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# Fluency Development Within and Across the Grades in IM K–5 Math™, part 1: Addition and Subtraction

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We appreciate these questions for a couple of reasons: 1) We took great care in thinking through fluency development in the design of the curriculum, and we are eager to share how we did it, and 2) We realize that the way we have incorporated fluency development is different from most curricula, and it may be difficult to recognize.

To outline how the IM K–5 Math™ curriculum focuses on fluency, we have a four-part series of blog posts. This first post is a deep dive into our Grade 1 course to show how students develop addition and subtraction fluency within 10. The next post will highlight the development of procedural fluency with addition and subtraction algorithms. The final 2 posts will show the progression of fluency development in multiplication and division.

We hope that this series of blog posts serves to bring to light the intentional design that supports students’ progress toward fluency over time, both within and across the grades. Also included are ways for teachers track and assess students’ math fluency development over the course of the school year.

We define procedural fluency as “using procedures flexibly, accurately, efficiently, and appropriately.” (Adding It Up: Helping Children Learn Mathematics from the National Research Council, 2001) In the IM K–5 Math™ curriculum, the representations, strategies, and algorithms that are used are purposefully designed to build a coherent progression where conceptual understanding and procedural fluency develop in parallel. This progression develops within and across grade levels.

The progression of procedural fluency with addition and subtraction of whole numbers spans Kindergarten through fourth grade. Each of these grade levels has specific expectations for procedural fluency that are interrelated.

The coherence of the curriculum is designed for students of any given grade to work towards accurately and efficiently operating with one set of numbers in the context of building conceptual understanding within a larger set of numbers. This coherence is evident in the way IM K–5 Math intends for students to come to know their facts at any given grade level.

1. Students learn the meaning of the operations and the relationships between them.
2. Students start to learn their facts.
3. Students relate more complicated facts to simpler ones.
4. Students know their facts.

## Fluency with Addition and Subtraction within 10

To better understand the intentional design of the curriculum, we will share how we thought about fluency development in grade 1. Let’s look at how students come to know their facts within 10.

#### 1. Students learn the meaning of the operations and the relationships between them.

The foundation for fluently adding and subtracting within 10 is built in kindergarten, when students first learn about the meaning of addition and subtraction through story problems. The fluency focus of kindergarten is to learn to accurately and efficiently add and subtract within 5, which supports the grade 1 work of composing and decomposing numbers within 10.

In Unit 5 of our Kindergarten course, students compose numbers to 10 in ways that make sense to them. For example, in Lesson 13, Activity 2, students are given a number less than 10, and work with a partner to determine the part needed to make 10. They then write an equation for each 10 they make using the frame 10 = ___ + ___. Students are encouraged to use their fingers, counters, or 10-frames to help them determine the parts needed to make 10.

During the synthesis of this activity, it is suggested that teachers highlight the different strategies used based on the given number. This important step supports students’ development of flexibility. As they hear other students sharing their strategies, they may learn a strategy that is more efficient, and add that strategy to their repertoire.

#### 2. Students start to learn their facts.

With a foundational understanding of addition and subtraction, students begin to know a few facts. Unit 1 of our Grade 1 course begins with activities and centers focused on addition and subtraction within 10. The work in this unit allows teachers to assess students’ understanding of addition and subtraction, as well as their fluency with facts within 5, a kindergarten goal. To develop fluency within 10, students start with activities where they add or subtract 1 or 2, which not only encourages students to use strategies such as counting on or counting back, but also helps them relate addition and subtraction to counting.

Here are some examples of how students continue to practice sums within 10 during warm-up routines throughout Units 1 and 2. As displayed below, by Unit 2, students are noticing and using relationships between facts to mentally find differences.

The centers within our materials play an important role in developing fluency. Centers afford students opportunities to practice by recalling facts, computing mentally, and writing true equations. In the examples that follow, students work with either cubes or number cards to make 10 with one part given. As students share their thinking and write the equations, they begin to learn new facts.

#### 3. Students relate more complicated facts to simpler ones.

According to NCTM, “effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.” (Principles to Actions NCTM, 2014) It’s no secret that IM aims to help students develop conceptual understanding in mathematics. In IM K–5 Math™, the overall design builds conceptual understanding while simultaneously developing students’ procedural fluency. Students practice their facts in ways that allow them to show or explain their thinking. They analyze and learn to use representations, strategies, and algorithms in a purposeful and coherent way.

An example in the curriculum of the parallel development of procedural fluency and conceptual understanding is the way students transition from knowing a few facts to relating the facts they know to more complex facts. In Unit 3 of our Grade 1 course, students begin by naming the sums within 10 that they know. In Lesson 1, Activity 2, students practice their facts using cards with different addition expressions, such as 2 + 5 and 4 + 4. They sort them into the ones they can do right away and the ones they are still working on.

During the activity synthesis of this lesson, teachers name the year-long fluency expectations for addition within 10, and support students in reflecting on the facts they know and the facts they still need to practice. This process supports students in building positive math identities and encourages them to connect new learning to what they already know.

As the unit continues, students deepen their understanding of the commutative property, equivalence, and the relationship between addition and subtraction, and apply this understanding to use simpler facts to find more complex ones.

#### 4. Students know their facts.

Over time, after a variety of experiences where they have shared their developing thinking, learned new facts, and practiced relating more complex facts to simpler ones, students begin to know more facts.

Throughout Units 4–8 in our Grade 1 course, students continue to practice their addition and subtraction facts during centers and warm-up activities. They also have multiple opportunities to reflect on which facts they know and which facts they are still learning.

The example below shows this practice continuing through the final unit of the course. In Lessons 1 and 2 of Unit 8, students make note of the sums and differences they know from memory, and create cards to practice the facts they don’t know yet.

## Assessing Fluency

Because fluency is developed over time and practice takes place in many different ways in our curriculum, student progress with fact fluency may seem difficult to measure. To address this challenge, IM K–5 Math™ includes monitoring sheets for each grade level that align with the section goals of each unit. The example below shows the progression of grade 1 monitoring sheets over the course of the year that support teachers with assessing students’ progress toward fluency within 10.

## A Thoughtful Approach to Developing Students’ Math Fluency

To our current and prospective IM K–5 Math™ teachers, we hope we were able to display the care that went into designing a curriculum that intentionally distributes opportunities for students to practice their math facts in activities, centers, practice problems, and warm-ups. (Read more about distributed practice.) Furthermore, with monitoring sheets that align with the fluency requirements for each grade level, teachers can track and assess student progress.

Our approach to math fluency is aligned with our vision “to create a world where all learners know, use, and enjoy mathematics.” Throughout the journey towards becoming fluent with operations, students are given opportunities to connect what they know to what they are learning. By pairing these opportunities with joyful, shared experiences among a community of learners, students can collaboratively build on the strategies they know and develop useful and efficient computation methods, leading to fluency.

### Next Steps

How do students develop fluency with addition and subtraction within 5? Within 20? We encourage you to look at the curriculum for similar progressions in Kindergarten and Grade 2.

Watch this video where lead writer Dionne Aminata shares the intentional design of the curriculum and students show their accuracy, efficiency, and flexibility with math facts.

In case you missed it, read part 2 on fluency addition and subtraction and part 3 and part 4 on fluency multiplication and division. You can also learn more about fluency in IM K–5 Math™ on the IM website.

Download our Fluency Development ebook, and use it as a resource to support students’ confidence in their execution of skills with a purpose.

##### Dionne Aminata

Dionne Aminata has served over 20 years in education as a teacher, instructional coach, math content specialist, curriculum writer, and curriculum implementation specialist, and has taught at the elementary, middle school, undergraduate, and graduate levels.

At Illustrative Mathematics (IM), Dionne is one of the Lead Authors of the IM K–5 Math curriculum, playing a key role in embedding curricular supports for culturally responsive teachers. She successfully supported LAUSD’s beta pilot of IM K–5 Math in 2020–21, and now serves as the Senior Director of Strategic Initiatives at IM in the Marketing department. In her current position she uses her marketing and curriculum implementation expertise to oversee research and outreach efforts to IM's community of users. As a part of this work, she supports math leaders nation-wide to successfully implement IM K–12 Math through the use of impact measurement tools, cycles of observation and feedback, and protocols to collaborate with peers to discuss problems of practice and share best practices.

In 2020, she launched MathTrust, a consulting firm dedicated to supporting educators to seek the brilliance of Black students and trust all students to do the math. In 2022, she joined the Council of Chief State School Officers (CCSSO) as a Math Advisor for the Math Collaborative, a network of state math leaders dedicated to affecting change in systems and policies in order to promote mathematics as a subject for all students.

Dionne earned her undergraduate degree in Sociology and Business at UCLA. She has an MBA In Entrepreneurship and Global Marketing from Babson College, and a Masters in Math Education from CUNY Brooklyn College.