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Secondary Magazine - Issue 31: Focus on

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 01 April 2009 by ncetm_administrator
Updated on 27 April 2009 by ncetm_administrator

Secondary Magazine Issue 31

Focus on...negative numbers

Negative numbers first appear in China in The Nine Chapters on the Mathematical Art, which dates from the period of the Han Dynasty (202 BC – 220 AD) but may well contain much older material. The Nine Chapters used red counting rods to denote positive coefficients and black rods for negative.

Negative numbers were resisted in Europe until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits.

In 1759, the British mathematician Francis Maseres wrote that negative numbers "darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple".
Listen to In Our Time with Ian Stewart talking about negative numbers with Melvyn Bragg here.
Learners can benefit from having access to multiple representations of positive and negative numbers and the use of such representations to support foundations of understanding that can reduce opportunities for misconceptions… Read more of the Mathemapedia entry Addition and Subtraction of Negative Numbers, including the ‘tokens’ model, here.

Teachers usually introduce rules to help pupils remember results or steps in methods. However, few are always true and many are never fully developed so that pupils understand the context of a rule… [for example] ‘Two minuses make a plus’ -5 × -3 = +15 but-5 + -3 ≠ +8. This rule is an inaccurate simplification of a generalisation. Incorrectly applied ‘rules’ on signs and operations are the source of many errors for secondary pupils in work on number and algebra, usually because the ‘rule’ is learned without understanding and they do not take into account the different contexts of the operations of multiplication and addition, and the positive and negative states.
How might it be improved? Where it is considered that rules might be useful, they should be unambiguous and developed with the pupils. The unthinking use of rules should be discouraged. 
From page 14 of the Ofsted report Mathematics: understanding the score.

Adding makes bigger? It is quite natural for young children to associate ‘making bigger’ with adding. It is important for the teacher to value the pupil's generalisation and then at the same time reinforce the mathematics skill of testing out a hypothesis to see if it works for all circumstances.

Why is -2 × -3 = 6? I was trying to agree an explanation/justification for this with an intelligent non-mathematician: we weren't satisfied with our answer, can anyone help? Perhaps asking students to accept this sort of thing, with vague or unconvincing justifications, puts off many bright students, with questioning minds, at an early age?

Join the discussion here.
Is 10 bigger than -10? Find out how this question arose from a national lottery scratch card here.  
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