What is fluency? As a literacy specialist in my early career, I equated the word “fluency” with the ability to read with automaticity, accuracy, and prosody (expression using the patterns of rhythm and sound). Simply put, a fluent reader expresses herself fluidly and with expression. I understood fully that fluency is not the same as comprehension, and yet it contributes immensely to the ability to get to the higher levels of thinking required to comprehend well.
Several years later, as I shifted toward a mathematics specialization, I discovered that there is a mathematical equivalent to reading fluency. Much like reading fluency helps children succeed in higher levels of comprehension, mathematical fluency plays a foundational role in helping children succeed in problem solving. Therefore, just as students who struggle with reading fluency find comprehension tasks difficult, students who struggle with mathematical fluency find problem solving tasks difficult. For me, thinking about this in the form of an analogy helps illuminate the relationship:
Reading Fluency : Comprehension :: Mathematical Fluency : Problem Solving
So…what is mathematical fluency, and how can we develop it as a foundation that leads to stronger problem solving? This topic has many layers, and we’ll be revisiting it often over the next several weeks. For now, let’s take a look at what it is not.
Mathematical fluency cannot be reduced to memorization. This limited view of fluency puts undue pressure on students to focus on rote recall of facts. The mathematics community has agreed upon four qualities that comprise mathematical fluency:
- Accuracy: finding the correct answer
- Efficiency: using strategies or methods that allow for ease and flow while solving
- Flexibility: knowing more than one approach to solve a problem and then selecting one that provides for efficiency
- Appropriateness: knowing when to apply a particular procedure, strategy, or method
Students who are mathematically fluent are able to choose flexibly from a variety of methods and strategies to solve problems, and they solve them accurately and efficiently.
Mathematical fluency is not limited to knowing and using basic facts. Rather, it is much broader, involving concepts such as number sense, math facts, multi-digit operations, fraction operations, problem solving — and the list goes on (see below for more on this). Children need opportunities to develop fluency with the concepts on which they are focused at any given time. Students at the elementary level should have opportunities to develop fluency in at least four areas:
- Number Sense: thinking about numbers and number relationships fluidly and flexibly.
- Math Fact Fluency: adding numbers through 10+10 and the related subtraction facts; multiplying numbers through 10×10 and the related division facts.
- Operational Fluency: using strategies to add, subtract, multiply, and divide whole numbers, decimals, fractions, and integers.
- Problem Solving Fluency: selecting from and applying a variety of strategies and methods with the goal of solving a contextual or mathematical problem.
Mathematical fluency does not just happen spontaneously. Fluency develops over time as students are given multiple opportunities to focus on mathematical relationships and manipulations, often as a result of engaging in deliberately designed and sequenced experiences. Here are a few examples to get you started (more to follow in the coming posts):
- Number Sense: use ten frames to help students understand relationships among numbers, both large and small. Ten frames needn’t be restricted to numbers within ten – they can be used to explore multi-digit numbers, decimals, and integers, as well! At KP Mathematics, we call this the “infinite ten frame,” and I’ll share more during this fluency series.
- Math Fact Fluency: use card and dice games to allow students to practice their math facts in ways that are enjoyable and build confidence. For example, Double War is a simple card game where pairs of students each flip over two cards and either add or multiply the two numbers. Each student calls out his sum or product. The student with the greater (or lesser) sum or product takes all four cards. Repeat until all cards are used.
- Operational Fluency: implement daily Number Talks with your students. This robust-yet-simple daily routine helps students develop number sense and mental math skills while focusing on developing operational fluency.
- Problem Solving Fluency: engage students in daily problem solving, giving them tasks that require them to use strategies other than standard algorithms. For example, engage young children in solving a division problem before teaching them “how” to divide. Observe how they approach the process. Provide them with tools and manipulatives to guide their thinking. Resist the temptation to jump in – let them struggle. They will amaze you!!!
If you’re wanting more ideas in each of these categories, tune in for the next few weeks as we unwrap each category and share specific classroom-tested ways to develop fluency with your students.
For now, let’s carry on this conversation. What is your understanding of math fluency? What do you do in your classroom to develop fluency among your students? Please leave your comments in the boxes below.