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.

taking away a positive integer

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:

taking away a negative integer

our mind can only take so much at a time

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.

#### Author

I'm a math teacher, researcher, writer, and facilitator of professional development for teachers. Email me at mathforteaching@gmail.com.

### 7 Responses to “Subtracting integers using numberline – why it doesn’t help the learning”

1. Hi Erlina
Great post. Thanks. I searched for “numberline why?” on google = 1.6 million hits. Your site was I think on page one. Thanks to you.
I am a medical graduate and did maths in UK and Australia to year 12. My wife in Thailand. Neither of us used this numberline system.
My 10 y.o. daughter was struggling with this tonight. The question (UK curriculum) was poorly written in my opinion and also in my opinion difficult to understand. My wife, an arts graduate, worked it out. The daughter understands negative numbers and can easily do the maths question you have above -5 – (-3, and 92-100..
These fashions in education are not helping anyone. We have had to teach our daughter to intensively memorize the times table ourselves. Funnily enough without memorizing it, you need a calculator to do many maths problems. Now I partly understand why many retail assistants have no simple maths, at all. Learning properly he times table is not given enough emphasis in many school systems at present.
My friends kids in Australia at expensive private schools forced to pay for Kumon tuition to get their maths up to scratch. I have got my kids doing Khan Academy. I am so grateful to Mr Khan, and it goes to Uni level.
People in standard schools in developing countries are going to leap over those in developed countries and using developed countries’ systems.
Best regards

• Thank you!

2. “Subtraction should be seen as a shorthand way of writing the addition of a negative value. For example, when a student sees “5 – 3″ they should be thinking “5 + (-3)”. This is a more formal and accurate definition of subtraction.”
This definition of subtraction and a corresponding definition of division are often used, but I think of them as ways to simplify the assumptions rather than as more accurate. Addition, subtraction and unary minus are independently important, but in an axiomatic treatment we can start with only two of them. (It may be even simpler to build addition & unary minus from subtraction.) Taking away works for ordinal subtraction and set difference, where Noratorious’s approach does not. (Suppose we append negatives to the ordinal numbers so that a+(-b)=a-b. Then (omega + (-2)) + 2 = (omega – 2) + 2 = omega + 2, which is not omega. Either (-2)+2 has to be nonzero or we give up the associative law. (Conway’s number system has negative ordinals but the addition is not the usual ordinal addition.))

3. I must respectfully disagree with your assessment of the use of a number line to teach subtraction. I find it to be immensely useful.
As a counter-argument, the concept of subtraction as “taking away” has the psychological effect of creating a bad feeling about math in general. When this feeling begins at such a young age, it is likely to grow and fester into an adult who can barely balance her checkbook without shuddering.
I am a certified one-on-one math tutor with 14 years of experience, and I find that students like the number line better. Here’s why:
For visual learners, this tool helps them see subtraction as a distance, which becomes a more valuable interpretation in applications and future math classes.
For kinesthetic learners, the act of hopping across a number line helps cement the concept and method in the student’s mind; they can experience subtraction, and will therefore understand it more deeply.
The difference in methods is not as obvious for auditory learners; however, they benefit from the combined use of visual and kinesthetic styles in this method of teaching.
In order to teach this method, it must be properly explained to students what, exactly, we mean by subtraction and negative numbers. When we subtract we are not simply performing the opposite of addition; in fact, we are performing addition with a negative number. Subtraction should be seen as a shorthand way of writing the addition of a negative value. For example, when a student sees “5 – 3″ they should be thinking “5 + (-3)”. This is a more formal and accurate definition of subtraction.
With this definition, understanding a number line becomes simple. All one must remember is that if they are adding a positive number, they jump to the right, and if they are adding a negative number, they jump to the left.
When subtracting a negative number, we first apply this definition of subtraction to simplify the value before applying it to the number line. For example, 7 – (-4) would become 7 + -(-4) or simply 7 + 4. Now we know to begin at 7 and jump 4 places to the right.
As for your list of things to remember, using the method I have outlined, here is all a student must do when given a subtraction expression:
1. Rewrite the expression as addition of a negative value.
2. Cancel any double negatives.
3. Start at the first value.
4. Jump to the right if the second value is positive and to the left if it is negative.
For example, consider the expression -3 – (-7).
1. We begin by rewriting the expression as addition of a negative:
-3 + -(-7)
2. Next we cancel the double negative:
-3 + 7
3. Now we find the position on our number line labeled “-3″.
4. Beginning at this position, we jump 7 spaces to the right, because the 7 is positive.
We land on 4 and have our solution.
Thank you for posting this blog. I hope you understand I am posting this reply with the deepest respect.