This is the second in a series of instructional articles to support teachers in implementing the 8 Effective Teaching Practices outlined in the book Principles to Actions: Ensuring Mathematical Success for All. This article discusses teaching practice 6: Build Procedural Fluency from Conceptual Understanding.
Ensuring students have a deep understanding of mathematics ideas and number fluency is essential for all other aspects of mathematics the gateway to upper level mathematics. Currently, there is a lot of discussion and conversation about fluency in mathematics. The term appears nine times within our current state standards for mathematics beginning in kindergarten through grade six.
So what do we mean when we say a student does something “fluently?” Many people think that students need to know the standard method for adding, subtracting, multiplying and dividing to demonstrate fluency. But merely knowing and following the steps of the standard method does not demonstrate fluency. To be fluent in mathematics, one must demonstrate competency in three areas- efficiency, accuracy and flexibility.
Let’s take a look at the three components of fluency and what each represents. Efficiency implies that the task is completed in a quick, timely manner. The calculations should be automatic, not requiring think time. Accuracy implies free from error. Flexibility implies the freedom to choose from various methods depending on the numbers involved. The student has the flexibility in selecting the method to complete the task, as long as it is efficient and accurate. These three components together describe what it means to be fluent in mathematics.
To look more deeply at each of these components, consider these multiplication problems:
- 8 X 5
- 7 X 19,999
We would not expect a student who is mathematically fluent to use the same method to solve these multiplication problems. For example, we would hope that a 5th grade student would have the first answer memorized as a basic fact. Would memorization be the method for the remaining problem? Absolutely not!
Fluent students would rely on their conceptual understanding of place value to complete the remaining problem in an efficient and accurate manner. They would round 19,999 to 20,000 and then multiply by 7. Next, they would know to subtract 7 to get the final answer. This is a more efficient approach for solving this problem rather than using the standard method. Students who use this approach are more efficient, accurate and flexible in their understanding of multiplication, place value, and number sense and are demonstrating their fluency in multi-digit multiplication. To see additional examples of students demonstrating flexibility, see the Math Reasoning Inventory which provides some video interviews of students who break numbers apart and use the standard method to multiply 15 x 12.
Although fluency is the goal, specific milestones can be identified along the way. This is why we see fluency appear in Maine’s College and Career Readiness Standards for Mathematics multiple times beginning in kindergarten through grade 6. These milestones help students build towards fluency over time.
The key to supporting students in developing fluency is offering opportunities for multiple strategies in problem solving. All teachers should actively embed opportunities for students to offer multiple strategies for problem solving. Educators, who actively employ teaching practice 6, support “students in developing the ability to understand and explain their use of procedures, choose flexibly among methods and strategies to solve contextual and mathematical problems, and produce accurate answers efficiently” (Principles to Actions, 2014, p. 46). In addition, educators who use teaching practice 6 support students’ conceptual understanding and builds confidence in knowing the best way to solve mathematical problems.
Illustrative Mathematics and youcubed provide a variety of tasks and Inside Mathematics provides tasks and classroom videos of resources that can be used by teachers to support fluency and to offer opportunities to students for problem solving. You will also find an article at youcubed.org titled Fluency without Fear which provides research-based evidence on development of fluency with math facts. Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement describes ten instructional shifts, the associated research, and a summary of what one should expect to see in an effective mathematics classroom. Five of the ten shifts are particularly supportive of fluency. Educators across the state can support fluency by making the five shifts below a routine part of their instructional practice:
- Incorporating ongoing cumulative review into every day’s lesson.
- Using multiple representations of mathematical entities.
- Taking every available opportunity to support the development of number sense.
- Tying the math to such questions as: “How big?”, “How much?” and “How far?” to increase the natural use of measurement throughout the curriculum.
- Making “Why?,” “How do you know?” and “Can you explain?” classroom mantras.