To set the stage for figuring out fluency for rational numbers and algebraic equations, find the solution to each of these problems using a method other than what you learned as the standard algorithm:

If you had difficulty strategizing beyond using the standard algorithm, you are in good company. It is common for adults to have multiple strategies for adding 48 + 49 or multiplying 12 × 25, but once the problem moves beyond whole numbers, that flexibility and strategy selection diminishes. Here are possible alternatives to the standard method for these examples:

Reflect on how you thought about these problems. How did you decide which strategy to use? Did you start with one strategy and shift to another? Fluency involves selecting strategies that are efficient for the numbers given in the problems. While some problems lend to a single option, other problems have several efficient methods. For example, 3(x + 5) = 6x + 9 could have been solved about as efficiently by first applying the distributive property on the left side to get 3x + 15 = 6x + 9. Standard algorithms are not always efficient, either. For example, a fluent person will not use the standard algorithm to solve 14.99 + 7.07. Real fluency in mathematics is the ability to select efficient Page 3  |  Top of Articlestrategies; to adapt, modify, or change out strategies; and to find solutions with accuracy. Fluency involves decision-making, reasoning, and reflecting on how the process is going, instead of simply replicating steps or procedures for doing mathematics.

Procedural fluency has several subsets. Basic fact fluency attends to fluently adding, subtracting, multiplying, and dividing single-digit numbers. Computational fluency refers to the four operations. Procedural fluency encompasses both basic fact fluency and computational fluency (see Figure 1 ), along with other procedures like solving proportions for missing values and solving systems of equations.

 
FIGURE 1 The Relationship of Different Fluency Terms in Mathematics

Procedural fluency is defined as solving procedures efficiently, flexibly, and accurately (Kilpatrick et al., 2001; National Council of Teachers of Mathematics [NCTM], 2014 ). These three components are defined as follows:

Strategies are not the same as algorithms. Strategies are general methods that are flexible in design; algorithms are established steps implemented the same way across problems.

To focus on fluency, we need specific, observable actions that we can look for in what students are doing in order to ensure they are developing fluency. We have identified six such actions. The three components and six Fluency Actions, and their relationships, are illustrated in Figure 2 .

 
FIGURE 2 Procedural Fluency Components, Actions, and Checks for Reasonableness

Source: 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.

Three of the six Fluency Actions attend to reasonableness. Fluency Actions and reasonableness are described later in Part 1 , but first, it is important to consider why this “bigger,” more comprehensive view of fluency matters.

There are important reasons to focus on real fluency. First, the traditional teach-the-algorithm-only approach has not worked. Memorization approaches result in lower achievement (e.g., Braithwaite et al., 2018; Lortie-Forgues et al., 2015; Newton et al., 2014; OECD, 2010, 2016 ). The Programme for International Student Assessment (PISA), a study of about 250,000 15-year-olds, analyzed learning strategies across 41 countries and found that students’ use of memorization/rehearsal strategies is almost universally negatively associated with Page 5  |  Top of Articlelearning (OECD, 2010 ). Conversely, we have strong evidence that a focus on reasoning strategies improves student achievement (c.f., Baroody et al., 2016; Brendefur et al., 2015; Siegler et al., 2010; Torbeyns et al., 2015 ). Having fluency with fraction operations predicts success in algebra and in mathematics in general through high school (Bailey et al., 2012; Siegler et al., 2012; Torbeyns et al., 2015 ). In solving algebraic equations, students who learned strategies and compared methods achieved greater gains in conceptual knowledge, procedural knowledge, and flexibility, as compared to control students (Rittle-Johnson & Star, 2007; Rittle-Johnson, Star, & Durkin, 2012 ). Importantly, a fluency focus is about equity and access. Equipping students with options for working fluently with rational numbers and algebraic equations not only improves their performance, it also develops a positive mathematics identity and sense of agency because students are positioned as capable of deciding how to solve a problem.

The six Fluency Actions are observable and therefore form a foundation for assessing student progress toward fluency. Let’s look at each of these actions in the context of rational numbers and algebraic equations.

FLUENCY ACTION 1: Select an Appropriate Strategy

Selecting an appropriate strategy does not mean selecting the appropriate strategy. Many problems can be solved efficiently in more than one way. Here is our operational definition: Of the available strategies, the one the student opts to use gets to a solution in about as many steps and/or about as much time as other appropriate options.

Sometimes a standard algorithm is the most efficient option to solve a problem, but sometimes it is not. Teaching students to select appropriate strategies involves helping students notice features of a problem that help to decide which is more efficient: a reasoning strategy or a standard algorithm. For example, when adding integers, noticing that the signs are different is a feature that indicates Make a Zero strategy may be a good option (e.g., −22 + 39 can be thought of as −22 + 22 + 17). Or, when multiplying decimals, noticing that a number is one-tenth away from a whole suggests compensation may be efficient (e.g., 5.9 × 8 can be adjusted to 6 × 8, then subtract 0.1 × 8, or 0.8). While written out here, these methods can be done mentally. Importantly, students can only select appropriate strategies if they have learned more than one way to solve a problem! With fraction addition, for example, there are more ways than just adding the wholes and parts and then simplifying. Figure 3 shows two other strategies that are useful options that students should understand and be able to use.

Students need to learn different strategies so that they can choose efficiently appropriate strategies. Throughout this book, we name strategies so that we can talk about them. But strategies overlap. For example, both strategies illustrated in Figure 3 are also Partial Sums strategies. The focus must be on the ideas (not the naming of the strategy).

 
FIGURE 3

Note to Figure 3: Two Appropriate Strategies for Adding

FLUENCY ACTION 2: Solve in a Reasonable Amount of Time

The time it takes to solve a problem will vary with the complexity of the numbers or length of an expression. Students should be able to work through a problem without getting stuck or lost. Appropriate strategies can be carried out in inefficient, unreasonable ways. For example, students may add −5.7 + −4.8 by counting on in efficient or inefficient ways, as illustrated in Figure 4 . But how efficient a strategy is depends on experience. The first example in Figure 4 is initially a reasonable method that reinforces the meaning of adding negative numbers. With more experiences, jumps can be chunked to solve the problem more quickly.

 
FIGURE 4 Different Ways to Use a Counting Strategy to Add −5.7 + −4.8

FLUENCY ACTION 3: Trade Out or Adapt a Strategy

As students’ number sense and understanding of strategies advance, they are able to adapt and trade out strategies. For example, to solve for the missing value in the proportion
a student might first think cross products, but then notice that
can be simplified to
At this point, they notice the relationship between the two denominators (×3) and use that relationship to solve for x.

This Fluency Action is one of three connected to reasonableness. If a strategy isn’t going well, then the strategy might need to be adapted or traded for another option. This is key to solving equations for an unknown, wherein a common first step, like applying the distributive property, can make a problem messier. A fluent student will stop and trade out the method for another approach:

First effort:

Second effort:

FLUENCY ACTION 4: Apply a Strategy to a New Problem Type

This Fluency Action means that you can take a strategy, like Make Tens, that was originally used with whole numbers, apply it to decimals or fractions (Make a Whole), and then connect it to negative numbers as Make a Zero. The importance of this Fluency Action with rational numbers cannot be overstated. Students benefit from seeing how a strategy learned with fractions also works with decimals (and vice versa).

FLUENCY ACTIONS 5 AND 6: Complete Steps Accurately and Get Correct Answers

These two Fluency Actions are about accuracy, which has been the primary focus of teaching and assessing mathematics, often at the expense of the other actions. But attending to accuracy is important. For example, a wrong answer may be due to an error in how to enact a strategy or an error may be due to a computational mistake. Take a look at the three wrong answers in Figure 5 and decide if the error is related to the strategy or to a computational mistake. In the first error, the student is not understanding integer subtraction and/or how to implement Count Back strategy. In the second example, the student understands how to use Compensation, but makes an error in solving −30.6 + 0.2 (treating the negative side of the number line like the positive side where decimals count up as they go to the right). In the third example, the student makes a counting or adding error.

 
FIGURE 5 Incorrect Solutions for −17.6 − 12.8

Fluency Action 6 is one of three connected to reasonableness. Within this action is noticing if your answer makes sense. In Figure 5 , the answer of −4.8 is not reasonable because taking away a whole number should result in a smaller number, not a larger number. The other two results are both reasonable though incorrect. While reasonableness has been woven into the discussion of Fluency Actions, it is critical to fluency and warrants more discussion.

REASONABLENESS

A fluent student determines if their strategies and results are reasonable. It takes time to develop reasonableness, and it should be practiced and discussed often. Students can develop reasonableness by practicing these three moves (a match to the Fluency Actions 1, 3, and 6 and illustrated in Figure 2 ).

Let’s explore how reasonableness plays out for Jessie in solving −65.5 − −49.8.

Provide students Choose, Change, Check reflection cards to encourage and support those thinking about reasonableness (see Figure 6 ). You can adapt these cards into anchor charts for students to use while working on problems and refer to them during discussions.

 
FIGURE 6 Choose, Change, Check Reflection Card for Students

Icon sources: Choose by iStock.com /Enis Aksoy; Change by iStock.com /Sigit Mulyo Utomo; Check by iStock.com /Indigo Diamond.

This resource can be downloaded at resources.corwin.com/FOF/rationalnumbersalgequations .

Fluency is based on conceptual understanding. As students expand their work to include negative numbers and variables, it continues to be important to start with contexts that build meaning and visuals to support reasoning. Manipulatives and visuals support conceptual development (Cramer et al., 2002; Monson et al., 2020 ). For example, number lines are an important representation for numerous reasons. Number lines support:

Area and set models also support understanding (Van de Walle et al., 2019 ). For example, fraction circles, fraction strips, Unifix or multilink cubes, base-10 blocks, two-color counters, and algebra tiles help students make meaning of quantity and the action of the operation while also helping them reason about the results. Contexts and visuals also support reasoning strategies and appear through Part 2 of this book. Although the focus of this book is on reasoning strategies—not on developing initial conceptual understandings—here is a short list of what these critical conceptual underpinnings are.

EQUIVALENCIES

Cutting across all rational numbers and algebraic equations is the critical importance of understanding equivalency. Knowing equivalency is both a conceptual foundation and an automaticity. As a conceptual foundation, it means that students “see” that four-tenths means all of these things:

There are infinitely many other equivalencies for four-tenths. Students also understand that if they have any fraction and they want to represent it as a decimal, the goal is to find an equivalent fraction with a denominator of 10, 1 100, and so on. To find the decimal equivalent to
for example, students might ? reason that there is no equivalent fraction in the form
so they look for an equivalency in the form
which turns out to be

Area models and number lines can help students see the relative size of the numbers: for example, finding multiple names for this value on a number line:

Equivalency is at the heart of proportional reasoning—for example, recognizing that when a recipe is in a ratio of four parts water to one part sugar, then there are infinitely many options, including eight cups of water to two cups of 1 sugar and one cup water and 4 cup sugar. In algebraic equations, students must know which changes they can make that maintain equivalence.

FRACTIONS

Fractions are numbers. This may sound obvious, but many students come to understand fractions as a number over a number or shaded parts over total parts. Researchers include the following significant fraction concepts (Cramer & Whitney, 2010; Empson & Levi, 2011; Lamon, 2020 ):

DECIMALS

Decimals are sometimes considered “easier” than fractions, perhaps because the algorithms we use with decimals parallel whole number algorithms. But the conceptual understanding of decimals is at least as challenging as fractions (Hurst & Cordes, 2018; Lortie-Forgues et al., 2015; Martinie, 2014 ). Number sense with decimals is therefore dependent on an understanding of place value and fractions. The following are important conceptual understandings to support student fluency:

Using appropriate language (tenths and hundredths) throughout the process further supports both the connection between fractions and decimals and the fundamental idea that these numbers are parts of a whole.

NEGATIVE NUMBERS

Negative numbers are introduced with integers—the whole numbers and their negatives or opposites—instead of with fractions or decimals. But students must understand that a number like −5.25 is to the left of −5 (the opposite of where it is located with positive numbers). The following are important conceptual understandings to support student fluency:

ALGEBRAIC EQUATIONS

Like rational numbers, algebraic equations involve new notations. Beyond notations, students must be able to apply properties and the order of operations in new ways to determine missing values. Significant conceptual foundations include:

 
FIGURE 7 Concept Map for Systems of Linear Equations

Source: Proulx, J., Beisiegel, M., Miranda, H., & Simmt, E. (2009 ). Rethinking the teaching of systems of equations. Mathematics teacher, 102(7), 526–533.

Automaticity is the ability to complete a task with minimal attention to process. Automaticities help students select strategies, move between strategies, and carry out a strategy. To begin, automaticity with basic facts warrants special attention. Students must know their facts (i.e., be automatic) and know the strategies that come with basic fact fluency (Bay-Williams & Kling, 2019 ). These strategies grow into useful strategies for rational numbers, including whole numbers, fractions, and decimals. Figure 8 shows this relationship for addition and subtraction fact strategies.

 
FIGURE 8 How Addition and Subtraction Fact Strategies Grow Into General Strategies

Multiplication fact strategies also generalize, as shown in Figure 9 . Negative numbers fall across these categories and simply have an additional step of determining the sign of the answer.

There are automaticities beyond the basic facts that students need to be able to employ reasoning strategies: for example, using 25s, 15s, and 30s; doubling; halving; using fraction equivalents within fraction families (e.g., halves, fourths, and eighths); and making conversions between common decimals, percentages, and fractions.

 
FIGURE 9 How Multiplication Fact Strategies Grow Into General Reasoning Strategies

All strategies and algorithms are based on properties of the operations. And navigating strategies requires adeptness at various skills.

PROPERTIES

It is important to note that knowing properties does not equal using properties. It is not useful to have students name the associative property. It is absolutely necessary that students use properties in solving problems efficiently. For example, in the problem −8 × 3.5, it is not important that students say, “I am going to factor −8 into −4 × 2 and apply the associative property.” What is important is that they realize they can reassociate numbers and get the same Page 16  |  Top of Articleproduct: (−4 × 2) × 3.5 = −4 × (2 × 3.5) = −4 × 7 = −28. This is the Halve and Double strategy, which uses the associative property.

The distributive property is critical for fluency with multiplication. The examples in Figure 10 show Break Apart to Multiply (left) and Compensation (right). Both are based on the distributive property.

 
FIGURE 10 Using the Distributive Property to Solve Multiplication Problems

The distributive property is at the heart of algebraic reasoning. Consider the relationship between these two visuals to make sense of the distributive property:

In the first case, the result is unknown, and the distributive property is used to multiply partial products. In the second case, the result is typically given. For example: 4(x + 7) = 120 and the goal is to figure out the missing part. Important for algebraic reasoning is to know that the distributive property is bidirectional. It is often presented as multiplication: 4(x + 7) = 4x + 28, but it is often Page 17  |  Top of Articleuseful going the other direction (Ramful, 2015 ). For example, consider how a student might solve this problem:

4x + 28 = 4(2x + 6)

One option is to factor on the left side:

4(x + 7) = 4(2x + 6)

Thus, x + 7 = 2x + 6 and x = 1

In other words, the distributive property doesn’t just mean “distribute”; it means distribute or factor.

The identity property is also central to procedural fluency. The multiplicative identity (1) is useful in multiplicative situations. For example, to solve 1.28 ÷ 0.4, you can think of it as the fraction
and multiply by
in order to have a whole number divisor:
and from this fraction, see that the answer is 3.2. Both the additive identity (0) and the multiplicative identity (1) are utilized in solving equations. For example, multiplying by the reciprocal is an efficient way to solve the equation

UTILITIES AND OTHER FOUNDATIONS

Beyond utility with the properties of the operations, there are a few other special relationships that are utilized extensively. They include knowing: (1) distance from a 10, (2) flexible decomposition, and (3) part–part–whole relationships.

In addition to knowing these utilities, there are other foundations that support reasoning. These skills are not to be gatekeepers, though when students are not adept at these skills, that is what they become. Thus, students must have adequate time for quality practice—practice that is focused, varied, processed, and connected. Figure 11 briefly summarizes important foundations and offers a list of games and routines, included online, that you can use for ongoing, meaningful experiences with the whole class, or for interventions with a subset of students.

 
FIGURE 11 Foundations and Activity List for Quality Practice

These games (and this chart) can all be found at this book’s companion website at resources.corwin.com/FOF/rationalnumbersalgequations .

Just like computation, estimation has strategies and the use of those strategies should be flexible. For multiplication and division of fractions or decimals, students might use any of these methods:

Teaching strategies beyond the standard algorithms is necessary for fluency, but time is limited. Thus, we must ask ourselves, which strategies are worthy of attention? Let’s just take some pressure off here. The list is short, and we must help students see that they are not necessarily learning a new strategy, but they are applying a strategy they learned with basic facts or whole numbers and transferring it to fractions or decimals. In the anchor book, we propose Seven Significant Strategies for the operations, which are shared in Figure 12 .

 
FIGURE 12 Reasoning Strategies for Rational Numbers and Algebraic Equations

Notice that these are ways to reason and not representations. Representations are not strategies—representations support the use of strategies! For example, with proportional reasoning, ratio tables, tape diagrams, and double number lines are representations. These representations support reasoning. Reasoning strategies include generating equivalent ratios, using unit rates, and using scale Page 20  |  Top of Articlefactor (or within/between ratio). When a student says, “I used a ratio table,” ask how they used it—then you will learn what reasoning they used.

The answer to this question is the focus of Parts 2 and 3. Here, we offer general background and guidance to support implementation of the activities in Parts 2 and 3.

EXPLICIT STRATEGY INSTRUCTION

Let’s unpack the phrase explicit strategy instruction. The Merriam-Webster dictionary defines explicit as “fully revealed or expressed without vagueness” (“Explicit,” 2021 ). In mathematics teaching, being explicit means making mathematical relationships visible. A strategy is a flexible method to solve a problem. Explicit strategy instruction, then, is engaging students in ways to clearly see why and how a strategy works, eventually learning when it is a useful strategy. Learning how and when to use strategies empowers students, developing a positive mathematics identity and a sense of agency. Importantly, explicitly teaching a strategy does not mean turning the strategy into an algorithm. Strategies require flexible thinking.

QUALITY PRACTICE

Quality practice is not a worksheet! Quality practice is focused on enacting a strategy, varied in the type of activity, processed by the student to make sense of what they did, and connected to what they are learning. The variety of practice in this book includes worked examples, routines, games, and independent tasks. Worked examples are excellent opportunities for students to attend to the thinking involved with a strategy. Three types of worked examples are useful and focus on different aspects of fluency:

Also, comparing two correctly worked examples is effective in helping students become flexible in choosing methods (Durkin, Star, & Rittle-Johnson, 2017; Star et al., 2020 ). Throughout the modules are dozens of examples, which can be used as worked examples (and adapted to other similar worked examples). Your worked examples can be from a fictional “student” or be authentic student work.

Routines are 5- to 10-minute whole-class interactions that can be repeated many times with different numbers. Routines in one section of this book can often be applied to other sections/topics, so when you find one that your students love, adapt it! Games provide enjoyable practice but are particularly valuable because they invite discussion and peer teaching. Independent activities are designed for individuals to work at their own pace and have independent think time. Collectively, these four ways to practice provide ongoing opportunities for students to become adept at using the strategies.

Most of these activities can be adapted to use for other numbers, strategies, operations, or topics. For example, a routine can be changed from adding integers to adding fractions, or a game that focuses on adding can be adapted to multiplication. Resources like game boards and recording sheets are downloadable and editable files. Knowing this means that you have many more activities than you find on a first count.

ASSESSING STRATEGY USE

Assessing strategy use is hard to do with traditional quizzes and tests. Here are three effective alternatives.

Or a prompt can focus on identifying when that strategy is a good idea:

 
FIGURE 13 Observation Tool for Proportional Reasoning Strategies

This resource can be downloaded at resources.corwin.com/FOF/rationalnumbersalgequations .

With interviews and observations, you don’t need to assess every student the same day. Some days, you collect data on some students; other days, you collect data on other students. For instructional planning, you don’t need to talk to every student to gain important insights to inform what strategies need more attention.

As soon as students know more than one strategy for a given operation, it is time to integrate routines, tasks, independent activities, and games that focus on choosing when to use a strategy. That is where Part 3 of this book comes in. As you read in Fluency Action 1, students need to be able to choose efficient strategies based on those they’ve learned.

Do not wait until after all strategies are learned to focus on when to use a strategy—instead, weave in Part 3 activities regularly. Each time a new strategy is learned, it is time to revisit activities that engage students in making choices from among the strategies in their repertoire. Students must learn what to look for in a problem to decide which strategy they will use to solve the problem efficiently based on the numbers in the problem. This is flexibility in action, and thus leads to fluency.

Part 1 has briefly described factors that are important in developing fluency, and these factors are necessary for implementing the modules. We close Part 1 with five key factors to figuring out fluency:

Time invested in strategy work has a big payoff: confident and fluent students (and that is the “best product”). That is why we have so many activities in this book. Teach the strategies as part of core instruction and continue to practice throughout the year, looping back to strategies that students might be forgetting to use (with Part 2 activities) and offering ongoing opportunities to choose from among strategies (with Part 3 activities).

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