Friday / September 13

# Math Fluency Practice is not a Worksheet!

Short description: Students do not achieve fluency in mathematics by completing endless pages of basic facts problems. Learn what mathematical fluency really means and how worked examples, math routines, and math games will help your students develop it.

Have you ever googled the phrase ‘math fluency practice’ and then clicked on images? Here is what you might find:

We find this disheartening, if not tragic. Fluency practice worksheets fail in (at least) three ways.

1. They limit the perception of fluency as “accuracy” of basic facts,
2. They over-scaffold instructions, which robs students of the opportunity to think,
3. They are boring. Math never has to be this boring.

But, what does fluency then really mean and what should be done instead?

### What is Fluency in Mathematics?

There are three major components to all procedural fluency: efficiency, flexibility, and accuracy, each of which can be broken down into observable ‘fluency actions.’

Source: J. Bay-Williams & J.J. SanGiovanni, Figuring Out Fluency in Mathematics Teaching and Learning: Moving Beyond Facts and Memorization, Grades K-8, Copyright 2021 by Corwin. Adapted with permission from D. Spangler & J. Wanko (Eds.), Enhancing Classroom Practice with Research behind Principles to Actions, copyright 2017, by the National Council of Teachers of Mathematics. All rights reserved.

Not only must real fluency contain all of these components and actions, it also applies to all operations and all number types: whole numbers, rational numbers, algebraic expressions, and beyond. In short, fluency is more than quickly recalling facts and smoothly performing algorithms. Look at one of the worksheets your search delivered and ask yourself “Which of these fluency components and actions does it support and how well does it support them?” Our bet is that it doesn’t support most of them. Worksheets like these tend to have three serious shortcomings:

1. They pay exclusive attention to Accuracy. Because real fluency practice must balance all three components of fluency, that means effective practice will attend to selecting appropriate strategies, adapting them as needed, and applying a strategy to a new problem type, and so on. Note that the efficiency and flexibility actions focus on student thinking. Procedural fluency is about thinking, not about answer getting.
2. They give instructions with (mis)directions. There are actually (at least) three fatal flaws in worksheet instructions. First, they may say, “Solve using ___________ method.” Nothing takes fluency thinking out of fluency practice faster than telling students how to do the problem. Second, there are sometimes reminders, like, “Remember to subtract the ones column first.” This is, by the way, incorrect mathematics unless the purpose is to practice the U.S. Standard algorithm, in which case, it is not fluency practice, but rather practice to master a specific algorithm. To be clear, mastery ≠ fluency. Third, you often see the instruction, “Show your steps” or some version of this. Well, fluency includes efficiency. If the learner can solve the problem in their head, they should.
3. They are boring. Emotions Matter. How does it feel to see a page of dozens of problems to compute? Did you feel a little anxiety? Does it seem appealing or is it uninteresting? Long pages of computation don’t feel good to most any learner. Emotions are connected to achievement. Math anxiety is connected to achievement. Clipart and artsy borders are not going to make the worksheet any less distasteful or more engaging to students. Even if your students were only working on the accuracy component of fluency, the impact of assigning them lengthy practice worksheets could be hurting their progress, not helping it.

### What is Fluency Practice?

Put simply, fluency practice is an opportunity to practice all of the fluency actions described in the diagram, with the goal of eventually being able to make efficient choices to solve a given problem.

Here are three instructional strategies that attend to the often neglected fluency components (efficiency and flexibility).

### 1. Analyzing Worked Examples:

Using worked examples moves the attention to strategy choice and focuses on student thinking. Correctly worked examples highlight effective strategies. Pose a problem and give students time to digest what the student did.

Then, ask,What did Jojo do? Why does this method work? Is this a good method for this problem? When might you use this idea?” You can post these questions and small groups can discuss their responses. These carefully sequenced questions focus on efficiency (is this a good method?) and flexibility (when is this strategy useful?).

### 2. Fluency Routines:

Different routines can focus on various elements of fluency. For example, you might want to focus on selecting an appropriate strategy. Once students have learned a standard algorithm, they may feel it is the method they should use. Real fluency practice helps students decide when they need that algorithm and when the problem can be solved more efficiently with a different strategy. The routine here simply has students take a step back and consider if they will use the standard algorithm or not.

Routine: “That One”

Materials: This routine does not require any materials.

Directions:

1. Pose a few problems to students.
2. Have students discuss with a partner which problems are good candidates for solving with an algorithm and which are not.
3. Have students explain their decisions. Be sure to avoid insinuating that any problem should or must be solved with an algorithm.

Example Set:

 472 – 376 519          – 304 856– 448

### 3. Games:

Games often times result in much more practice than a worksheet. And, they open up the opportunity to incorporate effective learning strategies, such as thinking aloud, using sentence frames, and engaging in peer tutoring. As students play a fluency game (i.e., one designed to work on the actions described above), they practice strategies and you are able to listen and learn about students’ progress towards fluency.

 Game: Strategy Spin Materials: Strategy spinner and set of cards with problems (could be a cut up worksheet!) Directions (in brief): Players take turns spinning the strategy spinner. The player looks through the cards to find a problem they want to solve using that strategy and talks through the strategy to ‘win’ the card. Repeat with next player.

Alternatively, a spinner can focus on whether or not to use the standard algorithm. For online learning, use virtual spinners like Wheel Decide (https://wheeldecide.com/).

<A> What about those worksheets? You can adapt those worksheets you found to address more components of fluency. For example, change the instructions to ask students to solve using an efficient strategy  or to use at least three different methods as they work through the page. You could show a few problems already solved and ask students to analyze the worked example. Of course, these worksheets must include sets of problems that lend to different strategies. This worksheet re-set would eliminate those ‘free downloads’ that ask students to use the standard algorithm for problems like this: 305 – 289. These worksheets aren’t free. They come with a high cost of misdirecting our efforts to truly develop procedural fluency.

Jennifer M. Bay-Williams is a professor of mathematics education at the University of Louisville, where she teaches preservice teachers, emerging elementary mathematics specialists, and doctoral students in mathematics education.

John J. SanGiovanni is a mathematics supervisor in Howard County, Maryland, where he leads mathematics curriculum development, digital learning, assessment, and professional development. Together they are the authors of the forthcoming Corwin Mathematics book Figuring Out Fluency in Mathematics Teaching and Learning: Moving Beyond Basic Facts and Memorization, Grades K-8