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In order to be fluent in mathematics, you need to be efficient. Typically, when approached with the idea of efficiency, our instinct is to get the task done and to get it done quickly. While efficiency in mathematics can involve both of those things, we first need to ensure that students have the opportunity to be flexible and to employ appropriate strategies.

What does efficiency look like in mathematics?

Of the three parts of fluency in mathematics, efficiency is the most complex. This is because it involves:

1. Solving in a reasonable amount of time.
2. Selecting the appropriate strategy.
3. Accurately and readily using that strategy (Bay-Williams & SanGiovanni, 2021).

When we consider basic fact fluency in this context, we need to step back and see what it means to be efficient. The first step is the ability to solve in a reasonable amount of time. Most research says that when students are fluent in basic facts, they are able to solve a given fact in less than three seconds. Essentially, they are no longer picking a strategy and using it but have done so enough that they can simply recall the fact. So, how do we get there?

Building toward efficiency: Experience first, recall next

The truth is that students need time and rich, visual experience to spot the patterns. Ten-frames support students with facts up to 10 as well as the partitions of 10. Using frames with two distinct colors helps students to see the partitions of numbers more clearly. For more ideas around working with ten-frames, check out the addition and subtraction strategies, as well as the “Why Is Subtraction So Hard?” 30-minute webinar recording, on the resource tab.

Double ten-frames and bead racks are our go-to tools for visualizing facts over 10. Also check out our resources for supporting operational fluency with whole numbers.

The second element of efficiency requires students to become flexible with the patterns of numbers and to build their understanding of the operations. This may seem like an overgeneralization, but, often, where a student struggles to recall their facts, it’s probably because the strategy they feel confident in using is inefficient. The only way to help them move away from this is to teach them new strategies and allow the students time to experience them.

Most people associate mastery of math fact fluency starting in the second grade with addition and subtraction, but it actually starts in Kindergarten. Let’s look at these standards and what it might look like for students:

• K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings or sounds (e.g., claps). Other methods include acting out situations or giving verbal explanations — using expressions or equations as necessary.
• K.OA.4 For any number from one to nine, find the number that makes 10 when added to the given number. Students may use objects or drawings and then record their answer with a drawing or equation.
• K.OA.5 Fluently add and subtract within five.

These standards are building toward fluency together. Let’s take another look — this time from the perspective of what the student might be doing:

• K.OA.1 Play with addition and subtraction in a variety of ways and show your thinking using a drawing or an equation (aka a math sentence).
• K.OA.4 Notice a pattern when you’re making 10, using drawings or objects to help you, and then show your thinking using a drawing or an equation.
• K.OA.5 Use what you learned earlier to add numbers within five quickly and correctly.

The last standard of efficiency in mathematics is a product of the others. If a child needs more support to become fluent, we should ask ourselves: Did they get to play in enough ways to help them make sense of the mathematics and become flexible? Were they noticing patterns? What other sensemaking opportunities can we provide?

In conclusion, we need to start thinking about mathematical fluency in early education. If our students reach upper elementary and need support in achieving fluency, instead of asking them to memorize facts, think about what they already know and can draw from and where they might need support. We can then embed that in their current learning, using routines, games and strategies to support fluency.

If you would like to learn more about how to implement these ideas around counting, addition and subtraction to support early fluency, check out our strategies and our webinars.