Announcer> This program is dedicated to Martin Gardner, an American writer and mathematician.

In Scientific American , he writes a column called "Mathematical Games," challenging readers with puzzles, tricks, and other diversions.

He has convinced many adults that numbers can be perpetually entertaining.

He says the best way to make mathematics interesting to students is to approach it in a spirit of play.

He has also explained the works of Lewis Carroll in The Annotated Alice , and has written several books for older children about the limitless world of numbers.

♪ Oh I can believe ♪ ♪ in everything I read ♪ ♪ A little boy blue ♪ ♪ a lady with a shoe ♪ ♪ and a dish run off with a spoon ♪ ♪ Well Hickory Dickory Dock ♪ ♪ a scholar he's late to school ♪ ♪ And I like a cow ♪ ♪ I jump over the moon ♪ ♪ Child> One, two, three... Rick> Fingers are one of the first things most of us learn to count.

They're always handy.

Child> One, two, three... Rick> Toes are likewise accessible.

Learning to count is one of the major feats of early childhood.

In fact, children can see quickly that our world is full of numbers.

Newborn babies are immediately defined by a number of pounds and a number of ounces.

And very soon as these children learn to speak, they also need to develop the ability to deal with numbers.

Mathematics are an essential part of life today.

So very young children must grapple with the complex process of learning to count.

Counting is not just a new idea, it's also a whole new set of words.

A strict sequence for those words, and a set of strange designs called "numbers."

Adults have to help kids with all of this.

And yet the teaching of numbers poses some questions itself.

How much can a child learn?

How fast?

How early?

Are there any limits?

Psychologists are looking for answers to these questions.

Jean Piaget, the Swiss psychologist, wrote extensively about the cognitive abilities of children.

In one of his books, The Child's Conception of Number , first published in 1941, he hypothesized that young children are initially unable to grasp some basic mathematical concepts.

His experiments indicated to him that children understand some ideas only gradually.

The children must progress through several stages of development before they can deal with certain concepts.

Piaget points out that "Both counting on the fingers and the exchange of one object for another, are indications of the considerable part played by correspondence in the synthesis of number."

In 1978, Rochel Gelman and C.R.

Gallistel wrote a book called The Child's Understanding of Number .

They challenged some of Piaget's methods and proposed some conclusions based on their own research.

They observed that many young children will count objects without being asked to do so.

Even before those children know the conventional words for numbers.

Some children created their own words for numbers.

Some used alphabet letters and some used number words, but in a jumbled sequence.

There seems to be little doubt.

Children are interested in counting and are eager to learn.

Mother> Okay.

There's one.

Say one.

(child speaking unclearly) Good.

Now let's put another one.

And that's two.

(child mimicking mother) One.

Two.

Good.

Now one more.

That's one.

(child speaking) Two.

(child exclaiming) Three.

Rick> The rhythmical quality of numbers has made them well suited for poetry.

For ages, people have been introducing the names and sequence of numbers by reciting counting rhymes.

Parents and teachers have always known that children remember verse more readily than prose.

One of the oldest counting rhymes in print is in Mother Goose's Melody .

The first known collection of Mother Goose rhymes, published in 1781.

"One, two, three, four and five.

I caught a hare alive; six, seven, eight, nine and ten.

I let him go again."

Countless different counting rhymes have been heard in nurseries around the world.

Probably the most familiar one in English is "one, two, buckle my shoe," which is included in most Mother Goose collections today.

The popularity of these counting rhymes undoubtedly led to the publication of counting books.

Whole volumes devoted to the simple display of numbers with various objects to count.

The books aren't as effective as actual counting of real objects, but they do offer a good way to reinforce the counting concept.

They also can introduce the shapes of our numerals and their sequence.

Modern counting books are much like modern alphabet books in their presentation of numerals and a corresponding set of objects or creatures to count.

Tana Hoban's simple book called Count and See , is an admirable example of an elementary counting book.

A good way to start teaching numbers.

Her simple black and white photos are models of clarity, and the numbers on the pages opposite the photos are simple and distinct.

For nine, there are nine fireman's hats, all marked with the nine.

She also includes dots on the pages with the numbers.

Dots that reinforce the number of objects.

And Mrs. Hoban doesn't stop at ten.

She shows 11 spools of thread, a dozen eggs, on up through 15 cookies.

Then she jumps by five to 20... melon seeds, 30 bottle caps, 40 peanuts, 50 nails.

And finally, on the last page, a challenge.

How many peas in their pods?

In their book Introduction to Children's Literature , Joan I. Glaser and Gurney Williams III, point out that "The clarity of Hoban's book makes it useful for preschool readers."

Clarity is an essential quality in number books for young children.

In the text, Children and Books , Zena Sutherland and May Hill Arbuthnot state that "The same sort of clarity needed in ABC books is important in counting books: clear depiction of numerals, close relation of text to pictures, and easily identifiable objects if the objects are intended to be counted."

This is Numbers by John J. Reiss.

It's a bright and vivid introduction to numerals.

One is for one boy, two is for two socks, and for two shoes.

Not all the examples are as well presented as these.

For instance, on the "three" page, there's one traffic light which has three different colored lights in it.

But there's really only one complete traffic light.

A better example could have been found.

Reiss' designs for the most part, however, are outstanding.

Sutherland and Arbuthnot point out that "When considering counting books for very young children, the best books are those that have plenty of open space to set off numerals and objects, those in which the numerals are large and clear."

Like Reiss' 14 bananas or his 15 grapefruit.

He illustrates every number until 20 and then counts by tens to 100.

At which point he gives the clever example of 100 centipede legs.

The final page of Reiss' book shows a tiny child bombarded by rain drops, 1000 in all.

As with alphabet books, there are essentially three kinds of counting books.

First, those with a potpourri of examples like Reiss' Numbers .

Secondly, those with related examples.

And finally, those with examples that tell a sequential story.

Helen Oxenbury's popular Number of Things, is an example of a potpourri counting book.

The humorous illustrations include one lion, two cars, six acrobats, and even 30 penguins.

A controversial, albeit colorful counting book is this Brian Wildsmith's 1, 2, 3.

On each two-page layout, there's a number on the left with the name of that number written twice beneath it.

On the opposite page, there's a typically Wildsmithian design.

These may be simple in intent, but they are often geometrically complex.

There are indeed only two circles on this "number two" page, but the divisions of each circle can be misleading.

Charlotte Huck discusses this Wildsmith book in her text, Children's Literature In The Elementary School .

She admits "Brian Wildsmith's 1, 2, 3's is handsome but confusing.

The designs within the basic shapes are distracting and might well be miscounted."

The number ten reveals how the designs can be a problem.

First of all, Wildsmith expects children to count very different kinds of shapes as equal objects.

Counting the parts of this train, the black smoke circles are worth equal value, with the large, complex rectangle that makes up the body of the locomotive.

Also, the rectangular smokestack, which is divided equally into two colors, is to be counted as one object, not two.

Glaser and Williams also berate the Wildsmith work as confusing.

They suggest that it would be "better used with children who have a good grasp of number concepts from 1 to 20."

1, 2, 3 To The Zoo by Eric Carle is a bright and captivatingly simple counting book.

Its related examples, all animals, are presented as passengers on a train headed for a zoo.

One elephant, two hippos, three giraffes, etc.

As each carload of animals is introduced, there's a reminder of all the previous cars lined up along the bottom of the page.

Charlotte Huck points out, "The cumulative train at the bottom of each double page spread gives the child added practice, in knowing how many cars have passed."

Mr. Carle also rewards observant readers with a playful mouse somewhere on each carload of animals.

There's a rousing finale on a foldout page at the end, which shows the whole train load of animals in their new home.

They're ready to be counted all over again.

If you're teaching numbers and you're tired of the zoo, you can always move on to a new location and a new book.

Rodney Peppe's Circus Numbers is a spirited look at a number of things under the big top.

It's a related example counting book with one ringmaster, two horses, three elephants, and four jugglers, among others.

Peppe reinforces the number of concepts by including stars, which correspond to the numbers being illustrated.

The star groups are easy and helpful until Peppe gets to 20, where he arranges the stars in a hoop for the 20 doves to fly through.

Researchers have found that young children can be confused by objects arranged in a circle.

Kids often forget where they started to count.

Objects and straight lines are easier to count, easier to understand.

Simple counting books can also introduce basic mathematical concepts.

Look at One More and One Less with pictures by Giulio Maestro.

He introduces a silly bird as one, and then he explains that one and one more is two.

A cat enters as the second creature.

Although natural animosity might lead a reader to expect a little conflict here, there's none.

The cat just slyly keeps his eyes on the bird.

On each successive set of pages a new creature arrives, and the one more concept is continued until there are ten white grinning animals.

Then they begin to leave, and Maestro states that one less than ten is nine.

The illustrations are clear and appropriate.

It's always obvious which animal is arriving or departing.

In a clear and easily understood fashion.

Maestro introduces the ideas of addition and subtraction.

Another counting book with many little creatures, as related examples, is Robert Crowther's The Most Amazing Hide and Seek Counting Book.

Crowther has devised a paper pop-up trick for introducing the various numbers.

Here are five water lilies and when the paper tab is moved, six goldfish jump out of hiding.

The beauty of Crowther's pop-up gimmicks is that each is a surprise.

Counting becomes a real game of discovery.

And The Most Amazing Hide and Seek Counting Book also encourages hand and eye coordination.

Careful sharing of this book can also aid in the development of the literacy skills, which John Stewig identifies in his text, Children and Literature.

Those skills are first, the ability to describe.

Next, the ability to compare.

And thirdly, the ability to value one example over another.

Crowther's pop-ups can be used to encourage all these skills.

It can be compared with many others.

For example, this page of 30 leaves with six special leaves that lift to reveal 40 worms, 50 slugs, 60 Beatles, and so on can be closely compared with the presentation of similar insects in Billions of Bugs by Harris Petie.

The Petie book is for advanced counters, but they'll likewise enjoy the simple surprises in the Crowther book, Petie jumps from one praying mantis eating a grub to ten walking sticks hiding in a shrub.

He counts by tens to 100, where he shows 100 yellowjackets nesting in the ground, then 200 fireflies flashing all around.

Then he counts by hundreds to 1000, where he has drawn a thousand butterflies.

It might be interesting to point out the different ways that butterflies are shown in this book, and in Crowther's pop-up volume.

Asking kids to compare these examples helps to develop simple verbal skills, or as Stewig calls them, literacy skills.

In 1971, Gail Kredenser and Stanley Mack put together this book called One Dancing Drum .

They claimed it was a counting book for children and parents who are tired of puppies and chickens and horses.

In their book readers count the number of instruments played by a wild and exuberant band of musicians.

Five tumbling tubas and six persnickety piccolos are just 11 of the instruments which gather for a concert under the leadership of an energetic yellow band leader.

Each group of musicians is a different color, so they're still countable even when the whole group is assembled, including the ten crashing cymbals.

One Dancing Drum is a good example of a counting book, which uses the gathering concept as a sequential storyline.

Sequential counting books are more common than sequential alphabet books, because it's much easier to deal with numbers in sequence than with the 26 letters in our alphabet.

And often you can tell a story with just one number or ten instead of 26 letters.

Consider 10 Bears In My Bed , a more recent counting book by Stan Mack, who worked on One Dancing drum .

He calls this Ten Bears book "a good night countdown."

It's based on an old counting song.

A little boy finds ten bears in his bed and dispatches them one at a time, by telling them all to "roll over, roll over."

Each departing bear takes a toy with him, but the boy is not concerned.

He is glad when there are none left, and he says "good night," before starting to dream about all the playful bears.

For some reason, bears are an unusually popular subject for counting books.

This is Teddy Bears 1 to 10 by Susanna Gretz.

She starts with one teddy bear, then two old teddy bears, and eventually tells a story about how the bears are washed, dried, cleaned and dyed.

Eventually there are ten bears home for tea.

The simple and charming book can be a bit baffling however.

On each page there's always an extra unexplained bear.

If there were four teddy bears in the wash, why are there five on the clothesline?

There are all kinds of crazy animals in Fritz Eichenberg's book Dancing In The Moon .

It includes a silly rhyme for every number from 1 to 20, including "four pandas resting on verandas," "six kangaroos selling news" and "eight llamas wearing pajamas."

Eichenberg's illustrations are hilarious, the animals expressions unforgettable.

Check out these "16 pigs dancing jigs."

Eichenberg dedicates the book to Albert Einstein, who liked children, animals and numbers.

Eichenberg's illustrations are recognized as outstanding examples in the art of picture books.

While we're considering unusual animals, we must also look at Nicola Bayley's counting book, entitled One Old Oxford Ox , based on an Old English counting rhyme.

It features a tongue twister for every number and a detailed painting for each example.

Here are some alligators who are identified as "five frippery Frenchmen foolishly fishing for frogs."

They seem to have the glow of success in their eyes.

The "six sportsmen shooting snipes" are actually red foxes with expensive firearms.

The surrealistic illustrations can be enjoyed by adults as well as by children.

In fact, there are some especially striking works that are sophisticated counting books for people of all ages.

Anno's Counting Book is one of those.

It is fascinating and unusual from first page to last.

The first pages show a winter landscape with a pale blue stream and an equally pale blue sky.

The number being illustrated is zero.

Oddly, few counting books include zero, despite its importance in our counting system.

On the pages for the number one, there are several items to count.

One house, one snowman who's holding a flag marked one, one little girl, one skier, one pine tree, one cloud, one of each thing.

It's January and it's still snowy.

To the left of the picture, there is also one block to count.

The blocks increase with each number.

When you turn to two, there's less snow, but one more of everything else.

Now there are two buildings, two men, two trucks, and it's 2 o'clock on the church steeple.

And there's another block on the left.

As the numbers increase, the months and the seasons change.

Buildings are built.

People and animals arrive.

The counting continues.

On the pages for 12, it's snowing again.

There's a Christmas tree by the church, and 12 reindeer are flying across the sky.

It's December.

Seymour Chwast and Martin Stephen Moskof created still another number book entitled Still Another Number Book .

They have illustrated it in the simple modern style using flat areas of bright colors.

It begins with one boat on a purple ocean, and then establishes a sequential style that continues throughout.

For each number, take five, for example.

There are things to count in this case dogs.

But they aren't all presented on the same page.

They are spread out over five pages, and the numerals at the top of these pages try to reinforce the concept that the number five represents a set of five separate items... dogs.

This Still Another Number Book is yet one more counting book for kids who have already gained an understanding of basic number concepts.

Maurice Sendak created this counting book called One Was Johnny in 1962.

It's included in his popular collection called "The Nutshell Library."

One Was Johnny is a sequential story about a boy named Johnny who lived by himself.

But with each number, a new creature arrives to disturb Johnny's solitude.

It's a gathering story at first.

Eventually, Johnny is crowded by eight interlopers, including a robber who took an old shoe.

Johnny can't stand it and gets rid of the whole crowd by simply counting backwards from nine to one.

Just by the way, One Was Johnny subtly reinforces the idea of reading as a pleasure.

Johnny apparently wants to be by himself so he can read a book.

There aren't many children's books that show children reading books.

There are several concepts included in The Very Hungry Caterpillar .

It was created by Eric Carle, the same author/illustrator who did 1, 2, 3 To The Zoo , which we mentioned earlier.

The Very Hungry Caterpillar is considerably more complex than that trainload of animals.

This is the story of a caterpillar who emerges from his egg on page two, and quite literally, eats his way through the book.

In a typically Carle-ish fashion, the middle pages of the book are specially cut.

The little green larva eats for a week, then fat and satisfied, he spins a cocoon.

And on the last page completes his metamorphosis.

Carle covers the concepts of numbers, days of the week, various fruit, and the unusual life cycle of a caterpillar.

Joan Glaser and Gurney Williams are a bit uneasy about the book, however.

They worry that "Young readers may lose count because the holes reveal yesterday's meal and today's and part of several upcoming feasts, all showing at once."

The Very Hungry Caterpillar might be visually confusing for a small child.

But the novelty of the book's design will probably delight any counting kid who's ready to start playing with numbers.

A playful attitude would be helpful with this book too.

Ten What?

is called a mystery counting book by Russell Hoban and Sylvie Selig.

It presents an odd story of one urgent message and two secret agents on a covert mission.

For each number, there are several sets of examples in bizarre illustrations.

Zena Sutherland and May Hill Arbuthnot recommend Ten What?

They explain that "There is a clutter of objects in Russell Hoban's Ten What?

But as drawn by Sylvie Selig, it's an inspired clutter."

Speaking of clutter, this is One to Ten Count Again , written and illustrated by Elaine Livermore.

She challenges her readers to find a number of animals in each of her intricate pen and ink drawings.

It's a difficult task, and certainly not for young children who may be baffled by examples like this.

Readers were asked to find four mules in the rain, in this picture.

Three of them are legitimately visible, but the fourth is only a hind end in the lower left corner.

It's difficult to understand the reasoning behind this counting book, and its intended audience is hard to imagine.

A counting rhyme called "Over In The Meadow" is the basis for two colorful counting books, both with that same title.

The first, published in 1957 with pictures by Feodor Rojankovsky, was written by John Langstaff.

His rhymes describe a series of animal families.

14 years later, Ezra Jack Keats created his version of Over In The Meadow .

He claimed that he adapted the rhyme from an original verse by Olive A. Wadsworth.

Like the Langstaff version, the Keats book features illustrations of several sets of creatures, all mothers and children, who live "over in the meadow."

There are no numerals in either version, but the number of animals is presented clearly and simply on each page.

There's a family in Two Lonely Ducks too.

This is a counting book by Roger Duvoisin.

He introduces the title characters and explains that this lonely drake and duck decide "it is high time we raise a family and have ducklings swimming behind us."

The duck lays a set of ten eggs, one at a time, in the corner of a barn.

Readers can count the eggs as they accumulate in the nest.

But there's more to this book.

Readers also get to count the days as the duck sits on the eggs.

And finally, as the ducklings hatch one by one, readers get to count them too.

The book includes all these opportunities for counting and ends with several repetitions of the numbers one through ten.

It's a good book for stressing the sequence and the sounds of the numbers.

Even young children will be able to follow the simple story of the two lonely ducks, who end up as a dozen.

The story of Noah and his ark is a favorite subject for children's books, and the Emberleys use it as the basis for their unusual counting book called One Wide River To Cross .

The text was adapted by Barbara Emberley from an old folk song, and the striking woodcuts were created by her husband, Ed.

The music for this song is included in the book also.

The rhythmic qualities of counting obviously lead readers toward music.

As you sing, the animals come in one by one at first, and Mr. Emberley cleverly chooses four mythological beasts as those unfortunate ones who get on the ark without mates.

Then, with each page, the animals come in larger and larger numbers.

Some of the descriptive verses are surprisingly silly.

These "animals came in eight by eight, some came in by roller skate."

The usual two by two aspect of the Noah story is ignored.

The ark fills up fast.

The illustrations are crisp and clear, and they can cause a lot of counting practice.

Some tiny readers who know the traditional story might miss the two only angle, however.

Once children have learned to count and to understand the simpler aspects of numbers, you might want to consider sharing a more sophisticated counting book like Moja Means One .

The Swahili counting book by Muriel and Tom Feelings Mrs. Feelings introduces each of the numbers from one to ten in the Swahili language, from Moja to Kumi.

At the same time, Mr. Feelings' illustrations show various aspects of African culture.

For one Moja, the text explains that snowy Kilimanjaro is the highest mountain in Africa.

The word mountain is printed in red to distinguish it as the object to be counted.

Charlotte Huck writes about this book in her text.

"Older children will find this a beautiful presentation of African life, but younger children will find it a confusing counting book.

The number six is represented by six persons in different kinds of dress, but only five kinds of clothing are identified."

There are many more counting books available than this sampling that we've just looked at.

You may find others, old as well as new, that can help illuminate the concept of numbers to young children.

The process of counting is perhaps best taught by using actual objects that can be handled.

But counting books are a wonderful way to reinforce the sounds and the sequence of numbers, as well as the shapes of the numerals.

They're also a good way to show children how beautiful and entertaining books can be, and there's no question the more often children are exposed to numbers and their names, the quicker they're going to be learned.

And learning the numbers is what really counts.

Child> Four, five.

Six, seven.

Eight, nine, ten.

Mother> Good.

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