Fix the Pathway. Not the Practice.

The Five-Stage Model for Building Fact Fluency

For more than a century, helping students learn basic number facts has been a central goal of elementary mathematics. Yet many students still lack accurate and flexible recall. Too often, when fluency is weak, instruction defaults to repetition. But practice alone does not build durable learning. Without understanding, it becomes rehearsal of fragile knowledge.

Fluency develops through a structured progression. At ORIGO Education, we employ a five-stage framework: Prepare, Introduce, Reinforce, Practice, and Extend. Each stage builds on the last, moving students from concrete experience to abstraction and generalisation.

 

To illustrate the model, I’ll use the Make-10 addition strategy, as it is one of the most widely recognized and explicitly cited strategies in contemporary mathematics standards.

 

1. Prepare

Before introducing any new strategy, teachers must ensure that students possess the prerequisite knowledge and skills needed for success. For the Make-10 strategy, students should:

• Have a deep conceptual understanding of addition
• Be able to read, write, and count numbers to at least 20
• Be able to subitize quantities and recognize visual number patterns
• Have had experience using five-frames and ten-frames

While the first three are non-negotiable, the fourth is strongly recommended. A well-structured core program gives kindergarten students opportunities to fill five-frames, helping them see that five is the sub-base of ten. Over time, they use ten-frames to represent numbers greater than five.

 

2. Introduce

At this stage, students use hands-on counters to physically make a ten. A meaningful context anchors the idea to something familiar. It connects new learning to prior experience.

For example, suppose we want to find the total cost of two items.

 

Students use double ten-frames and counters to represent the value of each item.

 

This representation intentionally leverages the prior understanding that nine is five and four more, and most importantly, that nine is one less than ten. Language is key. As students move the counters, we want them to verbalize their thinking, such as: “Nine and four is the same value as ten and three.”

 

To see this process in action, you can view a short demonstration on my YouTube channel, The Number from Down Under, which models how to teach the Make-10 strategy using double ten-frames and counters (watch here).

 

3. Reinforce

This stage allows students to assimilate and internalize the strategy. It serves as a bridge. It connects the concrete and pictorial models of the introductory phase with the abstract symbols used during practice.

 

A simple folding card from ORIGO (above) can help remind students of the thinking they developed earlier with counters and frames. When the card is opened, they see both parts (for example, nine and four). When folded, they see the ten and the remaining three, visually reinforcing the relationship they discovered.

But it isn’t enough to use this folding card on its own. To truly reinforce the strategy, students need an activity that bridges the concrete and the abstract.

In the task below, students write numerals on the faces of two blank cubes. They roll the cubes and search for the matching equation that will help them find the sum. They then record the answer for both related facts.

 

For example, rolling a 9 and a 5 reinforces that 10 + 4 has the same value as 9 + 5.

Activities like this are not conceptual, so they cannot be part of the Introducing stage, but they are not yet practice either. They deliberately bridge the visual model and the symbolic representation.

 

Stage 4: Practice

Only after conceptual understanding and reinforcement are in place does practice serve its intended purpose. Practice at this stage focuses on developing efficient recall, not introducing or repairing understanding.

Effective practice is:

• Frequent (daily)
• Brief (3–5 minutes)
• Purposeful (strategically targeted)

Short, distributed practice sessions are more effective for building fluency than longer, massed blocks, particularly when practice is strategically targeted rather than repetitive.

Integrating facts across strategy clusters, such as Count On, Use Doubles, and Make-10, supports the development of flexible networks of knowledge rather than isolated recall.

Games, card routines, and quick-response prompts are all well suited to this stage, provided they are aligned with previously established strategies.

 

5. Extend

This is where true fluency is revealed.

The final stage, and perhaps the most overlooked, is extension: applying the same strategies to greater numbers, fractions, and decimals.

If students can successfully use a strategy to calculate 9 + 4 = 13, why not encourage them to use that same thinking later to solve 29 + 15 = 44?

The chart below shows how all the basic addition strategies can be extended across the grades. Strong mathematics programs such as ORIGO Stepping Stones and ORIGO Pathways ™️ (In development) will have these progressions naturally embedded in their developmental sequence.

 

Conclusion

Fact fluency is not a topic to be covered and tested. It is a foundation to be expanded, even in classrooms short on time. The problem isn’t that students forget their facts. It’s that we too often stop teaching for understanding too soon.

When fluency is taught this way, it doesn’t fade over summer. It endures because it’s built on understanding.