Let’s address fluency in math by defining what fluency is, why it matters, and how the three stages of fluency are defined by Florida in the B.E.S.T. Standards for Mathematics.
What is mathematical fluency?
“When we are fluent in a language, we can respond and converse without having to think too hard. The language comes naturally, and we do not use up space in our brain thinking about what word to use. Fluency comes from using the language in multiple settings, from trying things out, and failing and trying again.” – Dr. Nic, Creative Maths
This approach to fluency in any language applies to the language of mathematics, too.
In mathematics, fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem-solving to achieve automaticity. Students connect conceptual understanding (Stage 1) with strategies and methods (Stage 2) and use the methods in a way that makes sense to them (Stage 3).
When students go through these stages to build fluency, they gain an understanding of the operations and the strategies and methods in their toolbox for solving them, and they become strategic thinkers who can efficiently compute arithmetic.
Fluency is often misunderstood as being able to quickly compute basic math facts, regardless of conceptual understanding, otherwise known as memorization. But being fluent in mathematics is more than memorization, accuracy, and speed.
Accuracy goes beyond memorizing a procedure to get the right answer; it involves understanding the meaning of the procedure, applying it carefully, and checking to see if the answer makes sense. Emphasizing speed can discourage flexible thinking. True fluency is built when students are permitted to stop, think, and use strategies that make sense to efficiently solve a problem.
Why is mathematical fluency important?
By building fluency in math, students can efficiently use foundational skills to solve deeper, more meaningful problems that they encounter in the world around them. Fluency contributes to success in the math classroom and in everyday life.
For example, math fluency is useful for:
- adding scores while playing a game
- using mental math to decide the best buy while shopping at a grocery store
- estimating a percent when determining a tip for a delivery driver
- and so much more!
Throughout everyday life, fluent math thinkers use strategies and methods that they understand to efficiently compute operations and check that their answers are reasonable.
“While being fluent with math facts doesn’t make word problems easy, it does reduce the number of cognitive resources needed to tackle the computation portion of the process, allowing those resources to be allocated to other components of the process.” –Differentiated Teaching
3 Stages of Fluency Defined by Florida’s B.E.S.T. Standards for Mathematics
Let’s examine the three stages of fluency as defined by Florida’s B.E.S.T Standards for Mathematics.
Stage 1: Exploration
- Students investigate arithmetic operations to increase understanding by using manipulatives, visual models, and engaging in rich discussion.
- Models help build on prior learning and make connections between concepts.
- Exercises classified as Stage 1 will prompt students to use a model to solve.
Stage 2: Procedural Reliability
- Students utilize skills from the exploration stage to develop an accurate, reliable method that aligns with the student’s understanding and learning style.
- Students may need the teacher’s help to choose a method, and they are learning how to use a method without help.
- Students choose any method to solve problems independently. Then students are asked to describe their method to ensure that they understand the method and why it works.
Stage 3: Procedural Fluency
- Students build on their conceptual understanding from Stages 1 and 2 and use an efficient and accurate procedure to compute an operation, including the standard algorithms.
- Students are no longer asked to describe their method because they are proving that they can solve accurately and without assistance.
Note: Embedded within Stages 1-3 is Automaticity. Automaticity is the ability to act according to an automatic response which is easily retrieved from long-term memory. It usually results from repetition and practice.
How do math programs and curriculum incorporate fluency?
When looking for a new math curriculum, districts should consider math programs that use a variety of models (Stage 1) and strategies (Stage 2) as well as standard algorithms (Stage 3) to teach math.
Practice problems should encourage the use of various methods to solve problems as well as student explanations of the methods they choose to use (Stage 2). Student exploration, collaboration, and peer discussion will also aid students in the development of their mathematical thinking.
Programs that integrate foundational mathematical thinking and reasoning skills will help students become mathematical thinkers who can strategically choose efficient methods to solve problems.
By acquiring mathematical fluency, students will have a greater cognitive capacity to solve more complex problems in the real world.