I once asked a student whether 3 was a factor of 48.
He paused, thought for a while, then said: “Well, I know 8 × 6 is 48, so 6 is a factor of 48. And 3 is a factor of 6, so 3 must be a factor of 48.”
He didn’t retrieve an answer. He built one — moving through known relationships, using factors, multiplication, division and structure to reason his way to a conclusion. That is strategic fluency: not flashy, not always fast, but powerful.
If that number looks familiar, it should — it’s the same 48 from a piece a little while back, the one about all the ways a class could make it: 6 × 8, 3 × 16, 12 × 4, 24 × 2. That piece was about flexibility, how many paths lead to the same number. This one’s about something related but different: what a student does when they don’t have a path yet, and has to build one out of facts they already trust.
That’s the gap I want to talk about.
The Gap Between Knowing Facts and Using Them
In math education, we often talk as though there are two main goals. First, students need to know their facts. Then, once they know their facts, they can move on to rich problem solving.
There is truth in that. Students do need facts, recall, automaticity — enough knowledge stored in long-term memory that every new problem doesn’t overload them.
But there is a missing middle.
Knowing a fact is not the same as knowing what to do with it.
A student might know that 6 × 9 = 54 when it appears in a times tables list, but struggle to recognise the same relationship in a worded problem about six groups of nine. Another student might know multiplication facts in isolation, but not see them inside an area model, a fraction problem, a ratio table or an algebraic expression.
This isn’t simply a recall issue — it’s a fluency issue, though not fluency understood only as speed. It’s what I’d call strategic fluency.
Strategic fluency is the ability to use mathematical facts, relationships and structures flexibly and purposefully. It is the bridge between knowing facts and using them well. It includes recall, but goes beyond it. It includes automaticity, but is not limited to it.
A student with strategic fluency can compose and decompose numbers, use known facts to derive unknown ones, choose an efficient pathway, change strategy when needed, and recognise the same mathematical structure in different contexts. They can look at a problem and think, “What do I know that could help me here?”
That last question matters. Because in real mathematics, facts rarely arrive in neat columns — they arrive hidden inside contexts, diagrams, relationships and decisions.
Recall vs. Automaticity vs. Strategic Fluency
This is why it’s helpful to separate three ideas that are often blended together.
Recall is retrieving a fact from memory. A student remembers that 7 × 8 = 56.
Automaticity is when that recall becomes fast and effortless. The student knows 7 × 8 = 56 without hesitation, freeing up working memory for more complex thinking.
Strategic fluency is the ability to use facts and relationships flexibly. A student might not immediately know 6 × 9, but they know 6 × 10 is 60, so they subtract one group of 6 to get 54. Or take the student from the start of this piece, reasoning through whether 3 is a factor of 48 — not recalling, but building.
Another student once looked at 6 × 70 and said, “I can just use 6 × 7 × 10.” Again, that’s more than recall. The student is seeing the structure inside the number — using place value, turning a larger calculation into a known fact and scaling it.
This is the kind of fluency we need to notice more often. Because too often, the students who look fluent are simply the students who are quickest in the most familiar format.
Give them a list of multiplication facts and they perform beautifully. But place the same facts inside a worded problem, an area model or a multi-step task, and some begin to struggle. They know the fact, but they do not yet recognise where it lives.
Why This Matters for Later Math
This matters because later mathematics depends heavily on recognising structure. Fractions require students to see relationships between parts and wholes. Ratios require multiplicative comparison. Algebra requires students to see equivalence, inverse operations, factors and generalised relationships. Geometry requires students to move between number, space and representation. Problem solving requires students to decide which facts and structures are useful in the first place.
If students only learn facts as isolated answers, their fluency can become brittle. It works when the question looks familiar. It breaks when the mathematics changes clothes.
Strategic fluency makes knowledge more transferable. It helps students see that 6 × 9 is not only an answer on a times table chart. It is six groups of nine. It is an array. It is an area. It is 6 × 10 minus 6. It is 3 × 18. It is 9 × 6. It is connected to 54 ÷ 6 and 54 ÷ 9. It may later appear inside simplifying fractions, scaling a recipe, finding area, solving equations or reasoning about proportion.
The fact matters. But the network matters too.
This is why I am cautious when fluency is reduced to speed alone. Speed can be a sign of fluency, but it is not the whole of fluency. A student may be quick because they understand deeply. Another may be quick because they have memorised a narrow pathway. A third may be slower, but much more strategic.
Speed Isn’t the Whole Story
I have seen many students with quick recall rush into larger problem-solving questions and simply start performing operations with the numbers they see. Add these two. Multiply those. Divide by something. Hope it works.
I have also seen students who were not especially fast at recall slowly and persistently work through a complex problem with care. They reread. They draw. They test. They ask what the numbers mean, choose a pathway, and check whether their answer makes sense.
The second student may not look as fluent if we only value speed. But in many ways, they are showing the kind of fluency mathematics actually requires.
This doesn’t mean automaticity is unimportant — it’s very important. Students who have to reconstruct every basic fact from scratch will often find higher-level mathematics exhausting. Working memory is limited. If all of a student’s attention is spent calculating 6 × 7, there may be little left for interpreting the problem, choosing a strategy or reasoning about the result.
Automaticity gives students breathing room. But automaticity is not the destination. It is fuel.
The question is what students can do with it. Can they use known facts to solve unknown ones? Can they choose an efficient strategy, or recognise when a problem is multiplicative rather than additive? Can they move between a fact, a diagram and a context? Can they recover when memory fails, and explain why a relationship must be true?
These are strategic fluency questions, and they’re worth asking often.
How to Build Strategic Fluency in the Classroom
One simple way to build this in the classroom is to shift some of our questions from answer-getting to relationship-finding. Instead of only asking “What is 6 × 8?”, we can also ask:
“How could you use that fact to find 12 × 8?”
“What if one factor doubled, or halved?”
“What related division facts do you know?”
“How could you show it with an array?”
These questions slow the fact down just enough for students to see inside it.
Another useful approach is to give students an answer and ask them to build the mathematics around it. Put 72 on the board. Some students might write 8 × 9. Others might write 6 × 12. Someone might see 4 × 18. Another might notice 144 ÷ 2. Someone else might build 70 plus 2, or 6 groups of 12, or double of 6 × 6.
Suddenly the answer is not the end of the task. It is the beginning of a network.
That is often where students who do not usually see themselves as “fast” begin to shine. They can contribute a relationship, notice a pattern, build from someone else’s idea. They can see that mathematics is not just about being first — it is about making connections.
This is not a small thing. For many students, their mathematical identity is shaped early. If they think fluency means instant performance, they may decide very quickly whether they belong. The quick students belong. The slow students do not.
But if fluency includes strategy, flexibility and sense-making, more students can see themselves as mathematical thinkers.
The student who decomposes cleverly belongs. The student who asks a clarifying question belongs. The student who notices a relationship belongs. The student who takes a slower but more thoughtful pathway belongs.
This does not lower expectations. It raises them.
We are not saying, “It is fine if students do not know facts.” We are saying, “Facts matter so much that students need to know how to use them.” That is a higher bar.
A student with strategic fluency isn’t merely carrying a bag of memorised answers — they’re building a connected system. They understand that numbers can be taken apart and put back together, that operations are related, that one fact can unlock another. They are less likely to freeze when a question looks unfamiliar because they have more than one way in.
Three questions worth asking often
What else do you know if you know 6 × 8? This turns one fact into a starting point rather than an endpoint.
Where might this fact appear in a worded problem? This helps students recognise a fact even when it’s wearing different clothes.
What would make this calculation easier? This builds the instinct to look for structure before reaching for a method.
This is the missing middle. Between recall and reasoning. Between facts and problem solving. Between knowing and using.
If we want students to become confident mathematicians, we need to build that middle deliberately. We need to give students enough practice for important facts to become familiar and automatic. We also need to give them enough rich, varied experience for those facts to become flexible and meaningful.
We need to value efficient answers, but also clever pathways. We need to celebrate the student who knows quickly, and the student who thinks strategically. We need to ask not only, “Do they know it?” but also, “Can they use it?”
That is the heart of strategic fluency. It is not speed versus understanding. It is not facts versus reasoning. It is not memory versus creativity. It is the place where those things begin to work together.
And that is the kind of fluency students need most.
Further Reading
When Answers Are Instant, Thinking Matters More
Fluency Should Feel Like Confidence, Not Pressure
The Research: Why a Right Answer Isn’t Always Understanding
