Consider the following that I took from a student’s book. Is this something you have seen before? What do you think the student’s reasoning is?
What’s the Problem?
In my experience, if there is one rule that students of all ages and abilities remember (and can recall proudly when prompted), it’s that two minuses make a plus. This, of course, is all well and good when multiplying and dividing, but can lead to a whole world of problems when applied, without prejudice, to addition and subtraction problems. Hence, the answer to the above question is often given as 9, with the logic being that those two nasty minus signs magically turn themselves into a lovely little plus. It’s as simple as that!
What’s the Solution?
Rules, without understanding, can be a very dangerous thing in mathematics, and are the root cause of many of the misconceptions that students possess. The problem is that once rules are ingrained, they are very difficult to weed out. To solve this nasty problem, you have to get the students early, and get them thinking about the questions themselves. I have found that young students respond quite positively to thinking about a bowl of soup.
Imagine you have a lovely bowl of soup that has a current temperature of 10 degrees. We also have ice cubes, each of which takes 1 degree off the temperature, and we have fire cubes (bear with me here!), each of which adds 1 degree to the temperature. Now, what happens if we add 3 ice cubes to our soup? Well, our soup gets 3 degrees colder, and we have 10 + – 3 = 7. And if we now dip our hand in and take out 4 fire cubes, we have 7 – + 4, which must equal 3 degrees. Once students have got their heads around this concept, it is time to progress to the likes of 4 – – 5 (a soup of temperate 4 degrees, has 5 ice cubes removed, so the temperature goes up to 9 degrees), and the potentially deadly -4 + – 5 (a chilly soup of temperature -4 degrees has a further 5 ice cubes added and goes down to -9 degrees).
I have heard similar techniques involving money, sandcastles, and even witches! I know the scenarios lack a certain degree of credibility, but I personally prefer that to a rule which can be mercilessly misapplied by students left, right and centre.
Resources to help
Check out these resources and collections on TES which may help tackle to problem:
Collection: Tarsia – Number Resources: http://www.tes.co.uk/article.aspx?storyCode=6107408
Negative Numbers – http://www.tes.co.uk/teaching-resource/Negative-Numbers-6062741/
N8 – Using Directed Numbers in Context: http://www.tes.co.uk/teaching-resource/N8-Using-directed-numbers-in-context-6086839
Negative Numbers (MEP – Year 7 – Unit 15): http://www.tes.co.uk/teaching-resource/Negative-Numbers-MEP-Year-7-Unit-15-6051096
Tarsia – Negative Numbers (Level 5): http://www.tes.co.uk/ResourceDetail.aspx?storyCode=6106865