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In this article, we delve into how to teach division fluency and the strategies that promote understanding and flexibility in calculation.
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What does it really mean to know how to divide?
«Let’s solve 158 ÷ 3. I take 15, how many times can 3 fit in 15? 5. Now, what’s next? Do I bring down the 3 or do I subtract first?”
You might have encountered this situation before: when tasked with solving a division problem on paper, you need a moment to recall the precise steps of the algorithm. This illustrates that relying solely on one algorithm for learning an operation may prevent us from fully understanding it.
A solid understanding of division is crucial for tackling later concepts, such as percentages, fractions, and proportionality, as well as for various everyday situations.
Dividing involves a range of skills that extend beyond merely executing an algorithm. A student who truly grasps division should be able to arrive at the correct answer efficiently—both fluently and accurately—while also understanding what they are doing and why.
So, how do we approach this? What strategies and materials can we use, and when should we stop depending on them?
Division is akin to an iceberg: what we see on the surface represents only a small fraction of its complexity. We invite you to delve into this article and uncover the educational depths involved in learning division.
What is division?
To share or to make groups— that is the question.
Imagine you have 24 pieces of candy and you want to share them equally among 6 children. How many pieces of candy will each child receive? This scenario illustrates the most common understanding of division: equitable sharing.
However, division can also be viewed as making packages. For example, how many packs of 4 balls can we make with 24 balls? This perspective adds another dimension to division, leading to a richer and deeper understanding of the concept.
How to solve division: from concrete to abstract
Understanding division involves recognizing that there isn’t just one method to approach it. The algorithm many of us learned is only one of several available strategies. Fluency in an operation means understanding different approaches to solving it and having the judgment to select the most suitable one based on the context and the involved numbers.
To foster this flexibility in the classroom, we create a wide range of strategies that emphasize comprehension, and we practice these strategies multiple times to build agility.
The main strategies we focus on for division are:
- Distribution Strategy
- Decomposition strategy
In the case of the distribution strategy, we follow a learning sequence based on the CRA model (Concrete, Representational, Abstract) to ensure a deep understanding. This involves three stages:
- Start with hands-on manipulation using various materials (Concrete).
- Represent our manipulations on paper (Representational).
- Move to abstract representations, such as algorithms (Abstract).
Distribution strategy: the vertical division format
The first strategy we use to introduce the concept of division is the distribution strategy. It provides a clear understanding of the written division calculation, leading to the standard algorithm.
Educational path of the distribution strategy
The initial step in the educational path of the distribution strategy involves manipulating objects that can be shared or grouped. We do this through the action of division; that is, we propose situations that help the students to make equal distributions. For example, share cards, snap cubes, etc.
As students engage in sharing, we encourage them to explore more efficient methods. Instead of sharing items by 1s, for example, they could share by 5s or 10s.
After this first contact, and after repeating this process several times, we invite them to make a first approach to abstraction by representing on paper what they have done using manipulatives. For example, we might present a situation like this: “How can we share 158 cards among 3 players so that everyone receives the same number of cards?”
From this question, children can share their distribution strategies and narrate their thought processes, documenting the steps in their logbooks.
They might begin by sharing 10 cards to each player, leaving them with 128 cards. Feeling more confident, they could then share 20 cards each, reducing the remaining cards to 68. They would continue this process: sharing 20 cards again, leaving 8, and finally sharing 2 cards each. Ultimately, each player would receive 52 cards, with 2 cards left over.
This process helps them clearly understand that things don’t happen by chance and allows them to understand the reasoning behind each step. While this method is transparent, it can be slow and inefficient.
Once students have successfully solved several division operations using this method, we encourage them to step out of their comfort zone and adopt more efficient distributions, focusing on written representations.
Gradually, we remove the scaffolding to optimize their distribution, similar to how we approach the standard algorithm.
Our goal is for students to solve divisions efficiently and meaningfully, while also understanding the rationale behind each number in the operation.
A key factor in achieving this proficiency is practice. Developing automaticity and creating algorithmic strategies will help students master these concepts and improve their ability to select the most effective method in various contexts.
Decomposition strategy for division
Concurrently, another strategy we work on for division is the decomposition strategy. This is a more sophisticated strategy based on the premise of solving divisions through mental math.
While mental math is beneficial for addition and subtraction, it can also be advantageous for division.
This strategy involves breaking down the original operation into simpler divisions that can be calculated mentally.
It can be supported by visual models that help students better understand the process. In this regard, although not strictly equivalent, the rectangular model can be a useful representation, as it helps visualize how a number can be broken down into smaller parts.
In exact divisions, the rectangular model allows students to anticipate or test possible decompositions, helping them explore different options and find the most efficient breakdown. However, finding the optimal decomposition is not immediate — it takes time, practice, and a process of trial and refinement to develop mental calculation skills.
For example, let’s solve 158 ÷ 3. First, we decompose 158 into 120 + 30 + 8. Next, we divide each of these numbers by the divisor, which is 3. Therefore, we calculate: 120 ÷ 3 = 40, 30 ÷ 3 = 10, and 8 ÷ 3 = 2 R2. Thus, the result is 52 R2.
We must keep in mind that, to achieve efficient decomposition, it’s advisable to select numbers in such a way that, ideally, only one of them results in a remainder.
The remainder is crucial, as it naturally reflects real-life situations. In fact, the remainder is often the answer to situations where it’s asked how many items cannot be evenly shared. Additionally, as students progress to upper elementary math, the concept of remainders will help them connect to decimals and divisibility.
Deducing results from known facts: The key to thinking like a mathematician
While developing various strategies, students should also practice deducing results from known facts. Mathematics is fundamentally a deductive science, and enhancing this skill fosters students’ reasoning abilities along with essential skills, such as making connections, formulating conjectures, and thinking like mathematicians.
A student familiar with known facts can deduce the result of 158 ÷ 3 from previously learned results, such as 150 ÷ 3 (50). From that, the student can infer that: 153 ÷ 3 (51), 156 ÷ 3 (52), and 159 ÷ 3 (53).
This student deduced the result through exact divisions, without any remainders. Since 156 ÷ 3 equals 52 and 159 ÷ 3 equals 53, 158 ÷ 3, we can see that 158 ÷ 3 is almost 53 (52 R2). This approach demonstrates not only an understanding of deduction but also a flexible and practical application.
Students often find that this strategy is quite useful for solving calculations more quickly. Therefore, it is essential for them to practice this approach regularly to build confidence and fluency.
The importance of making estimates
Finally, we should not overlook estimation calculations. While it is crucial for calculations to be accurate and precise, fluency in making estimates is also necessary.
Making good estimates before performing calculations helps students choose the most appropriate strategy based on the situation—such as using paper and pencil or calculating mentally—and the numbers involved—whether they are close together or far apart.
Additionally, having a solid estimate enables students to assess whether their final result is correct.
How to achieve fluency and judgement in the use of strategies
Developing fluency in an operation relies on understanding different approaches to tackling problems.
In the classroom, we have a variety of strategies that coexist, and it is important for students not only to grasp these strategies but also to develop the judgment needed to select the most suitable one based on the context and the numbers involved in each operation.
To achieve this goal, we dedicate significant time to thoroughly build each strategy, ensuring their understanding. However, theory alone is insufficient; practice is also essential. Therefore, we propose a range of activities and environments to enhance calculation agility.
Although this process takes time, the development of each strategy usually occurs within a single school year. For example, the step towards optimizing distribution typically does not extend beyond the 4th grade.
What does expand is the complexity of the numerical range we work with. As the numerical range increases, manipulatives are reintroduced with the purpose of guiding students through another cycle of abstraction before gradually phasing them out.
Ultimately, the aim is to establish a strong foundation that enables students to advance toward more abstract and efficient mathematical processes, equipping them with knowledge that helps them adapt to various situations. We must not forget about practice, which is crucial for consolidating and developing automaticity of the skills we have developed.
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