Operational Fluency

Apart from understanding the logic behind arithmetic operation procedures, students should also achieve mastery in performing them. Instead of the term, mastery, we will use the term fluency which gives a better understanding. Fluency implies both speed, accuracy & efficiency in computations.

Operational Fluency

Fluency is an integration of Efficiency, Accuracy and Flexibility

1. Flexibility

a. Flexibility comes from understanding the concept underlying a procedure. For example place value representation in understanding carry & borrow procedures. It is closely connected to number sense.

b. If students cannot figure out a problem in one way, they know another way to approach it.

2. Accuracy

a. “Deliberate practice” with understanding and clarity of computational procedures leads to correct implementation of computations & procedures

b. Correct Implementation of the procedures & performance of computations, leads to the ability to get the correct answers.

3. Efficiency

a. Efficiency is the ability to solve a problem with the least effort and thus save time when it is critical.

b. 35 + 56 – Adding 30 & 50 gives 80. Adding 5 & 6 gives 11. 80 & 11 make 91.

c. 21 X 32 using number bonds (20 X 30 -> 20 X 32 -> 21 X 32 )

Practice of number sense concepts and techniques ultimately leads to flexibility in procedures.

The Practice of Drilling

Currently, the general understanding in schools is that fluency is achieved by a lot of practice.

But as we will see, fluency needs practice with understanding. Without understanding, mere practice becomes boring and mostly does not lead to any learning but on the contrary, leads to a loss of motivation to learn.

This practice is called “drilling” in schools, possibly mirroring the idea of a military drill. But my teacher, PKS used to say that drilling most likely produces holes!

Happy Drills

An alternative to “drilling” is the idea of giving “happy drills” to students. Math has a special area called “recreational math” (Section 22) which deals with math games, puzzles and explorations. These activities are fun to do and learn through them.

The use of games & puzzles engages students in practicing with joy and without realizing that they are practicing math skills. These activities provide mathematical skills while at the same time provide mental pleasure.

They are easy & mentally engaging and accessible to students in Primary School. This is especially important in the Primary classes where students are not mature enough to motivate themselves to work hard for an easy tomorrow. Hence these should form a part of the mathematics curriculum.

Using Fingers

Another issue in many schools is the idea that math is a mental subject and hence memory is important. A related idea is that use of fingers should be discouraged.

Teachers who discourage the use of fingers have not understood their value. What needs to be understood is that math has to be “constructed” in the mind and not “memorized”. Using fingers is an excellent way for “constructing math” in our minds.

Fingers are available with us anywhere and anytime! Using fingers will also give visual & kinaesthetic pictures of numbers & procedures and deepen understanding of what is happening in that operation. This will provide visual and kinaesthetic images which will help “remember” the facts.

Consequently, the ability to mentally perform these computations will also develop. As the visual images are internalized, the need for using fingers will gradually reduce.

While looking at ways of improving fluency, we will be focusing on many techniques which use fingers, for improving number sense as well as performing computations.

Strategies to Avoid in Primary Grades

Some commonly used strategies should be completely avoided in primary grades as they lead to math anxiety and math avoidance.

1. Memorization (before understanding and without understanding)

Memorisation should be replaced by simultaneous practice of understanding and meaningful and contextual practice, with use of number sense strategies.

2. Timed Tests

In Primary Grades, the objective should be that students learn, even if some of them take more time. Once they understand & practice, they will gain fluency & speed which are necessary in the Middle School.

Math needs understanding. Understanding takes time, especially for students in the Primary Grades as it is a new skill that they are learning.

Fluency - A Broader View

Interestingly math educator & author Sunil Singh proposes a broader view of math fluency by adding two more elements to the usual matrix of conceptual, procedural & factual. These are historical & contemporary fluencies.

Historical fluency is understanding, knowing & having curiosity for — the history of mathematics, how math developed through the ages. Contemporary fluency is knowing what is happening in the world of mathematics today.