The Importance of Number Sense

We need to know letter sounds and their many combinations in order to be able to read. In the same way, we need to understand numbers and their relationships in order to make sense of math. Number sense refers to a student’s fluency with thinking flexibly about numbers, using a multitude of strategies for problem-solving (and knowing which strategy is best for a particular problem and why), and being able to make simple, grade-appropriate computations mentally. 

Thinking Flexibly about Numbers

Leon is a 3rd grader. He is working on memorizing multiplication facts up to 10. Though he has most of his facts memorized, he always gets stuck on 9 x 9. He knows how to skip count, so every time he needs to recall 9 x 9, he skip counts. His teacher notices that this takes him a long time. She asks him if can recall the fact 10 x 9, which he can do easily. She then asks him how he could get from 10 x 9 to 9 x 9. He realizes he can subtract 9 from 90 to get to 81, which is the product of 9 x 9.

His teacher has shown him one way to think flexibly about that particular problem. Students who are able to independently apply this type of thinking to math problems, by either using number facts they already know or other strategies to make the math more accessible, are able to think flexibly about numbers.

The Case for Multiple Strategies

Most parents these days remember learning only one strategy for most arithmetic: stacking. While some older students still use stacking, it is almost never taught in the younger grades. This is because solving a problem using stacking requires almost no number sense at all. If you memorize the steps involved in stacking without realizing why those steps are needed, you arrive at the answer without any knowledge of the process. Learning stacking before understanding how it works is like learning how to drive with the car on auto-pilot. Students may get to their destination, but they won’t have made any progress in developing their skill.

Instead, math students are now exposed to many different strategies, which vary depending on grade level and math fluency. Students are taught these strategies so that they can either use what appeals most to their learning style, or choose a strategy based on what they think will help them find their answer most efficiently. Drawing pictures, using manipulatives like base ten blocks, and making jumps on a number line are just a few of the strategies introduced early on in elementary school.

When It’s Okay to “Solve in Your Head”

Every kid knows the dreaded reminder to “show your work”. Most of the time, showing work is crucial, not just as proof to the teacher that a student knows what they’re doing, but as a way to keep track of one’s thinking. Too often, a student will go through an entire multi-step problem in their head, refusing to write any steps down, when they suddenly realize they need to blow their nose. Or their pencil breaks. Or they wonder what’s for dinner. When they finally bring their mind back to the problem, they realize they have completely lost track of what they were doing, and need to start from scratch. Had they written down their work in a clear and organized way, they could’ve avoided this frustrating setback.

But is there such a thing as showing too much work? It depends on the student and their mastery of number sense. A second grader with limited number sense trying to solve a 2-digit subtraction problem is better off writing down every single step. A middle schooler with fluent number sense, working on a multi-step word problem involving several operations, is probably better off making small calculations in their head, and keeping track of the larger steps on paper. As a rule of thumb, no student should complete more than two calculations in their head, or they’re more likely to lose track or make a careless error. And if your student has a particular difficult with number sense, dysgraphia may be the culprit.