Mastering Multiplication: The Science and Practice of Retrieval-Based Learning

A few weeks ago, I walked into a math classroom and immediately ran into a pair of students sitting on the floor by the door.  I quickly realized that the teacher had pairs of students all around the room.  Each pair of students used flashcards to test each other’s multiplication facts. This brought memories of how I learned multiplication. I wondered if this activity had merit and was something that we should see more of. 

Math teachers have long searched for the most effective multiplication strategies and ways to help young students master foundational skills. Among these skills, multiplication fact fluency is a critical building block for mathematical success. 

Retrieval Practice: Why It’s the Clear Multiplication Strategy Winner

Recent research offers compelling evidence about which learning methods work best for helping elementary students develop this essential skill, with retrieval practice emerging as a clear winner over traditional approaches.

Before we discuss the research, it is necessary to touch on the difference between math fluency and automaticity.

Is Math Fluency and Automaticity the Same?

The difference between Math Fluency and Automaticity lies in the depth of understanding and flexibility in problem-solving.

What Is Math Fluency?

Math Fluency, as described by the National Council of Teachers of Mathematics, refers to a student’s ability to efficiently and accurately solve problems using various strategies. It includes:

• Accuracy – Getting the correct answer.
• Efficiency – Using strategies that make problem-solving quicker.
• Flexibility – Choosing the best strategy based on the problem and adapting when necessary.
• Conceptual Understanding – Knowing why a method works, not just how to use it.

Math Fluency Definition

A simple definition of math fluency is: A fluent student can approach problems in multiple ways and select the best method for the situation.

What Is Math Automaticity?

Math Automaticity refers to the quick, effortless recall of basic facts without needing to think through the process. It is a component of fluency but does not involve strategic thinking. Automaticity is:

• Fast and accurate recall of facts like multiplication tables (e.g., knowing 7 × 8 = 56 instantly).
• Developed through repetition and practice until a response becomes second nature.
• Necessary for freeing cognitive resources so students can focus on higher-order problem-solving.
  

Key Difference:

• Automaticity is about speed and effortless recall.
• Fluency is about applying knowledge flexibly and efficiently.
  

Math Automaticity Example:

• A student with automaticity might instantly recall that 6 × 7 = 42.
• A student with fluency might recognize that if they forget 6 × 7, they can break it down into (6 × 5) + (6 × 2) to still arrive at 42.
  

Both are essential, but fluency is the ultimate goal, as it enables students to think critically and apply math in meaningful ways.

Why Multiplication Fact Automaticity Matters

When students can quickly and accurately recall multiplication facts (like 3 × 4 = 12), they free up valuable mental resources for tackling more complex mathematical problems. This skill forms the foundation for division, calculating percentages, multi-digit multiplication, and numerous real-world problem-solving situations.

Students who struggle with basic fact recall often face significant challenges as math complexity increases. Their working memory becomes overloaded when they must simultaneously calculate basic facts while trying to apply higher-order reasoning—ultimately affecting their overall math performance. As Bryant et al. (2008) noted in the research, this working memory overload “eventually will negatively influence the overall math scores.”

With so many different choices in developing activities to enhance elementary multiplication fundamentals, it is essential to look at some evidence that might help steer our decision-making in a positive direction.

Improving Multiplication Automaticity: Retrieval Practice Outperforms Traditional Methods

A study published in Applied Cognitive Psychology by Ophuis-Cox, Catrysse, and Camp (2023) provides strong evidence for the effective use of retrieval practice in developing multiplication fact fluency. The researchers conducted a carefully designed experiment in authentic elementary classroom settings, comparing two common approaches:

1. Retrieval Practice: Students used flashcards to actively recall multiplication facts from memory
2. Restudy: Students chanted multiplication tables out loud (a common traditional approach)

The study involved students experiencing both learning methods but with different multiplication tables. This clever design helped control for individual differences in learning ability.

The findings were clear: retrieval practice led to significantly better performance in both short-term (five minutes after practice) and long-term (one week later) tests compared to the restudy condition. What’s particularly noteworthy is that students maintained their performance gains over time, contradicting previous research suggesting that classroom-based retrieval practice might only show short-term advantages.

What is Retrieval Practice?

Imagine that you just moved to a new city and are trying to learn how to get around the city.  How we go about learning our new surroundings may be different. For example, you may memorize street names or look for specific structures that dictate your actions.  In both situations, your brain has created a map you can reference immediately because you were required to “retrieve” that specific information multiple times. The more you follow a path, the more automatic it becomes.  Learning is the same.  The more times you recall the information, the more likely you are to remember it when you need it.  Retrieval Practice is the deliberate process of trying to remember specific information, like multiplication facts.

Why Retrieval Practice Works Better than Restudy

The study supports a growing body of research showing that actively recalling information from memory (rather than passively reviewing it) strengthens learning pathways in several ways:

1. It creates more substantial memory traces 
2. It mimics the actual skill needed during problem-solving
3. It provides immediate feedback about what students do and don’t know
4. It makes learning more effortful, which paradoxically leads to better retention

As the researchers explain, “Retrieval practice forces the brain to work harder to reconstruct the information, strengthening the neural pathways associated with the multiplication facts. This active engagement is significantly more effective than a passive review.”

While Restudying, chanting, and singing will improve student multiplication automaticity, the question is how much faster will students become automatic using the flashcard method.

Enhancing Retrieval Practice Strategy through Collaboration

While the original study focused on individual practice, the principles easily extend to collaborative learning environments. We know that combining effective instructional strategies, like retrieval practice and peer collaboration, can amplify the benefits substantially.

Benefits of Collaborative Retrieval Practice:

1. Peer Teaching Reinforces Learning: When students explain concepts to each other, they strengthen their own understanding
2. Productive Struggle: According to Rittle-Johnson et al. (2017), when students work together to recall information, they engage in “productive struggle” that strengthens neural connections
3. Reduced Math Anxiety: Boaler and Staples (2008) found that collaborative mathematics environments fostered more positive attitudes toward mathematics compared to traditional or technology-centered approaches
4. Development of Meta-Cognitive Skills: Gillies (2016) demonstrated that structured peer collaboration promotes metacognitive thinking and self-regulation—critical skills for mathematical fluency

A meta-analysis by Puzio and Colby (2013) found that collaborative learning approaches yielded consistently higher achievement gains in mathematics compared to individual or technology-only approaches, with effect sizes ranging from 0.19 to 0.33 standard deviations.

How to Implement  Collaborative Retrieval Practice in the Classroom

Here’s a structured approach to implementing collaborative retrieval practice for multiplication facts:

1. Pair Students and Prepare Materials

• Organize students into pairs
• Provide each pair with a set of flashcards for the target multiplication facts
• Ensure the problem appears on one side and the answer on the other

2. Structure the Collaborative Retrieval Process

• Partner A shows the problem side of the flashcard to Partner B
• Partner B attempts to recall the answer from memory
• Partner A provides immediate feedback (confirmation or correction)
• Partners switch roles for the next flashcard

3. Activity: The 3× Table Challenge

Here’s how this might look in practice:

1. Pair Up: Students form pairs (Partner A and Partner B)
2. Flashcards Ready: Each pair receives a set of flashcards for the 3× table
3. Round 1:
   • Partner A shows “3 × 4 = ?” to Partner B
   • Partner B tries to answer (“12”)
   • Partner A confirms (“Yes!”) or provides the correct answer, showing the back of the card
4. Role Reversal: Partners switch roles for the next card
5. Multiple Rounds: The pairs repeat this process through the entire set multiple times
6. Quick Check-in: After several rounds, have pairs quiz each other without the cards

For a 5-minute session, a good structure would be 3.5 minutes of flashcard practice, 1 minute of no-card quizzing, and 30 seconds for discussion or reflection.

Tips for how to schedule

Try to incorporate spaced practice by 

• Schedule shorter practice sessions distributed over time
• Revisit previously learned facts while introducing new ones
• Gradually increase the challenge by mixing facts from different tables

The Power of Peer Collaboration vs. Digital Tools in Elementary Math

While digital tools like Kahoot can increase the attention of students, research consistently demonstrates that student collaboration offers significant advantages for mathematical learning. Here are just a few: 

• A meta-analysis by Puzio and Colby (2013) found that collaborative learning approaches yielded consistently higher achievement gains in mathematics compared to individual or technology-only approaches, with effect sizes ranging from 0.19 to 0.33 standard deviations.
• The benefits of peer collaboration extend beyond just academic achievement. According to research by Slavin (2014), collaborative learning environments foster deeper conceptual understanding through verbalization and explanation of mathematical concepts. When students explain multiplication facts to peers, they engage in what Rittle-Johnson et al. (2017) call “productive struggle,” strengthening neural connections more effectively than immediate feedback from digital platforms.
• Gillies (2016) found that structured peer collaboration, like the partner approach outlined in this article, promotes metacognitive thinking and self-regulation—critical skills for mathematical fluency that are less developed when students interact primarily with technology. Students working together develop communication skills and mathematical vocabulary simultaneously with content mastery.
• Furthermore, a longitudinal study by Boaler and Staples (2008) demonstrated that students in collaborative mathematics environments showed greater persistence when facing challenges and developed more positive attitudes toward mathematics compared to those in traditional or technology-centered learning environments. This emotional component is particularly valuable for students who experience math anxiety when learning multiplication facts.

While digital tools can provide immediate feedback and gamification elements, they cannot replicate the multi-dimensional benefits of human interaction. The quality of peer explanations during collaborative practice significantly predicts learning outcomes, creating a learning experience that technology alone cannot match.

The collaborative flashcard approach described in this article harnesses these research-backed benefits while maintaining the structured retrieval practice that cognitive science has proven effective for long-term retention.

Practical Applications for Elementary Teachers

This research offers clear, actionable guidance for improving multiplication fact learning:

• Replace some of the time spent on choral recitation with collaborative retrieval practice
• Structure classroom activities to ensure students practice retrieving facts from memory repeatedly
• Provide immediate feedback after retrieval attempts
• Space practice sessions over time rather than concentrating them
• Use peer collaboration to enhance engagement and deepen learning

Retrieval-Based Learning Beyond Multiplication Facts

While these studies focused specifically on multiplication fact fluency, retrieval practice and collaborative learning principles apply to other foundational math skills and even beyond mathematics. Similar benefits have been observed for vocabulary learning, science concepts, and other academic content.

Conclusion

The evidence is clear: incorporating retrieval practice—especially in collaborative settings—into math instruction can dramatically improve students’ multiplication fact fluency. This research-backed technique empowers students to achieve mastery, reduces classroom frustration, and builds a solid foundation for more advanced mathematical concepts.

Moving beyond traditional methods like choral recitation and embracing the power of active recall and peer interaction, we can help students develop fluency with basic facts that will support their mathematical journey for years to come. As the research demonstrates, this approach leads to more vigorous initial learning and better long-term retention, ultimately setting students up for mathematical success.

Want to Improve Math Instruction At Your School?

At Learning-Focused, we specialize in providing effective teacher training and instructional improvement PD for teachers, coaches, and school and district leaders. 

Reach out today to discuss how we can better equip your educators to help struggling students and improve student learning and achievement for all.

Ophuis-Cox, F. H. A., Catrysse, L., & Camp, G. (2023). The effect of retrieval practice on fluently retrieving multiplication facts in an authentic elementary school setting. Applied Cognitive Psychology, 37 (6), 1463 – 1469. https://doi.org/10.1002/acp.4141 

Boaler, Jo, and Megan Staples. “Creating Mathematical Futures through an Equitable Teaching Approach: The Case of Railside School.” Teachers College Record, vol. 110, no. 3, 2008, pp. 608-645.

Gillies, Robyn M. “Cooperative Learning: Review of Research and Practice.” Australian Journal of Teacher Education, vol. 41, no. 3, 2016, pp. 39-54.

Puzio, Kelly, and Glenn T. Colby. “Cooperative Learning and Literacy: A Meta-Analytic Review.” Journal of Research on Educational Effectiveness, vol. 6, no. 4, 2013, pp. 339-360.

Rittle-Johnson, Bethany, et al. “The Importance of Prior Knowledge When Comparing Examples: Influences on Conceptual and Procedural Knowledge of Equation Solving.” Journal of Educational Psychology, vol. 109, no. 3, 2017, pp. 471-485.

Slavin, Robert E. “Cooperative Learning and Academic Achievement: Why Does Groupwork Work?” Anales de Psicología, vol. 30, no. 3, 2014, pp. 785-791.

Webb, Noreen M. “Information Processing Approaches to Collaborative Learning.” The International Handbook of Collaborative Learning, edited by Cindy E. Hmelo-Silver et al., Routledge, 2013, pp. 19-40.