Your friend who can reliably estimate which brand of peanut butter is the least expensive even though they come in differently sized jars probably has good number sense. Likewise, the fifth grader who makes a calculation error on a test but then notices that the answer doesn't look right. Number sense is useful throughout life and generally makes all things mathematical much easier. The question is, how can we help children (or teenagers, or adults) develop number sense?

In the most basic terms, people who think more deeply about math are more likely to develop and intuitive sense of how numbers act. But that just begs the question of how we can encourage children (who are often reluctant) to think deeply about math? One method is to expose them to a wide variety of problems, especially problems that are genuinely interesting. Here's an example. Pick a number from 1 to 9. Now multiply that number by 3. Finally, multiply it by 37,037,037. What happens? Why? (Can you figure it out? Hint: what is 37,373,737 times 3?) Thinking about problems like this, and especially talking about them can be a great way to develop number sense. You can find more interesting math problems like this by doing an internet search for terms such as “math magic” and “surprising math”.

Certain games are good ways to develop number sense, too. “24” comes to mind. “24” is really a family of games designed specifically to hone mathematical thinking, and they're well worth checking out. I've had good luck getting even reluctant learners to enthusiastically participate. Other games might not have been designed with math in mind, but they rely heavily on probability and therefore can provide great exposure. Examples of this type of game include tabletop strategy games made by the Games Workshop or Wizards of the Coast. An older, better known game that can also serve this purpose is Risk.

Math problems that have a serious bearing on real life rather than the silly, unrealistic problems that most students are usually exposed to can be stimulating too, especially for older kids. Studying compounding interest and the way credit cards, bank accounts and other financial instruments work come to mind as an example. Edward Zaccaro's book 25 Real Life Math Investigations is another rich source for this type of problem.

A related strategy is to study the mathematics of interesting physical phenomenon. For instance, if you have the opportunity to build and launch rockets with kids, it's possible to use all sorts of math in exciting ways (combined with physics, of course). For example, you can use algebraic formulas to calculate how high you expect the rocket to fly and trigonometry to actually measure the path it takes. The topic you investigate doesn't need to be as spectacular as rocketry- there are a wide variety of amenable subjects (primarily from physics). Using the density formula while designing model boats is one of my favorite projects using math.

Developing a strong number sense isn't quick or easy, but it also isn't magic. Anyone can do it with practice and it is certainly worthwhile. Done right, it might even be fun.