booth (1984, 1988), Greeno (1982), Kieran (1988, 1992), Lins (1990), and Matz (1980), attributed many of the fundamental difficulties experienced by beginning algebra students to their failure to identify equivalent forms of an algebraic expression. According to Kieran (1988), structural knowledge means being able to identify ‘all the equivalent forms of the expression’. Linchevski and Vinner (1990) argued that this definition should be modified to include the ability to discriminate between the forms relevant to the task – generally one or two forms – and all the others. Booth (1981, 1984, 1988) emphasized that students construct their algebraic notions on the basis of their previous experience in arithmetic. Thus, their algebraic system inherits structural properties associated with the number system they are familiar with. She suggests (Booth, 1988) that the students’ difficulties in algebra are in part due to their lack of understanding of various structural notions in arithmetic.

The above statement is part of the literature review in the paper Structure sense: the relationships between algebraic and numerical context by Liora Linchevski and Drora Livneh. They investigated the question: Do misinterpretations of the mathematical structure by beginning algebra students reflect difficulties they already have in arithmetic? And, if these difficulties do exist in arithmetic, are they systematic, or unsystematic?

Category: Algebra

Historical objections against the number line

What is structure sense?