I have reasons to suspect that for a good percentage of students, the end of their mathematics career begin when they are introduced to subtracting integers. Well, for some, it’s when the x‘s start dropping from the sky without warning. In this post, let’s focus on the first culprit – subtracting integers.
One of the most popular tools for teaching addition and subtraction of integers is the number line. Does it really help the students? If so, why do they always look like they’ve seen a ghost when they see -5 – (-3)?
Teachers introduce the following interpretations to show how to subtract integers in the number line: The first number in the expression tells you the initial position, the second number tells the number of ‘jumps’ you need to make in the number line and, the minus sign tells the direction of the jump which is to the left of the first number. For example to subtract 3 from 2, (in symbol, 2 – 3), you will end at -1 after jumping 3 units to the left of 2.
The problem arises when you will take away a negative number, e.g., 2- (-3). For the process to work, the negative sign is to be interpreted as “do the opposite” and this means jump to the right instead of to the left, by 3 units. This process is also symbolized by 2 + 3. This makes 2 – (-3) and 2 + 3 equivalent representations of the same number and are therefore equivalent processes.
But only very few students could making sense of the number line method that is why teachers still eventually end up just telling the students the rule for subtracting integers. Here’s why I think the number line doesn’t work:
The first problem has to do with overload of information to the working memory (click the link for a brief explanation of cognitive load theory). There are simply too many information to remember:
1. the interpretation of the operation sign (to the left for minus, to the right for plus);
2. the meaning of the numbers (the number your are subtracting as jumps, the number from which you are starting the jumps from as initial position);
3. the meaning of the negative sign as do the opposite of subtraction which is addition.
To simply memorize the rule would be a lot easier than remembering all the three rules above. That is why most teachers I know breeze through presenting the subtraction process using number line (to lessen their guilt of not trying to explain) and then eventually gives the rule followed by tons of exercises! A perfect recipe for rote learning.
The second problem has to do with the meaning attached to the symbols. They are not mathematical (#1 and #2). They are isolated pieces of information which could not be linked to other mathematical concepts, tools, or procedures and hence cannot contribute to students’ building schema for working with mathematics.
But don’t get me wrong, though. The number line is a great way for representing integers the relationships among the numbers but not for teaching operations.
Click link for an easier and more conceptual way of teaching how to subtract integers without using the rules.