Arithmetic Operation Metaphors

Numbers & Operations Emerge from Our Life Experiences

Ideas about numbers emerged out of patterns we observe in the environment around us. Similarly, even arithmetic operations must have emerged out of observing "changes in patterns" in our environment.

The idea of Addition would have emerged out of seeing a quantity of grain or produce or livestock increase when a new amount is added to them.

Conversely the idea of Subtraction would have emerged out of seeing the same quantities decrease when they are either taken out for consumption or giving to others.

The idea of Division would have emerged out of the daily activities of sharing & dividing food or grains within the community or home.

The origins of Multiplication are more difficult to decipher. It is likely to have emerged when humans saw "growth" of living beings around them over a period of time.

Living beings grow in size but also retain their "shape" to a large degree. A baby elephant still looks like a full-grown one. We can recognize a childhood friend even when we meet her after many years.

This idea called "scaling" must have resulted in the idea of multiplication.

We will deal with each of these aspects in further chapters.

Life Situations Vs Computations

Arithmetic operations can be seen in two perspectives.

1. The "life situation" they are related to

   1. Each of the 4 operations can be related to several "life situations"

   2. Identifying a specific life situation is a "conceptual skill" requiring "understanding"

2. The "procedure" or "algorithm" or "computations" for performing that operation

   1. This is a "procedural skill" requiring "practice with understanding"

But when we hear the term "arithmetic operations" almost all of us think only of computations with numbers involving the four operations. Not much thought is given to understanding the different life situations which result in these arithmetic operations.

Operational Metaphors

Each of the arithmetic operations emerges from several different life situations. To help students remember the various life situations, we invent the idea of "operational metaphors."

These metaphors are short phrases which act as a bridge between the life situations and the related operations. They help connect a known idea (a familiar life situation) to an unknown idea (an arithmetic operation). They hep relate a "new" idea to a "previously learnt" idea.

An Illustrative Example

Let us do a thought experiment. We know that 3 + 5 = 8. Now try to think of a life situation or a math story which would result in the operation 3 + 5 =8.

In variably most people would come out with stories like "Ram had 3 pencils and Lakshman had 5 pencils. How many pencils do they have together?" OR "3 birds were sitting on the branch of a tree. 5 other birds joined them. How many birds were there on the branch?"

A little thinking would reveal that most of these problems can be seen as reflecting a "put together" life situation. If asked to tell a life situation which is not "put together" most would not be able to tell a story.

Now consider another story - "Ram had 3 pencils. Lakshman has 5 pencils more than Ram. How many pencils does Lakshman have?"

Most people would immediately be able to see that this story also results in 3 + 5 =8. But why were they not able to think of this life situation? One reason is that conceptually this situation is more "abstract" than the "put together" situation.

Another reason is also that they were not exposed to such situations in their school.

We can call this a "more than" situation.

So, we see that addition operation can result out of 2 very different life situations; "put together" as well as "more than" situations.

This is true of all the four operations which can be viewed from two perspectives; a conceptual aspect and a procedural/ computational aspect.

Importance of Arithmetic Metaphors

We use the term "arithmetic metaphors" to describe the life situations. Life situations are "abstract" ideas through which arithmetic operations can be viewed and presented. . With an understanding of metaphors, children learn to classify word problems into problem types and then solve problems according to structured solution methods for each problem type.

Metaphors Facilitate Abstract Thinking

Metaphor is the basic tool through which abstract thinking is possible. One of the main consequences in cognitive science is that abstract concepts are understood in terms of more concrete concepts, typically through metaphor.

A language metaphor is a phrase which indirectly relates to a life situation. Similarly, an "arithmetic metaphor" relates a life situation to a particular arithmetic operation. Hence "put together" and "more than" are arithmetic metaphors which relate to the operation of addition.

The other 3 operations (subtraction, multiplication & division) also come out of several different life situations. In the next chapter we give a summary of the various metaphors associated with all the four arithmetic operators. Subsequent chapters will deal with each of the metaphors in a detailed manner.

Before doing the computations in a problem, students need to make sense of the context of the life situation in which the problem is situated.

Many Procedures are Based on Metaphors

Many of the procedures themselves are the outcome of the underlying metaphors.

For example, the procedure of "count all" is a result of the "put together" metaphor and the procedure of "count on" is a result of the "more than" situation.

Focus on Computations rather than Concepts

Unfortunately, even our school curriculum focuses mostly on the "computational" aspects of operations. This could be the result of an "Indian math tradition" which mainly focuses on computation.