D
Donna Boucher
Guest
- Thread Author
- #1
Flashcards have traditionally been used to practice multiplication facts: see the fact, say the product, move on to the next card. But when multiplication is first introduced, students need much more than fact recall. They’re making an important shift from additive thinking to multiplicative thinking. With addition, students think about quantities being joined or combined. Multiplication introduces a different structure—equal groups of equal size. Students need lots of experiences building and representing those relationships so they can see what makes multiplication different from addition.
This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.
A multiplication fact such as 5×35\times3 represents that multiplicative relationship, but on a flashcard, it’s completely abstract. So before you toss those old multiplication flashcards, try repurposing them! Instead of using the cards only to quiz students on products, use them as prompts for building, drawing, and making sense of multiplication.
When students see 5×35\times3 on a flashcard, they’re looking at symbols. A student may even know that the product is 15 without really understanding the multiplicative relationship the equation represents. Just as a child can memorize how to spell a word without knowing what the word means, a student can memorize a multiplication fact without understanding what the fact means. Knowing that 5×3=155\times3=15 isn’t the same as understanding that it represents 5 equal groups of 3.
Give students counters and ask them to model the fact, however, and there’s suddenly much more to think about. For 5×35\times3, students can build 5 equal groups with 3 counters in each group.
As they work, ask questions that focus their attention on the structure:
How many groups do you have? How many counters are in each group? Are the groups equal? How many counters are there altogether? How does your model show 5×3?
Those conversations help students connect the abstract multiplication equation to the quantities and relationships it represents.
Once students have built the multiplication fact with counters, have them draw a representation of what they built.
That’s where the simple recording sheet comes in. Students choose a multiplication flashcard, use counters to model the fact, write the equation, and then draw a representation of their model.
The drawing is an important part of the process because students are moving from objects they can physically manipulate to a visual representation of the multiplicative relationship. They might draw equal groups or organize their counters into an array. Either way, the representation should preserve the important structure: a specific number of equal groups with the same number in each group.
Over time, repeated experiences building and drawing multiplication situations help students develop mental images they can use to reason about multiplication without always needing the counters.
This is what makes repurposing flashcards so useful. The flashcard still provides a multiplication fact to practice, but recalling the product isn’t the only goal.
Students are connecting three different representations of the same mathematical idea: the abstract equation on the flashcard, the concrete model they build with counters, and the representation they draw.
That connection is especially important as students transition from additive to multiplicative thinking. If multiplication is introduced primarily as facts to memorize, students can miss the structure that makes multiplication fundamentally different from addition. Building and drawing equal groups gives them experiences they can connect to the symbols.
Once students understand the routine, this makes an easy math workstation. Put out a stack of multiplication flashcards, counters, and recording sheets, and students have repeated opportunities to connect abstract multiplication facts to concrete and visual models.
It’s also a great activity for small-group instruction. Choose the facts students are ready to work with and pay attention to the models they create. A student who can tell you that 5×3=155\times3=15 has demonstrated fact recall. A student who can build and explain 5 equal groups of 3 has demonstrated something more—an understanding of the multiplicative structure represented by the fact.
And because students are drawing what they build, their recording sheets give you another window into their thinking. You can see whether the representation actually matches the equation rather than simply checking whether they wrote the correct product.
Have addition flashcards tucked away somewhere, too? The same CRA process works with them: students choose a fact, model it with counters, and draw a representation.
But there’s an important distinction. The process may be the same, but the mathematical structure isn’t. With addition, students are representing quantities being joined or combined. With multiplication, they’re representing equal groups. Giving students opportunities to build and draw both types of situations can help make that difference more visible.
I’ve created free recording sheets for both multiplication and addition, so grab some counters and put those old flashcards back to work!
Download the free multiplication recording sheet.
Download the free addition recording sheet.
The post Repurposing Flashcards: Representing Multiplication appeared first on Math Coach's Corner.
This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.
A multiplication fact such as 5×35\times3 represents that multiplicative relationship, but on a flashcard, it’s completely abstract. So before you toss those old multiplication flashcards, try repurposing them! Instead of using the cards only to quiz students on products, use them as prompts for building, drawing, and making sense of multiplication.
Take Multiplication Facts Back to Concrete
When students see 5×35\times3 on a flashcard, they’re looking at symbols. A student may even know that the product is 15 without really understanding the multiplicative relationship the equation represents. Just as a child can memorize how to spell a word without knowing what the word means, a student can memorize a multiplication fact without understanding what the fact means. Knowing that 5×3=155\times3=15 isn’t the same as understanding that it represents 5 equal groups of 3.
Give students counters and ask them to model the fact, however, and there’s suddenly much more to think about. For 5×35\times3, students can build 5 equal groups with 3 counters in each group.
As they work, ask questions that focus their attention on the structure:
How many groups do you have? How many counters are in each group? Are the groups equal? How many counters are there altogether? How does your model show 5×3?
Those conversations help students connect the abstract multiplication equation to the quantities and relationships it represents.
Move from Concrete to Representational
Once students have built the multiplication fact with counters, have them draw a representation of what they built.
That’s where the simple recording sheet comes in. Students choose a multiplication flashcard, use counters to model the fact, write the equation, and then draw a representation of their model.
The drawing is an important part of the process because students are moving from objects they can physically manipulate to a visual representation of the multiplicative relationship. They might draw equal groups or organize their counters into an array. Either way, the representation should preserve the important structure: a specific number of equal groups with the same number in each group.
Over time, repeated experiences building and drawing multiplication situations help students develop mental images they can use to reason about multiplication without always needing the counters.
More Than Fact Practice
This is what makes repurposing flashcards so useful. The flashcard still provides a multiplication fact to practice, but recalling the product isn’t the only goal.
Students are connecting three different representations of the same mathematical idea: the abstract equation on the flashcard, the concrete model they build with counters, and the representation they draw.
That connection is especially important as students transition from additive to multiplicative thinking. If multiplication is introduced primarily as facts to memorize, students can miss the structure that makes multiplication fundamentally different from addition. Building and drawing equal groups gives them experiences they can connect to the symbols.
Turn It into a Math Workstation
Once students understand the routine, this makes an easy math workstation. Put out a stack of multiplication flashcards, counters, and recording sheets, and students have repeated opportunities to connect abstract multiplication facts to concrete and visual models.
It’s also a great activity for small-group instruction. Choose the facts students are ready to work with and pay attention to the models they create. A student who can tell you that 5×3=155\times3=15 has demonstrated fact recall. A student who can build and explain 5 equal groups of 3 has demonstrated something more—an understanding of the multiplicative structure represented by the fact.
And because students are drawing what they build, their recording sheets give you another window into their thinking. You can see whether the representation actually matches the equation rather than simply checking whether they wrote the correct product.
Repurpose Those Addition Flashcards, Too!
Have addition flashcards tucked away somewhere, too? The same CRA process works with them: students choose a fact, model it with counters, and draw a representation.
But there’s an important distinction. The process may be the same, but the mathematical structure isn’t. With addition, students are representing quantities being joined or combined. With multiplication, they’re representing equal groups. Giving students opportunities to build and draw both types of situations can help make that difference more visible.
I’ve created free recording sheets for both multiplication and addition, so grab some counters and put those old flashcards back to work!
Download the free multiplication recording sheet.
Download the free addition recording sheet.
The post Repurposing Flashcards: Representing Multiplication appeared first on Math Coach's Corner.