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Unlike accuracy and efficiency, flexibility feels more concrete. Accuracy is the ability to get the “right” answer, and efficiency is the ability to do it quickly (we’ll talk more about that in the next post). But what does it mean to be flexible?

Strong Foundations

I’m sure many of you can come up with a saying or quote about the importance of a strong foundation. Whatever it is, it most likely applies when we think about basic facts as well. Before I explain further, let me share an example.

To this day, I still struggle to recall 12 x 8. I’m not sure why, but as of right now, it’s not an automatic fact for me. My knee-jerk reaction is to say 72 almost every time, even though I know that 9 x 8 is 72. Instead, I typically use one of three strategies to help me:

1. I decompose the 12 and multiply 10 x 8 and 2 x 8, which are 80 and 16 respectively. Then I add 80 and 16 together to get 96.
2. I know that 5 x 12 is 60 and 3 x 12 is 36. I also know that 3 + 5 is 8 (effectively decomposing the 8), so I know I can also add 60 + 36 to get 96.
3. I skip count by 12 from 60, so… 72, 84, 96.

My strategies reinforce that while I haven’t mastered 8 x 12, I am fluent with my basic math facts. This is because I have used a derived strategy — a strategy that utilizes other mastered facts to help me solve the original equation.

Derived Strategies

According to Bay-Williams & Kling (2019), there are three different phases of fact fluency:

• Phase 1: Counting
• Phase 2: Deriving
• Phase 3: Mastery

Phase 1 can look like counting on or back when working toward fluency in addition or skip counting (even when it’s inefficient) when working toward fluency in multiplication. We talk about how students become efficient counters in our early numeracy webinars. Check out the Resources section of Math for Teachers.

Phase 2, which is the one I mentioned, is a little different. In this phase, students have some of what Bay-Williams & Kling call “foundational facts.” These are facts that many of us might relate to “friendly numbers,” such as adding/subtracting 0, 1 and 2 or doubles facts. Another important fact is adding 10 to a number or combinations to 10. To quote Rachel Lambert in Rethinking Disability in Mathematics (2024), “It helps students to have foundational facts efficiently in their minds to be able to use derived facts.”

In the example above, you can see I’ve used strategies of decomposition to help support my ability to solve 12 x 8. Now, this will look different in addition and subtraction. Take a second to think of the fact 9 + 8. Have the answer? Now come up with two to three ways to solve it, with at least one being a “counting” strategy and another being derived strategies.

Do you feel like your students could come up with derived facts to help support them?

Supporting Students

As always, we go back to the bigger question: How do I support my students?

First, remember that before students can derive facts through strategies, they need to have foundational facts. To get to those facts, they need to have had lots of experience with counting. If your students are struggling with those foundational facts, take some time and have them count. Choral counting is a great strategy to help students notice patterns in counting. Here's a .

Another way to support students is to give them opportunities to share their problem-solving of basic facts in multiple ways — while making them visible to all students. Rachel Lambert shares a vignette about working with two boys who, after remote learning during COVID, needed support with multiplication:

“Throughout, I wrote the language that I used out loud. I paired the language with visuals of the towers [we were building]. I also wrote out the strategies they were using for multiplication. We noticed and named the strategies they used both times.”

Allowing students to build concrete models that they talk about, draw and then relate back to an equation helps them not only become fluent but also build meaning about the operations. This reinforces their ability to work with the numbers flexibly.

These activities don’t have to be lengthy moments for students. Five minutes every day to engage them in building, sharing, drawing and naming strategies will help them become flexible, build those derived facts and eventually master the facts.

I hope you enjoyed reading this blog. Check out our series on counting for K-2 to learn more about the importance of counting, particularly in helping students become fluent in math facts.