(ProQuest: ... denotes formulae omitted.)IntroductionStatus QuoIt is a well known fact from the History of Mathematics that negative numbers were not widely accepted by mainstream European mathematicians until the Eighteenth Century, and then only after much controversy (Katz, 2009). By contrast, the arithmetic of rational numbers was known in essentially its modern form to Diophantus in the Third Century (Heath, 1910; Klein, 1968) and the Ancient Egyptians possessed a system of unit fractions more than two thousand years earlier (Chace, 1979). It is therefore unsurprising that negative numbers are regarded as difficult to teach (Hitchcock, 1997; Hefendehl-Hebeker, 1991) or, as the Report of the Cambridge Conference on School Mathematics put it, "Perhaps no area of discussion brought out more viewpoints that the question of how the multiplication of signed numbers should be introduced" (Cambridge, 1963). Comparatively, the properties of rational numbers are greeted with far less trepidation, especially when armed with a suitable physical model. This difference in difficulty is also seen in the order of contemporary school curricula, with negative numbers, regarded as more abstract, generally delayed until secondary school, while fractions, viewed as more concrete, can be taught as early as lower primary.While this may represent the status quo which has existed for many years, mathematics educators are naturally resistant to the idea that certain topics are difficult to teach, or can only be taught to the more able student. Indeed, each new generation of researchers seems to yield a fresh initiative for overcoming the barriers to entry into higher mathematics, such as the New Math in the 60's and 70's (Cambridge, 1963; Southampton, 1961) or the Reform Mathematics in the 80's and 90's (Cockcroft, 1982; Standards, 1989). Most recently, the 2000's and 2010's have seen the development of the Early Algebra movement, which aims to smooth out the abrupt jump from primary school arithmetic to secondary school algebra by selectively introducing certain kinds of algebraic thinking into primary schools (Carraher & Schliemann, 2007). Unlike the New Math and Reform Mathematics movements, both of which achieved mixed success, a key research finding already obtained by Early Algebra research is that young children are capable of a much higher level of abstract thought than previously believed (Mason, 2008). In particular, even children of lower primary school age have been found to be able to work with negative numbers, well before the point at which this topic is normally introduced (Bishop et al., 2011; Wilcox, 2008; Behrend, 2006).Concern over negative numbers is not confined to the Early Algebra movement, however, and the many research articles recently published on this topic show that important issues still remain unresolved (e.g. Almeida & Bruno, 2014; Bishop et al., 2014; Bofferding, 2014; Leong et al., 2014; Whitacre et al., 2012; Selter et al., 2012; Altiparmak & Ozdogan, 2010). All these articles feature classroom based research, where the mathematical content can be assumed fixed, and different teaching approaches are investigated. This article also aims to improve classroom teaching, but the exactly opposite way: it considers different approaches to the mathematics, and existing classroom based research is used to justify that what is proposed is worth trying with children. In other words, rather than searching for a pedagogy which matches the difficulty level of the mathematics, the difficulty level of the mathematics is lowered to match the available pedagogy. That changes to the mathematics should be possible at all is justified in the next section, which highlights unresolved mathematical issues in each of the three strands mentioned in the first paragraph, namely: the historical origins of negative numbers, their mathematical definition, and the physical models used in teaching. …
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Jan 26, 2017
Olivier P Faugeras 
Open Access
