Mental math is one of the most helpful skills one can acquire. Mental math frees you from feeling enslaved to a calculator. Mental math can make you feel powerful. Mental math saves time—it’s more efficient than paper-and-pencil arithmetic. Mental math is often “good enough” for an answer. And last but not least, mental math is cool: it is cool how the mental math works, it’s cool how helpful the mental math is, it’s cool how much time it saves, and it’s cool being cool.
Yet for many students, mental math is one of the least acquired skills. How can mental math be fostered and developed? Several dynamics come to mind.
- Basic Facts
Get the basic facts of +, –, x, and ÷ simply flat-out memorized. No matter where you stand on the use of electronic calculators, and no matter how early a student starts using a calculator, it’s still valuable, even crucial, to have basic facts memorized.
There are three key dynamics to memorization: repetition, repetition, and repetition. Simply repeating, repeating, and repeating a math fact is all that’s necessary for memorization. Such repetition can occur in many ways: saying/chanting basic facts (6×7=42, 6×7=42, 6×7=42), seeing basic facts, hearing them, writing them, picturing/imagining them, singing them, and staring at them. In our house, we posted basic facts tables on the bathroom wall, so when our kids were sitting on the throne (ahem!) the basic facts were in their field of vision. All of these ways can be part of the repetition-repetition-repetition necessary for memorization to occur. Keep in mind: memorization doesn’t make mastery harder; memorization makes mastery possible.
Teach estimation, that 69 x 32 is approximately 70 x 30, which is 7×10 x 3×10, which is 7×3 x 10×10, which is 21 x 100 — so 68 x 33 is approximately 2100. Likewise, 69 + 32 is approximately 70 + 30, which is 100, so 69 + 32 is about 100. And 71 – 32 is about 70 – 30, which is 40, so 71 – 32 is about 40.
Parents, it also is tremendously helpful for you to model out loud your own mental math with your kids. This is what’s called talking your walk, explaining what you’re doing while you’re doing it.
- Flexible Thinking
I believe the single most helpful dynamic for mental math (after basic fact mastery) is the ability to think of a given quantity in more than one way. Thinking in several and numerous ways. For example, consider the number 58. This quantity of 58 can be thought of as…
58 = 50 + 8 = 5 x 10 + 8 = 40 + 18 = 30 + 28
= 60 – 2 = 6 x 10 – 2 = 70 – 12 = 80 – 22
= 100 – 42 = 100 – 40 – 2 = 100 – 30 – 12 = 100 – 20 – 22
= 2 x 29 = 2 x (25 + 4) = 2 x (30 – 1) = 2 x (35 – 6)
= 5.8 x 10 = 0.58 x 100 = 58% x 100 = 0.058 x 1000
This ability to think in numerous ways about a single quantity represents the fluency and flexibility that students should cultivate. This fluency and flexibility is the opposite of rigidity. Students who are weak in math far too often have a rigid, narrow view of quantity. For example, they think of the quantity 58 as just that, simply and only 58, or possibly as a number that’s just “kinda big.” So, encourage and model flexibility in how to think about different quantities: 1/2 = 2/4 = 3/6 = 4/8; or 3/4 = 0.75 = 75%.
- The Distributive Property of Multiplication
Okay, I’m getting technical, but another frequent basis for mental math is based on what mathematicians call the distributive property: that with 6 x 58, the 58 can be thought of as (50 + 8), so now the original problem is 6 x (50 + 8.) The distributive property of multiplications says that the 6 distributes itself, or multiplies itself, times both the 50 and the 8, as shown here: 6 x 58 = 6 x (50 + 8) = (6 x 50) + (6 x 8) = 300 + 48 = 348. The distributive property is the basis for a whole lot of accurate, non-estimated mental math.
- Use Math with Games
Here are some great games that help develop some of the shortcuts of mental math. By the way, parents, these games are the kinds of games that you can and should play with your kids. And any game involving dice or cards usually involves a lot of counting, which leads to the need for shortcuts, which is mental math. Make math games an intentional part of your math homeschooling and remember: game time is a time for practicing math, not for teaching math. A child should know how to add before playing a game that requires addition. Game time is for practicing math.
Yahtzee: Scoring here often involves both multiplication and addition, with lots of opportunities for the older players (both parents and older siblings) to count out loud how they calculated their score.
Monopoly: Counting spaces on the board, adding up different dollar denominations to pay rent ($85 rent could be 1 twenty + 6 tens + 5 ones, or give a one-hundred-dollar bill and get back $15 in change, etc.).
Cribbage: This classic card game has a scoreboard, the cribbage board, marked in groups of five holes, which makes counting easier and teaches mental math shortcuts.
Parcheesi (the game of Sorry is a modern version): This ancient game, played by monarchs and commoners for thousands of years, also teaches addition, subtraction, and flexible thinking about quantities.
Backgammon: Ditto the above re: Parcheesi.
Math Dice: This modern game (sold by ThinkFun) is great for all operations of arithmetic.
- Use Math at Home
Show your kids how math is used at home. First, anything involving a tape measure, like carpentry and sewing, is great for learning fractions. Second, we taught our boys how to bake at an early age. Measuring cups show fractions and fraction equivalents and counting carefully and exactly is important. It’s easy to show that 3/4 cup of water is the same as 3 x 1/4 cup of water. I refer to baking as “math you can eat!” Third, when you take your kids grocery shopping, use mental math to keep track of prices.
Enjoy all of this and avoid tedium. Don’t ever make math a punishment. Not everything can be “fun,” but things can certainly be done with enjoyment—especially on the part of us parents!
Copyright 2018 The Old Schoolhouse® used with permission. All rights reserved by author. Originally appeared in the Winter 2016 issue of The Old Schoolhouse® Magazine, the family education magazine.