The biggest challenge to understand why number sense matters, and isn’t just something that kids have or don’t, is that if your focus on math instruction is to simply get to the correct answer, then it makes no sense to teach with a focus on number sense. It takes time, there’s a lot of exploring, and it’s not about finding efficient ways to solve problems but rather creative ways to solve problems. That being said, I think it makes no sense to teach math strictly from a procedural perspective. But I didn’t always think that way!

When I first started tutoring math in college, I was a firm believer that kids were just missing some content and if they just had the missing skills filled in, they’d be set. I was very successful in traditional mathematics education despite having weak number sense. I could memorize procedures easily, was always one of the first people to finish the timed drills, and even passed AP Calculus in high school by mimicking answers – with absolutely no clue how it all fit together or what it meant. In college, I saw some of the teacher education courses where they were working with manipulatives and I remember thinking I would never have studied math if I’d had to use those. Once my classes got rid of the numbers and turned to pure mathematics I was thrilled! But, despite my success in math class, I couldn’t even think about calculations without a pencil in my hand, I learned math by studying the answer keys, and I hadn’t understood a word problem since third grade!

It’s hard for me to believe how much my views on mathematics have changed. Once I started working with kids who struggled in math, I noticed that the issues they were having in “basic” math were similar to the issues I had in “higher-level” math with one major distinction – they thrive on visualizing and building mathematics – while I have to fight with my brain to get it to engage. It took me several years to figure out how to design a lesson to physically build 3^{5}. I was floored by the richness of the conversations that the students were having while they worked on building a representation of the problem! The hardest part for them was they kept trying to build the answer.

So what is number sense? Most people agree it’s hard to define and that it involves mental math, flexible thinking with numbers, and estimation. It underlies learning the basic facts, place value, and comparing numbers.

Here’s a great example of thinking flexibly with numbers from youcubed.org. This approach still hurts my brain a little, but that just means I still have room to grow!

18 x 5 from YouCubed on Vimeo.

I am a fan of including numbers sense activities across the grades, and not limit it to K-2 instruction. A popular approach to including numbers sense instruction is using number talks.

Just search online for “number talks” and you’ll find a wealth of information. The basic concept is that students find an answer to a math problem using mental math, and think about other strategies as they wait for their classmates. And then solutions are discussed. If you’d like to dig deeper, here are some great options for books:

- Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10
- Number Talks: Whole Number Computation, Grades K-5
- Number Talks: Fractions, Decimals, and Percentages

Another activity for building number sense is to work with number systems from different cultures. I love to work embed the history of math into class because it helps students see that there are different representations and how much it can change. It’s hard to find history of math books that don’t get bogged down into the math and the history, but here are some fun books on number systems:

- The History of Number Systems: Place Value
- Can You Count in Greek?: Exploring Ancient Number Systems, Grades 5-8
- Count Like an Egyptian – A Hands-on Introduction to Ancient Mathematics (HS)

It’s important to check a student’s understanding of place value because it is one of the best examples of how number sense can impact understanding. For instance, I will ask a student to show me 534 blocks with base ten blocks. Usually, they can mimic the answer of what it should look like with the blocks. Then I rearrange the order of the blocks to 3 tens sticks, 4 cubes, and 5 five hundreds sheets. The ones who struggle with number sense will simply answer that there are 345 blocks. There’s no real connection between the actual number of blocks, the representation of place value using the blocks, or preservation of number. I set those blocks aside, have them show me 345 blocks and then compare them to 534 blocks. Most students immediately catch what they have done, but I have worked with several students who have to go through the process a few times.

Another example is when students are regrouping numbers for addition and subraction, I will ask them what the 1’s digit is worth wherever it appears in their problem. Invariably, student who struggle with number sense, despite using base ten blocks to learn regrouping, will answer that it’s worth 1 regardless of it’s place value.

One of the things that becomes obvious about teaching number sense, is that you can’t rely on written answers to check understanding. You have to hear what students are thinking. And, bonus – talking about math with peers and teachers helps to clarify understanding and increase retention of the material.

Estimation is another example of a number sense activity. Estimation180 is full of great resource to integrate activities into your daily routine. Estimation is a skill that is used outside of the classroom and often comes more naturally to students if they think about what they would do in the real world instead of in math class. And it encourages critical thinking, helping students learn to think about get the gist of an answer rather than relying on exact information. It leads you into questioning if an answer makes sense, rather than just blindly following a procedure. And we certainly need more of that type of thinking!