Subtraction Metaphors 1

Children normally find subtraction problems more difficult than addition problems. Teachers normally attribute it to the difficulty of the subtraction procedures involving "borrowing". But the real reason is that in contrast to addition, which arises from 2 metaphors, children have to identify 4 metaphors which lead to subtraction. 

"Separating" Situations

In Separate situations, one collection is removed physically from another collection by removing, taking away, giving away, throwing away, filtering etc. What are the things which can be separated?

Apart from discrete things (like pencils, erasers, tokens) even weights, volumes, areas, lengths, time, age, money etc can be separated.

Examples can be given from the life experiences of children themselves; part of the rice or milk being poured out into another container, the toffees left in the box after some being distributed etc.

Measurable units can be used in word problems, in younger grades, without converting within units. The conversions themselves can be taught in higher classes. Right now the intention is to make students think about things they can separate.

Grouping - Separating but not physically

"Separate" could also mean making a subgroup without physically taking away. It could be separating red pencils out of a collection of pencils and just made a sub group.

Three Inter-related situations

In "Separate" situations we can see 3 inter-related mathematical situations. Students need to understand all these situations using real life examples. For better clarity we use the term "take away" instead of "separate."

1. Total - Taken Away = Remainder

2. Total - Remainder = Taken Away.

3. Taken Away + Remainder = Total which is an addition situation

Situation (3) shows that "Separate" situation is the reverse of "Join" situation.

Ideally both Join and Separate metaphors should be done concurrently so that children can see that they are related. What is joined can be separated and what is separated can be joined.

"Less Than" situations

Here, one collection is described to have some things "Less Than" a given collection. The following problem is an example.

Ram got 65 tokens. Lakshman has 5 tokens less than him. How many tokens does Lakshman have?

Three Inter-related situations

Here also there are 3 inter-related situations. Though they look complicated in written text, once mathematically understood through activities, they are easy to remember.

1. Greater Amount - Difference = Lesser Amount

2. Greater Amount - Lesser Amount = Difference.

3. Lesser Amount + Difference = Greater Amount which is an addition situation

Language Complexity

Like in the case of "more than", in real life the idea of "less than" is also conveyed by many other phrases, depending on the context, like "older /younger", "hotter/ colder", "bigger/ smaller", "longer/ shorter", "taller/ shorter", "heavier/ lighter" etc. Teachers should train children in using vocabulary appropriate to the context.

In temperature situations "less colder" translates to "higher temperature"!

The words used "word problems" in various contexts have been summarised in Chapter 9.7.

Tricky phrase- "More Than" & "Less Than"

We also have situations where the term "more than" in a word problem actually translates into a "less than" idea requiring subtraction. Take the following example.

Ram got 65 tokens. Ram has 5 tokens more than Lakshman. How many tokens does Lakshman have?

Similarly we can have a term "less than" in a word problem which actually indicates an addition situation.

Hence "key word" rules like "subtract" if the word "less" occurs in the problem can easily be misleading.