Mental Math: Three Basic Principles

“We know that algorithms are amazing human achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.”

— Pam Harris, Math Is Figure-Out-Able Podcast

Whether you work with a math curriculum or take a less-traditional route to learning, do not be satisfied with mere pencil-and-paper competence. Instead, work on building your children’s mental math skills, because mental calculation forces a child to understand arithmetic at a much deeper level than is required by traditional pencil-and-paper methods.

Traditional algorithms (the math most of us learned in school) rely on memorizing and rigidly following the same set of rules for every problem, repeatedly applying the basic, single-digit math facts. Computers excel at this sort of step-by-step procedure, but children struggle with memory lapses and careless errors.

Mental math, on the other hand, relies on a child’s own creative mind to consider how numbers interact with each other in many ways. It teaches students the true 3R’s of math: to Recognize and Reason about the Relationships between numbers.

The techniques that let us work with numbers in our heads reflect the fundamental properties of arithmetic. These principles are also fundamental to algebra, which explains why flexibility and confidence in mental math is one of the best predictors of success in high school math and beyond.

Your textbook may explain these properties in technical terms, but don’t be intimidated by the jargon. These are just common-sense rules for playing with numbers.

Principle 1: Work with Parts and Wholes

We can take numbers apart and put them together in different ways. For example:

57 = 55 + 2

or 50 + 7

or 40 + 17 or…

Mathematicians call this principle the associative property.

Principle 2: Do It in Any Order

We can rearrange a calculation to make it easier. For example:

1 + 2 + 3 + 4 + 5

= (1 + 4) + (2 + 3) + 5

= 5 + 5 + 5 = 15

Mathematicians call this principle the commutative property.

Principle 3: Work in Chunks

We can do multiplication in parts and then put the parts back together at the end. For example:

6 × 23

= (6 × 20) + (6 × 3)

= 120 + 18 = 138

Mathematicians call this principle the distributive property.

Two Key Strategies

The principles are the basic rules of how numbers work. But our mental math strategies give us a plan for tackling problems.

Use Friendly Numbers

“Friendly” numbers are any numbers your student finds easy to work with.

For small-number calculations, the friendly numbers may be 5, 10, 20, and so on. Number complements, or pairs that combine to make a friendly number, can be useful in solving many calculations.

For multiplication, friendly numbers include the doubles, multiples of 10, and the square numbers. Plus, everyone will have a few multiplication facts that easily stick in their mind, though these vary from one person to another.

Estimate and Adjust

We use friendly numbers to get close to the answer we need, and then we make adjustments to get the exact value.

For example, we may choose to get up to a friendly multiple of ten, and then figure out whatever else we need to add:

36 + 7 = (36 + 4) + 3

Or we may subtract a comfortable friendly number, then tweak as needed:

53 − 19 = (53 − 20) + 1

For multiplication, we may remember that five is half of ten, so we can multiply by a very friendly number and then work backwards:

18 × 5 = (18 × 10) ÷ 2

Or we can multiply by a number that’s a bit too much and then take off the extra bits.

99 × 12 = (100 × 12) − 12

Over the next several posts, we’ll look at how these key mental math strategies apply to specific math problems your child will face, from simple addition to ratios and proportional reasoning.

Practice Makes Fluent

Look for opportunities for your children to practice mental calculation skills.

Keep in mind that “mental math” is doing calculations with your mind, using logic and the basic principles of how numbers work. But it doesn’t have to be done all in your mind. Students can use paper and pencil (or whiteboard and markers) to make notes along the way.

If you use a textbook, try reading the problems and letting children answer aloud. Oral work has another advantage: young children need not be limited by their still-developing fine motor skills. My sons, in particular, were able to advance through math topics orally that they would never have had the patience to write down.

And play math games, which help children develop flexible fluency. Check out my Tabletop Math Games Collection for great ways to practice math skills from preschool to high school.

Read the Whole Series

Check out all the posts in my Mental Math Series:

• Mental Math Is the Key to Algebra
• Mental Math: Three Basic Principles
• Coming Soon: Early Addition…