Musings: Mental Math Is the Key to Algebra

“If you stay with meaningful mental arithmetic longer, you will find that your child, if she is average, can do problems much more advanced than the level listed for her grade. You will find that she likes arithmetic more.

“And when she does get to abstractions, she will understand them better.

“She will not need two or three years of work in primary grades to learn how to write out something like a subtraction problem with two-digit numbers. She can learn that in a few moments of time, if you just wait.”

—Ruth Beechick, An Easy Start in Arithmetic

What Do You Mean by Mental Math?

Mental math is doing calculations in your head, with perhaps the aid of scratch paper or a whiteboard to jot down notes along the way.

But you cannot simply transfer the standard pencil-and-paper calculations to a mental chalkboard. That’s far too complicated.

For example, to solve 7 multiplied by 19, you don’t imagine something like:

“7 × 9 is 63. Write down the 3, and carry the 6, etc.”

Instead, you might think:

“19 is almost 20, which is easier to work with. How much is 20 sevens? Two 7s are 14, so twenty 7s would be 140. But then I need to take back the extra 7, so it’s 133.”

To do mental math, you must figure out a way to simplify your calculation into something your mind can handle.

How Is This the Key to Algebra?

The key to mastering algebra is learning to take a hard problem and make it simpler. When we solve an algebra equation, such as…

90 − x = 2x + (50 + 88)/2

…we start with something complex. And then we make that result simpler, bit by bit, until we get down to the simplest possible form of the equation, which is:

x = 7

The rules we use to simplify algebraic calculations are the same rules we use to simplify calculations in mental math. They are the basic principles of arithmetic, the basic structures of how numbers work.

And the more our children practice these structures in mental math, the better prepared they will be to recognize the same principles in algebra.

Give Us an Example

Let’s look more closely at the calculation for 7 multiplied by 19.

To solve it, I used two key principles of algebra.

The principle of substitution says that if two quantities are equal, we can replace one of them with the other one whenever we wish.

19 = 20 − 1

So 7 × 19 is the same as 7 × (20 − 1).

Then the principle of the distributive property says we can break up a multiplication calculation and do it in parts.

7 × (20 − 1) is the same as (7 × 20) – (7 × 1).

Students who do a lot of mental math get to know these principles so well that when they meet them in algebra, it’s like running into an old friend.

Where Does Memorizing Math Facts Fit In?

Children do not have to memorize their math facts before doing mental math.

There is always more than one way to simplify a calculation, so children can choose a method that uses facts they already know.

For example, to calculate 7 + 8, children might think of:

7 + 3 + 5

or 8 + 2 + 5

or double-7, plus one more

or double-8, then take off the extra one

…or anything else except counting one-by-one. It’s too easy to lose track when you count.

As our kids work with mental math, they will naturally begin to remember many of the connections and relationships between numbers. Even better, they’ll develop a creatively robust way of thinking about these number relationships that is much more valuable than mere memorized facts.

How Can I Help My Children Learn Mental Math?

One of the best ways to practice mental math is to play math games. The friendly competition of a game provides a non-stressful excuse for manipulating numbers.

And talk to your kids. As you play the game with them, explain how you’re thinking about the numbers. Ask about how they figure things out.

Check out my blog page to find great math games for all ages: My Best (Free) Math Games for All Ages. For more options, check out the game books at my Playful Math Store.

Another good resource is the weekly # MathStratChat. Check out their archive page to find a topic at your child’s level.

What If I Don’t Understand Math Myself?

Start where you are and just play with the numbers. You’ll learn and discover new relationships as you go along.

If you use the # MathStratChat archives, be sure to compare your answers with what other people posted. It’s amazing how many different ways there are to think about numbers.

Pam Harris and Kim Montague, the minds behind the # MathStratChat also host the Math Is Figure-Out-Able podcast. That’s a great resource for learning to think about math.

Finally, stay tuned to this website. Over the next several weeks, we’ll dig into mental math in detail and teach you all the key strategies for reasoning your way through arithmetic calculations.

Conclusion

Mental math (and algebra) is about reasoning, using the things we know to figure out something we don’t know.

This is an important skill in all areas of life, and well worth the effort to help our children learn.