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Discover how to learn multiplication with strategies that go beyond memorizing tables, fostering understanding, fluency, and flexibility.
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What does it mean to know how to multiply?
“2 × 7 = 14, 2 × 8 = 16, 2 × 9 = 18.” Do you remember? Many of us learned multiplication tables this way. One by one, in order. Like learning the names of rivers or capitals. But does reciting the tables from memory mean we know how to multiply?
When we reduce learning multiplication to simple memorization, we miss great opportunities to understand it deeply. And understanding what multiplication means is key to advancing to later concepts such as division or powers.
A student who understands multiplication should arrive at the correct result efficiently (fluent and accurate) and understand what they are doing and why.
But how do we get there? What strategies do we present? What is the learning process for these strategies? Should the tables be memorized?
As you can see, multiplication is like an iceberg: what we see on the surface is only a small part of its complexity. We invite you to dive into this article and explore the instructional depths behind learning multiplication.
What is multiplication?
“6 + 6 + 6 + 6” is the most popular and widespread meaning of multiplication: repeated addition. It is a model that consists of the repeated addition of the same quantity of elements.
However, multiplying can also be understood as calculating the number of elements if we arrange them in a rectangle. This idea is known as an array. For example, 6 × 4 represents the total number of elements arranged in a rectangle with four rows and six columns.
How we solve a multiplication: from concrete to abstract
The first thing to understand is that there is not just one way to multiply.
The algorithm that we all learned is not the only one. Being fluent in an operation also means knowing the various ways to solve it and having the judgment to choose the most appropriate one according to the context and the numbers involved.
To ensure this flexibility, we build a wide range of strategies in the classroom, focusing on their understanding, and we practice them frequently to develop fluency.
In the case of multiplication, we build the algorithm from two models that help us reach it visually and transparently: the array and the area model.
This strategy follows 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).
Multiplication tables: memorize or develop automaticity?
The star content of the middle grades. Should they be memorized? How should they be learned?
Multiplication tables should be answered quickly, either by memorizing them or by deducing them quickly from a few basic results. Being unable to answer quickly, and even more so, lacking efficient strategies to derive the results, forces children to spend effort on basic calculations, reducing their ability to focus on more advanced or complex concepts.
However, what matters is how we learn them. Multiplication tables are not presented but built one by one, relying heavily on deduction.
This is not enough. The automatization of multiplication tables do not develop immediately after they are learned. To achieve this, the classroom encourages the practice of multiplication tables through problem-solving (challenges), pattern finding, and numerous interactive activities. The goal is to provide students with the tools and strategies they need to develop automaticity with multiplication facts and quickly solve those they have not yet mastered.
If you want to learn more about learning multiplication tables, you can check out this article.
The array: the path to the standard algorithm
The array is a strategy we use to progressively build the standard algorithm and understand the commutative property of multiplication.
The repeated addition model is useful for the early stages of multiplicative thinking but has limited scope. We must complement it with the array and connect it to area calculation (and the area model) to demonstrate the commutative property.
The array views multiplication as elements arranged in a rectangle. For example, 17 × 4 represents the total number of elements arranged in 4 rows and 17 columns.
Additionally, this approach allows for working with number decomposition, a skill that students have already practiced with other operations and will continue to develop when learning division.
Learning progression from array to area model
The first step in introducing the array in the learning trajectory is to solve multiplication problems using a rectangular drawing divided into rows and columns.
For example, to calculate 17 × 4, draw a rectangle with 17 columns and four rows. Initially, students might find it challenging to solve this multiplication directly. However, they can figure out the solution by applying strategies they already know, such as decomposing numbers.
Even if they don’t know the 17 times table, they do know the 4 times table. Therefore, they can decompose 17 into 8 + 9 and divide the large rectangle into smaller rectangles in the area model. By adapting 17 × 4 to (8 × 4) + (9 × 4), they get the same result, but with more manageable numbers.
But this decomposition is not the only one. One of the most important aspects is ensuring students observe different ways to decompose a multiplication. 8 + 9 is one option, but 10 + 7 is also an option.
The area model facilitates understanding of the procedure and provides students with resources in case they get lost in any of the steps.
The visual representation —the rectangle with rows and columns— accompanying the operation is only a temporary support. Once students correctly solve several multiplications following this model, they should progress toward more abstract representations. First, by removing the support of rows and columns, and later, by condensing the representations until reaching the multiplication diagram and, then, the standard algorithm.
Let’s see how we condense a slightly more advanced multiplication, such as 12 × 15. Since we are introducing two-digit multiplication, we need to return to concrete representations to support understanding of the procedure.
Therefore, we represent a rectangle with 12 rows and 15 columns and apply the same criteria: decompose each number into more manageable quantities. The 12 into 10 + 2 and the 15 into 10 + 5. Next, systematically, we multiply all the values together and add the results:
(10 × 10) + (10 × 5) + (2 × 10) + (2 × 5) = 100 + 50 + 20 + 10 = 180
In this way, once students have correctly solved several multiplications like this, they are encouraged to do it without the support of rows and columns. When they achieve this, they can move to the multiplication diagram until reaching the maximum condensation: the standard algorithm for multiplication.
Deriving results from known facts: the key to thinking like a mathematician
Alongside the construction of strategies and practice with multiplication tables, students in the classroom must also develop key skills such as deriving results from facts they already know. Mathematics, by definition, is a deductive science.
This fosters their reasoning ability and other essential skills, such as making connections, formulating conjectures, and thinking like real mathematicians.
A clear example of this skill occurs in multiplication when a student can derive, for example, the result of 12 × 15 by playing with doubles and halves.
If the double of 15 is 30 and the half of 12 is 6, they can derive the result of 12 × 15 from 30 × 6, which is 180.
This student has been able to adjust the operation to simplify it into a simpler one that they already know.
Students soon discover that this strategy is beneficial for quickly solving calculations. Therefore, they must practice it at different times to gain 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 develop fluency and judgment 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 construction of and fluency with using the area model does not extend beyond 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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