Procedural Knowledge

We have seen that factual & conceptual knowledge is learning the math content at two different levels.

Math also consists of computations with numbers and procedures to solve problems. These constitute procedural knowledge and are qualitatively different from both the above.

They are more about lots of practice with understanding and developing muscle memory & fluency.

Here are some examples of procedural knowledge and the understanding and practice needed to master them.

1. 2, 3, 5, 7 are prime numbers less than 10.

a. Understanding the logic of the procedures necessary for deciding if a number is prime.

b. Practicing the necessary computation procedures several times so as to attain accuracy & speed (called fluency)

c. Applying that procedure to all numbers less than 10 and locating the prime numbers

d. Remembering a list of prime numbers by repeated use & recall.

2. 8 X 7 = 56

a. Understanding the logic of the procedures necessary to find out any multiplication fact.

b. Practicing the necessary computation procedures several times so as to attain accuracy & speed (called fluency)

c. Remembering the multiplication tables by repeated use & recall

3. Odd + Odd = Even

a. Understanding the logic of the procedures necessary to find out any addition fact involving Odd & Even numbers

b. Practicing the necessary computation procedures several times so as to attain accuracy & speed (called fluency)

c. Remembering the addition facts involving Odd & Even by repeated use & recall

There are important points to note in this kind of knowledge.

Procedures involve both computational skills (ability to recall 8 X 7) as well as algorithms (steps for multiplying a 3-digit number by a 2-digit number).

Both need to be practiced many times in order to achieve mastery.

Procedures involve both mental & physical practice. Hence there is both mental visualisation and muscle memory.

Attaining fluency is the purpose of procedural knowledge. Fluency is a combination of understanding, speed & accuracy. Practicing procedures without understanding can lead to errors where a problem is slightly different.

Procedural Knowledge also needs conceptual understanding

Most math teachers refer to practice as “drill”.

Mathematics is a discipline where procedures need to be flexibly applied depending on the context.

A typical example is a child who writes the answer to 75 – 49 as 34! He only remembers that the smaller number (5) has to be subtracted from the bigger number (9).

Subtracting 49 from 75 also needs mastery of the “regrouping” procedure, which in turn needs a thorough understanding of place value.

With understanding & practice, the procedure is “remembered” by the brain like a movie and the procedure unwinds automatically, without any special effort, as the student is performing the procedure.

As my mentor Shri P K Srinivasan put it, practice without understanding is drill which only produces holes instead of understanding.

You can see that procedural knowledge is easier than conceptual knowledge but harder than factual knowledge.