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Contents
What does it really mean to know how to subtract?
What is 76 minus 28? When we think of subtraction, we often envision the standard vertical representation of the operation.
However, subtraction encompasses much more than that. Mathematics involves not only deep content understanding but also problem-solving, making connections, reasoning, and communicating through various representations.
Returning to our initial subtraction problem: Do you already know the result of 76 minus 28? How did you arrive at that answer? Could you have approached it another way?
These questions allow us to assess each student’s mastery of subtraction. Ultimately, they reveal whether a student truly understands subtraction. A student who has mastered subtraction should reach the correct answer efficiently—both quickly and accurately—and understand what they are doing and why.
So, how do we develop this understanding? What strategies do we present? What materials should we use? And when should we move away from them?
As you can see, subtraction is like an iceberg: what is visible on the surface is just a small part of its complexity. We invite you to dive into this article and explore the deeper aspects of learning subtraction.
What is subtraction?
How many do we have left if we have 8 candies and eat 5?
This word problem illustrates the most common understanding of subtraction: the action of taking away, which is essentially the opposite of addition.
Yet, subtraction also involves separating items from a group and finding the distance between two numbers. For instance, it answers the question: “What is the distance between 28 and 76?”
How do we solve a subtraction?: From concrete to abstract
First, it’s essential to understand that there is no single method for performing subtraction. The standard algorithm we all learned is not the only option available. Fluency in a mathematical operation also means knowing various methods for solving it and having the discernment to choose the most appropriate one based on the context and the numbers involved.
To ensure flexibility in problem-solving, we teach a wide range of strategies in the classroom, focusing on understanding and frequent practice to achieve efficiency.
The primary strategies we use for subtraction are:
- Jumping along the number line
- Subtraction by decomposition
Each strategy follows a learning sequence based on the CRA (Concrete, Representational, Abstract) model to foster understanding. We move through three stages:
- We begin with manipulation using different materials (Concrete).
- We represent on paper what we did using manipulatives (Representational).
- We move on 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 serves as a valuable visual tool that can easily be extended to larger ranges and also lays the groundwork to work with integers, rationals, and irrationals in the future.
This strategy helps us develop mental math skills, solve simple operations efficiently, and move away from finger counting.
Furthermore, it allows us to understand subtraction through two models:
- Taking away model: We visualize ourselves at a point from which we want to take as many steps back as the items we want to remove.
- Distance between two numbers model: This model clearly illustrates the distance between two numbers by representing the two numbers we want to subtract and counting the jump distance between them.
Didactic path for the strategy of jumping along the number line
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.
In the early stages of learning this strategy, students often count the beads one by one. However, we soon encourage them to adopt more efficient counting strategies, such as using color changes (every 5 or every 10).
Once students have successfully solved a few subtraction equations using the bead string (depending on each child), we encourage them to transition to abstraction by representing on paper what they were doing using manipulatives.
As students gain proficiency in jumps (both using manipulatives and on paper), we progressively expand the range of beads: first working with the 10-bead string, then moving to 20, and finally reaching 50.
To continue compacting representations and encouraging students to step out of their comfort zones, we invite them to transition to the two-color line after they have correctly solved several subtraction equations using concrete representations. In this approach, we maintain the differentiation of colors and marks in their representations while not focusing on specific numbers.
Students practice various jumping techniques, which allows them to achieve proficiency and optimize their skills. For example, the subtraction problem 76 – 28 can be solved in two ways:
However, other methods likely exist.
And that’s not all! Eventually, students will integrate these concepts so well that they will no longer require paper representations; they will be able to solve such problems mentally.
Subtraction using decomposition strategy
In 2nd grade, we develop a strategy focused on improving the written calculation of subtraction: the decomposition strategy. This approach helps us understand the standard algorithm in a clear and transparent way.
To master this strategy, it is essential to recognize that in mathematics, each digit of the numbers involved in an operation represents a specific quantity. This concept is fundamental.
For example, the number 76 consists of 7 tens, which equals 70, and 6 ones. To reinforce this understanding, we explore the place value system in depth, introducing various materials such as the abacus at the beginning, followed by base-10 blocks later on.
Didactic path for the subtraction using decomposition strategy
The first step in applying the decomposition strategy is subtraction using manipulatives, such as base-10 blocks.
This hands-on approach allows children to understand subtraction as the act of taking away or separating items from an initial quantity. For example, let’s consider a simple subtraction: 26 – 12. We can decompose 26 into 2 tens and 6 ones. If we take 1 ten away from the 2 tens, we are left with 1 ten. Next, if we take 2 ones away from the 6 ones, we are left with 4 ones. Thus, the result of 26 – 12 is 14.
After mastering the basic concept, we can introduce more complex subtraction equations where we cannot always take away the desired quantity due to insufficient amounts.
For instance, consider 76 – 28. Here, we cannot take away 8 from 6 ones. Therefore, we need to encourage students to plan their approach and borrow from the tens to gain more ones.
Instead of decomposing 76 into 7 tens and 6 ones, we should express it as 6 tens and 16 ones by converting one ten into ones. This way, we can properly solve the operation. If we subtract 8 from 16 ones, we are left with 8 ones. Then, if we take away 2 from 6 tens, we are left with 4 tens. Therefore, the final result is 48.
It is clear that this process helps students understand that events do not occur randomly and allows them to grasp the origin of each step. However, despite its transparency, it can be a slow and inefficient method.
Once students have solved several subtraction equations using this approach, we should encourage them to take their first step toward abstraction by representing on paper what they did using manipulatives. For instance, while solving subtraction equations, they can draw base-10 blocks with crosses and sticks to provide visual support.
Once students have successfully solved a number of subtraction equations this way, we must encourage them to step out of their comfort zones by gradually reducing the use of concrete representations. We will slowly withdraw these supports until they are able to represent the standard subtraction algorithm independently.
Deducing results from known facts: The key to thinking like a mathematician
In addition to constructing strategies, students must practice and develop essential mathematical skills in the classroom, such as deducing results from known facts. Mathematics is, by definition, a deductive science.
This approach not only enhances students’ reasoning abilities but also fosters other crucial skills, such as making connections, formulating conjectures, and thinking like true mathematicians.
In the context of subtraction, this skill is evident in the ability to simplify operations. For instance, a student who is proficient with known and derived facts can deduce that 76 – 28 is equivalent to 78 – 30.
This adjustment, based on the principle of number translation, preserves the same difference between the values while transforming the operation into a simpler one. This flexibility not only demonstrates mastery of number sense but also showcases the ability to optimize calculations.
Students soon realize that this strategy is useful for solving calculations more quickly. Therefore, they should practice it regularly to build confidence and fluency.
The importance of making estimates
Finally, we must not forget the importance of estimation. While accurate calculations are essential, it is equally important to be fluent in making estimates.
Good estimates, made before solving an operation, help in selecting the most appropriate strategy based on the situation—whether students have access to paper and pencil or need to solve problems mentally—and the nature of the numbers involved, whether they are close together or far apart.
Furthermore, having made a solid estimate allows students to determine whether the result they obtained is reasonable and correct.
How to develop fluency and judgment in the use of 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.
References
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