Procedural Fluency

I know I said last time that I would write about long division. That will have to wait. I feel some urgency to write a bit about fluency in mathematics.

The Utah Core State Standards in mathematics require students to acquire fluency in basic math facts as well as in using other mathematical procedures. The chart below lists the fluency targets in the core and when they are to be achieved.


What does fluency mean? For some, perhaps most of us, when we hear fluency we think of speed. Those who are fluent are those who can do the most facts the fastest. However, fluency is defined in Adding it Up: Helping Children Learn Mathematics, the seminal work on research in what works in elementary mathematics teaching and learning as follows: “Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing then flexibly, accurately, and efficiently.” The core does define fluency as being fast and accurate when using mathematical procedures, however speed is not the major factor.

Now, none of that should be read to mean that students shouldn’t memorize math facts. As shown in the chart above, kindergartners should know addition facts with sums up to 5 and subtraction facts with minuends up to five by memory by the end of kindergarten. Likewise, third graders are expected to know from memory the products of all combinations of two one-digit numbers (up to 9 x 9) by the end of third grade. Getting the facts down is important to later success in mathematics, including Algebra and beyond. Memorizing, however, is just one part of fluency.

In order to really be fluent, students must understand the mathematics of the facts they are expected to memorize. Fluency comes at the endpoint of clear learning progressions that are well documented in the core standards that lead to conceptual understanding. By spending a great deal of time working with and reasoning about numbers and the operations that combine them students come gradually to that fluency. Click here for a You-Tube video that explains this concept a little more. As students work toward being fluent with addition, subtraction, multiplication, and division they should have multiple experiences, guided by the standards in the core, that help them along the path. They should have plenty of opportunities to use concrete and pictorial representations of math problems.

As students work toward addition and subtraction fluency they may also use such  mental math strategies as:

  • Counting on: 8 + 4 = □ (8 …9, 10,11,12)
  • Counting back: 12 – 4 = □ (12…11, 10, 9, 8)
  • Making tens: 5 + 7 = □ (5 = 2 + 3 so 3 + 7 = 10 therefore 10 + 2 = 12)
  • Doubles: 6 + 6 = □
  • Doubles plus/minus one: 6 + 7 = □ (6 + 6 + 1 or 7 + 7 – 1)
  • Decomposing a number leading to a ten: 15 – 7 = □, so 15 – 5 = 10, therefore 10 – 2 = 8)
  • Working knowledge of fact families/related facts: 3 + 9 = 12 so 12 – 9 = □

As they work toward fluency in multiplication and division they may use such mental strategies as:

  • Doubles (2 x 2 = 2 + 2)
  • Double and double again (4 x 2 = (2 x 2) x 2)
  • Halve, then double (6 x 8 = (3 x 8) + (3 x 8))
  • Doubles plus one more set (3 x 7 = (2 x 7) + 7)
  • Add one more set (6 x 7 = (5 x 7) + 7)
  • Decomposing into known facts (i.e., use facts you know to solve the ones you don’t)
  • Halves (12 ÷ 2 = 6)
  • Multiplying by zero and one
  • Patterns in 9’s
  • Fact families
  • Number bonds

Other experiences include gaining a firm understanding of place value and the properties of operations. These understandings are particularly important when students begin working with multi-digit numbers in any of the four operations.

Now, just a word about timed tests. Timed tests are widely used through the state as a means of developing fluency in math facts. Timed tests, however, have become somewhat controversial. Dr. Jo Boaler, a mathematics education professor at Stanford has conducted research showing that “students as young as five years old are given timed tests-even though these have been shown to create math anxiety in young children.” ( .

Dr. Cathy Seeley, former president of the National Council of Teachers of Mathematics, in her book Faster Isn’t Smarter, states, “While computational recall is important, it is only part of a comprehensive mathematical background that includes more complex computation, an understanding of mathematical concepts, and the ability to think and reason to solve problems. Measuring this one aspect of mathematics— fact recall—using timed tests is both flawed as an assessment approach and damaging to many students’ confidence and willingness to tackle new problems.” She further states, ” Overemphasizing fast fact recall at the expense of problem solving and conceptual experiences gives students a distorted idea of the nature of mathematics and of their ability to do mathematics. Some students never survive this experience and they turn away from mathematics for years, sometimes forever. Having experienced timed tests when they were students, many adults believe that accurate, fast computation is the most significant part of mathematics. When pressed, many of these adults who dislike or fear mathematics attribute these negative feelings to experiences from their school years, especially the use of timed tests.” (

Fluency in mathematics comes about from a clear understanding of mathematical ideas. It comes as students reason about numbers and use them in a variety of contexts and with a variety of representations. Teachers should use a wide variety of experiences to help students memorize the basic facts and to be fluent with other procedures. Mental math strategies are extremely helpful. Number Talks can help students develop those mental strategies. See for some video examples of number talks in classrooms. Games are also useful. Just google “math fluency games” to find many on-line examples.  Certain software products can also help.  Think creatively and mathematically!


About MathDavidUtah

Husband of 1 gorgeous woman. Father to four intelligent boys. Father in law to three. Grampy to 7 (Charlie, Meili, Camden, Grayson, Marick, Diem, Remi)! Ham radio callsign K7CMG. Educational Leadership Specialist at the Utah State Board of Education.
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6 Responses to Procedural Fluency

  1. Michael Coonen says:

    I love your comment about how timed tests can result in negative feelings by students and later by adults. I teach third grade so I am in the midst of teaching multiplication fluency. I truly believe that fluency in subjects like multiplication facts must be taught rigorously using recitation and speed competition BUT, it must be balanced by praise for the attempts by slower students and just as importantly, it should be balanced by daily instruction and assessment that emphasizes a slow, deliberative process for solving problems. Accuracy is the emphasis. I teach and ask my students to recite back to me, ” I want your best work” and they respond “not your fastest”. I am a former engineering manager for 20 years before I entered teaching and I tell the students “would you want to drive over a bridge that an engineer raced to design or one that took his time and made sure it was built right?” My greatest priority in math is for all my students to leave class with a positive attitude about their ability to do math. What ever time it takes for them to find solutions to complex problems or simple multiplication facts is acceptable (of course within reason). I believe this trumps their problem solving rate.
    Thanks for your blog,
    Mike Coonen
    Third Grade Teacher
    Iron Springs Elementary
    Cedar City, UT


    • dsmith1953 says:

      Thanks, Michael for your comment. I am very impressed with what you are teaching your students. I would love to come and visit your classroom! I’ll be down that way in January. Perhaps we can set something up.


  2. April Leder says:

    I love this David! Marilyn Burns has said that timed tests are not instructional tools. In other words they are not a means for teaching students their facts. If teachers simply give timed tests day after day without teaching students strategies to improve their score, then they are not teaching. After teaching strategies such as those David has listed, I give a timed test as an assessment. Then I create cards for the facts that each child needs to practice. We practice those facts for about two weeks before retesting and going through the process all over again. If as teachers we put all our emphasis into memorizing the facts without any understanding, are we really teaching our students to be mathematicians?


  3. Cheryl Nielson says:

    Amen! Amen! Amen! Just like reading words very quickly without being able to retell what you’ve read isn’t considered reading fluency, mindless memorization and recall of math facts without understanding how to use them and what they mean is not learning math!


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