# Subtracting integers using numberline – why it doesn’t help the learning

January 12, 2010 4 Comments

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. For some, it’s when the *x*‘s start dropping from the sky without warning. For 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 introduces 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 there are two problems with this process which could be the reason despite the use of this visual tool, subtraction of integers are still taught and learned by rote. Only very few students could making sense of the process in the number line so teachers eventually end up just telling the students the rule for subtracting integers.

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, and;

4. finding an expression that also represents the process in the case of taking away a negative integer.

To simply memorize the rule would be a lot easier that is why most teachers I know breeze through presenting the subtraction process using number line and then 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. 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 or the signed 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.

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.

As for your complaints about this method, here are my responses.

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. 🙂

Thank you for your comment. I’m sure your method works. You may want to check my post on subtracting integers. https://keepingmathsimple.wordpress.com/2010/05/11/who-says-subtracting-integers-is-difficult/. It uses the idea that ‘-‘ means difference and distance between the number in the numberline.

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