A lot of online resources on integers are about operations on integers especially addition and subtraction. Most of these resources show visual representations of integer operations. These representations are almost always in the form of jumping bunnies, kitties, frogs, … practically anything that can or cannot jump are made to jump on the number line. Sometimes I wonder where and when in their math life will the students ever encounter or use jumping on the number line again. If you want to know why I think number line might not work for teaching operations, click link – Subtracting integers using number line – why it doesn’t help the learning.
Of course there may be other culprits apart from rote learning and the numberline model. Maybe there are other things that blocks students’ understanding of integers especially doing operations with them.
Before integers, students’ life with numbers had been all about whole numbers and some friendly fractions and decimals. So it is not surprising that they would have made some generalizations related to whole numbers with or without teachers help. I pray of course that teachers will have no hand in arriving at these generalizations and that if indeed students will come to these conclusions, it should be by the natural course of things. Here are some dangerous generalizations.
These generalizations are very difficult to unlearn (accommodate according to Piaget) because based on students experiences they all work and are all true. Now, here comes integers turning all of these upside down, creating cognitive conflict. In the set of integers,
- when a number is added to another number it could get smaller (5 + -3 gives 2; 2 is smaller than 5)
- the sum of any two numbers can be smaller than both of the addends (-3 + -2 gives -5; -5 is smaller than -3 and -2)
- when a number is taken a way from another number, it could get bigger (3 – -2 = 5, 3 just got bigger by 2)
- you can get an answer for taking away a bigger number from a smaller number (3 – 5 = -2)
- when a number is multiplied by another number, it could get smaller (-3 x 2 = -5)
- when a number is divided by another number, it could get bigger (-15/-3 = 5)
On top of these, mathematics is taught as something that gives absolute result. So how come things change?
You may be interested to read my article on Math War over Multiplication. It’s also about overgeneralization.
Feel free to share your thoughts about these.