Quick: multiply 17 x 12 in your head.
How did that feel?
You may have approached it several different ways and got an answer of 204. Or you may have felt lost, not even tried it, or felt like you are bad at math.
I’m here to tell you a secret: the difference maker here is number sense.
Number sense is one of the main differentiators between those who are ‘good’ at math and those who struggle.
As a math teacher, I see a lack of number sense often in students who struggle. These students can’t determine if their answers make sense; they lack a gut feel about their work. If a student plugs in -5 + 5 and gets an answer of -25, they don’t realize they made a mistake – they trust the calculator because “the calculator is always right”. But students with number sense will realize that while they are trying to do an addition problem, the result is more like a multiplication problem. They are likely to check their work and arrive at the correct answer.
Students who lack number sense develop a handicap that is increasingly difficult to overcome as they progress in their learning. It is crucial that we as educators develop and nurture number sense in our students.
Number sense is made of two components. The first is number fluency or fluidity: knowing what numbers mean, their magnitude and relationship to each other, and how they are affected by operations. The second is number flexibility: the ability to break down and rearrange numbers to create different forms with the same value (10=7+3 and 6+4).
Why is this so important?
Because these skills allow the compression of ideas into a set of building blocks that can be used whenever needed. Individuals who excel at mental math use these strategies often — you just can’t see what’s going on in their head.
Let’s go back to the problem of multiplying 17 x 12 in your head. Many struggle trying to do the algorithm they were taught. However, if you recognize that you can break this into two multiplication facts that you already know, the problem becomes a lot easier. While there is no right way to break up this problem, here are a few options:
Number sense develops early: once children learn to count, they can tell you which bowl has more M&M’s in it, they understand the idea that 5 is less than 9. This gut feel for numbers develops over time with practice and experience.While there is only one right answer, there is no one right way to solve a problem! This is the magic of math. There are actually an infinite amount of ways to calculate this, including the algorithm. But many students think there is ONLY one way. They don’t realize you can be creative breaking the numbers up! I have had students look at me when explaining this concept and say, “You can do that!?”
Without number sense, students travel on a trajectory to struggle in math – in school, and in life. When students begin to learn math formally, they begin to lose this sense-making requirement. At some point they learn something they haven’t made sense of and they decide to suspend the requirement so that they can follow a ‘rule’ and keep pace with what they are learning. They learn that math doesn’t have to make sense, you just have to follow a bunch of rules. Eventually, they lose their “gut feel”’ and never again ask, “Does this make sense?”.