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.
How hard is the unlearning? (This is a real question, even if I seem to be arguing for one approach. What have teachers & education researchers found?) It’s useful to know these things about positive integers even if they don’t generalise. Special cases continue to be important as students advance: for example, statements specific to positive numbers are part of the foundation of calculus. The proof that there is ‘no’ solution to x^2+1=0 is in a way a good lead in to complex numbers. (Why extend the real line if it already contains a solution?) Does the unlearning sometimes lead to good questions? A teacher might have the opportunity to mention that matrix multiplication and ordinal addition are not commutative.
If you realize the level of mathematics enjoyed by the public ed teacher, then you will find the answers to the misrepresentation of integers. Most have taken a test to become certified without higher college math courses…therefore are lacking the connections that math models can provide.
Hey,
This is a inquiry for the webmaster/admin here at math4teaching.com.
May I use part of the information from this post above if I provide a backlink back to this site?
Thanks,
Peter
sure and thanks for the interest
I think the most common mistake is that if multiplying 2numbersthey must make a larger number. Kids really have a hard time understanding this. I think it is because prior to integers all they knew was whole numbers. So now teachers have to find a way to get that generalization out of kid’s heads.