Arithmetic Operations

The Importance of the Place Value Concept

The logic of all the operation procedures can be understood with an understanding of the Place Value System and visualisation of standard and non-standard representation of numbers. If teachers understand the underlying logic, they would be able to demonstrate the procedures in a better way and also help individual students who have specific problems.

Once the logic of the procedures are "understood" it is also easy to gain fluency by memorizing them.

A Summary of the Logic of Operations

The concepts used in the 4 operations can be visualised in terms of situations given below. We are using 1 and 2-digit numbers just for illustration. The concept can easily be extended to bigger numbers.

Please remember that these concepts are different from “metaphors” which we dealt with in earlier chapters.

We are avoiding certain terms, as mentioned above, because these terms have come to be associated with certain operations and hence prevent a deeper understanding. We will explain this in further chapters in appropriate contexts.

In the subsequent chapters, we will take each arithmetic operation and see how the above concepts are used while developing algorithms for performing them.

Number computations in a real-life context

When introducing number operations to children, we need to remember that though the objectives of the operations are clear to adults, they are not clear to the children. Hence just learning operations without a “real life” context can be meaningless & boring to children. It is better to introduce simple meaningful situations through which the operations emerge.

For example, addition can start with a simple story like “You have 5 toys. On your birthday you received 2 toys as gifts. How many toys do you have now?” This story can be role modelled in the class and discussed. The answer 7 emerges naturally from the modelling. We can then tell the students that it can be written in math language as 5 + 2 = 7!

The classroom discussions can lead to a deeper understanding of the operation. Questions like what other items can be added can be asked?

By starting to listen and speak about mathematical situations in real life, the transition for students to reading and writing word problems would become much easier.

Computations Vs Algorithms

While learning arithmetic operations, students also need to understand the difference between a computation and an algorithm.

In solving a problem like 34 X 57 the student needs to know all the associated multiplication & addition facts. This is the computation aspect.

The student should also know the procedure (algorithm) – which numbers to multiply first, where & how to write the results of the operation, which number of carry over etc etc.

When a student is found to have problems in computations such as above, the teacher should find out if the problem is in knowing the computational facts or in remembering the procedure. This would enable an effective remedial intervention to be planned.

Mental Math

Before we plunge into various arithmetic operations, understanding the meaning & role of "mental math" is very important.

Mental math is not doing arithmetic operations "in the mind." It is doing arithmetic operations "using the mind." For example while adding 49 & 53, it is the mental ability to see that 49 & 53 are the same as 50 & 52 which is 102.

To that extent, using a paper & pencil is not banned in mental math. The purpose of using paper & pencil should be to "aid your memory & thinking" rather than to perform the standard algorithm for addition.

Mental math and number sense go hand-in-hand.

Vertical Arrangement of Several Numbers

All the 4 operation algorithms involve, at least in some stages, an arrangement of several multi-digit numbers vertically. There are also cases where the problem is given in a horizontal manner and has to be rearranged vertically.

Let us take the example of 34 + 5 + 1003.

The three numbers need to be first arranged vertically with their place values aligned in columns. The arrangement should be as shown below.

  5

  34

1003

For convenience, many teachers instruct children to write the place values H, T & U on the top. A student would require a good understanding of place value to do this rearrangement correctly. A visual understanding of place value in terms of sheets, strips & pieces would certainly help students to master this step.

A teacher can even use an exercise like this to check the understanding of place value of her students.

Importance of Estimation

Along with the algorithms, students should also develop skills of "estimating" the approximate answers.

This is an important tool in checking the "reasonableness" of answers and errors in calculations.

The various strategies for estimation have been dealt with in another chapter.