Teaching Procedures

Procedures are step-by-step procedures to be followed in a computation.

Let us take the example of finding the result of 35 + 57.

1. Where do we start addition from?

a. We start from the lowest place value, which in this case is the Unit's place. The reason why we start at the lowest place value is part of conceptual knowledge.

2. Find 5 + 7 which are the digits in the unit's place.

a. This addition can be done in several ways & modes. Adding all strategy or Counting on strategy. Counters or fingers can be used. With practice students should be able to do such additions mentally.

3. The answer is 12 which is a 2-digit number. What do we do when we get a 2-digit number as the total?

a. We need to perform a "carry over" operation.

b. Before we perform this, we need to understand a "hidden" operation which is happening here.

c. 5 & 7 add to “twelve”. If we were using bundles & sticks to explain this problem, we would be adding 5 sticks & 7 sticks to get twelve sticks. Using our understanding of place-value, we “regroup” them into one bundle & two sticks. This is written as 12 in the place value notation.

4. Why do we need a “carry-over” procedure? This is part of conceptual knowledge. In decimal place value notation, the digit in any place cannot exceed 9.

5. Carry over to where? Carry Over to the next higher place value. In this problem, to the ten’s place.

6. Which of the digits, in this case 1 or 2, should be “carried over” to the ten’s place?

7. If the understanding in step (3c) is clear then there would not be any confusion that the “ten” or “bundle” or 1 in the number 12 is the digit which needs to be carried over.

8. The 1 which has been “carried over” has to be totalled along with the 3 & 5. We are actually adding 10, 30 & 50.

9. Hence the result of this addition is 90. This is written as 9 in the ten's place.

10. So the answer to 35 + 57 is 92.

In this problem, there were only 2 computations. One is 5 + 7 in the unit's place. Other was 1 + 3 + 5 in the ten's place.

Apart from these computations, the student had to follow a step-by-step procedure which has been listed above.

Many of the steps need an understanding of certain concepts like "regrouping" & "carry over".

There could be many variations in this procedure depending on the original problem.

There could be "carry over" operations in more than one place value.

Hence the procedure and its different variations need to be understood by students, along with the underlying concepts.

The best way to do this would be to demonstrate each step of the procedure using bundles & sticks. The steps where a conceptual understanding is necessary can also be pointed out.

To master the procedure for adding two 2-digit numbers, the student needs to go over the procedure step by step with the understanding of the underlying concepts.

This is called "deliberate" practice or practice with understanding. Such practice is best done, in the initial stages, in the classroom with the guidance of the teacher.

With such practice, the procedure will unwind in the mind of the student like a movie enabling her to perform the procedure fluently.

If a student does not understand the underlying concepts, the procedure just becomes a set of rules to be memorized. All the different variations also need to be memorized.

There are plenty of computational procedures in math. If all the different rules are again memorized without understanding, the result would be an "information overload" causing confusion in the mind of the student.

This is a sure recipe for math anxiety, math avoidance and even math phobia.