The Fishtank Math curriculum aims to devote appropriate attention to the three aspects of rigor, one of which is procedural fluency. But what do we mean when we say fluency?
We think the National Council of Teachers of Mathematics (NCTM) sums it up best:
“Procedural fluency is the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another.”
NCTM goes on to explain how teachers can support students in developing this crucial skill set:
“To develop procedural fluency, students need experience in integrating concepts and procedures and building on familiar procedures as they create their own informal strategies and procedures. Students need opportunities to justify both informal strategies and commonly used procedures mathematically, to support and justify their choices of appropriate procedures, and to strengthen their understanding and skill through distributed practice.”
The most important takeaway for us is that procedural skill and fluency is more than rote memorization or execution of algorithms. Instead, it is ownership of a skill or procedure that is rooted in conceptual understanding. When students demonstrate fluency with a skill or procedure, they are able to execute it not only with speed and accuracy, but also with flexibility to meet the demands of new contexts and situations.
Algorithms versus Strategies
So, how do we achieve this kind of procedural fluency? First, two important definitions from the NBT Progression:
A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly
Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another
In certain ways, fluency is explicitly called for in the Common Core State Standards, such as in 5th grade when students learn to “fluently multiply multi-digit whole numbers using the standard algorithm” (5.NBT.B.5). Here, students will need to practice one of those “commonly used procedures” mentioned in NCTM’s definition of fluency above, which can be used to solve any and all computations of a particular type.
However, the standard algorithm may not always be the most efficient and appropriate way to solve a problem. In some cases, a strategy as opposed to an algorithm may be a better choice. While a strategy may not work for all computations, it may be the more efficient way to solve certain problems by converting them into even easier computations.
Take the following example:
28 × 25
Using the standard algorithm, a student's work may look like the following:
The student arrived at the correct answer. However, is there another way they could have solved more efficiently, perhaps without even the need for paper and pencil? Consider each factor’s relationship to benchmark/friendly numbers. How could that help them solve?
There are a few different strategies students might use. For example, they might use a strategy we refer to as “adjusting one number”. Since 28 is close to 30, a student might adjust 28 to be 30, then multiply 30 × 25 to get 750. Then, they account for the adjusted factor afterward by subtracting two groups of 25, or 50, arriving at a solution of 700.
Or, perhaps students use the strategy of “breaking factors into smaller factors” and change this computation to 7 × 2 × 2 × 5 × 5 . Multiplying pairs of 5s and 2s together, we get 7 × 10 × 10, or 700.
Lastly, they might use a strategy called “doubling and halving”, which involves adjusting both factors in the first place so that no later adjustment needs to occur. So, they might halve the factor 28 and double the factor 25, changing this to an equivalent computation of 14 × 50. Doing this again produces the computation of 7 × 100, which is 700.
No matter which strategy is used, it’s easy to see how each of them is a much more efficient path to a solution than stacking like units vertically, computing partial products, adding those partial products together, and regrouping where necessary, as called for in the standard algorithm.
Also note how these strategies might not be applicable to all computational problems. For example, it’s hard to think of how you’d use these strategies for the problem 17 × 23. But, when they are an appropriate choice for a computation, these strategies and others sure are handy.
Number Talks as a Way to Foster Computational Strategies
Seeing how valuable these computational strategies are, the next question that might arise is, “So how do we develop these strategies with students?”. Enter Number Talks!
Number Talks were popularized by Sherry Parrish and are typically a sequence of related computations that allow students to apply strategies from earlier computations to later ones. They can be done in 5 to 15 minutes and can be facilitated with the whole class or in a small group with the teacher.
Let’s return to our previous example of the computation 28 × 25 to see what a Number Talk might look like.
Depending on which strategy is being focused on, the string of computations in the Number Talk may be different. For example, to encourage the use of the “adjusting one number” strategy, students might be presented with the following computations:
2 × 25
3 × 25
30 × 25
28 × 25
Imagine a class of students solving this Number Talk. You might hear a student explain their thinking in the following way:
S: I knew that 2 times 25 is 50 because it’s like having two quarters, which is 50. Same with 3 times 25, three quarters is 75. 30 times 25 is just ten times that much, so it’s 750. But to solve 28 times 25, I already knew what 30 times 25 was so that’s just two more groups of 25 than I want, so I can take away 50 and get 700!
Of course, students might use these facts in different ways from how the student above did. And it’s important to value all students’ contributions, even, and perhaps especially, if they are different from what you might expect.
Consider what a Number Talk might look like that would encourage the use of the “breaking factors into smaller factors” strategy or the “doubling and halving” strategy. Also consider how these strategies transfer to other types of numbers, too. For example, what might a Number Talk look like that encourages students to use the “adjusting one number” strategy to solve 1.2 × 2.9, or to use the “doubling and halving” strategy to solve 1½ × ⅔?
Number Talks in Fishtank Math Curriculum
Neither standard algorithms nor computational strategies alone will result in procedural fluency. Students need both in order to solve accurately, efficiently, and flexibly, choosing the correct solution path based on what’s most appropriate for a given computation.
Our core curriculum includes some instruction on computational strategies but is largely focused on helping students develop a conceptual understanding of algorithms. Thus, we have integrated Number Talks into our Grades 3-5 Fishtank Plus curriculum as part of our fluency activities to help students develop and practice computational strategies and build a toolbox they can use to solve any computational problem.
Fishtank Plus users can access our Number Talks Teacher Tool, which provides teachers with guidelines for how to facilitate a Number Talk; explains the various strategies aligned to each Number Talk, such as “breaking factors into smaller factors” or “doubling and halving” discussed above; and models how to record student thinking.
For each unit, teachers are encouraged to use problems from our Number Talks bank that are aligned to computations relevant to that particular unit. Teachers also have the flexibility to pick and choose problems aligned to any strategy that they may want to emphasize, even if it’s not directly aligned to the work of the unit. They can even design their own Number Talks, using the wealth of information provided with the Teacher Tool and bank to truly customize instruction for their students.
With the Number Talks resources at their disposal, teachers have both flexibility and support to build students’ accurate, efficient, and flexible fluency.
Sarah Britton is Fishtank Learning's Curriculum Director for Elementary Mathematics. Ms. Britton began her career in education through Teach For America Massachusetts, where she taught 7th and 8th grade mathematics. She then joined the staff at Teach For America as a manager of teacher leadership development, supporting and coaching their new math teachers of all grade levels. She has a Bachelor's degree in mathematics from Union College and a Master's degree in curriculum and teaching from Boston University.