When students compute, are they executing steps or reasoning from relationships? Most of us were taught the former. But the latter is where mathematical power lives.

Feel the Structure

We all want our students to become fluent with facts. There are many ways we can try to get them there. Some focus on strategy development and building mathematical relationships. Others focus on speed and memorization. One way builds confidence and connections. The other creates anxiety.

To build fluency, we need to understand what it is.

If fluency is represented by this walnut.

When students are able to apply strategies to multiple contexts to solve problems, they are fluent. They are also working their way toward numeracy: having the confidence to use basic maths at work and in everyday life.

Let’s look at some ways to solve 27 x 4:

What’s the underlying structure of each? They all break the factors into friendlier numbers and multiply, then put them back together. Decomposing and Composing. All with the underlying structure of the Distributive Property. One uses the Place Value structure as well (top left). Another also uses Equivalence as a structure (bottom left). But all use the structure of distributive property. Naming the property isn’t something students in grades 3 -5 need to focus on. They just need to understand what’s happing and apply it.

Sometimes teachers go with timed tests rather than focusing on strategies because they think that the strategies are replaced by fluency. It’s not really the strategies, though. It’s the underlying structure of these strategies. And fluency doesn’t replace the structures, it compresses them.

In my next post, I’ll discuss the three big structures that encompass all of K-5 mathematics.