I have been thinking about negative integers and wondering how anyone I taught ever managed to understand them. Many did, but when I am asked, by KS2 and KS3 teachers, ‘Is there a good task for discovering the rules of negative numbers?’, I find it impossible to answer. There seem to be a few popular approaches that are supposed to lead to discovering the rules so I shall start by summarising them:
- Working with physical number line so that students have to walk along it in a positive or negative direction in order to ‘add’ directed numbers. In this model ‘addition’ means accumulating journeys. Great fun is often had shouting out instructions to the people doing the walking. However, I don’t think this method leads to discovering anything new because students already have to know that numbers have direction, and adding means ‘do them one after the other’.
- Thinking about temperatures going up and down. This is similar to working with the physical number line but usually vertical rather than horizontal. Again, this is not going to lead to discovery because, as above, addition means ‘doing one thing and then another’.
- Talking about a situation in which people end up in debt. Or any negative situation (e.g. games that involve removal as well as accumulation and hence lead to negative scores) that has to be resolved somehow in a way that we would recognise at compensating a negative value with a positive value. Again, discovery is not the right word to use because students already have to have the idea that positive and negative values can cancel each other out.
So maybe ‘discovery’ is not the right word, and what these approaches do is to model situations in which negative numbers can be understood. Subtraction in these models can be seen as ‘difference’ or ‘distance between’, and these models make it possible for ‘subtraction’ to be achieved by adding the two absolute distances from zero. A formal rule for subtracting negative numbers is unnecessary because they can be seen as a ‘distance from zero’.
Some teachers have adapted the number line model so that there are two meanings for the negative sign: one meaning is that the walker has to turn through 180° and the other meaning is that the walker has to walk backwards. I leave it to you to work out how to use these adaptations to introduce subtraction into the model in a way that leads to the formal rule for subtracting negative numbers. There is a helpful elaboration of this learning in Ryan and Williams (2007). To make this work requires strong rapport with your students and absolute clarity in your own head about what is going on.
However, none of these models seem to easily offer the idea that multiplying two negative numbers gives you a positive answer. I have been told that some students can be persuaded that removing a debt of £10 (–10) three times (× –3) is equivalent to giving somebody £30, but that idea seems very confusing because (a) no one would ever do that and (b) the debtor could end up with £30 rather than £20!
The legitimate historical reason is that this rule has to be true so that arithmetic ‘works’ once negative numbers were introduced, so the rule cannot be discovered, it has to be ‘given’, and is hard to model.
|×||+ve numbers||–ve numbers|
|+ve numbers||+ve||–ve because the process can be interpreted as ‘so many lots of a negative quantity’|
|–ve numbers||–ve because the process can be interpreted as ‘so many lots of a negative quantity’||has to be positive because multiplying negatives by positives are negative, so multiplying by negatives must be different, i.e. positive|
I realise that this approach might seem shocking to some people who want mathematics to be concrete, but this is a form of reasoning used often in mathematics where there are only two possible states. For example, combining signs has the same structure as switching lights on and off, rotations of order 2 and reflections.
From all these experiences, the rule emerges that is usually stated as ‘two minuses make a plus’ and you probably know how misused this can be! A classic error is: –5 – 6 = 11 because there are two minuses. Various curriculum projects had ways of handling this kind of confusion, by making the direction of each number clear in superscripts. Thus –5 – 6 could be –5 – +6 or –5 + –6, the first version indicating a distance between two numbers (but it matters which direction you go in), the second being an accumulation of two directed numbers. Another way is to put directed numbers in brackets: –5 – (+6) or –5 + (–6). A further way is to see the expression as a sequence of operations that are equivalent to –11: either, the distance from +6 to –5 or, the equivalent single operation to subtracting 5 and then subtracting 6 from something.
You may wonder why I am complicating something students can be told. I am beginning to wonder myself!
It is because of what happens for many students when they are trying to understand algebraic expressions – and it is not a pretty sight. Augustus De Morgan, a 19th century mathematician noted for his teaching as well as his own work, and a founder member of the London Mathematical Society, devoted a chapter of his book about the teaching of mathematics to the problems of variables and unknowns that might turn out to be negative. His tutor George Peacock had been even more worried and wrote a two volume book about them!
The main problem is that they saw a – b as only yielding a permissible answer if a > b. For a < b they suggested all kinds of ways of reformulating the expression so that it makes sense. Algebraically, we do not know whether a < b or a > b and want a way to manipulate expressions that works in either case. This work was all done less than 200 years ago, so we are asking our students to accept something that is neither intuitive nor provable when we expect them to operate correctly in algebra. De Morgan’s solution is to regard a – b as meaning the sequence of operations: + a – b, and if b turns out to be bigger than a, then the effect is equivalent to –c for some c, i.e. + a – b = –c. This is the same relation between the three values a, b and c as saying b – a = c. The two algebraic equations are equivalent because of the essential part-part-whole relationship being described – the fundamental additive relation. For students who have a strong number line image of negative numbers this makes visual sense because the situations in which both cases a > b and b > a can be illustrated and seen to be positive and negative versions of the same distance.
But many students have a problem grasping what –b can possibly mean if b is a variable that can take positive or negative values. Their logic is: ‘if b turns out to be negative, then –b is positive and should be written as just b’. Or ‘if the equation contains the term –b that is telling us that b is negative’. The first time I heard students explaining this to me I had one of those classroom moments in which I had no idea what was going on – something sounded like logic and I could not grip it at all. The ‘–’ here is changing the direction or telling us to subtract (depending on context) not making b itself negative.
I know one secondary school maths department who have agreed on whether the sign ‘–’ is referred to as ‘negative’ or ‘minus’ in order that students will have a coherent experience throughout the school, but even that coherence does not emphasise the role of ‘–’ as a direction changer – turning round as you walk along the number line does this.
According to Ryan and Williams’ (2007) large scale study 27% of 14 year olds make mistakes related to these confusions when working with numbers, so we can assume they carry these into their algebra work, and maybe some extras because of the role of letters. Or maybe they just learn the rules of algebra and don’t even think of the underlying variables and unknowns. Somehow some of our students learn these correct interpretations as we did. What about the 27%?
My next blog will be more fun I promise!
All the best,
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.
Ryan, J., & Williams, J. (2007). Children’s Mathematics 4–15: Learning From Errors And Misconceptions. McGraw-Hill Education (UK).
De Morgan, A. (1831, edition 1910). On the study and difficulties of mathematics. Open Court Publishing Company.
Peacock, G. (1830). A treatise on algebra. J. &J. J. Deighton.