Arithmetic Operation Metaphors

< 8.7 Fundamental Theorem of Arithmetic | Topic Index | 9.2 Summary of Operation Metaphors >

Ideas about numbers emerged out of patterns we observe in the environment around us. Similarly even arithmetic operations must have emerged out of observing &ldquo;changes in patterns&rdquo;.

Addition &amp; subtraction ideas would have emerged by seeing quantities being &ldquo;joined&rdquo; or &ldquo;separated&rdquo;. Division would have emerged out of the daily activities of sharing &amp; dividing in the community or home. The origins of multiplication are more difficult to decipher and we will deal with it in a separate chapter. But undoubtedly multiplication must also have emerged out of changes that we see around us.

In general we can say that arithmetic operations emerged from observing various &ldquo;life situations&rdquo;

But when we hear the term &ldquo;arithmetic operations&rdquo; almost all of us think only of computations with numbers. This includes adding, subtracting, multiplying or dividing 2 or more numbers. Not much thought is given to understanding the different life situations which result in these arithmetic operations. We need to realise that the same arithmetic operation can result from &ldquo;different&rdquo; life situations. This is the &lsquo;conceptual&rsquo; aspect of arithmetic operations.

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 &ldquo;Ram had 3 pencils and Lakshman had 5 pencils. How many pencils do they have together&rdquo; OR &ldquo;3 birds were sitting on the branch of a tree. 5 other birds joined them. How many birds were there on the branch?&rdquo;

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

Now consider another story – &ldquo;Ram had 3 pencils. Lakshman has 5 pencils more than Ram&rdquo;. How many pencils does Lakshman have?&rdquo;

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 &ldquo;abstract&rdquo; than the &ldquo;put together&rdquo; situation. Another reason is that they were not exposed to such situations in their school.

We can call this a &ldquo;more than&rdquo; situation.

So we see that addition operation can result out of 2 very different life situations; &ldquo;put together&rdquo; and &ldquo;more than&rdquo;.

Unfortunately even our school curriculum focuses mostly on the &ldquo;computational&rdquo; aspect of operations. This could be the result of an &ldquo;Indian math tradition&rdquo; which mainly focuses on computation.

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. For want of a better word, we use the term &ldquo;arithmetic metaphors&rdquo; to describe the life situations. A language metaphor is a phrase which indirectly relates to a life situation. Similarly an &ldquo;arithmetic metaphor&rdquo; relates a life situation to a particular arithmetic operation. Hence &ldquo;put together&rdquo; and &ldquo;more than&rdquo; are arithmetic metaphors which relate to the operation of addition.

The other 3 operations (subtraction, multiplication &amp; 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.

We will deal with operation metaphors before procedures since many of the procedures themselves are the outcome of the underlying metaphors. For example the procedure of &ldquo;count all&rdquo; is a result of the &ldquo;put together&rdquo; metaphor and the procedure of &ldquo;count on&rdquo; is a result of the &ldquo;more than&rdquo; situation.

< 8.7 Fundamental Theorem of Arithmetic | Topic Index | 9.2 Summary of Operation Metaphors >