How should students understand the subtraction operation?

Studies show that students whose understanding of subtraction is only to take away will have difficulty learning other mathematical concepts.

There are three ways by which subtraction can be understood: (1) Taking Away, (2) Difference, and (3) Inverse relation to addition operation. Pupils’ first experience with subtraction involves taking away. The dash sign means minus and minus is taken to mean ‘take away’.

Subtraction as ‘taking away’

Here are formats of subtraction tasks that involve taking away:

1. Marco has 12 twelve marbles. He gave 5 to his friend Precy. How many does he have left? (This problem is represented by the equation, 12 – 5 = ____.)

   

2. Marco has 12 marbles. He gave some to his friend, Precy. If he had 7 marbles left, how many did he give to Precy? (This problem is represented by the equation, 12 – ____ = 7.)
3. Marco gave his friend 5 marbles. If has 7 marbles left, how many did he have at the start? (This problem is represented by the equation,  ____ – 5 = 7.)

Problem situation number 3  require subtraction representation but is actually an addition problem because the solution involve adding 5 and 7 instead of doing subtraction.

For most students this is all they understand about subtraction — to take away. This is probably because the use of subtraction in many daily life situations use this meaning. To compound this situation, many of the subtraction tasks in textbooks are also of this type. Very few, if there is any, will include problems that supports the development of the other meaning of subtraction.  And when for a long time all one know about subtraction is to take away, it would be very hard to accept other meanings. Studies show that students whose conception of subtraction is only to take away will have difficulty learning other mathematical concepts.

In mathematics, the minus sign is also taken to mean “difference”. In calculating for distance for example, the minus sign means difference. If  the number  line is used to teach subtraction, 4-10 is the distance from coördinate 4 to 10. This is represented by 4 – 10. To subtract ^–4 from ^–10,  the distance  ^–10 to ^–4 is represented by ^–10 – ^–4. If to subtract is to take away, how will you explain to students how to get the result? Others textbooks and teachers explains this by defining the minus sign “to turn around”. This definition is not mathematical even if you can trace it back to the third meaning of subtraction as inverse relation to addition. In the first place, where in their mathematical life will they ever use all those interpretations of “+” and “–“ as turn left, turn right, and change direction? In real life yes, they will have to do turn left, turn right, and change direction in driving a car. But I have already digressed too much from my main concern in this discussion. Anyway, studies show that teaching operations with signed numbers using the number line this way is not pedagogically effective not because students don’t learn how to add and subtract integers after the lesson but because students forget everything after the test. I have another way of teaching subtraction using number line in my post Who says subtracting integers is difficult?. Check it out and please tell me what you think.

Subtraction as ‘determining the difference’

All I want to say in this post is that as early as Grade 1 and perhaps even in their kindergarten grade, pupils should already be exposed to more problem situations that involve subtraction as determining the difference. The following are sample problems:

1. Marco has 12 green marbles and 5 purple marbles. How many more red marbles does he have than green marbles? Pupils will solve this by matching the red and green marbles. The equation representation will be 12 – 5 = 7.

   

2. Precy is playing along the beach making holes in the sand. Marco said he wants to hide one marble each in a hole. Precy made 17 holes but Marco has only 12 marbles. How many holes will remain empty?

These problems require matching and comparing and not taking away.

Subtraction as ‘inverse relation of addition’

Children get a sense of the third meaning of subtraction as inverse relation to addition as they solve subtraction and addition tasks.  They know that they can undo any addition task by subtracting the same and vice versa.

Thank you for reading. Please share if you think this is informative.

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