David M. Schwartz

My friend Merri Rudd, a contra dance caller from Albuquerque, sent me this picture. Some of you “dog people” might like it — and who knows, maybe it will inspire some folks to love the book as much as LuckyDog does. (More about Merri and LuckyDog)

Since I mentioned that Merri is a contra dance caller, I should say that I am an avid contra dancer. I can hear you asking, “What is contra dance?” I could answer, “It’s a bit like square dance except that it’s done in lines instead of squares,” but that wouldn’t be a very satisfying explanation because contra dancing is really VERY different from square dancing and not just because of the geometry. Contra dancers get asked about their dance form so often that some have posted definitions and explanations on the web. Here’s a site with several long and one short explanation.

But no collection of words can really explain a dance form, and words certainly can’t capture the terrific music (which is always live at contra dances), so why don’t you just come out and join me on the dance floor?! Most contra dances are kid-friendly, though they are not usually kid-oriented. More about dances in your area.

Now… did you think I was going to sign off without a math connection? Contra dance abounds with “math moments.” Here is one I just experienced at “Labor Day Dance Away,” a fabulous weekend of dancing that took place in the San Bernardino Mountains of southern California. In a contra dance, the dancers start out standing in lines, but as they move through different figures the geometry changes. In one of the dances last weekend, groups of four dancers formed circles, and our caller, Cis Hinkle from Atlanta, told us to rotate the circle to the left “three-quarters of the way around and a little bit more.” What a delightful, kinesthetic way for a child to learn fractions, I thought to myself. I can just imagine the discussion that might grow out of a question like, “What fraction is a little bit more than three-quarters of the way around but still less than all the way around?” A contra dancer I know, Bernie Scanlon, a math instructor at Bakersfield College, gives workshops for teachers in using dance to teach math. And check out this Science News article for another take on the math-contra dance connection.

See you on the dance floor!

DMS

When food is served, math can be dished up at the same time. Children will be eager to help themselves to both the meal and the math.

Most days, four people dine at the Archibold home, but today company’s coming. There will be five for lunch. The frittata is out of the oven. Soon it will have to be cut by 7-year-old twins Evvy and Mariah.

Mom: How many pieces should we make?

Evvy: Five.

Mom: Why?

Evvy: There are five people.

Mom: Do you think everyone will eat the same amount?

Mariah: No. Adults eat more.

It is decided that the three adults will each get two pieces while the children will have one. Of course, the girls have to figure out how many pieces are needed all together.

Mariah (after thinking): Eight.

Dad: How do you know?

Mariah: There are three adults, so that’s one-two… three-four… five-six… and the kids have one each, so that’s seven and eight.

Dad: So how can we cut the frittata into eight pieces?

Out come paper and pencils so they can draw up their plans for dividing the frittata into “eighths” (a term adults can use without explanation, since it is obvious from context). Several division schemes are considered before the idea of two rows and four columns emerges as the winner. But another possibility is suggested:

Mom: Suppose we wanted 12 pieces. How would you divide the pan?

Soon the girls are inventing their own scenarios and solving them without prompting. Thinking ahead, Mom shows Evvy and Mariah the cake they will have for dessert. She complicates the problem in a new way.

Mom: Let’s eat half of it today and save the rest for tomorrow. How should we cut it?

Even something as simple as setting the table can be a math lesson for young ones: Count the number of plates, glasses and cutlery to be carried to the table. How does it change when there are guests? With one knife, one spoon, one fork and so forth for each setting, the young tablesetter is dealing with the basic math concept of matching, or “one-to-one correspondence.”

When Evvy and Mariah had to divvy up the frittata so that the three adults each got two and the two children each got one, they were actually being exposed to algebra! A middle-school math teacher might have expressed it more abstractly, but by the time these girls enter middle school, they will easily be able to understand the frittata problem as an algebraic equation.

When they tried out different combinations of rows and columns, they were learning not only about area and geometry, but basic multiplication visualized in a grid. Their parents did not proclaim “Right!” or “Wrong!,” but instead asked the children to justify their thinking, a strategy that helps kids think clearly about whether their answers make sense.

In the few minutes before lunch, Evvy and Mariah solved real life, real food problems with math. What could be tastier? Math Moments are an appropriate side dish to any meal.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

Every morning, Michael Pease drives his daughters to school. It’s a seven-minute drive and Michael makes sure the minutes are used well by playing mental math games with Maddy, 7, and Jessie, 11.

“Who wants a head problem?” he asks, turning out of the driveway. Both girls shout,“I do!” He starts with Maddy.

“Take the number of sides of a hexagon… double it… take two from that… take half of that. What do you get?”

“Five,” blurts out Maddy.

“Give me five!” he says, extending a hand over his shoulder to the backseat. Now it’s Jessie’s turn. She’s older, so the math gets harder.

“Take the number of people in the car (three)… Raise it to the fourth power… Add the digits…Take the square root… Multiply by 13. What do you get?” Jessie pauses about two seconds.

“Thirty-nine!”

Michael believes that turning mental math into an enjoyable daily game helps both girls excel in math. “My goal is to help them feel confident and successful,” he explains, “and to see math as fun, useful and meaningful.”

Any family can invent games that transform car trips into math-rich experiences. The Pease girls came up with “Target Number.” They pick a number (like 100) and then add and subtract the numbers they find on road signs until one player hits the target exactly.

Beth Hook’s family gets mileage out of the rising price of gas. “On our drive to school, my kids and I write down the price,” says Beth. “The next day, we find out how much it has gone up. Next week, we’ll figure out the seven-day rise.” To do the math, Beth taught her children some mental math techniques. When gas cost $1.99 per gallon, you have to “count on” by one cent to get to $2. When it rose to $2.07 per gallon, it was 7 cents above $2. To get the total price increase, just add 1 plus 7.

Easy! Taking it another step, a parent might ask, “If the price keeps rising this fast, what will it be a year from now?” Banish the thought!

Howie and Marcy Black have turned “Twenty Questions” into a family math game. “Who has a number between one and 100?”

“I do,” comes a voice from the backseat.

“Is it less than 50?” asks another child.

“Yes.”

“Is it odd?”

Yes.

“A prime number?”

“A what?” asks 7-year-old Tania. Her older brother, Paul, explains that a prime number is divisible only by one and itself.

“No.”

“Is it divisible by three?”

And so on. The kids are entertained, they are learning math… and they get to their destination without once asking “Are we there yet?”

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

Susan Jarema doesn’t have to look far to find Math Moments. Her two children, Maya, 6, and Colin, 4, provide them on a daily basis with their imaginative questions: “How many slugs are in Canada?” “How many stars are in the universe?” “How many termites are in a termite hill?” “How many bacteria are inside me when I’m sick?”

Moments before bedtime one evening, Colin wonders how many seconds he has lived. Susan can’t resist pulling out a calculator to answer the question: More than 100 million seconds! Not to be outdone in bedtime extensions, Maya suggests this scenario: “If we had 27 kids and they each fought with each other once, how many fights would that be?”

Numbers like million, billion, trillion and googol (a one with a hundred zeros) are fascinating to children of all ages. Parents can draw upon them to help kids understand the basis of our number system: each additional zero multiplies the value 10 times.

Here’s one way to get a handle on big numbers:

• Look for a thousand of some common object (blades of grass, for instance, or tufts of carpeting).
• Write the numeral 1,000.
• Now imagine a thousand of these thousands (1,000 X 1,000) to get a million:1,000,000. A thousand million (1,000 X 1,000,000) is a billion: 1,000,000,000. A thousand billion (1,000 X 1,000,000,000) is a trillion: 1,000,000,000,000. Don’t be surprised if your child wants to know how many grains of sand are in the world! (One answer: not as many as stars in the universe!)

It is helpful to have “benchmarks” that make large numbers concrete. One family has discovered a thousand bricks in their patio; 10,000 seats in their section of a baseball stadium; 100,000 people living in their suburb. For one million, they taped a large piece of graph paper with 1,000,000 tiny squares onto a wall of their garage.They now have a way to visualize big numbers that often appear in books, newspapers, TV, radio – or in everyday conversation.

Inevitably, children will wonder about infinity. It’s not a number because it doesn’t represent a particular amount, but it is a mathematical concept that stretch es the imagination. In his parents’ bathroom, Colin notices that two mirrors placed in front of each other create a pattern of reflections that seems infinite because “it goes on forever and ever and ever and ever…” Luckily for Susan, on this evening he doesn’t try for infinite bedtime extensions.

Instead, he kisses her goodnight and sweetly says,“Mom, I’ll love you ’til infinity.”

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

“How much does it weigh?”

“I have no idea!”

“How much do you think it weighs?”

Nine-year-old Brien cannot believe his mom is asking him the weight of a cow at the county fair. But Chris Nugent knows a Math Moment when she sees one. She doesn’t expect her son to know the cow’s weight, but she knows he can use some math to come up with a reasonable estimate. Finally, he compares the cow to his mother and decides that the cow weighs five times as much as she does. Being compared to a cow might not flatter his mom, but Chris likes Brien’s approach to the problem. He announces the cow’s weight.

He’s way off. The cow’s owner tells him that the Holstein weighs about 1,500 pounds – much more than five of Brien’s mom, as she quickly points out.

Brien does better with the pigs. Guessing that six would weigh the same as the cow, he predicts 250 pounds per pig.

“Close, young fella, close!” says the farmer, enjoying the game.

Across the nation, millions of people converge on county and state fairs every summer to see farm products, crafts, games and stage shows. With a little prompting from parents, children can include math in the mix. Carnivals and theme parks don’t have prize pigs, but they offer similar opportunities for mathematical amusement. Waiting with his tray at the cafeteria, Brien starts folding a napkin and Chris points out that he has folded it into eight equal parts. What is each part called? Brien knows they are eighths. His mom asks him what to call four of those eighths? Then she takes another napkin and folds it into fourths. How many of Brien’s eighths equal one of Chris’s fourths? What would he call the portion of the napkin with one fourth and one eighth? Enough fractions for now, it’s burger time! Later, a ride on the Ferris wheel raises the question of how many people the wheel holds. Brien realizes it’s a multiplication problem. He must multiply the number of cars (16) by the number of people in each car (6). Chris shows him how to make easy work of it: 16 is 10 plus 6, so he can multiply each of those numbers by six and add the results.“Sixty,” Brien says tentatively, “plus 36… is 96. Easy!”

Now Brien is getting into the spirit of his mom’s mathematical mentoring. “How many screws are in the roller coaster?” he wonders aloud. But neither of them stops to figure it out as they rush to the Big Top to catch a magician’s act. By the time they head to their car in the parking lot, Brien and Chris are bushed, but Math Moments keep cropping up. “If we count the cars in this row as we walk by, and multiply by the number of rows…”

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

Math is everywhere, even inside a watermelon, as four children discovered at a vacation cabin in the mountains. It all began when a family friend arrived with a plump watermelon from a roadside stand.

Placing it on the dining room table, he wondered aloud, “How many seeds do you think it has?” That simple question led to an unplanned math project more complex and more enjoyable than anyone could have predicted. To start,everyone ventured a guess.

“Ninety-five,” Grace Linderholm, 10, said confidently. “More,” said her sister, Amelia Gurley, 12. “Way more, like 250.”

Daria and Adam Fixler, 9 and 12 respectively, proffered their own guesses: 65 and 82. Once the adults had piped in, there were nine guesses, ranging from 65 to 280.

“That’s quite a wide range,” observed Grace and Amelia’s dad Owen. “What should we do next?”

“Eat it!” exclaimed Amelia.

Adam and Daria’s mom, Rae, sliced through the melon, revealing its glistening red flesh speckled with seeds. The children were again intrigued by the original question.

“Don’t eat it yet,” Grace implored. “Let’s estimate the seeds.”

Grace understood the distinction between estimating and guessing. When the children had merely looked at the melon and called out numbers, they were guessing.To estimate, they will need more information about the seeds and they will have to apply math.

Estimation is an important math skill used by everyone from young children to advanced students and scientists. Parents can encourage estimation with questions like “How many candies are in the jar?” or “How many people are in the stadium?” or “What time will we get to Grandma’s?” It’s a good idea to guess first, then discuss possible strategies that would refine the guesses into estimates.

At the cabin, the watermelon project becomes a brainstorming session with each child offering a different strategy for estimating the seeds. All agree on one thing: the process has to include consumption of the melon! Finally, with some parental persuasion, it is decided that half of the watermelon will be saved in the fridge for tomorrow’s picnic, and the other half will be divided into four wedges, each to be cut into four pieces.

Everyone will count the seeds in one piece as it is eaten. From this data and a little multiplication, the group can estimate the total for the entire fruit. But some of the kids want to know how close their estimate will be to an actual count.After all, the seeds are not evenly distributed through the melon. Will counting only nine sections be enough? To find out, they also decide to count every seed.

Minutes later, only rind and seeds are left on the plates. The counts for nine sections are averaged, then multiplied by 32 (the number of sections in the whole watermelon). The group arrives at an estimate of 580 seeds for the full watermelon. Everyone is surprised at how much higher it is than the initial guesses. Tomorrow, once the other half of the watermelon has been eaten and its seeds counted, they will do a reality check on their estimate.

At the picnic, the final count is 640. Some are disappointed that it is 60 seeds more than their estimate, but Daria and Adam’s father, Craig, puts it in perspective: “It’s really not bad. You were off by only 10 percent. Congratulations!” The two older children do some mental math and agree with him before heading off to go swimming.

Craig’s congratulations are certainly deserved and not only because the estimate was close. These kids attacked a simple question – “How many seeds?” – and turned it into a math project that involved high-order thinking, problem- solving and lots of computation. Best of all, they had fun.

Parents on the lookout for Math Moments will find no shortage if they ask questions – realistic or ridiculous – and encourage kids to find answers by estimating. The process can help sharpen math skills – and appetites.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

A family road trip offers many possibilities for side trips and sightseeing. For the Hart children – Josh, 12, Kallie, 9, and Marissa, 6 – a journey to Yellowstone National Park also includes opportunities for excursions into a landscape of mathematical learning. Math Moments abound on the American roadside.

In Minnesota, the Harts stop to ponder the proportions of a famous steel and concrete figure of Paul Bunyan and his blue ox Babe, which towers over the mortals who stop to admire. How large are the outsized figures? Kids love guessing games, and this can be a quick estimation activity: Josh is almost 5 feet tall, and Big Paul looks to be as high as four of Josh … so the lumberjack must be a little less than 20 feet. (Checking the sign after estimating reveals that the technique works: the statue is 18 feet high.) Now spread the fun (and the math) around: Kallie, at 52 inches, can be a benchmark to gauge Babe’s oxenly dimensions. And how many Marissas, end to end, would it take to span the big bovine’s horns? Now that’s a longhorn!

A few hundred miles down the road, the Harts are in Gillette, Wyo., touring a working coal mine. Never mind the coal – what fascinates the kids are the behemoth mining trucks. A tire, available for inspection, can bring up words like “diameter,” “radius” and “circumference.”

Josh’s height is just about equal to the tire’s radius, so what is its diameter? How far does the truck move in a single tire revolution? To figure it out, compute the circumference (multiply the diameter by 3.14, or pi, possibly with a calculator; or mentally multiply by 3 to approximate). Compare that with the family minivan!

In the Black Hills of South Dakota, everyone gets a chance to pan for riches at Big Thunder Gold Mine. The yellow flakes may be small, but there is a lode of math to be mined. Pans come in diameters of 8, 10, 12 and 14 inches, and in the gold miner’s lexicon, there are 12 Troy ounces to the pound (instead of the cus- tomary 16). A parent could wonder aloud, “So which is heavier,a Troy ounce or a U.S. ounce?”

A similar math principle comes from examining the panning sieves that miners use to screen material. The holes range in size from 1/4-inch (“4 mesh”) to 1/100inch (“100 mesh”). Which size has smaller holes? A simple Math Moment spent considering that question will lead to a key principle of fractions: the larger the denominator,the smaller the amount.The 1/4 holes are large compared to the tiny 1/100-inch holes, which thus screen out the most.

An impressive numerical comparison comes up when the family learns that in 35 years of operation, Big Thunder mine produced a mere 10 ounces of gold from 250,000 pounds of excavated ore! (The Holy Terror Mine across the stream averaged 26 ounces of gold per ton of ore. Which mine did better?) But today, Big Thunder is producing: the Hart kids get to take home a few gleaming yellow specs they find in their pan bottoms.

With its geysers and grizzlies (seen at a safe distance), Yellowstone is the highlight of the trip, but Old Faithful is less faithful than expected. It doesn’t erupt at perfectly-timed intervals. Instead, say the rangers, the time between blasts ranges from 35 to 120 minutes with an average of 94 minutes – more math vocabulary! Still, geologists can predict the next eruption with accuracy. So, to see it again in 95 minutes, what time should we return?

In visiting America’s natural and human-made wonders, children encounter fascinating facts and figures that add to their appreciation of the sights. Don’t let those numbers pass unnoticed. By asking questions that lead to mathematical thinking, parents and children can enjoy Math Moments that add another dimension to the family vacation.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

In his camouflage trousers and brown sweatshirt,Ty Evenich, 6, looks like he’s getting ready for a hunting trip. Which, in a way, is what he’s doing. He and his great-aunt Debbie are standing over a sewing table, pinning the hem of an apron.The last time Ty went hunting with his dad, he decided he wanted to help cook breakfast at the camp. But there was no apron even close to his size.

Aunt Debbie places a pattern, folded in half, atop the fabric. At first puzzled by its size and shape, Ty wonders aloud why it looks the way it does. Finally, his confusion turns to understanding.

“I get it. When you open it up … it will be bigger,” he says.

“How much bigger?” Debbie asks.

“Twice as big.”

She tells him that they need a 1-inch hem, and folds some fabric to demonstrate. She sets her “hem gauge” (a 6-inch ruler with sliding pointer) to 1 inch and shows Ty how to use it.

“So how much smaller will the apron be after we sew this hem all the way around?” Debbie runs a finger 1 inch from the outside edge of the pattern.

Ty almost says “1 inch,” but seems to know it isn’t right.Then he realizes the finished apron will be 1 inch shorter on each side.“Two inches!”

“You got it!” Debbie says, tousling his hair. She loves to see children think through her queries.

Almost any home craft project – sewing, knitting, beading, macramé and myriad others – can also be math projects. If adults ask questions and point out math connections, they will provide valuable lessons in measuring, computation, estimation, geometry and other skills.

While Ty pins the hem, his sister Shelby, 8, works on a quilt square, part of a 4-H project. Quilting can be a splendid introduction to geometry. The more complex the quilt pattern, the more sophisticated the math. Even children who don’t get to work on actual quilts can read delightful picture books that tie quilting to math.

Shelby is holding up two fabric pieces that she is about to sew together.

“What shape are they?” Debbie asks.

“Rectangles,” Shelby says with confidence.

“What shape will you have after you sew them together?”

This is harder.After a minute at the sewing machine, Shelby has the answer, fascinated to discover that two oblong rectangles can make a square. (An older child might be asked how to predict when two joined rectangles will make a square.)

Back at the sewing table,Ty has finished pinning the hem and, after folding the inner edge under,Aunt Debbie runs the apron through the machine.

The next step is planning the pocket. The pattern calls for one long pocket across the front, seamed down the middle.

“Where should we put the seam?” Aunt Debbie asks.

“In the middle,” says Ty.

“But where is the middle?”

He thinks, but not for long: “I don’t know.” Debbie won’t take that answer: “Think harder,Ty!”

Shelby comes to the rescue. Without a moment’s hesitation, she folds the fabric in half.

“Right there!” she pronounces with big-sister superiority, pointing to the folded edge.

Ty is still unsure. He looks to his great-aunt, who doesn’t resolve the issue but instead takes a piece of chalk and marks the folded edge. She opens the pocket.

“Is the chalk down the middle?” Revelation dawns across Ty’s face: “Yessssss!”

In minutes, he is donning his new apron and savoring the thought of flipping flapjacks. But his great-aunt is just as happy to know she has facilitated many Math Moments that will serve Ty well long after he outgrows the custom-made apron.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

Elizabeth Todd and her mother walk through their neighborhood almost every day. At almost 4 years old, Elizabeth loves to look for special things along the way: the carved figure of an Indian on a nearby roof, purple flowers in a particular garden, a bench tucked under the trees, a bright yellow antique car parked in a driveway. But most of all, Elizabeth loves to look for numbers. She finds them on mailboxes, fences, garage doors, license plates and street signs. There is no limit to the number of places to find numbers, and no limit to the pleasure Elizabeth gets from finding and naming them. Her mother, Cathy Sharp, has been nurturing Elizabeth’s number nature for a long time.

“When we find them, we read them and talk about them,” says Cathy. Elizabeth doesn’t need any prompting. She is pointing and reading the house numbers posted on a wooden retaining wall outside a neighbor’s home. “One–eight–four–eight,” she says with obvious satisfaction.

Elizabeth does more than just read numbers. She and her mother count all kinds of things, including their steps. Sometimes, it’s just “One, two, three, four, five …” as they walk along. Other times, they play a game.“One,” says Cathy. “Two,” says Elizabeth.“Three,” says Cathy. “Four,” replies Elizabeth.And so forth.A little later, they may reverse roles. Elizabeth can usually count to 20 without a problem, but when Cathy stops by a mailbox bearing the number 12 and asks her daughter to name the number, the girl says,“One-two.”

“And what do we call it if it’s written ‘one-two’?”

“One-two,” says Elizabeth.

Cathy tells her it’s also called “twelve.” Then she asks, “What’s bigger than 12?” To her mother’s surprise, Elizabeth points to another 12, in larger numerals, on the side of the house! Her mother laughs. “Now that’s confusing,” she admits. But, because Elizabeth has a strong foundation in basic counting and because she encounters numbers daily in an enjoyable way, Cathy knows that her daughter will soon be reading two-digit numbers and understanding concepts like “bigger than.”

Variations on simple counting games abound and parents can adapt them to their child’s level.“What number comes after (or before) this one?” … “What number is two more than this?” … Looking at a three-digit number – 268, for example – you can play with place value:“If we changed this 6 to a 7, what would the number be called? What if we put a 1 in front of this 268?” (To make it more visual, carry number cards.) Even multiplication (and division) can come into play:“What’s twice (or half) as much as this 12?” Like reading, appreciating the joys of numbers should start at a very early age. Charles Richards and his mom count his feet and hands when it’s boots and mittens time. “We count fingers and toes when it’s fingernail and toenail clipping time,” says the two-year-old’s mother.

Counting goes in two directions for Charles. “We count down with the microwave oven as it finishes heating the bottle: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, BEEEEPP! Charles always looks forward to the beep part.”

When Charles’ older brother was 18 months and in the hospital recuperating from surgery, his parents discovered a way to keep him happily occupied.

“We rode in the elevators counting up and down along with the flashing numbers. It occupied hours,” his mother recalls.

The simple act of naming numbers can occupy many pleasurable hours in a young child’s life. Children introduced to number concepts in this way will probably be eager to take on more advanced math challenges as they grow. Parents need merely to ensure that, in their family, everybody counts.

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.

When Phyllis Licht decided it was time to replace her lawn with a lush garden, she knew where to turn for expert advice: her 8 year-old grandson. “I love plants,” says Sam Licht who, with his father Mark, had already created a butterfly garden that is the envy of their neighborhood, “and I knew I could give Grandma a nice garden instead of her un-nice garden.”

Sam’s grandmother didn’t have a large space to work with – all the more reason that Sam had to plan carefully. First, he measured the area that was to be planted, including landscape features that could not be altered – walkways, fences and a concrete pillar. Scaling down the dimensions to notebook size, he sketched the area, then drew in the plants.

“I had to think about the spacing of each plant,” Sam recalls.“So if a border was 56 inches and I put in a butterfly ginger that needs 8 inches, then I was down to 48 inches.” If he then had four variegated Scheffleras that took 3 inches apiece, he had to subtract another foot, and so on. Sam marked the position of each plant on his sketch.

He didn’t know it but, as a third-grader, Sam was practicing not only basic computation, but also real-life mathematical thinking – exactly the kind of math that bedevils many American students. In a recent international study,American teens ranked 24th among those from 29 industrialized nations in practical math applications. Ironically, Sam does not enjoy math in school, but in the context of a garden, he says,“It’s not like math. It’s just like fun. I don’t mind that kind of math.”

Natalie Milnikoff enjoys math at school, at home, in the car – anywhere she finds it. Her love of the subject is not surprising, since her parents have been working it into her everyday routines since Natalie was very young. Now, at 11, she’s eager to help with projects around the house, and they often involve math.

When Natalie’s father, Vladimir, decided to make a patio of terra-cotta tiles, Natalie knew that a math problem was on its way. Each tile is a 16-inch square (16” x 16”) and the finished patio will be an 18-foot square (18’ x 18’). Natalie figured out how many tiles to buy.

“First, I multiplied 18 feet by 12 to find out how many inches long each side is.Then, I divided by 16 to see how many tiles would fit on that side,” she explains. Natalie had to square that number to get the total number of tiles needed, allowing for space between the tiles.

With a long measuring tape, she helped her dad mark the boundaries of the patio, and with a builder’s square, she checked to be sure they were angled right, which is to say at right angles.The project took a few weeks to complete, but the results were beautiful, useable and, for Natalie, educational.

Nearly every home-improvement project involves solving mathematical problems through measurement, estimation, computation, geometry, proportional thinking – or all of the above:

• How many cans of paint will we need to cover the deck? Figure the area of the deck and read the label to see the coverage of each can.
• Where should we nail the hanger to center this picture on the wall? Measure the width of the wall and divide it in half.
• How many yards of soil should we order for the flower bed? Multiply the bed’s length by width by height in feet and convert to cubic yards, with 27 cubic feet equaling one cubic yard (for “extra credit” your child can figure out why!).

Many children enjoy helping their parents around the house, and “doing the math” can be one of the enjoyable ways they can be helpful. As Sam Licht would say, “It’s not like math. It’s just like fun.”

Math Moments™ creator David Schwartz spends much of his time finding unusual, whimsical ways to make math and science come alive for kids and teachers, both through writing and through speaking at schools and conferences. He has written nearly 50 books for kids, including How Much Is a Million? and the “Look Once, Look Again” series. For more information about David’s math and science adventures, check out his Web site, http://davidschwartz.com.

Share Your Math Moments

David Schwartz would love to include your family’s Math Moments in this column. Send your stories and photos, along with your name and mailing address, to mathmoments@davidschwartz.com. David will award a signed copy of one of his books to those whose submissions he uses in this column.