Mathematics is a basic skill required to perform several operations of daily life. Proficiency in this subject can be explained using several terms such as lucidity or fluency, automaticity, comfort, etc. Though these all seem to mean the same, a thin line tells one apart from the other. Let’s focus on fluency and automaticity and understand the difference between the two.

While acquiring math skills or language skills, one needs to accept and remember certain concepts as they are. For example, a multiplication table needs to be memorized; there is no shortcut to that! The meaning of math operations needs to be absorbed as is, there is no question of asking why. The educator may train the student on the usage of math operations, but the crux should be ingrained to get the correct outcomes. Based on the way the skill is acquired and demonstrated, a math geek may be termed as fluent or the one demonstrating automaticity.

**I. Math fluency**

Kilpatrick (2008) proposed that math fluency is one of the proficiencies a mathematically competent person possesses^{[1]}. He placed this among other determiners of math proficiency such as strategic competence, adaptive reasoning, productive disposition, and conceptual understanding. Thus, to make a math-based decision or to perform operations, math fluency is a must.

In a generalized manner, math fluency is summed up as procedural fluidity or computational ease(Russell, 2000)^{[2]}. If a person is able to apply strategy quickly, compute the operation correctly, and has the confidence that his strategy is correct and so is the result, he is said to be a math fluent person.

To be math fluent, a person requires sound number sense^{[3]}, knowledge of how to apply operations, and the correct strategy. For instance, if a question demands the student to distribute 20 books among 4 students equally, it is the math fluency that compels him to use division and do the required calculation. Thus, applying a math concept confidently and correctly and achieving an outcome effortlessly sums up math fluency.

**What math fluency looks like**

Math fluency is a broader term than math automaticity. It is needed to identify the problem, strategize solutions and perform the operation with confidence. Not only that, the skill allows the person to devise more than one strategy and be flexible in approach. The skill takes up different forms as the kids progress upon building more math facts and be comfortable using them in academic and daily life. **For instance,**

- Strong number sense is a sign of math fluent arithmetic beginner. Subitizing skills and picking correct value for the given number is one of the aims of a preschooler and primary school student.
- Math fluency for a high school learner means ability to do adaptive reasoning and flexibly solve problems. They may show comfort in calculating change while shopping, guessing the speed or distance of ball while playing or even suggest ways to solve a problem.
- In adult life, the computational ease is demonstrated in making financial decisions, comparing prices of commodities, evaluating job opportunities, etc.

Most of the thinking and acting upon information or using geometry, or algebra in real life happens effortlessly when the speed, accuracy, and appropriateness in response while applying maths is attained at the early learning stage. So, being fluent has a lot to do with the concept application part or early ingraining of number sense^{[4]} or symbol-quantity association.

**How to improve math fluency**

Since math fluency is an application-based outcome, the ways to improve should be mentally stimulating. Giving different situations to tackle and to drive thinking to analyze problems from different perspectives and visualize solutions broadly represent the ways to improve math fluency. In a practical sense, the process involves:

- Doing activities like making anchor charts where kids tabulate all possible strategies
- Solving puzzles to help brain revisit concepts and apply them seamlessly
- Working the way backward, i.e. from solution to conceptualizing problem
- Role-play to introduce application of maths in real-life situations

**Emergence of fluency in maths**

What is the process through which anybody attains math fluency? The answer to this question also explains the main difference between math fluency and automaticity. A lot of effort and involvement goes into attaining ease in solving math problems.

A person becomes computationally strong by doing regular practice. It is a common observation that when kids are introduced to fractions, decimals, and percentages, they tend to become confused. The reason is the requirement of more practice to appreciate the fact that half is nothing but 50% or 0.5. Hence, asking kids to solve questions and present them in different ways such as fractions, decimals, percentage, etc. lead to factual fluency.

Giving real-life situations or driving thinking to apply facts is another suitable way of building fluency. Let’s take the example of a percentage again to solve a simple problem. When you give a word problem like, ‘Suppose the 50% of the wall requires 2 hours to paint. How much time will be required to paint the complete wall?’ You can have your versions of such problems too. It is nothing but driving the mind to adaptive reasoning and the answer they provide emerges from strategic soundness and computational ease. Hence, discussions, and encouraging kids to think beyond classroom situations help impart math fluency.

**II. Automaticity**

Math Automaticity is nothing but giving prompt replies to math questions or recalling math facts without realizing the process involved. It is quite similar to attaining reading fluency. While reading fluency is the fundamental of reading where you don’t register reading the text word for word and absorb the idea as a whole, automaticity does the same in the case of numeric calculations. Blurting out the answers of multiplication tables or math facts charts after mugging up is a classic example of math automaticity.

**What math automaticity looks like**

Automaticity in math is confidently answering questions based on quick facts. Doing mental math calculations in a series, solving order of operations problems without using pen and paper exemplifies the concept of automaticity. It is different from fluency because there is no strategizing involved. Learners simply recall and present the answer from the fact learned by rote memorizing. Common examples include:

- Recalling multiplication tables
- Solving basic math operations like addition, subtraction, multiplicaton or division mentally
- Remembering and effortlessly applying all formulas learnt
- Mastering conversion of units
- Quick subitizing for small sets of items, etc.

**Ways to achieve math automaticity**

Automaticity is achieved from regular practice and repeated memorizing. One has to go through the same concept like counting, or doing other complex operations again and again and almost daily at the start to become seamless in mental calculations. Hence, self-practice, playing quizzes, identifying learning gaps, and working on them till attainment of perfection, etc. are some of the ways to acquire automaticity in math.

**Where does automaticity help?**

In academic life, automaticity helps in solving problems within a limited time period. Every basic skill is further compounded to move to the next level. A simple example is to apply addition to find the product of two numbers or do it by recalling the multiplication fact. If you choose to add four 6 times or multiple 4 by 6, you can produce the answer to the latter faster, right? Hence, automatically solving basic operations to apply them to more complex problems can help in solving problems quickly and to score good grades.

In practical instances, too, going home using a chosen path, remembering the miles involved and hence the time of travel required, and similar situations employ math automaticity. It is needed to work like a pro and to make swift decisions that constitute daily life activities.

**Automaticity emerges from…**

By considering all the examples and explanations mentioned above, one can easily infer that automaticity results from practice, repetitions and is largely based on experiences created during the learning process. Hence, it requires constant correction, mechanical repetitions, revisiting the concepts, and reflective thinking.

**Summing up the major differences…**

Math Fluency | Math Automaticity |

Total sum of computational fluency, adaptive reasoning, strategic competence, conceptual knowledge, productive disposition, and appropriate response | Mostly comprises computational fluency and quick response generation |

Primarily strategy-based; offers an indication of ease of application | Primarily operations-based; demonstrated through quick concept recall |

Emerges from strategic thinking and applying concepts clearly | Emerges from practice, repetitions, testing, and re-practicing the concept till perfection |

**Conclusion**

Both math automaticity and fluency are needed to be a confident problem-solver. These strengths emerge from deeper involvement and practice. Automaticity is a contributing factor to attaining math fluency. Hence, while fluency is the total outcome, automaticity is one of its various components. Fluency is the ultimate destination or is an ongoing process; automaticity is one of the milestones or the building blocks. None can be figured out separately.

**References:**

- Schoenfeld, Alan & Kilpatrick, Jeremy. (2008). Toward a theory of proficiency in teaching mathematics. The International Handbook of Mathematics Teacher Education. 2.
- Russell, S. J. (2000). Principles and Standards: Developing Computational Fluency with Whole Numbers.
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