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In this article, we focus on addition and the key insights and strategies for developing deep understanding and fluency in both exact and estimated calculations.
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What does it really mean to know how to add?
The addition, along with subtraction —as its inverse operation— is the first basic operation learned in elementary school.
From the earliest math lessons to our daily lives, addition appears in many situations: counting objects at home, paying for groceries at the supermarket, or keeping score in games.
However, we should not confuse skill in executing an algorithm with knowing how to add. Doing mathematics is much more than that. In addition to developing a deep understanding of content, mathematics involves solving problems, making connections, reasoning, and communicating using different representations.
Knowing how to add means arriving at the correct answer efficiently (with fluency and accuracy), understanding what we are doing, and correctly applying the steps of each strategy.
But how do we get there? What strategies do we present? What materials do we use for each one? When should we stop using the materials?
As you can see, addition is like an iceberg: what we see on the surface is only a tiny part of its complexity. We invite you to dive into this article and explore the instructional depths behind learning addition.
What is addition?
If we have 48 candies in a bag and add 28 more, how many candies do we have in total?
This example of a word problem shows us just one of the concepts of addition: the action of adding new elements to an initial quantity.
But addition is also combining two or more groups of elements. For example, if we have 28 cubes and add 48 more, how many cubes do we have in total?
How do we solve an addition, from concrete to abstract?
The first thing to understand is that there isn’t just one way to add. The standard algorithm we all learned is not the only option. Being fluent in an operation also means knowing various ways to solve it and having the judgment to choose the most appropriate one based on the context and the numbers involved.
To ensure this flexibility in the classroom, we build a wide range of strategies, focusing on conceptual understanding and practicing them repeatedly to achieve efficiency.
The main addition strategies we work on are:
Number line jumps
Addition by decomposition (Expanded form)
Each strategy follows a learning sequence based on the CRA model (Concrete, Representational, Abstract) to ensure comprehension. We move through three key stages:
We start with hands-on manipulation using different materials (Concrete).
We represent on paper what we did with the materials (Representational).
We move to abstract representations, such as algorithms (Abstract).
Jumping along the number line strategy
The first strategy we introduce in 1st grade for addition and subtraction is Jumping along the Number Line.
The Number Line is a valuable and visual tool that can be easily extended to work with ranges as large as we want and introduce, in the future, integers, rational, and irrational numbers.
This strategy helps us develop mental math, solve simple operations efficiently and move beyond counting on fingers.
Learning progression for the Jumping along the Number Line strategy
The initial step in the didactic sequence of the jumps strategy is to practice jumping forward and backward with a bead string to develop number sense.
When first learning this strategy, it’s normal for students to count the beads one by one. However, we should soon encourage the use of more efficient strategies to speed up counting, such as color changes (every 5 or every 10).
Once they have correctly solved some addition operations with the bead string —depending on each student’s pace— we should invite them to take a first step toward abstraction, representing on paper what they did using manipulatives.
As they gain fluency with jumping —both with manipulatives and on paper— we gradually expand the bead ranges: first they work with the 10-bead string, then move to 20 and finally reach 50.
To continue condensing their representations and taking them out of their comfort zone, once they have correctly solved several addition equations with concrete representations, invite them to move to the two-color line. That is, we continue to differentiate the colors and marks in the representation but without showing the concrete numbers.
Gradually, we eliminate colors and marks until we finally reach the Empty Number Line.
Students practice jumping in many different ways, allowing them to master and optimize this strategy.
For example, the addition 48 + 28 can be solved in two ways:
But, there are certainly other ways!
And this doesn’t end here. There will come a time when students will have internalized it so well that they won’t need the paper representation, because they will be able to solve the addition mentally.
Adding with the Decomposing Strategy
In 2nd grade, we introduce another way to solve addition: the Decomposing strategy.
To master this strategy, the first thing to understand is that in mathematics, each digit in the numbers involved in an operation represents a specific quantity. This is fundamental.
For example, 48 is a number composed of 4 tens (40 units) and 8 units. That’s why we work with the place value system in depth, introducing different materials such as the abacus in the early stages and, later on, base-10 blocks.
Learning progression for the Decomposing strategy for addition
The first step in working with the Decomposing strategy is adding using manipulatives. In this case, using base-10 blocks.
Through these materials, children can easily observe what they are doing when they add: combining elements. For example, we start with an addition like 26 + 12. First, they decompose el 26 into 2 tens and 6 ones, and the 12 as 1 ten and 2 ones. If they combine tens with tens and ones with ones, they can clearly see that the result is 38 (3 tens and 8 ones).
Now, we should take advantage of this manipulative stage to present more challenging addition operations, which involve regrouping.
For example, in the addition problem 48 + 28, first, encourage students to represent 48 as 4 tens and 8 ones, and the 28 as 2 tens and 8 ones. Starting with the tens, we solve 40 + 20 = 60. Then, although the order is not important, we group the ones: 8 + 8.
Using the manipulatives, students observe that when they combine 10 units they form one rod. Therefore, instead of expressing the sum as 8 + 8, can rewrite it as 10 + 6 (one rod of 10 units and 6 remaining units).
The final sum we get is 60 + 16 = 76.
It’s clear that this process helps students understand that things don’t happen randomly, allowing them to understand the reason behind each step. However, despite being very transparent, it is also slow and inefficient.
That’s why, once students have solved several addition operations this way—at their own pace—we should invite them to take a first step toward abstraction, representing on paper what they previously did using manipulatives. For example, while solving addition operations, they can draw base-10 blocks as bars and crosses, to have visual support.
When they have correctly solved several addition operations this way, we should move them out of their comfort zone, encouraging them to reduce their use of concrete representations. This way, progressively, we remove the supports until reaching a representation of the standard algorithm for addition.
Developing fact fluency with single-digit addition
Often, when we talk about learning based on deep understanding, it seems that we completely demonize speed in response. However, there are certain content that should be part of students’ automaticity.
One of these is developing fact fluency with single-digit addition. This means recalling within a reasonable time the results of simple addition operations such as 7 + 3 = 10, 8 + 5 = 13, 9 + 6 = 15, etc.
The fact that students cannot respond quickly and, even more importantly, do not have efficient strategies to derive results means they must devote too much effort to basic calculations, which reduces their ability to focus on more advanced or complex aspects.
It is very useful to be able to respond quickly, either because the results are memorized or because they can be derived quickly from a few known facts without too much effort.
Deriving results from known facts: The key to thinking like a mathematician
Alongside building strategies, students must practice and develop key mathematical skills in the classroom, such as deriving results from facts they already know. Mathematics, by definition, is a deductive science.
This not only promotes their reasoning ability but also strengthens other essential skills, such as making connections, formulating conjectures, and thinking like real mathematicians.
In addition, this skill is reflected in the ability to simplify operations. For example, a proficient student in known facts and derived facts can figure out that 48 + 28 is equivalent to 50 + 30, and then subtract 4 from the result.
This ability to convert an addition with regrouping into a simpler one demonstrates that they have become flexible and efficient. They have not only simplified the calculation to operate more comfortably, but they have also known how to compensate for the modification to get the correct answer.
Students soon discover that this strategy is very useful for solving calculations more quickly. That’s why they should practice it at different times to gain confidence and fluency
The importance of making estimates
Finally, we must not forget the importance of estimation. Yes, our students should be able to arrive at correct answers in exact calculations, but they should also develop fluency in estimating.
Making good estimates before solving an operation allows them to choose the most appropriate strategy according to the circumstances, whether with paper and pencil or mentally. The choice of strategy is also influenced by whether the numbers are close to or far from each other.
Additionally, making a good estimate will help them determine whether the result obtained is correct or not.
How to develop fluency and judgment in using strategies
One of the key elements in developing fluency in operations is knowing various approaches to tackle them.
In the classroom, all the strategies we have developed coexist. Students must not only understand these strategies but also cultivate the judgment needed to choose the most appropriate one based on the context and the numbers they are working with in each operation.
To achieve this, significant time is dedicated to thoroughly constructing each strategy to ensure understanding. However, theory alone is insufficient; practice is essential. For this reason, we propose a variety of spaces and activities to enhance calculation agility.
While this process requires time, the development of a strategy typically does not take more than a single school year. For instance, the transition from using a bead string to the empty number line occurs throughout the first year of elementary school, allowing students to reach 2nd grade with the ability to make jumps directly on the empty line.
What changes is the complexity of the numbers we work with. As the numerical range expands—such as moving from 10 to 20 or from 20 to 50—manipulatives are reintroduced to facilitate another cycle of abstraction, after which we progressively phase them out.
The primary aim is to establish a strong foundation for students, enabling them to advance toward more abstract and efficient mathematical processes. This foundation should equip them with the knowledge necessary to adapt to various situations. Now, it’s time to practice in order to develop fact fluency or consolidate what we have built. We will discuss this further later on.
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