Elementary Problem Solving: The Early Years

[Rescued from my old blog. To read the entire series, click here: Elementary Problem Solving Series. Photo by Studio 757 via flickr.]

You can begin to teach your children algebraic thinking in preschool, if you treat algebra as a problem-solving game. Young children are masters at solving problems, at figuring things out. They constantly explore their world, piecing together the mystery of how things work. For preschool children, mathematical concepts are just part of life’s daily adventure. Their minds grapple with understanding the three-ness of three blocks or three fingers or one raisin plus two more raisins make three.

Wise homeschooling parents put those creative minds to work. They build a foundation for algebra with games that require the same problem-solving skills children need for abstract math: the ability to visualize a situation and to apply common sense.

Teaching Children to Think

Even without “thinking skills” workbooks, thinking and plotting come naturally to my kids. Children confront a world that refuses to bend to their wishes, yet they find ways to accomplish their goals. They tinker with the Lego model until that stubborn piece finally stays in place, they pester their older sister until she puts down her book and plays with them, or they fix a broken holster with duct tape and string—and who knows? Maybe they even use chewing gum. Whatever the problem, my kids will figure out something.

I will bet your children are just the same. It’s human nature. As George Polya, one of the last century’s great math teachers, wrote, “Solving problems can be regarded as the most characteristically human activity.”

We need to help our students apply that same type of crafty common sense to math. But how?

Polya continued, “If you wish to learn swimming you have to go into the water, and if you wish to become a problem solver you have to solve problems.”

In other words: Students learn to solve math problems through doing it—practice, practice, and more practice. If you’d like to practice problem-solving with your children, try one of the following games, which encourage the same sort of abstract thinking they will need for algebra.

Tell Me a Story…

One of the best mental math games relies on adult/child conversation, a proven method for increasing children’s reasoning skills. As soon as your child can count past five, give him simple, oral story problems to solve:

“If you have two cookies
and I give you two more cookies,
how many cookies will you have then?”

The way to a child’s mind is through the taste buds—children can easily visualize their favorite foods. Use mainly edible stories at first, then try stories about other familiar things: toys, pets, cars, shopping…

Don’t limit your story problems to the child’s grade level. If he can make a picture in his mind, he will be able to work with it. As you both get used to the game, occasionally throw in something more difficult: bigger numbers, division with a remainder, or an answer that comes out negative. See what your student can do with a tough problem, and you might be pleasantly surprised. Even a toddler has ideas about how to split three hot-dogs between two people. Of course, his idea might be that he takes the extra one. That does solve the problem, efficiently if not charitably.

Try not to give away the “right” answer. Instead, ask your child to explain the problem back to you. As he puts the problem in his own words, he will often see a solution. Pretend to be Socrates, asking questions that guide him toward the answer.

Most importantly, take turns. If I ask my daughter a story problem, she gets to give me one. And I have to try to solve it, even if she uses made-up numbers like 80-hundred and a gazillion. This is a game, not an oral quiz.

Oral story problems are not just for young children. Students of all ages benefit from the practice of working math in their heads. As your children grow, let the stories grow with them. Soccer games, horse stories, or space adventures will keep middle-school students figuring.

Can You Guess My Secret?

By the time they reach school age, children are ready to try a more abstract challenge. Don’t stop playing with stories, but add another math game to your repertoire. Introduce the idea of variables, or unknown numbers. I call them “Mystery Numbers.”

Elementary textbooks slip a few pages of stealth algebra into even the earliest years of schoolwork with “missing addend” problems. Your children have seen these:

  • 5 + [] = 7

Because the numbers are small, children can do them in their heads. Without recognizing it, they engage in algebraic thinking: “If 5 + [] = 7, that means [] must be whatever is left from 7 if I take away the 5. So the secret number is 2.”

Solve for mystery numbers in subtraction and multiplication problems, too. Encourage your children to do plenty of these problems, both orally and written out. The textbooks never have enough of these, so you will have to make up your own. You can use question marks or blank boxes to stand for the secret number.

And sometimes use a letter symbol, to ease your children’s transition to algebra:

“I know a secret number.
I can’t tell you its name,
so I’ll call it N for number.
If you had N plus three more, that would be five.
Can you tell what my secret number is?”

Don’t forget to take turns. If you let the kids make up problems for you, it becomes a game instead of busywork. Children love trying to outwit Mom and Dad!

6 thoughts on “Elementary Problem Solving: The Early Years

  1. We do something with a similar objective to the ‘guess my secret’ game. I’ll say something like ’10 – cheetah = 6,’ and then ask what cheetah is. My daughter loves animals so we’ll go through a whole ream of animals with different questions and she’ll enthusiastically answer them. At some point it usually gets silly and it’s ‘bubble-blowing unicorn – 8 = 16″ But it’s a favourite game.:)

  2. THANKS! You’re exactly what Iwas hoping to find. I did horribly in algebra when I went to school because the way it was taught. I homeschool too and I was looking for a way to teach this that I can understand it myself. You are a great help!

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