##### “We have to break from the notion that learning mathematics must be a linear and procedural endeavor mastered through rote practice and memorization. Instead, we must recognize and emphasize that interconnected concepts lead to stronger foundations in mathematics and stronger personal and mathematical identities.”

Because of the problem-based structure of our curriculum, students’ ideas play an integral role in the progression of learning in IM K–5 Math™. Starting with the belief that all students are capable learners with prior skills and funds of knowledge, we designed our curriculum to give teachers opportunities to gauge what students already know by allowing them to play with mathematical ideas before formalizing concepts. Students learn math by doing math, even to develop procedural fluency with algorithms.

During the IM K-5 Math Beta Pilot, Kaylee Rivera-Ramirez interviewed her mom as a part of an assignment in Meghan Codere’s 3rd grade class. Students were invited to ask a family member how they learned to subtract.

Starting the addition and subtraction unit in this way celebrates the knowledge of students’ families, promotes positive math identities, and provides opportunities for teachers to build cultural knowledge.

In our last post we shared how the curriculum design supports the development of fact fluency. Focusing on the fluency requirement in grade 1—fluently add and subtract within 10—we highlighted places in our Grade 1 course where students learn and practice their facts, building from the facts they know and their conceptual understanding of numbers and operations. In this post we shed light on how the curriculum supports students to develop procedural fluency with addition and subtraction algorithms across grades 2–5.

**Fluency with Addition and Subtraction Algorithms in Grades 2–5**

Beginning in grade 2, as students develop fluency with facts within 20, there is a coinciding progression towards developing fluency with the standard algorithm for addition and subtraction of multi-digit whole numbers by the end of grade 4.

In IM K–5 Math™, the process for developing fluency with an algorithm is as follows:

- Students operate in ways that make sense to them.
- Students analyze and try strategies and algorithms based on place value understanding, the properties of operations, and the relationships between operations.
- Students know and use the standard algorithm.

With this process, students begin with what they know and build from there. They make sense of new strategies, representations, and algorithms before being asked to use them. Below we highlight the progression towards procedural fluency with the addition and subtraction algorithms in the curriculum across grades 2–5.

**1. Students operate in ways that make sense to them.**

When students enter grade 2, they have a developing understanding of place value, the properties of operations, and the relationship between addition and subtraction. They apply this knowledge to accurately and efficiently add and subtract within 100. As they work towards this goal, they also learn to use strategies based on place value, the properties of operations, and the relationship between addition and subtraction to add and subtract within 1,000.

This progression of fluency development in our Grade 2 course can be most clearly seen by looking at the warm-up activities throughout the year. Students are encouraged to think flexibly and use strategies that make sense to them. Teachers can use these instructional routines to understand student thinking and assess math fluency.

#### 2. **Students analyze and try strategies and algorithms based on place value understanding, the properties of operations, and the relationships between operations.**

After several units of deep study, building numeracy and flexibility with strategies for operating with large numbers, students begin to analyze and try written methods. In Unit 7 of Grade 2, students use base-ten blocks and diagrams to connect what they know about place value, the properties of operations, and the relationship between addition and subtraction to equations. They use these tools to represent adding hundreds to hundreds, tens to tens, and ones to ones, and learn that when they add or subtract, they may need to compose or decompose a ten, a hundred, or both.

In the example below, students analyze two different strategies to find 358 + 67. They make connections between composing a ten and a hundred using a base-ten diagram, and using a written method.

These concepts continue to be developed in grade 3, when students practice using algorithms that are based on place value and support the progression towards adding and subtracting larger numbers and decimals. In our Grade 3 course, students start with algorithms that show expanded form, and then move toward algorithms that are more efficient. Students may come to know and use the standard algorithms for addition and subtraction at different times. IM–5 Math™ is structured so conceptual understanding can be developed alongside, or even after, students have learned how to execute procedures.

The examples below are from our Grade 2 and Grade 3 courses. They show how students examine algorithms by making connections to the strategies they know. Students use their understanding of place value and properties of operations to make sense of the algorithms and then compute.

**3. Students know and use the standard algorithm.**

In grade 4, students build from the written methods learned in grades 2 and 3 to understand and use the standard algorithm to add and subtract multi-digit numbers. In Unit 4 of our Grade 4 course, students note the similarities and differences between the algorithm in expanded form and the standard algorithm. They make sense of the standard algorithm by analyzing it before being asked to try it for themselves.

**Assessing Fluency**

Beyond reviewing student computations, teachers can assess students’ procedural fluency with an algorithm in a number of ways. Students can be tasked with analyzing the (correct or incorrect) work of a fictional student, or students can be asked to compare and contrast two different computational methods.

In the Grade 4 example below, students attend to potential errors in using the algorithm, particularly when it is necessary to decompose or compose a base-ten unit multiple times, as in the case when subtracting from a number with zeros. Students consider different strategies for approaching multi-digit subtraction, including leveraging the relationship between addition and subtraction.

Across the grades you can see how students progress from using strategies that make sense to them to using strategies and written methods based on generalizable concepts. The process for developing procedural fluency with algorithms is evident in the look-fors on the monitoring sheets that teachers can use to track student progress.

**Building Flexibility with Procedures**

The development of fluency with addition and subtraction in kindergarten through grade 4 is critical for operations with decimals in grade 5. While developing an understanding of place value to the right of the decimal, students learn that they can apply what they know about properties of operations, place value understanding, and the relationship between addition and subtraction with whole numbers to flexibly add and subtract decimals to the hundredths place. They use place value understanding to decide whether sums and differences are reasonable and to ensure that the digits in the numbers are aligned correctly when using the standard algorithm.

Throughout this progression, as students learn to add and subtract multi-digit numbers using the standard algorithm with accuracy and efficiency, they continue to practice and gain flexibility with mental math during warm-up routines. This practice not only helps students build and maintain numeracy, but it also reminds students that the standard algorithm may not always be the most appropriate or efficient strategy.

**A Coherent Pathway Towards Accuracy, Efficiency, and Flexibility**

The progression towards fluency with addition and subtraction in IM K–5 Math™ is carefully developed to empower students to connect what they know to a coherent story about addition and subtraction concepts and procedures.

Along the way, we support the development of positive mathematical identities by reminding students of the knowledge they bring to each lesson as they engage in warm-up routines. These routines allow them to flex their mental math muscles and continue to use strategies that make sense to them. This consistent practice, in tandem with a coherent progression of learning, provides students with the tools they need to compute fluently. Not only do they become equipped to use procedures accurately, efficiently, and flexibly, but they also gain the ability to determine, from the variety of strategies at their disposal, which ones are most appropriate for a given problem.

### Next Steps

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.

We invite you to take a look at the curriculum to see how it supports fluency development. In our next blog post we will discuss the progression towards fluency with multiplication and division procedures and algorithms.

You can also learn more about fluency in IM K-5 Math™ on the IM website.