# Building Fluency and Number Sense

I think the most important thing elementary school teachers can do in math class is to build students’ fluency with rational numbers.  The research shows that weak number sense and fluency underlies many difficulties students have with math (Geary, Bow-Thomas, & Yao, 1992).

Just to be clear, I don’t mean having students memorize their multiplication tables or race to answer questions the fastest.  I mean having them be fluent with numbers similar to the way we think about being fluent in a language.

Here’s a definition of fluency from NCTM’s Principles and Standards for School Mathematics: “Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently” (p. 152).

To me, that means I need to ask:

• Can they work flexibly with numbers?  Can they decompose and recompose different numbers easily and in a variety of ways?
• Can they use mental math to solve problems or do they always need to resort to pencil and paper and a traditional algorithm?
• Do they have efficient ways to solve problems?

I’ve heard a lot of complaints recently about attempts to teach fluency to students–mostly related to complaints about the common core. Parents don’t understand why we are teaching students these new ways to add or subtract instead of the just showing them the traditional algorithm they learned in school.

The thing is that many of these strategies aren’t new. Students who have strong number sense and fluency have been developing these strategies on their own. What’s new is that we are now explicitly teaching all students these strategies. A parent who attended one of my workshops explained it nicely:

• This makes more sense to me than to me than how I learned math.  I am an Engineer with 5+ years of calculus and I find the thought process to solve the problems the kids are working on is much closer to how I think, but I had to figure it out on my own.

I’ll talk more about some of the ways we can develop fluency in students, but if you want to read about it now, check out:

O’Loughlin, T. (2007). Using Research to Develop Computational Fluency in Young Mathematicians. Teaching Children Mathematics14(3), 132-138.

# Teaching Time

I started the race at 7:54 am and crossed the finish line at 10:27 am.  How long did it take me to run the race?

This problems, often labeled as elapsed time problems, are difficult for students to solve.  One reason is that students often have trouble coordinating between the two different units–hours and minutes.  In fact, in one study, only 58% of eighth-grade knew that 150 minutes is equal to 2 1/2 hours (Jones and Arbaugh 2004).

I recently witnessed this first hand in a third grade class. The students really struggled with these types of problems. Many students wanted to subtract 754 from 1027. I didn’t know what to do. So I went to the research and found some work by Juli Dixon that used open number lines to help students reason about these types of problems.

Open number lines can be used by students to count up, count down and find distances between numbers.

Here’s one way to use the open number line to solve the problem above.

Now, that’s not the first problem I had students begin with, but I wanted you to get a sense of how the open number line works.

Dixon also notes that you should allow students to use the number line in ways that make sense to them. You should not prescribe one way of using it to solve a problem, as there are multiple ways that work. Students can share the different strategies with the class.

As I’ve mentioned before, I’m all for diagrams that allow students to reason about problems. In this case, I love that the diagrams allow for students to record what is happening as it is very hard to keep track of all the steps. I’ll talk more about how I built up to this another time, but I’m curious to hear what you have done with your students.

Want to know more?  Check out the articles below:

Dixon, Juli K. “Tracking Time.” Teaching Children Mathematics, 19 (August 2008): 18-24

Jones, Dustin L., and Fran Arbaugh. “What Do Students Know about Time?” Mathematics Teaching in the Middle School 10 (September 2004): 82–84.

Monroe, Eula Ewing, Michelle P. Orme, and Lynnette B. Erickson. “Working Cotton: Toward an Understanding of Time.” Teaching Children Mathematics 8 (April 2002): 475–79.