What do you think of when you think about fact fluency? Do you think about your students who can get answers quickly? Do you think about the ones who are able to answer all the questions correctly on a test? Fact fluency is about much more than speed and accuracy. Fluency also includes flexibility and appropriate strategy use (nctm.org).
Flexibility tends to be the most difficult part of fact fluency to teach, and often feels very unnatural. Teaching for flexible thinking means that teachers must make a mindset shift from teaching for answers to teaching for thinking. However, adding the opportunity for students to think flexibly and critically will forever transform the way your students learn and understand math.
If you are ready to focus on flexible thinking in your classroom this year, here are some simple shifts you can make to get started.
Make Connections Between Math Facts
Facts are connected, but too often they are taught as isolated pieces of information. We can always use what we know in order to solve facts that we don’t know. This is the idea behind near doubles facts, for example. To solve 6+7, we could think, “6+6=12 and 1 more makes 13.”
The fact webs below show examples of how we can encourage students to look for connections when multiplying or adding. This is a great thinking task for small groups! It’s empowering for students to know they are capable of solving anything. See this post on Instagram here.
Another way to reinforce the connectedness of math facts is through number strings. A number string is a string of related math problems. There are various ways to use number strings, but here is one of my favorite ways, where the number string is used to reinforce a strategy or way of thinking.
To create a number string, it’s essential to first consider the overall goal. Let’s suppose I’m teaching a strategy to encourage students to make friendly numbers then add the rest. I want students to recognize that they can solve 67+15 by making the 67 into a 70, then adding 12 more, so I might use a number string that looks something like the one below. I will begin with simple problems that encourage students to make a friendly number (like 10 or 20) and add the rest. Students will be slowly scaffolded until I get to the final problem, 67+15.
9+6 (goal is for students to realize they can make a 10 and add 5 more)
19+6 (goal is for students to realize they can make a 20 and add 5 more)
18+5 (goal is for students to realize they can make a 20 and add 3 more)
38+14 (goal is for students to realize they can make a 40 and add 12 more)
67+15 (final goal is for students to realize they can make a 70 and add 12 more)
Modify the Way You Ask Questions
Often in math, we communicate the message to our students that math is all about getting answers, even if this is not what we are trying to convey.
Consider these two scenarios.
Scenario 1: I am teaching a lesson on multiplication by 6. I pose a question to students, wait for them to think about it, then ask for answers. Students begin by solving 6×8. Then they solve 6×5. Each time they answer a question, I move on to the next one.
In this scenario, I reinforce the idea that the goal is to get answers, since I stop the conversation after a correct answer has been given.
Scenario 2: I am teaching a lesson on multiplication by 6. I pose the problem 6×8 to students and give everyone thinking time. I ask for an answer, to which a student replies, “48.” I ask him how he thought about it and he says, “I just knew it.”
I ask for other student volunteers to explain how they thought about it. One student says, “I knew that 5×8 is 40, so I added another group of 8 to make 48.”
Another student chimes in, “I know that 3×8 is 24, so I doubled it and got 48.”
As I listen to all of these ways of thinking, I record them on the whiteboard to inspire other students.
This simple act of asking for ways of thinking now communicates to students that our answers are not as important as the way we think.
This is the entire premise behind number talks. If the idea of implementing number talks feels intimidating, just remember it does not have to be a huge overhaul of how you do things in your classroom. It can be as simple as taking five minutes to lead a discussion like the one above, and is transformative for fact fluency.
Make Computation Concrete and Visual
In my opinion, there is nothing more important in math than making it visual for students. When we teach only abstractly (digits and symbols with no concrete manipulatives or visual models), we leave gaps in understanding for many of our students. Additionally, visuals are essential even for our students who do catch on quickly to abstract explanations because they force them to see things differently.
Let’s go back to the example above: 67+15. If we want students to use the strategy of getting to a friendly number then adding the rest, we can model this with manipulatives. The pictures below show what this might look like. First, we make 67 and 15 with base 10 blocks. Then we show how we can move 3 from the 15 over to the 67. Now we have 70 and 12 – typically an easier problem to solve mentally than 67+15.
As seen in the fourth picture, we can make connections between this concrete representation and other pictorial and abstract representations, showing students how they all connect.
Related: How to Teach Math Effectively Using the Concrete Representational Abstract Model
This is a powerful way to help students build their understanding, and it helps ensure that students don’t develop gaps in their understanding that will make math more difficult for them in the future.
Conclusion
Sometimes we get stuck, thinking that shifting the way we do things takes too much time, requires a complete overhaul of the way we teach, or is too difficult. But by making a few simple tweaks to our teaching, we can change the way our students see math and approach problems. Try these ideas this week and see how it goes!
